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Prepare Training Data for YOLO +Using Robotic In-Hand Observation and Synthesis +Hao Chen1, Weiwei Wan1∗, Masaki Matsushita2, Takeyuki Kotaka2 and Kensuke Harada13 +Abstract—Deep learning methods have recently exhibited im- +pressive performance in object detection. However, such methods +needed much training data to achieve high recognition accuracy, +which was time-consuming and required considerable manual +work like labeling images. In this paper, we automatically +prepare training data using robots. Considering the low efficiency +and high energy consumption in robot motion, we proposed +combining robotic in-hand observation and data synthesis to +enlarge the limited data set collected by the robot. We first used a +robot with a depth sensor to collect images of objects held in the +robot’s hands and segment the object pictures. Then, we used a +copy-paste method to synthesize the segmented objects with rack +backgrounds. The collected and synthetic images are combined to +train a deep detection neural network. We conducted experiments +to compare YOLOv5x detectors trained with images collected +using the proposed method and several other methods. The +results showed that combined observation and synthetic images +led to comparable performance to manual data preparation. +They provided a good guide on optimizing data configurations +and parameter settings for training detectors. The proposed +method required only a single process and was a low-cost way to +produce the combined data. Interested readers may find the data +sets and trained models from the following GitHub repository: +github.com/wrslab/tubedet +Note to Practitioners—The background of this study is a +requirement in lab automation – Using robots to arrange +randomly placed tubes automatically. Before sending test tubes +to an examination machine for gradient tests, humans need to +categorize and organize the tubes into specific patterns to fit the +machine’s internal design. Employing humans is difficult as the +tube arrangement requirements are time-varying. A preferred +solution is using robots to replace humans. The robots should +have a vision system to detect the tubes and a manipulation +system to perform physical arranging actions. They will be +used in busy seasons while deployed for other tasks in leisure +time. Deep neural networks like YOLO are effective for the +tube detection task. However, preparing the training data is +challenging and unsuitable for lab end users. Pre-trained neural +networks are options but have limited tube detection ability +and cannot deal with newly included tube types. The method +developed in this work helps solve the training data preparation +problem. With its support, the robot can automatically prepare +training data that has comparable quality to manually labeled +ones in a single-process and low-cost way. +Index Terms—Robotic data preparation, data synthesis, test +tube detection +I. INTRODUCTION +Recent advances in deep learning have led to a revolution +in object detection. Deep learning-based methods use deep +1Department of System Innovation, Graduate School of Engineering Sci- +ence, Osaka University, Toyonaka, Osaka, Japan. +2H.U. Group Research Inst. G. K., Japan. +3National Inst. of AIST, Japan. +∗Contact: Weiwei Wan, wan@sys.es.osaka-u.ac.jp +Fig. 1: Several examples of in-rack test tube detection. Each +grid includes two images. The left image is captured by a +vision sensor. The right image is the recognition result. The +data used for training the detection neural network is prepared +using the proposed method. +neural networks to learn features from training data. They +outperform traditional hand-crafted features with impressive +results. Despite these advantages, deep learning-based object +detection requires collecting a large amount of labeled data +for training, which is time-consuming and labor-intensive, and +has significantly hindered the scalability and flexibility of deep +learning-based applications. +Previously, researchers have developed several methods to +reduce data collection costs. For example, data augmentation +[1] enriched existing training data sets by applying random +transformations like image rotation or scaling. Data synthesis +[2][3] generated previously unseen data using simulation or +adversarial neural networks. The main challenge of the aug- +mentation or synthesis methods was the “domain gap” [4][5]: +arXiv:2301.01441v1 [cs.CV] 4 Jan 2023 + +ARXIV VERSION, 2022 +2 +Augmented data had less varied visual contexts. Synthesized +data was prone to discrepancies with the real world. Recently, +researchers have revisited using the copy-paste method [6] to +increase data. The method was effective in compensating for +the “domain gap” problem, exhibiting impressive performance. +There is no clear boundary between augmentation and synthe- +sis when using the copy-paste method to generate data. It was +mainly classified as an synthesis method [7][8], although some +studies considered it to be augmentation [9]. This paper calls +it a synthesis method to avoid confusion with transformation +and scale-based data generation. +The most tiring aspect of the copy-and-paste method is +how to neatly cut a large variety of target regions and paste +them onto a new background. Previously, researchers work- +ing on robotic manipulation have developed robotic methods +to segment novel objects from backgrounds. For example, +Florence et al. [10], Boerdijk et al. [11], and Pathak et +al. [12] respectively used robotic in-hand or non-prehensile +manipulation to change objects’ observation viewpoints and +segmented the objects based on the robot motion. Such sys- +tems could replace humans to segment goal object regions +under various conditions. Very recent studies [11][13] has +noticed the advantage, and increased data size and contextual +variety by pasting the objects segmented by robotic systems +onto random backgrounds. Despite their seminal proposals, +the need for copy-and-paste synthesis and the impact of data +volume and ratios remain undiscussed. +Based on the current research status, this paper further +delves into using robots to collect training data automatically. +Considering the low efficiency and high energy consumption +in robotic data collection, we propose combining robotic ob- +servation and copy-paste synthesis to reduce costs. We assume +a test tube detection task shown in Fig. 1 and use a robot with +a depth sensor to move and observe tubes. The robot collects +observation images and, at the same time, segments tubes +from the images for copy-paste synthesis. The observation and +synthetic images are used as training data for deep detection +neural networks. Especially for the synthesis routine, we value +the co-occurrence of tubes and racks, and paste tubes inside a +rack area to obtain contextual consistency. Also, we take into +account factors like tube-to-tube occlusions and foreground +changes caused by environment or visual difference to reduce +unrealistic synthetic results. The proposed method helps enrich +the data set and resolve the “domain gap”. It does not need +heavy robotic effort. +In experiments, we trained several YOLOv5x networks to +understand the performance of the proposed method. The +training data was collected using the proposed and several +other methods. The results confirmed data collected using the +proposed method do have claimed advantages. We also con- +ducted multiple ablation studies to look into the impact of data +volumes and ratios when training detection neural networks +using data collected with the proposed method. The results +provided a good guide on optimizing data configurations and +parameter settings for training detectors. +The contributions of this work are as follows. (1) We +develop an automatic data-collection method in which a robot +holds target objects and observes them. The method yields +observation images and target regions segmented from the +images. (2) We develop a copy-paste image synthesis method +to enrich the training data. The method pastes object regions on +various rack backgrounds to balance “domain randomization” +and “domain gap”. The rack backgrounds are also automati- +cally collected by the robot. (3) We examined combinations +of the observation and synthetic images and compared them +with other data sets to understand the impact of data volume +and ratios. +The remaining part of this paper is organized as follows: +Section II reviews related work. Section III presents the +hardware system and the proposed method’s workflow. Section +IV delivers technical details. Section V shows experiments and +analysis. Section VI draws conclusions. +II. RELATED WORK +We review the related work considering robotic data collec- +tion and data synthesis, respectively. +A. Automatic Data Collection Using Robots +Segmenting the object regions from a picture is the basis of +automatic data collection. Conventional methods used simple +backgrounds [14], known environments [15], or designed eas- +ily identifiable gadgets [16][17] to simplify object extraction. +The methods required careful preparation about scenes and +objects. +Robot-based methods leverage actuated robots to simplify +object segmentation. They can be traced back to early studies +in object recognition and 3D object modeling [18][19][20][21]. +These work took advantages of robotic manipulation sequence +to perceive objects from different viewpoints and segment +the objects from the background. From the robotic manipu- +lation perspective, such segmentation can be divided into two +categories: In-hand object segmentation and Interaction-based +segmentation. +1) In-hand object segmentation: Previous work on in-hand +object segmentation used known robot models and handcrafted +visual features to isolate in-hand objects from background +environments and robot hands. For example, Krainin et al. +[21] isolated in-hand objects’ point clouds by examining +the Euclidean distance to the robot model. Welke et al. +[19] segmented in-hand objects from images based on Eigen +background subtraction, disparity map, and hand localization. +These methods required manually preparing detectors for +various targets considering their visual features. +More recent studies used deep learning to reduce the +reliance on hand-crafted visual features for in-hand object +segmentation. For instance, Florence et al. [10] proposed a +self-supervised framework to segment in-hand objects. The +framework involved two steps that used the same training +and learning routine. In the first step, the authors generated +masks for the robot by considering combined depth and RGB +information, and trained a neural network model based on +the masks to differentiate the robot from the background. +In the second step, the authors masked the grasped object +and train neural network models to isolate the object from +the robot hand. Boerdijk et al. [13] used optical flow to + +ARXIV VERSION, 2022 +3 +respectively segment manipulators that were holding and not +holding objects. The segmented data set were used to train a +neural network for isolating manipulators and grasped objects. +2) Robot-object interaction: On the other hand, some re- +searchers took advantages of non-prehensile robot manipu- +lation like push to change object perspectives and segment +them based on robot motion cues [12][22][23]. For example, +Pathak et al. [12] designed a framework to continuously refine +a neural network model that generates object segmentation +masks through robot interaction. The model initially generated +hypothesis segmentation masks for objects. The masks were +refined based on the pixel differences of the images captured +before and after robotic interactions. The generating model +was updated along with the refined masks. Singh et al. [24] +proposed to segment unknown objects in a cluttered scene +while repeatedly using robotic nudge motions to interact with +objects and induce geometric constraints. Robotic interactive +segmentation often requires a static scene or surface to permit +interaction between robots and objects [25][26]. It is more +complicated compared with the in-hand object segmentation +as the object poses needs to be controlled and changed through +robotic manipulation. +A critical problem of the robotic methods is that they are +unsuitable for preparing a large amount of training data as +robots consume much time and energy to perform the physical +motion. Conducting thousands of robotic motion trajectories +to collect data is impractical. Also, the robots in the systems +are fixed, have limited views, and can only collect data in a +narrow range of scenarios. Neural networks trained using the +collected data may suffer from contextual (background) bias +and have bad generalizability [27][28]. +This study focuses on robotic data collection while con- +sidering leveraging data synthesis to reduce robotic usage. +We first ask the robot to hold a single tube and annotate the +tube’s bounding polyhedron by extracting in-hand point cloud +according to the robot’s tool center point (TCP) and hand +model. Then, we map the annotated bounding polyhedron to +2D image regions in the robot’s camera view for extracting the +tube region. The robot moves the tube to different positions +and rotations to obtain many varieties of 2D images and tube +regions. The images and tube regions are respectively used for +training and synthesizing new data in a later stage. +B. Data Augmentation and Synthesis +Data augmentation and synthesis are the two most well- +used methods to enrich training data. Data augmentation +generates new data by transforming the existing training data +with specific rules or learning-based methods. Data synthesis +generates synthetic data by merging existing data with others +or using computer simulations. Concurrent publications tend +to mix these nomenclatures. Therefore we conduct a uniform +literature review of them below without differentiation. +The copy-paste method is widely used for generating syn- +thetic data. It segments foreground objects from existing +images, possibly modifies them, and pastes them onto new +backgrounds [8][7][9]. The copy-paste method is easy to +implement and shows notable performance over using pure +real data. Previous studies showed that it was important to +carefully select the backgrounds when pasting objects. For +example, Divvala et al. [29] experimentally showed visual +context benefited object detection performance and reduced +detection errors. Dvornik et al. [30] showed that the correct +visual context when pasting object can improve prediction +performance while inappropriate visual context led to negative +results. Wang et al. [31] swapped objects of the same class +in different images to ensure contextual consistency between +objects and backgrounds and showed using the exsiting back- +grounds had better performance than random ones. Also, +the copy-paste method requires a data set containing many +possible views of the object that are easy to be cut out. It is +burdensome for humans to prepare them. +Graphical simulation is another popular method for synthe- +sizing training data. The benefits of simulation is that it allows +freely changing light conditions and materials to increase +variation. It also allows capturing many views of objects by +simply transforming virtual camera poses. For example, Hodaˇn +et al. [32] and Richter et al. [33] respectively used photo- +realistic rendering to synthesize images of 3D object models +and scenes. The methods required a lot of computational +resources to narrow down the domain gap between synthetic +and realistic data. Tobin et al. [34] proposed the concept of +domain randomization (DR). They randomized a simulator to +expose models to a wide range of environments and obtain +varied training data. Instead of photo-realistic rendering, the +method only required low-fidelity rendering results to reach +satisfying accuracy for medium-size objects. Carlson et al. +[35], Hinterstoisser et al. [4], Prakash et al. [36], and Tremblay +et al. [37] respectively used DR to narrow down the domain +gap. The authors randomly changed the context in simulation +so that “the real data was made to be just like another +simulation” [38]. Yang et al. [39] and Sundermeyer et al. [40] +respectively sampled viewpoints of 3D object models using +simulation and mixed the samples with real backgrounds to +reduce the human effort for preparing scenes with rich domain +randomness. Besides DR, Generative Adversarial Networks +(GANs) were also promising to reduce domain gap. For +example, Chatterjee et al. [41] designed a lightweight-GAN +to synthesize data for training plastic bottle detectors. +In this study, we leverage data synthesis to enrich the +training data. We develop a copy-paste based method to +attach tube cap regions separated from robotic observation +images to rack backgrounds and thus synthesize new images. +Various constraints like rack dimensions and tube occlusions +can be considered during the synthesis to reduce the domain +gap. The synthetic data is mixed with real-world data to +promote the performance of YOLO-based tube recognition +neural networks. It is also compared with other data collection +methods to understand the influence of data volume and data +combination ratio. +III. ROBOT SYSTEM AND WORKFLOW +A. Configurations of the Robot System +Fig. 2(a) shows our robot system used for preparing the +training data. A Photoneo Phoxi M 3D Scanner is used for + +ARXIV VERSION, 2022 +4 +Fig. 2: (a) The system configuration. (b) The test tubes and +rack in the view of the Phoxi M 3D Scanner. +capturing objects on the flat table. An ABB Yumi dual-arm +robot with a two-finger gripper is used to manipulate objects +in the system. A flat table is set up in the front of the Yumi +robot. The in-rack test tubes to be recognized are placed on +the surface. The Phoxi scanner is a structured-light based +depth sensor. It can capture gray images and point clouds +simultaneously. Each data point of a point cloud captured by +the Phoxi scanner have a one-to-one correspondence to a pixel +in a gray image. We can segment an object in the gray image +by considering its point cloud. +Especially, we install the Phoxi scanner on top of the robot +to obtain a top view of the racks and tubes. When recognizing +tubes in the rack, we select the tube caps as the primary +identifiers. There are two reasons why we prefer using the +tube caps for identification. The first one is that obtaining the +point cloud of a translucent or crystal test tube fails easily +due to limitations of the structured-light based depth sensors. +The second one is that the tube bodies are blocked by the +caps and also occluded by surrounding tubes when placed +in the rack and viewed from a top position. They are less +visible. However, despite the reasons and their merits, there is +a problem that different types of tubes may share a same cap +type. In this work, we assume the test tubes with the same +caps can be identified by their heights in the rack and analyze +the point cloud to differentiate them. +B. Workflow for Data Preparation +We prepare the training data using the robot system follow- +ing the workflow shown in Fig. 3. There are four dashed boxes +in the chart, where (a.1) and (a.2) have a blue background color +and represent the data collection component, (b) has an orange +background color and represents the data synthesis component, +(c) has a gray background and represents the resulted data. +The first blue dashed box (Fig. 3(a.1)) comprises three steps. +First, a human hands over an unknown test tube to the robot. +The tube is assumed to be grasped vertically by the robot after +handover, with the tube cap left above the robotic fingertips. +Second, the robot moves the test tube to the observation poses +prepared offline while considering avoiding self-occlusions. +The Phoxi sensor will capture the test tube’s gray image +and point cloud at each observation pose. Third, the system +segments the cap region out of the captured image based on +a mapping from its counterpart point cloud. The segmentation +result only includes the cap. The background will be removed +thanks to the point cloud mapping. The output of this dashed +box includes many cap region pictures. They are observed +from different views and thus have different illumination and +visual conditions. +The second blue dashed box (Fig. 3(a.2)) is similar to the +first one and also comprises three steps. First, a person places +a rack in the environment. Then, the robot pushes the rack to +random poses, capturing the rack’s gray image and point cloud +at each pose. Third, the system segments the rack region out +of the captured image based on the mapping from the rack’s +counterpart point cloud. The result of this dashed box includes +many rack region pictures. Like the caps, the rack region +pictures also have different illumination and visual conditions +since the data is captured from different view positions. +The orange dashed box shows the data synthesis process, +where the cap region pictures obtained in the first “Data +Collection” dashed box are pasted onto the rack region pictures +obtained in the second “Data Collection” dashed box for +synthesizing new images. Constraints like rack boundaries and +overlapping caused by perspective projection are considered +during the synthesis. The output of the dashed box will +be racks filled with many tube caps. The “Copy-paste data +synthesis” sub-block illustrates several examples of the output. +The final data preparation results include the images ob- +tained during collecting the tube cap data (observation images) +and the synthetic images. They are illustrated in the gray +dashed box (Fig. 3(c)). +Note that the above workflow is not completely automatic. +The sub-blocks with texts highlighted in a green color involve +human intervention. Also, before data collection, we need to +prepare the camera calibration matrix and test tube observation +poses. The camera calibration matrix transforms the point +cloud captured in the camera’s local coordinate system into +the robot coordinate system. Many existing methods exist for +obtaining the calibration matrix [42]. To avoid repetition, we +don’t discuss the details in this manuscript. The test tube +observation poses are a set of tube positions and rotations +for the robot to hold and capture observation images. The +developed method will generate robot joint configurations +considering the robot grasping and tube observation poses. +Section IV will present detailed algorithms on the generation. +IV. IMPLEMENTATION DETAILS +A. Observation Poses for Collecting Tube Caps +When collecting the tube cap data, the robot moves the +tube held in its hand to different poses for observation. The +observation poses are generated considering two constraints: +(1) Diversity of the captured cap data; (2) Occlusions by robot +links. Taking into account these two constraints allow us to +include the tube caps from many viewpoints and thus cover +lots of illumination and visual conditions. Meanwhile, they +help to prevent the robot links from occluding the grasped +test tubes and make sure the tubes are visible to the vision +sensor. +Fig. 4 illustrates the observation pose generation process and +how the two constraints are taken into account in it. First, we +sample the positions and rotations of a tube held by the robot +hand uniformly in the Phoxi depth sensor’s visible range. Tube + +(b) +_Tube +(a) +Phoxi M +Holder +3D Scanner +Rack +Yumi Robot +Purple +Tube +Flat Table +AB +Purple +Ring +Blue +White +In-rack +Tube +Tube +Tube +Test TubesARXIV VERSION, 2022 +5 +Fig. 3: Workflow of the proposed method. (a.1,2) Data collection component. (b) Data synthesis component. (c) Resulted data. +data captured under the sampled poses will have rich light +conditions and a large variety of visible tube edges for training +a recognition neural network. Especially, the tube rotations are +sampled according to the vertices of a level-four icosphere +[43]. An icosphere is a spherical polyhedron with regularly +distributed vertices. The vectors pointing to the vertices of an +icosphere help to define the rotations of a tube1. A level-four +icosphere has 642 vertices and thus leads to 642 vectors and +test tube rotation poses. Thanks to the visibility constraints, +we do not move a test tube to all of the rotation poses for +capturing data as the tube caps facing downward will not +be seen by the Phoxi sensor. We filter the 642 vectors by +considering their angles with the normal of the table surface +for placing a rack. The vectors with large angles from the +surface normal cannot be seen and will not be considered. The +spherical polyhedron in Fig. 4(b.1) illustrates the level-four +icosphere. Vectors pointing to the red vertices have more than +θ angles from the surface normal and are removed. The green +vertices are the remaining candidates. The purple tube bouquet +on the right side of Fig. 4(b.1) illustrate the tube poses implied +by vectors pointing to the remaining candidate vertices. +Next, we plan the robot motion to move the test tube held in +a robot hand to the sampled tube positions and rotations. We +assume a test tube is vertically grasped at the finger center of a +robot hand. Since a tube is central symmetric, many grasping +poses meet the assumption. The grasping hand may rotate +freely around the symmetry axis of the test tube, as shown +in Fig. 4(b.2). The rotation is compact and forms a SO(2) +group. For numerical analysis, we sample the rotation in the +SO(2) group with a rotation interval hyperparameter named +ω to obtain a series of discretized grasping poses. The hand +1A tube is centeral symmetric. We do not need to consider its rotation +around the central axis. The vectors pointing to the vertices of an icosphere +can thus define a tube pose. +illustrations in Fig. 4(b.2) are the grasping poses obtained with +ω = 60◦. The sampled grasping poses provide many candidate +goals for robot motion planning and thus increase the chances +of successfully moving and observing the tube. +When determining which exact candidate goal to move to, +we examine the occlusions from the robot arm links and +avoid choosing the grasping poses that lead to invisible tubes. +In detail, examining the occlusion is done by checking the +collision between a visual polyhedron and the robot arm links. +The visual polyhedron is computed using the camera origin +and vertices of the robot hand model, as illustrated in Fig. +4(c.1). The robot arm may occlude the tube and the vision +sensor fails to capture it when there is collision between +the visual polyhedron and the robot arm links. Fig. 4(c.2) +exemplifies such a case. +B. Using Annotation Masks to Segment Cap Pictures +Since the tube is handed over from a human and the +Phoxi sensor captures the cap data from many different +views, the captured tube point clouds change dynamically +and have noises. It is unstable to extract cap point clouds by +autonomously detecting them. Thus, instead of autonomous +detection, we prepare an annotation mask in the robot hand’s +local coordinate system to help extract the test tube cap’s point +clouds. The extracted point clouds will be back-projected to +the corresponding 2D grey image for segmenting a picture +of the cap region. Fig. 5 shows the details of this mask and +how it helps to segment the cap regions. The mask and back +projection enable us to precisely segment the cap regions while +avoiding including backgrounds. +To prepare an annotation mask, we move the robot hand +that holds a test tube to a fixed position under the Phoxi +sensor and trigger the sensor to capture a point cloud. We can + +(a.1) Collecting the tube cap data +(i) A human +(ii) The robot moves +Many pictures of the tube caps +(iii) Crop the tube cap +hands over +the test tube to new +based on the mapping +a test tube +observation poses +between the captured +to the robot +for capturing data +point cloud and image +(a.2) Collecting the rack data +(i) A human +(i) The robot +(iii) Crop the +Many pictures of the racks +places a +pushes the rack +rack from the +rack in the +to new poses for +environmental +environment +capturing data +background +(b) Data synthesis +(c) Resulted data (Images with known tube caps) +Pictures of tube caps + Pictures of racks +Images obtained +during collecting +Copy-paste data synthesis +the tube cap data +(Top view pictures +with tubes held in +Synthesized racks with +the robotic hands) +different backgroundsARXIV VERSION, 2022 +6 +Fig. 4: (a) Sampling observation positions. The green region +is the visible area of the Phoxi scanner. The red points are +the sampled positions. (b.1) Sampling rotations based on a +level-four icosphere. The left spherical polyhedron illustrates +the icosphere. The green vertices are the ends of feasible +vectors that have less than θ = 60◦ angles with the surface +normal. They imply the tube rotation poses shown on the right. +(b.2) The grasping poses for each sampled tube pose form a +SO(2) group. They are sampled considering an interval ω for +numerical analysis. (c.1) A visual polyhedron computed using +the camera origin and vertices of the robot hand model. (c.2) +The grasped object has a risk of being occluded by the robot +arm when there is a collision between the visual polyhedron +and the robot arm links. +Fig. 5: Workflow for extracting the cap picture using an +annotation mask. (a) Applying a mask described in the local +coordinate system of the holding robot hand to the captured +point cloud. (b) The extract point cloud is projected back to +the 2D grey image for segmenting a picture of the cap region. +(b.1) The back-projected results might be disconnected pixels. +(b.2) A bounding convex hull of the disconnected pixels is +computed. (b.3,4) The cap region is segmented based on the +bounding convex hull. +easily get the cap’s point cloud data by examining the area +on top of the holding fingers and obtain an annotation mask +by considering a bounding polyhedron of the data. However, +a single bounding polyhedron may not be general for others +since the captured point cloud is susceptible to light reflection +or perspective projection (self-occlusion). Thus, instead of a +single point cloud and polyhedron, we collect point clouds +from multiple views, merge them under the robot hand’s local +coordinate system, and compute a bounding box of the merged +result as an annotation mask. Fig. 6 shows an example. The +multiple views are sampled the same way as the observation +poses mentioned in the previous subsection. However, we do +not need to change the observation positions since we aim to +Fig. 6: (a) Capture data from different views. The tube cap’s +point clouds are obtained by examining the area on top +of the holding fingers. They are high lighted with colored +polyhedrons. (b) Merge the cap’s point clouds in (a) under +the robot hand’s local coordinate system, and compute a +bounding box of the merged result as an annotation mask. (b.1) +Raw bounding box. (b.2) The bounding box can be adjusted +interactively if needed. +obtain a bounding box mask in the hand’s coordinate system. +The views under various rotations could provide enough +superficial point cloud data to meet the requirements. Note +that the merged result may include noise point data induced by +reflections from the transparent tube body and lead to a mask +larger than the cap. We provide an interactive user interface +for manually adjusting the bounding box sizes and minimizing +the negative influences caused by the noises. The adjustment +is optional and may be performed when precisely segmenting +the cap region is demanded. +C. Copy-Paste Synthesis +We apply random scaling, blurring, brightness, and contrast +to the segmented tube caps and then paste them onto the +segmented rack background for data synthesis. During pasting, +we permit the overlap among the cap regions to approximate +tube-to-tube occlusion. After pasting, we randomize the en- +vironmental background (background of the rack) to narrow +further the domain gap between synthetic images and images +captured in the real world. +A critical maneuver here is that we consider the co- +occurrence of the test tubes and the rack and paste the tube +cap pictures onto a rack instead of random backgrounds like +[11]. We randomly sample positions inside rack pictures for +pasting tube caps and use a pasting number T to control the +clutter. Note that there is no need to exactly paste a tube cap +near the hole centers of a rack as the tubes tilt randomly inside +the rack holes. The visible cap regions may reasonably overlap +with a hole boundary or other holes. +For tube-to-tube occlusion, we consider the perspective +projection of a vision sensor and define an occlusion threshold +t to permit overlap among the visible cap regions. A vision +sensor’s perspective projection leads to mutual occlusions in +the rack at certain viewpoints. The occlusion threshold helps to +simulate the occlusion and defines the maximum percentage +that segmented cap pictures can overlap or occlude. Fig. 7 +shows how the t threshold works. It adds a constraint to +pasting, where a previously pasted cap picture “A” must have +less than t percentage overlap with the union of caps pasted +later. The B ∪ C ∪ ... component in the nominator of Fig. +7 implies the union of caps pasted after “A”. When a new +cap is randomized, it must be unioned with this component + +(a) +(b.1) +0 = 60° +Surface +Test +normal +tube +Positions +Rotations +(b.2) +(c.1) +(c.2) +Visual +Gripper +polygon +.09 = 3(a) +(b) +Annotation Mask +(b.1) +(b.2) +ZH +(b.3) +(b.4) +ZH(b) +(b.1) +(b.2)ARXIV VERSION, 2022 +7 +to ensure the t constraint on all previous “A” is not violated. +There are two noticeable points for t. First, its value could be +devised respectively considering the heights of specific tube +types. Second, its value is correlated with the pasting number +T. The maximum number of pasted tube caps in a rack that +meet the t threshold may be less than a given T. In that case, +we constrain the maximum number of pasted tube caps to the +smaller value to ensure t is not invalidated. +Fig. 7: (a) Using a threshold t to simulate cap occlusions. +“A” represents a previously pasted cap region. “B”, “C”, ... +represent the caps pasted after “A”. (b) Results with different +t values. +For the environmental background, we use the BG-20k data +set [44] to obtain high-resolution random background images +and change the background of a synthetic image with a 0.5 +probability. +V. EXPERIMENTS AND ANALYSIS +We carried out experiments to compare YOLOv5x [45] +detectors trained using data sets collected with the proposed +method and several other methods to understand the per- +formance. Table II shows the methods. The SR (Synthesis +by pasting to Racks) method pastes randomly selected cap +pictures onto rack backgrounds to synthesize training data. +It represents the synthesizing method used in this work. +The SB (Synthesis by pasting to BG-20k) method is an +alternative synthesis method. Instead of being pasted onto a +rack, randomly selected cap pictures are pasted to random +backgrounds selected from the BG-20k data set. The RO +(Robotic Observation) method is a byproduct of robotic cap +segmentation, where the robot holds test tubes for data col- +lection. We considered RO an independent method because +we wondered if the hand-held observation was enough for +training. We also combined RO, SR, and SB methods (the ** +row) to see if they help achieve a satisfying performance. The +RO+SR combination is exactly our proposed method in this +work. We especially proposed it since RO is a pre-process +of robotic cap segmentation. Using combined RO+SR does +not increase effort. Combining RO+SB or RO+SR+SB are +also candidate choices. They have the same cost as using +independent SR or SB data2. Finally, the CL (Crowd-source +Labeling) method is a conventional one that requires humans +to place racks with tubes under the robot and label the captured +images manually. Fig. 8 shows exemplary images collected +using the different methods. +2Synthesizing data is considered to be free as it only require computational +work. Thus, the costs of SR and SB depend on the RO process. +TABLE I: Summary of the data collection methods +Abbr. +Full Name +Description +SR +Synthesis by pasting to Racks +Caps on racks +SB +Synthesis by pasting to BG-20k +Caps on random background +RO +Robotic Observation +Tubes held in robotic hands +** +Combinations of above methods +SR+SB is the proposed one +CL +Crowd-source Labeling +Tubes in a rack on the table +Fig. 8: Exemplary images collected using the various methods. +(a) RO. (b) SR. (c) SB (d) CL. +A. Performance of Various Data sets +We collected various data sets with the methods and their +combinations, used the data sets to train YOLOv5x detectors, +and examined the performance of the trained detectors using +a testing data set for comparison. +The first data set is CL200. It is considered a baseline for +comparison. In collecting the data set, we collected 200 images +with random tube and rack states and labeled the tube regions +manually using LabelImg3. There are, in total, 5916 labeled +instances in the 200 images. +The second data set is SR1600. In order to collect it, we +first prepared many cap pictures using robotic observation. As +shown in Fig. 2(b), we assumed four different test tubes and +took advantage of the Yumi robot’s both arms to collect cap +data quickly. For each tube type, we handed over two same +ones to the two robotic arms for observation. Each arm moved +its held tube to 400 observation poses for data collection. See +Fig. 9(a) for example. Here, we set the hyperparameter θ and +ω to 30◦ and 360◦ (single grasping pose) and set the positions +to be evenly sampled on the table with a granularity of 0.1m +for generating the observation poses. In total, more than 400 +observation poses were obtained under the parameter setting +for each arm, and we used the first 400 for collecting images. +As a result, we obtained 400 observation images (800 cap +pictures since there are two tubes in each image, see Fig. +9(b) for example) for a single tube type and 1600 observation +images for all tube types. We segmented 3200 pictures of +cap regions from the observation images considering point +cloud mapping. Fig. 9(c) shows the collected point clouds with +highlighted caps (green). Fig. 9(d) shows the segmented cap +regions. Besides the cap regions, we collected 15 images with +racks (a single rack in each image) and segmented 15 pictures +of racks. We synthesized a data set of 1600 images by pasting +caps randomly selected from the 3200 cap pictures to racks +randomly selected from the 15 rack pictures (SR method). +During synthesis, we set the pasting number to be T = 30, and +set the occlusion threshold for the “Blue Tube” to be tblue = +0.4 and other tubes to be tothers = 0.15. We chose these +parameter settings because the “Blue Tube” was shorter and +susceptible to occlusion. We increased its occlusion threshold +3https://github.com/heartexlabs/labelImg + +(a) +IAN(BUCU. +Overlap(A, B, C, ...) +[A| +(b) +10 +t = 0.3 T = 20 +t = 0.6 T = 40a +dARXIV VERSION, 2022 +8 +to mimic frequent visual blockage from other tubes. Also, +we increased the variety of the segmented cap pictures by +applying random scaling (0.9 ∼ 1.1 of original picture size, +0.5 probability), random blur (3 × 3 kernel, 0.5 probability), +random brightness (0.9 ∼ 1.1 of original brightness, 0.5 +probability), and random contrast (0.9 ∼ 1.1 of the original +contrast value, 0.5 probability) using the Albumentations4 +library. The background of the rack was randomly chosen from +the BG-20k data set with a 0.5 changing probability. +Fig. 9: (a) The robot moves test tubes for observation. Both +arms are used. (b) Observation Image. (c) Point clouds cap- +tured by the Phoxi sensor. (d) Cap pictures segmented from +the observation image. +The third data set is RO1600. It is a semi product of robotic +cap segmentation and comprises the 1600 observation images +obtained during robotic observation. +The fourth data set is SB1600. In contrast with the SR1600 +data set, we pasted randomly selected caps directly to images +from the BG-20k data set for obtaining data. The pasted +caps might freely distribute on the image background. The +segmented racks were not used. The pasting number T and +occlusion threshold t are 35 and 0.15 respectively. There was +no difference on t for different tubes. The randomization were +performed in the same way as obtaining SR1600. +We also used combined methods to collect data sets +and study if the combination led to better results. The +combined data sets include RO1600+SR800, RO1600+SB800, +RO1600+SR400+SB400, SR800+SB800. Here, the superscript +number on the upper-right of a method name means the +number of images collected using the method. The “+” sym- +bol indicates that the data sets comprise data collected using +different methods. The RO1600+SR800 data set represents the +data collected using the proposed method. +The left part of Table II summarizes the various data sets. +They are used to train YOLOv5x detectors for comparison. +Before training, the YOLOv5x detectors for all data sets were +initialized with weights pre-trained using the COCO data set. +The images in all data sets were regulated into a resolution of +1376 × 1376. Each data set is divided into a training subset +and a validation subset according to a 4 to 1 data ratio. During +learning, the training subset was fed to the training program +with a batch size of 2, and the training program performed +validation per episode. The training process was stopped +when the mAPs (mean Average Precision) [46] for all objects +reached higher than 99.0% under a 0.5 IoU (Intersection over +Union). Here, we defined a detected bounding box to be +correct when its IoU with a ground truth cap bounding box +was larger than 0.5. +4https://albumentations.ai/ +For evaluating the performance of YOLOv5x detectors +trained using the various data sets, we collected a testing +data set with 100 images and labeled their ground truth +using the same method as CL. We used the trained detectors +to detect tubes in the testing data set. Like validation, we +defined a detected bounding box as correct when its IoU +with a ground truth cap bounding box is larger than 0.5. +We used the AP (Average Precision) metric to measure the +detection performance of a single object class and used the +mAP for all objects. Since the detector that met a single +satisfying validation was not necessarily the best, we trained +each detector twice and took the higher precision value on the +testing data set as the final evaluation result. +Table II shows the evaluation results. We obtained the +following observations and speculations from them. +i) Using the data set collected by robotic observation for +training exhibited the worst performance, as shown by +the 2nd row (RO1600). +Speculation: All images in the data set had a similar +robotic background. They suffered from a domain shift. +ii) The synthetic data sets do not necessarily lead to a good +AP, as shown by the 3rd (SR1600) and 4th (SB1600) rows. +The SR1600 data set exhibited higher performance than +the SB1600 data set. +Speculation: The copy-paste synthesis failed to cover +certain visual contexts; Pasting onto racks (SR) provided +more effective visual contexts and benefited the neural +network more than pasting onto random backgrounds +(SB). +iii) Combining the synthetic data sets with robotic observa- +tions is effective. It can be concluded by comparing the +5th, 6th, and 7th rows (RO1600+SR800, RO1600+SB800, +RO1600+SR400+SB400) with the 2nd, 3rd, and 4th rows +(RO1600, SR1600, and SB1600). The former rows had +higher mAP than the latter. +Speculation: The robotic observation data set additionally +provided helpful visual contexts. +iv) The 5th row (RO1600+SR800) had a 2.4% higher mAP +than the 6th row (RO1600+SB800). Especially, the AP of +the “Blue Tube” on the 5th row was 8.7% higher than +that on the 6th row. The AP of other tubes also had +0.1% ∼ 0.7% performance increase. +Speculation: Considering the rack as a local context +helped improve domain-specific performance; The short +“Blue Tube” could be easily blocked. The data set +collected using the SR method had more simulated oc- +clusions. They were important for recognizing the short +“Blue Tube”. +v) The 7th row (RO1600+SR400+SB400) exhibited slightly +higher mAP (0.3%) than the 5th row (RO1600+SR800). +Speculation: Pasting onto racks (SR) provided better +domain-specific features. Random backgrounds for the +tubes slightly benefited the neural network and were less +necessary if the goal context was limited. +vi) The 5th row (RO1600+SR800) is competitive compared +with the 1st row (CL200). The mAP was 0.7% lower. The +AP of the “Blue Tube” and “White Tube” were 2.5% and +1.1% lower, respectively. The AP of the “Purple Tube” + +(c) Point clouds +(d) Cap +(b) +picturesARXIV VERSION, 2022 +9 +TABLE II: Comparison of detectors trained using different data sets +AP +ID +Data Set Names +# Caps +Remark +Blue +Purple +White +Purple Ring +mAP +1 +CL200 +5916 +Multiple tubes / image +0.993 +0.995 +0.989 +0.984 +0.990 +2 +RO1600 +3200 +Two tubes / image +0.380 +0.923 +0.695 +0.630 +0.657 +3 +SR1600 +40000 +tblue=0.4 & tothers=0.15, T = 25 +0.955 +0.979 +0.871 +0.953 +0.940 +4 +SB1600 +56000 +t=0.15 (same for all tubes), T = 35 +0.808 +0.978 +0.812 +0.897 +0.874 +5 +RO1600+SR800 +23200 +See note 2 +0.968 +0.995 +0.978 +0.992 +0.983 +6 +RO1600+SB800 +31200 +See note 2 +0.881 +0.994 +0.971 +0.992 +0.959 +7 +RO1600+SR400+SB400 +27200 +See note 2 +0.969 +0.994 +0.986 +0.993 +0.986 +8 +SR800+SB800 +48000 +See note 2 +0.973 +0.993 +0.969 +0.985 +0.980 +Note 1: Largest AP and mAP values are highlighted in bold. +Note 2: The combined data sets are collected using the same parameters as respective ones. +was the same. The AP of the “Purple Ring” tube was +0.8% higher. +Speculation: The robotic observation and paste-to-rack +synthesis compensated for each other’s shortcomings; +There remained extreme cases that could be labeled +manually but failed to be covered by robotic observation +or synthesis, especially for the “Blue Tube”. +Several failure cases are visualized in Fig. 10 to provide +the readers an insight into our observations and speculations. +Fig. 10(a) and (b) exemplify the recognition results of detec- +tors trained using the 5th (RO1600+SR800) and 6th data sets +(RO1600+SR800). The latter one failed to recognize occluded +tubes as the training data set had fewer simulated occlusions. +The example is consistent with the observation and speculation +in iv). Fig. 10(c) and (d) exemplify cases that the detectors +trained using the 5th (RO1600+SB800) data set failed. In the +first case, shadows from other test tubes were cast on a blue +test tube cap. The detector failed to recognize the tube. In the +second case, the detector misrecognized a crystal tube body +as the “Blue Tube” cap due to the illusion caused by body- +and-rack overlap. The two failure examples are consistent with +the observation and speculation in vi). The synthetic data sets +do not involve shadows or tube bodies. The detectors trained +using them had worse performance in these cases than the one +trained using the crowd-sourced real-world data. +Fig. 10: (a) Detector trained using the 5th data set successfully +recognized all tubes. (b) Detector trained using the 6th data +set failed to recognize the occluded tube in the red circle. (c) +Detector trained using the 5th data set failed to recognize the +shadowed tube in the red circle. (d) Detector trained using the +5th data set misrecognized the tube body in the red circle as +a “Blue Tube”. +In summary, the results of the various training data sets +showed that combining data collected using the RO and SR +methods was effective. The conclusion was satisfying as the +RO method is a subset of the SR method. The workflow for +collecting them is simple and clean. However, we wonder if +the number of images in the RO data set could be reduced, +as it needs much manual handover to collect them. This +query prompted us to carry out the studies in the following +subsection. +B. Ablation Study +In this subsection, we conduct multiple ablation studies on +the combined RO+SR data set to further understand 1) the +influence of the data combination ratio and 2) the influence +of pasting number T and occlusion threshold t used for +generating synthetic data. +1) Influence of data combination ratio: The experiments for +studying the influence of data combination ratio are divided +into two parts. In the first part, we set the number of images +collected using the RO method to 800 and varied the number +of images collected using the SR method from 200 to 1600 +in a 2-fold ratio to understand the importance of the SR data. +The upper section of Table III shows the precision of detectors +trained using the varied data. The results indicate that the mAP +improved when the SR image numbers increased from 200 to +1600. The second part is similar to the first one. In this part, we +fixed the number of images collected using the SR method to +800 and varied the number of images collected using the RO +method from 200 to 1600 in a two-fold ratio to understand +the importance of the RO data. The lower section of Table +III shows the precision of detectors trained using the varied +data. The result indicates that the mAP improved when the +RO image numbers increased from 200 to 1600. +2) Influence of hyperparameters: Besides the data combi- +nation ratio, we also studied the influence of pasting number +T and occlusion threshold t used in the SR method. We set +both the RO and SR image numbers to 800 and observed +the performance of detectors trained with data sets collected +using different T and t values. Although we previously used a +different t value for the “Blue Tube”, we did not differentiate +the tubes here. Like the study on different data combination +ratios, this study also comprised two parts. In the first part, +we fixed t to be 0.1 and increased T from 10 to 40 with +a step length of 10. The upper section of Table IV shows +the precision changes under the parameter variations. The + +ring +ng0.94 +(d) +a +b +cARXIV VERSION, 2022 +10 +TABLE III: Influence of data combination ratio +AP +Data Set Names +Blue +Purple +White +Purple Ring +mAP +RO800+SR200 +0.958 +0.994 +0.973 +0.987 +0.978 +RO800+SR400 +0.964 +0.992 +0.975 +0.985 +0.979 +RO800+SR800 +0.966 +0.995 +0.979 +0.986 +0.981 +RO800+SR1600 +0.970 +0.994 +0.987 +0.987 +0.985 +RO200+SR800 +0.962 +0.992 +0.952 +0.978 +0.971 +RO400+SR800 +0.965 +0.992 +0.979 +0.978 +0.979 +RO800+SR800 +0.966 +0.995 +0.979 +0.986 +0.981 +RO1600+SR800 +0.968 +0.995 +0.978 +0.992 +0.983 +Note 1 Largest AP and mAP values are highlighted in bold. +Note 2 We used the following hyper-parameter setting tblue += +0.4 & +tothers = 0.15, T = 30 to collect the SB data sets. The values were +the same as the experiments in Section V.A. +results exhibited a significant increase from 10 to 30. However, +an even larger T had little influence on the recognition +performance. In the second part, we set T to be 30 and varied t +from 0.20 to 0.80 with a step length of 0.2. The lower section +of Table IV shows the precision changes under the parameter +variations. The results exhibited a clear precision increase on +the “Blue Tube”. We speculate that the reason was that the +“Blue Tube” was shorter and vulnerable to occlusions. A larger +t helped provide more occlusion cases in the training data set, +leading to a higher detection rate. The results also indicated +that the precision of the ”White Tube” and ”Purple Ring Tube” +irregularly changed as the t increased. They were taller and did +not suffer from occlusions. Adding occlusions for them caused +unexpected errors. For a complete observation, we recommend +interested readers to compare with the third row of the table’s +upper section to catch the changes starting from t = 0.1. The +T value of the upper section’s third row was the same as the +rows in the lower section. +TABLE IV: Influence of parameters used for synthesis +AP +Params. (T, t) +Blue +Purple +White +Purple Ring +mAP +(10, 0.10) +0.904 +0.995 +0.971 +0.973 +0.961 +(20, 0.10) +0.915 +0.995 +0.976 +0.985 +0.968 +(30, 0.10) +0.939 +0.995 +0.987 +0.992 +0.978 +(40, 0.10) +0.934 +0.994 +0.972 +0.983 +0.970 +(30, 0.20) +0.945 +0.995 +0.985 +0.989 +0.978 +(30, 0.40) +0.969 +0.995 +0.985 +0.994 +0.986 +(30, 0.60) +0.985 +0.995 +0.965 +0.967 +0.978 +(30, 0.80) +0.987 +0.995 +0.984 +0.988 +0.988 +* Largest AP and mAP values are highlighted in bold. +C. Further Analysis on Synthetic Data +We also studied the influence of cap variation and combina- +tion ratio on synthetic data sets (the data sets collected using +the SR, SB, or SR+SB methods). The goal was to understand +the best performance we could reach with synthesis. +First, we fixed the number of images collected by the SR +and SB methods to 800, respectively. We changed the number +of cap region pictures (equals to the number of observation +images multiplied by two) used for synthesis from 400 to 3200 +in a 2-fold ratio to study the influence of cap variation. The +previsions YOLOv5x detectors using the changing data sets +are shown in Table V. The results showed that the 400 row +had competitive precision compared to the 1600 or 3200 rows. +The number was enough to support a satisfying detector. The +cap variations were thus considered to have a low influence +on learning. +Second, we fix the number of cap region pictures to 3200 +and change the number of images collected using the SR +and SB methods, respectively, to study the influence of the +combination ratio. Like the ablation study in Section V-B1, +we divided the experiment here into two parts. In the first +part, we set the number of images collected by the SR method +to 800 and varied the number of images collected by the SB +method from 200 to 1600 in a 2-fold ratio to understand the +importance of the SB data. The upper section of Table VI +shows the precision of detectors trained using the varied data. +The number of SB images did not appear to be positively +correlated with the final detector’s precision, although the +largest mAP was observed when the number of SB images +was 800. In the second part, we fixed the number of images +collected by the SB method to 800 and varied the number +of images collected by the SR method to understand the +importance of the SR data. The lower section of Table VI +shows the precision of detectors trained using the varied data. +The result indicated that the mAP improved as the SR image +number increased to 800. There was no significant difference +when the image number increased from 800 to 1600. +TABLE V: The influence of #caps to synthesis +AP +#Caps +Blue +Purple +White +Purple Ring +mAP +400 +0.970 +0.994 +0.969 +0.984 +0.979 +800 +0.971 +0.993 +0.954 +0.976 +0.973 +1600 +0.971 +0.992 +0.980 +0.985 +0.982 +3200 +0.973 +0.993 +0.969 +0.985 +0.980 +TABLE VI: Influence of the SR and SB ratio +AP +Data Set Names +Blue +Purple +White +Purple Ring +mAP +SB200 + SR800 +0.975 +0.994 +0.943 +0.957 +0.967 +SB400 + SR800 +0.973 +0.990 +0.960 +0.970 +0.973 +SB600 + SR800 +0.967 +0.988 +0.951 +0.968 +0.969 +SB800 + SR800 +0.973 +0.993 +0.969 +0.985 +0.980 +SB1600 + SR800 +0.952 +0.978 +0.926 +0.972 +0.957 +SB800 + SR200 +0.951 +0.982 +0.925 +0.963 +0.955 +SB800 + SR400 +0.932 +0.969 +0.914 +0.952 +0.942 +SB800 + SR600 +0.966 +0.990 +0.930 +0.893 +0.945 +SB800 + SR800 +0.973 +0.993 +0.969 +0.985 +0.980 +SB800 + SR1600 +0.975 +0.993 +0.967 +0.986 +0.980 +Note 1 Largest AP and mAP values are highlighted in bold. +Note 2 We used the following hyper-parameter setting tblue += +0.4 & +tothers = 0.15, T = 30 to collect the SB data sets, and used the following +hyper-parameter setting T = 30, t = 0.15 (same for all tubes) to collect +the SR data sets. The values were the same as the experiments in Section +V.A. +Note 2 We used 3200 segmented cap region pictures for both methods. + +ARXIV VERSION, 2022 +11 +VI. CONCLUSIONS +In this paper, we proposed an integrated robot observation +and data synthesis framework for data preparation. The pro- +posed framework can significantly reduce the human effort in +data preparation. It required only a single process and was a +low-cost way to produce the combined data. The experimental +result showed that combined observation and synthetic images +led to comparable performance to manual data preparation. +The ablation studies provided a good guide on optimizing data +configurations and parameter settings for training detectors +using the combined data. +REFERENCES +[1] C. Shorten and T. M. Khoshgoftaar, “A survey on image data augmen- +tation for deep learning,” Journal of big data, vol. 6, no. 1, pp. 1–48, +2019. +[2] A. Gupta, A. Vedaldi, and A. 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Ye, “Object detection in 20 years: A +survey,” arXiv preprint arXiv:1905.05055, 2019. + diff --git a/09AzT4oBgHgl3EQfevxv/content/tmp_files/load_file.txt b/09AzT4oBgHgl3EQfevxv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4f41b086883b260dcd871154787120726cb170e --- /dev/null +++ b/09AzT4oBgHgl3EQfevxv/content/tmp_files/load_file.txt @@ -0,0 +1,1234 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf,len=1233 +page_content='ARXIV VERSION, 2022 1 Automatically Prepare Training Data for YOLO Using Robotic In-Hand Observation and Synthesis Hao Chen1, Weiwei Wan1∗, Masaki Matsushita2, Takeyuki Kotaka2 and Kensuke Harada13 Abstract—Deep learning methods have recently exhibited im- pressive performance in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, such methods needed much training data to achieve high recognition accuracy, which was time-consuming and required considerable manual work like labeling images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In this paper, we automatically prepare training data using robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Considering the low efficiency and high energy consumption in robot motion, we proposed combining robotic in-hand observation and data synthesis to enlarge the limited data set collected by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We first used a robot with a depth sensor to collect images of objects held in the robot’s hands and segment the object pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Then, we used a copy-paste method to synthesize the segmented objects with rack backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The collected and synthetic images are combined to train a deep detection neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We conducted experiments to compare YOLOv5x detectors trained with images collected using the proposed method and several other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results showed that combined observation and synthetic images led to comparable performance to manual data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They provided a good guide on optimizing data configurations and parameter settings for training detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The proposed method required only a single process and was a low-cost way to produce the combined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Interested readers may find the data sets and trained models from the following GitHub repository: github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='com/wrslab/tubedet Note to Practitioners—The background of this study is a requirement in lab automation – Using robots to arrange randomly placed tubes automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Before sending test tubes to an examination machine for gradient tests, humans need to categorize and organize the tubes into specific patterns to fit the machine’s internal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Employing humans is difficult as the tube arrangement requirements are time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A preferred solution is using robots to replace humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The robots should have a vision system to detect the tubes and a manipulation system to perform physical arranging actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They will be used in busy seasons while deployed for other tasks in leisure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Deep neural networks like YOLO are effective for the tube detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, preparing the training data is challenging and unsuitable for lab end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Pre-trained neural networks are options but have limited tube detection ability and cannot deal with newly included tube types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The method developed in this work helps solve the training data preparation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' With its support, the robot can automatically prepare training data that has comparable quality to manually labeled ones in a single-process and low-cost way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Index Terms—Robotic data preparation, data synthesis, test tube detection I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' INTRODUCTION Recent advances in deep learning have led to a revolution in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Deep learning-based methods use deep 1Department of System Innovation, Graduate School of Engineering Sci- ence, Osaka University, Toyonaka, Osaka, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Group Research Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=', Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3National Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' of AIST, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' ∗Contact: Weiwei Wan, wan@sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='jp Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 1: Several examples of in-rack test tube detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Each grid includes two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The left image is captured by a vision sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The right image is the recognition result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The data used for training the detection neural network is prepared using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' neural networks to learn features from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They outperform traditional hand-crafted features with impressive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Despite these advantages, deep learning-based object detection requires collecting a large amount of labeled data for training, which is time-consuming and labor-intensive, and has significantly hindered the scalability and flexibility of deep learning-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Previously, researchers have developed several methods to reduce data collection costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, data augmentation [1] enriched existing training data sets by applying random transformations like image rotation or scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Data synthesis [2][3] generated previously unseen data using simulation or adversarial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The main challenge of the aug- mentation or synthesis methods was the “domain gap” [4][5]: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='01441v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='CV] 4 Jan 2023 ARXIV VERSION, 2022 2 Augmented data had less varied visual contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Synthesized data was prone to discrepancies with the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Recently, researchers have revisited using the copy-paste method [6] to increase data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The method was effective in compensating for the “domain gap” problem, exhibiting impressive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There is no clear boundary between augmentation and synthe- sis when using the copy-paste method to generate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It was mainly classified as an synthesis method [7][8], although some studies considered it to be augmentation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' This paper calls it a synthesis method to avoid confusion with transformation and scale-based data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The most tiring aspect of the copy-and-paste method is how to neatly cut a large variety of target regions and paste them onto a new background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Previously, researchers work- ing on robotic manipulation have developed robotic methods to segment novel objects from backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Florence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [10], Boerdijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [11], and Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [12] respectively used robotic in-hand or non-prehensile manipulation to change objects’ observation viewpoints and segmented the objects based on the robot motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Such sys- tems could replace humans to segment goal object regions under various conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Very recent studies [11][13] has noticed the advantage, and increased data size and contextual variety by pasting the objects segmented by robotic systems onto random backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Despite their seminal proposals, the need for copy-and-paste synthesis and the impact of data volume and ratios remain undiscussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Based on the current research status, this paper further delves into using robots to collect training data automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Considering the low efficiency and high energy consumption in robotic data collection, we propose combining robotic ob- servation and copy-paste synthesis to reduce costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We assume a test tube detection task shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 1 and use a robot with a depth sensor to move and observe tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The robot collects observation images and, at the same time, segments tubes from the images for copy-paste synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The observation and synthetic images are used as training data for deep detection neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Especially for the synthesis routine, we value the co-occurrence of tubes and racks, and paste tubes inside a rack area to obtain contextual consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Also, we take into account factors like tube-to-tube occlusions and foreground changes caused by environment or visual difference to reduce unrealistic synthetic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The proposed method helps enrich the data set and resolve the “domain gap”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It does not need heavy robotic effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In experiments, we trained several YOLOv5x networks to understand the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The training data was collected using the proposed and several other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results confirmed data collected using the proposed method do have claimed advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We also con- ducted multiple ablation studies to look into the impact of data volumes and ratios when training detection neural networks using data collected with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results provided a good guide on optimizing data configurations and parameter settings for training detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The contributions of this work are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (1) We develop an automatic data-collection method in which a robot holds target objects and observes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The method yields observation images and target regions segmented from the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (2) We develop a copy-paste image synthesis method to enrich the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The method pastes object regions on various rack backgrounds to balance “domain randomization” and “domain gap”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The rack backgrounds are also automati- cally collected by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (3) We examined combinations of the observation and synthetic images and compared them with other data sets to understand the impact of data volume and ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The remaining part of this paper is organized as follows: Section II reviews related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Section III presents the hardware system and the proposed method’s workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Section IV delivers technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Section V shows experiments and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Section VI draws conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' RELATED WORK We review the related work considering robotic data collec- tion and data synthesis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Automatic Data Collection Using Robots Segmenting the object regions from a picture is the basis of automatic data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Conventional methods used simple backgrounds [14], known environments [15], or designed eas- ily identifiable gadgets [16][17] to simplify object extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The methods required careful preparation about scenes and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Robot-based methods leverage actuated robots to simplify object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They can be traced back to early studies in object recognition and 3D object modeling [18][19][20][21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' These work took advantages of robotic manipulation sequence to perceive objects from different viewpoints and segment the objects from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' From the robotic manipu- lation perspective, such segmentation can be divided into two categories: In-hand object segmentation and Interaction-based segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 1) In-hand object segmentation: Previous work on in-hand object segmentation used known robot models and handcrafted visual features to isolate in-hand objects from background environments and robot hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Krainin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [21] isolated in-hand objects’ point clouds by examining the Euclidean distance to the robot model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Welke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [19] segmented in-hand objects from images based on Eigen background subtraction, disparity map, and hand localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' These methods required manually preparing detectors for various targets considering their visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' More recent studies used deep learning to reduce the reliance on hand-crafted visual features for in-hand object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For instance, Florence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [10] proposed a self-supervised framework to segment in-hand objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The framework involved two steps that used the same training and learning routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the first step, the authors generated masks for the robot by considering combined depth and RGB information, and trained a neural network model based on the masks to differentiate the robot from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the second step, the authors masked the grasped object and train neural network models to isolate the object from the robot hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Boerdijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [13] used optical flow to ARXIV VERSION, 2022 3 respectively segment manipulators that were holding and not holding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The segmented data set were used to train a neural network for isolating manipulators and grasped objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2) Robot-object interaction: On the other hand, some re- searchers took advantages of non-prehensile robot manipu- lation like push to change object perspectives and segment them based on robot motion cues [12][22][23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [12] designed a framework to continuously refine a neural network model that generates object segmentation masks through robot interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The model initially generated hypothesis segmentation masks for objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The masks were refined based on the pixel differences of the images captured before and after robotic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The generating model was updated along with the refined masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [24] proposed to segment unknown objects in a cluttered scene while repeatedly using robotic nudge motions to interact with objects and induce geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Robotic interactive segmentation often requires a static scene or surface to permit interaction between robots and objects [25][26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is more complicated compared with the in-hand object segmentation as the object poses needs to be controlled and changed through robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A critical problem of the robotic methods is that they are unsuitable for preparing a large amount of training data as robots consume much time and energy to perform the physical motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Conducting thousands of robotic motion trajectories to collect data is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Also, the robots in the systems are fixed, have limited views, and can only collect data in a narrow range of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Neural networks trained using the collected data may suffer from contextual (background) bias and have bad generalizability [27][28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' This study focuses on robotic data collection while con- sidering leveraging data synthesis to reduce robotic usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We first ask the robot to hold a single tube and annotate the tube’s bounding polyhedron by extracting in-hand point cloud according to the robot’s tool center point (TCP) and hand model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Then, we map the annotated bounding polyhedron to 2D image regions in the robot’s camera view for extracting the tube region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The robot moves the tube to different positions and rotations to obtain many varieties of 2D images and tube regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The images and tube regions are respectively used for training and synthesizing new data in a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Data Augmentation and Synthesis Data augmentation and synthesis are the two most well- used methods to enrich training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Data augmentation generates new data by transforming the existing training data with specific rules or learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Data synthesis generates synthetic data by merging existing data with others or using computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Concurrent publications tend to mix these nomenclatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Therefore we conduct a uniform literature review of them below without differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The copy-paste method is widely used for generating syn- thetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It segments foreground objects from existing images, possibly modifies them, and pastes them onto new backgrounds [8][7][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The copy-paste method is easy to implement and shows notable performance over using pure real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Previous studies showed that it was important to carefully select the backgrounds when pasting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Divvala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [29] experimentally showed visual context benefited object detection performance and reduced detection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Dvornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [30] showed that the correct visual context when pasting object can improve prediction performance while inappropriate visual context led to negative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [31] swapped objects of the same class in different images to ensure contextual consistency between objects and backgrounds and showed using the exsiting back- grounds had better performance than random ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Also, the copy-paste method requires a data set containing many possible views of the object that are easy to be cut out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is burdensome for humans to prepare them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Graphical simulation is another popular method for synthe- sizing training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The benefits of simulation is that it allows freely changing light conditions and materials to increase variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It also allows capturing many views of objects by simply transforming virtual camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Hodaˇn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [32] and Richter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [33] respectively used photo- realistic rendering to synthesize images of 3D object models and scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The methods required a lot of computational resources to narrow down the domain gap between synthetic and realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Tobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [34] proposed the concept of domain randomization (DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They randomized a simulator to expose models to a wide range of environments and obtain varied training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Instead of photo-realistic rendering, the method only required low-fidelity rendering results to reach satisfying accuracy for medium-size objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Carlson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [35], Hinterstoisser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [4], Prakash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [36], and Tremblay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [37] respectively used DR to narrow down the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The authors randomly changed the context in simulation so that “the real data was made to be just like another simulation” [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [39] and Sundermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [40] respectively sampled viewpoints of 3D object models using simulation and mixed the samples with real backgrounds to reduce the human effort for preparing scenes with rich domain randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Besides DR, Generative Adversarial Networks (GANs) were also promising to reduce domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For example, Chatterjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' [41] designed a lightweight-GAN to synthesize data for training plastic bottle detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In this study, we leverage data synthesis to enrich the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We develop a copy-paste based method to attach tube cap regions separated from robotic observation images to rack backgrounds and thus synthesize new images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Various constraints like rack dimensions and tube occlusions can be considered during the synthesis to reduce the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The synthetic data is mixed with real-world data to promote the performance of YOLO-based tube recognition neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is also compared with other data collection methods to understand the influence of data volume and data combination ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' ROBOT SYSTEM AND WORKFLOW A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Configurations of the Robot System Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2(a) shows our robot system used for preparing the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A Photoneo Phoxi M 3D Scanner is used for ARXIV VERSION, 2022 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2: (a) The system configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) The test tubes and rack in the view of the Phoxi M 3D Scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' capturing objects on the flat table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' An ABB Yumi dual-arm robot with a two-finger gripper is used to manipulate objects in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A flat table is set up in the front of the Yumi robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The in-rack test tubes to be recognized are placed on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The Phoxi scanner is a structured-light based depth sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It can capture gray images and point clouds simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Each data point of a point cloud captured by the Phoxi scanner have a one-to-one correspondence to a pixel in a gray image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We can segment an object in the gray image by considering its point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Especially, we install the Phoxi scanner on top of the robot to obtain a top view of the racks and tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' When recognizing tubes in the rack, we select the tube caps as the primary identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There are two reasons why we prefer using the tube caps for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The first one is that obtaining the point cloud of a translucent or crystal test tube fails easily due to limitations of the structured-light based depth sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The second one is that the tube bodies are blocked by the caps and also occluded by surrounding tubes when placed in the rack and viewed from a top position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are less visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, despite the reasons and their merits, there is a problem that different types of tubes may share a same cap type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In this work, we assume the test tubes with the same caps can be identified by their heights in the rack and analyze the point cloud to differentiate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Workflow for Data Preparation We prepare the training data using the robot system follow- ing the workflow shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There are four dashed boxes in the chart, where (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) and (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) have a blue background color and represent the data collection component, (b) has an orange background color and represents the data synthesis component, (c) has a gray background and represents the resulted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The first blue dashed box (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1)) comprises three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' First, a human hands over an unknown test tube to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The tube is assumed to be grasped vertically by the robot after handover, with the tube cap left above the robotic fingertips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Second, the robot moves the test tube to the observation poses prepared offline while considering avoiding self-occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The Phoxi sensor will capture the test tube’s gray image and point cloud at each observation pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Third, the system segments the cap region out of the captured image based on a mapping from its counterpart point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The segmentation result only includes the cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The background will be removed thanks to the point cloud mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The output of this dashed box includes many cap region pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are observed from different views and thus have different illumination and visual conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The second blue dashed box (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2)) is similar to the first one and also comprises three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' First, a person places a rack in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Then, the robot pushes the rack to random poses, capturing the rack’s gray image and point cloud at each pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Third, the system segments the rack region out of the captured image based on the mapping from the rack’s counterpart point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The result of this dashed box includes many rack region pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Like the caps, the rack region pictures also have different illumination and visual conditions since the data is captured from different view positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The orange dashed box shows the data synthesis process, where the cap region pictures obtained in the first “Data Collection” dashed box are pasted onto the rack region pictures obtained in the second “Data Collection” dashed box for synthesizing new images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Constraints like rack boundaries and overlapping caused by perspective projection are considered during the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The output of the dashed box will be racks filled with many tube caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The “Copy-paste data synthesis” sub-block illustrates several examples of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The final data preparation results include the images ob- tained during collecting the tube cap data (observation images) and the synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are illustrated in the gray dashed box (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note that the above workflow is not completely automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The sub-blocks with texts highlighted in a green color involve human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Also, before data collection, we need to prepare the camera calibration matrix and test tube observation poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The camera calibration matrix transforms the point cloud captured in the camera’s local coordinate system into the robot coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Many existing methods exist for obtaining the calibration matrix [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' To avoid repetition, we don’t discuss the details in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The test tube observation poses are a set of tube positions and rotations for the robot to hold and capture observation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The developed method will generate robot joint configurations considering the robot grasping and tube observation poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Section IV will present detailed algorithms on the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' IMPLEMENTATION DETAILS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Observation Poses for Collecting Tube Caps When collecting the tube cap data, the robot moves the tube held in its hand to different poses for observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The observation poses are generated considering two constraints: (1) Diversity of the captured cap data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (2) Occlusions by robot links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Taking into account these two constraints allow us to include the tube caps from many viewpoints and thus cover lots of illumination and visual conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Meanwhile, they help to prevent the robot links from occluding the grasped test tubes and make sure the tubes are visible to the vision sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4 illustrates the observation pose generation process and how the two constraints are taken into account in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' First, we sample the positions and rotations of a tube held by the robot hand uniformly in the Phoxi depth sensor’s visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Tube (b) _Tube (a) Phoxi M Holder 3D Scanner Rack Yumi Robot Purple Tube Flat Table AB Purple Ring Blue White In-rack Tube Tube Tube Test TubesARXIV VERSION, 2022 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 3: Workflow of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1,2) Data collection component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) Data synthesis component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c) Resulted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' data captured under the sampled poses will have rich light conditions and a large variety of visible tube edges for training a recognition neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Especially, the tube rotations are sampled according to the vertices of a level-four icosphere [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' An icosphere is a spherical polyhedron with regularly distributed vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The vectors pointing to the vertices of an icosphere help to define the rotations of a tube1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A level-four icosphere has 642 vertices and thus leads to 642 vectors and test tube rotation poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Thanks to the visibility constraints, we do not move a test tube to all of the rotation poses for capturing data as the tube caps facing downward will not be seen by the Phoxi sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We filter the 642 vectors by considering their angles with the normal of the table surface for placing a rack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The vectors with large angles from the surface normal cannot be seen and will not be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The spherical polyhedron in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) illustrates the level-four icosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Vectors pointing to the red vertices have more than θ angles from the surface normal and are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The green vertices are the remaining candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The purple tube bouquet on the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) illustrate the tube poses implied by vectors pointing to the remaining candidate vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Next, we plan the robot motion to move the test tube held in a robot hand to the sampled tube positions and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We assume a test tube is vertically grasped at the finger center of a robot hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Since a tube is central symmetric, many grasping poses meet the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The grasping hand may rotate freely around the symmetry axis of the test tube, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The rotation is compact and forms a SO(2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For numerical analysis, we sample the rotation in the SO(2) group with a rotation interval hyperparameter named ω to obtain a series of discretized grasping poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The hand 1A tube is centeral symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We do not need to consider its rotation around the central axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The vectors pointing to the vertices of an icosphere can thus define a tube pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' illustrations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) are the grasping poses obtained with ω = 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The sampled grasping poses provide many candidate goals for robot motion planning and thus increase the chances of successfully moving and observing the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' When determining which exact candidate goal to move to, we examine the occlusions from the robot arm links and avoid choosing the grasping poses that lead to invisible tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In detail, examining the occlusion is done by checking the collision between a visual polyhedron and the robot arm links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The visual polyhedron is computed using the camera origin and vertices of the robot hand model, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The robot arm may occlude the tube and the vision sensor fails to capture it when there is collision between the visual polyhedron and the robot arm links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) exemplifies such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Using Annotation Masks to Segment Cap Pictures Since the tube is handed over from a human and the Phoxi sensor captures the cap data from many different views, the captured tube point clouds change dynamically and have noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is unstable to extract cap point clouds by autonomously detecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Thus, instead of autonomous detection, we prepare an annotation mask in the robot hand’s local coordinate system to help extract the test tube cap’s point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The extracted point clouds will be back-projected to the corresponding 2D grey image for segmenting a picture of the cap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 5 shows the details of this mask and how it helps to segment the cap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The mask and back projection enable us to precisely segment the cap regions while avoiding including backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' To prepare an annotation mask, we move the robot hand that holds a test tube to a fixed position under the Phoxi sensor and trigger the sensor to capture a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We can (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) Collecting the tube cap data (i) A human (ii) The robot moves Many pictures of the tube caps (iii) Crop the tube cap hands over the test tube to new based on the mapping a test tube observation poses between the captured to the robot for capturing data point cloud and image (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) Collecting the rack data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(i) A human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(i) The robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(iii) Crop the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Many pictures of the racks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='places a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='pushes the rack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='rack from the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='rack in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='to new poses for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='environmental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='capturing data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(b) Data synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(c) Resulted data (Images with known tube caps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Pictures of tube caps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Pictures of racks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Images obtained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='during collecting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Copy-paste data synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='the tube cap data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='(Top view pictures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='with tubes held in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='Synthesized racks with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='the robotic hands) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='different backgroundsARXIV VERSION,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2022 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4: (a) Sampling observation positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The green region is the visible area of the Phoxi scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The red points are the sampled positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) Sampling rotations based on a level-four icosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The left spherical polyhedron illustrates the icosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The green vertices are the ends of feasible vectors that have less than θ = 60◦ angles with the surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They imply the tube rotation poses shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) The grasping poses for each sampled tube pose form a SO(2) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are sampled considering an interval ω for numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) A visual polyhedron computed using the camera origin and vertices of the robot hand model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) The grasped object has a risk of being occluded by the robot arm when there is a collision between the visual polyhedron and the robot arm links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 5: Workflow for extracting the cap picture using an annotation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (a) Applying a mask described in the local coordinate system of the holding robot hand to the captured point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) The extract point cloud is projected back to the 2D grey image for segmenting a picture of the cap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) The back-projected results might be disconnected pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) A bounding convex hull of the disconnected pixels is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='3,4) The cap region is segmented based on the bounding convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' easily get the cap’s point cloud data by examining the area on top of the holding fingers and obtain an annotation mask by considering a bounding polyhedron of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, a single bounding polyhedron may not be general for others since the captured point cloud is susceptible to light reflection or perspective projection (self-occlusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Thus, instead of a single point cloud and polyhedron, we collect point clouds from multiple views, merge them under the robot hand’s local coordinate system, and compute a bounding box of the merged result as an annotation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 6 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The multiple views are sampled the same way as the observation poses mentioned in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, we do not need to change the observation positions since we aim to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 6: (a) Capture data from different views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The tube cap’s point clouds are obtained by examining the area on top of the holding fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are high lighted with colored polyhedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) Merge the cap’s point clouds in (a) under the robot hand’s local coordinate system, and compute a bounding box of the merged result as an annotation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) Raw bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) The bounding box can be adjusted interactively if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' obtain a bounding box mask in the hand’s coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The views under various rotations could provide enough superficial point cloud data to meet the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note that the merged result may include noise point data induced by reflections from the transparent tube body and lead to a mask larger than the cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We provide an interactive user interface for manually adjusting the bounding box sizes and minimizing the negative influences caused by the noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The adjustment is optional and may be performed when precisely segmenting the cap region is demanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Copy-Paste Synthesis We apply random scaling, blurring, brightness, and contrast to the segmented tube caps and then paste them onto the segmented rack background for data synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' During pasting, we permit the overlap among the cap regions to approximate tube-to-tube occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' After pasting, we randomize the en- vironmental background (background of the rack) to narrow further the domain gap between synthetic images and images captured in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A critical maneuver here is that we consider the co- occurrence of the test tubes and the rack and paste the tube cap pictures onto a rack instead of random backgrounds like [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We randomly sample positions inside rack pictures for pasting tube caps and use a pasting number T to control the clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note that there is no need to exactly paste a tube cap near the hole centers of a rack as the tubes tilt randomly inside the rack holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The visible cap regions may reasonably overlap with a hole boundary or other holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For tube-to-tube occlusion, we consider the perspective projection of a vision sensor and define an occlusion threshold t to permit overlap among the visible cap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A vision sensor’s perspective projection leads to mutual occlusions in the rack at certain viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The occlusion threshold helps to simulate the occlusion and defines the maximum percentage that segmented cap pictures can overlap or occlude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 7 shows how the t threshold works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It adds a constraint to pasting, where a previously pasted cap picture “A” must have less than t percentage overlap with the union of caps pasted later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The B ∪ C ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' component in the nominator of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 7 implies the union of caps pasted after “A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' When a new cap is randomized, it must be unioned with this component (a) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) 0 = 60° Surface Test normal tube Positions Rotations (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) Visual Gripper polygon .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='09 = 3(a) (b) Annotation Mask (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2) ZH (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='3) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4) ZH(b) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1) (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2)ARXIV VERSION, 2022 7 to ensure the t constraint on all previous “A” is not violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There are two noticeable points for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' First, its value could be devised respectively considering the heights of specific tube types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Second, its value is correlated with the pasting number T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The maximum number of pasted tube caps in a rack that meet the t threshold may be less than a given T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In that case, we constrain the maximum number of pasted tube caps to the smaller value to ensure t is not invalidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 7: (a) Using a threshold t to simulate cap occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' “A” represents a previously pasted cap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' “B”, “C”, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' represent the caps pasted after “A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) Results with different t values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For the environmental background, we use the BG-20k data set [44] to obtain high-resolution random background images and change the background of a synthetic image with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' EXPERIMENTS AND ANALYSIS We carried out experiments to compare YOLOv5x [45] detectors trained using data sets collected with the proposed method and several other methods to understand the per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Table II shows the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The SR (Synthesis by pasting to Racks) method pastes randomly selected cap pictures onto rack backgrounds to synthesize training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It represents the synthesizing method used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The SB (Synthesis by pasting to BG-20k) method is an alternative synthesis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Instead of being pasted onto a rack, randomly selected cap pictures are pasted to random backgrounds selected from the BG-20k data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The RO (Robotic Observation) method is a byproduct of robotic cap segmentation, where the robot holds test tubes for data col- lection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We considered RO an independent method because we wondered if the hand-held observation was enough for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We also combined RO, SR, and SB methods (the ** row) to see if they help achieve a satisfying performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The RO+SR combination is exactly our proposed method in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We especially proposed it since RO is a pre-process of robotic cap segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Using combined RO+SR does not increase effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Combining RO+SB or RO+SR+SB are also candidate choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They have the same cost as using independent SR or SB data2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Finally, the CL (Crowd-source Labeling) method is a conventional one that requires humans to place racks with tubes under the robot and label the captured images manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 8 shows exemplary images collected using the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2Synthesizing data is considered to be free as it only require computational work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Thus, the costs of SR and SB depend on the RO process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' TABLE I: Summary of the data collection methods Abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Full Name Description SR Synthesis by pasting to Racks Caps on racks SB Synthesis by pasting to BG-20k Caps on random background RO Robotic Observation Tubes held in robotic hands ** Combinations of above methods SR+SB is the proposed one CL Crowd-source Labeling Tubes in a rack on the table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 8: Exemplary images collected using the various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (a) RO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c) SB (d) CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Performance of Various Data sets We collected various data sets with the methods and their combinations, used the data sets to train YOLOv5x detectors, and examined the performance of the trained detectors using a testing data set for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The first data set is CL200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is considered a baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In collecting the data set, we collected 200 images with random tube and rack states and labeled the tube regions manually using LabelImg3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There are, in total, 5916 labeled instances in the 200 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The second data set is SR1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In order to collect it, we first prepared many cap pictures using robotic observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2(b), we assumed four different test tubes and took advantage of the Yumi robot’s both arms to collect cap data quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For each tube type, we handed over two same ones to the two robotic arms for observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Each arm moved its held tube to 400 observation poses for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 9(a) for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Here, we set the hyperparameter θ and ω to 30◦ and 360◦ (single grasping pose) and set the positions to be evenly sampled on the table with a granularity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1m for generating the observation poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In total, more than 400 observation poses were obtained under the parameter setting for each arm, and we used the first 400 for collecting images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' As a result, we obtained 400 observation images (800 cap pictures since there are two tubes in each image, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 9(b) for example) for a single tube type and 1600 observation images for all tube types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We segmented 3200 pictures of cap regions from the observation images considering point cloud mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 9(c) shows the collected point clouds with highlighted caps (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 9(d) shows the segmented cap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Besides the cap regions, we collected 15 images with racks (a single rack in each image) and segmented 15 pictures of racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We synthesized a data set of 1600 images by pasting caps randomly selected from the 3200 cap pictures to racks randomly selected from the 15 rack pictures (SR method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' During synthesis, we set the pasting number to be T = 30, and set the occlusion threshold for the “Blue Tube” to be tblue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4 and other tubes to be tothers = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We chose these parameter settings because the “Blue Tube” was shorter and susceptible to occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We increased its occlusion threshold 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='com/heartexlabs/labelImg (a) IAN(BUCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Overlap(A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=') [A| (b) 10 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='3 T = 20 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='6 T = 40a dARXIV VERSION, 2022 8 to mimic frequent visual blockage from other tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Also, we increased the variety of the segmented cap pictures by applying random scaling (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='9 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1 of original picture size, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 probability), random blur (3 × 3 kernel, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 probability), random brightness (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='9 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1 of original brightness, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 probability), and random contrast (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='9 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1 of the original contrast value, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 probability) using the Albumentations4 library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The background of the rack was randomly chosen from the BG-20k data set with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 changing probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 9: (a) The robot moves test tubes for observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Both arms are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) Observation Image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c) Point clouds cap- tured by the Phoxi sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (d) Cap pictures segmented from the observation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The third data set is RO1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It is a semi product of robotic cap segmentation and comprises the 1600 observation images obtained during robotic observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The fourth data set is SB1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In contrast with the SR1600 data set, we pasted randomly selected caps directly to images from the BG-20k data set for obtaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The pasted caps might freely distribute on the image background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The segmented racks were not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The pasting number T and occlusion threshold t are 35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There was no difference on t for different tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The randomization were performed in the same way as obtaining SR1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We also used combined methods to collect data sets and study if the combination led to better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The combined data sets include RO1600+SR800, RO1600+SB800, RO1600+SR400+SB400, SR800+SB800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Here, the superscript number on the upper-right of a method name means the number of images collected using the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The “+” sym- bol indicates that the data sets comprise data collected using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The RO1600+SR800 data set represents the data collected using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The left part of Table II summarizes the various data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They are used to train YOLOv5x detectors for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Before training, the YOLOv5x detectors for all data sets were initialized with weights pre-trained using the COCO data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The images in all data sets were regulated into a resolution of 1376 × 1376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Each data set is divided into a training subset and a validation subset according to a 4 to 1 data ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' During learning, the training subset was fed to the training program with a batch size of 2, and the training program performed validation per episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The training process was stopped when the mAPs (mean Average Precision) [46] for all objects reached higher than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='0% under a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5 IoU (Intersection over Union).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Here, we defined a detected bounding box to be correct when its IoU with a ground truth cap bounding box was larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 4https://albumentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='ai/ For evaluating the performance of YOLOv5x detectors trained using the various data sets, we collected a testing data set with 100 images and labeled their ground truth using the same method as CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We used the trained detectors to detect tubes in the testing data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Like validation, we defined a detected bounding box as correct when its IoU with a ground truth cap bounding box is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We used the AP (Average Precision) metric to measure the detection performance of a single object class and used the mAP for all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Since the detector that met a single satisfying validation was not necessarily the best, we trained each detector twice and took the higher precision value on the testing data set as the final evaluation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Table II shows the evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We obtained the following observations and speculations from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' i) Using the data set collected by robotic observation for training exhibited the worst performance, as shown by the 2nd row (RO1600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: All images in the data set had a similar robotic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They suffered from a domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' ii) The synthetic data sets do not necessarily lead to a good AP, as shown by the 3rd (SR1600) and 4th (SB1600) rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The SR1600 data set exhibited higher performance than the SB1600 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: The copy-paste synthesis failed to cover certain visual contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Pasting onto racks (SR) provided more effective visual contexts and benefited the neural network more than pasting onto random backgrounds (SB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' iii) Combining the synthetic data sets with robotic observa- tions is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It can be concluded by comparing the 5th, 6th, and 7th rows (RO1600+SR800, RO1600+SB800, RO1600+SR400+SB400) with the 2nd, 3rd, and 4th rows (RO1600, SR1600, and SB1600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The former rows had higher mAP than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: The robotic observation data set additionally provided helpful visual contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' iv) The 5th row (RO1600+SR800) had a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4% higher mAP than the 6th row (RO1600+SB800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Especially, the AP of the “Blue Tube” on the 5th row was 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='7% higher than that on the 6th row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The AP of other tubes also had 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1% ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='7% performance increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: Considering the rack as a local context helped improve domain-specific performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The short “Blue Tube” could be easily blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The data set collected using the SR method had more simulated oc- clusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They were important for recognizing the short “Blue Tube”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' v) The 7th row (RO1600+SR400+SB400) exhibited slightly higher mAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='3%) than the 5th row (RO1600+SR800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: Pasting onto racks (SR) provided better domain-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Random backgrounds for the tubes slightly benefited the neural network and were less necessary if the goal context was limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' vi) The 5th row (RO1600+SR800) is competitive compared with the 1st row (CL200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The mAP was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='7% lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The AP of the “Blue Tube” and “White Tube” were 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='5% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1% lower, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The AP of the “Purple Tube” (c) Point clouds (d) Cap (b) picturesARXIV VERSION, 2022 9 TABLE II: Comparison of detectors trained using different data sets AP ID Data Set Names # Caps Remark Blue Purple White Purple Ring mAP 1 CL200 5916 Multiple tubes / image 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='990 2 RO1600 3200 Two tubes / image 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='657 3 SR1600 40000 tblue=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4 & tothers=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15, T = 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='940 4 SB1600 56000 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15 (same for all tubes), T = 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='874 5 RO1600+SR800 23200 See note 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='983 6 RO1600+SB800 31200 See note 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='959 7 RO1600+SR400+SB400 27200 See note 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='986 8 SR800+SB800 48000 See note 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 Note 1: Largest AP and mAP values are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note 2: The combined data sets are collected using the same parameters as respective ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' was the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The AP of the “Purple Ring” tube was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='8% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Speculation: The robotic observation and paste-to-rack synthesis compensated for each other’s shortcomings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There remained extreme cases that could be labeled manually but failed to be covered by robotic observation or synthesis, especially for the “Blue Tube”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Several failure cases are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 10 to provide the readers an insight into our observations and speculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 10(a) and (b) exemplify the recognition results of detec- tors trained using the 5th (RO1600+SR800) and 6th data sets (RO1600+SR800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The latter one failed to recognize occluded tubes as the training data set had fewer simulated occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The example is consistent with the observation and speculation in iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 10(c) and (d) exemplify cases that the detectors trained using the 5th (RO1600+SB800) data set failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the first case, shadows from other test tubes were cast on a blue test tube cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The detector failed to recognize the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the second case, the detector misrecognized a crystal tube body as the “Blue Tube” cap due to the illusion caused by body- and-rack overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The two failure examples are consistent with the observation and speculation in vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The synthetic data sets do not involve shadows or tube bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The detectors trained using them had worse performance in these cases than the one trained using the crowd-sourced real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 10: (a) Detector trained using the 5th data set successfully recognized all tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (b) Detector trained using the 6th data set failed to recognize the occluded tube in the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (c) Detector trained using the 5th data set failed to recognize the shadowed tube in the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (d) Detector trained using the 5th data set misrecognized the tube body in the red circle as a “Blue Tube”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In summary, the results of the various training data sets showed that combining data collected using the RO and SR methods was effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The conclusion was satisfying as the RO method is a subset of the SR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The workflow for collecting them is simple and clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, we wonder if the number of images in the RO data set could be reduced, as it needs much manual handover to collect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' This query prompted us to carry out the studies in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Ablation Study In this subsection, we conduct multiple ablation studies on the combined RO+SR data set to further understand 1) the influence of the data combination ratio and 2) the influence of pasting number T and occlusion threshold t used for generating synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 1) Influence of data combination ratio: The experiments for studying the influence of data combination ratio are divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the first part, we set the number of images collected using the RO method to 800 and varied the number of images collected using the SR method from 200 to 1600 in a 2-fold ratio to understand the importance of the SR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The upper section of Table III shows the precision of detectors trained using the varied data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results indicate that the mAP improved when the SR image numbers increased from 200 to 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The second part is similar to the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In this part, we fixed the number of images collected using the SR method to 800 and varied the number of images collected using the RO method from 200 to 1600 in a two-fold ratio to understand the importance of the RO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The lower section of Table III shows the precision of detectors trained using the varied data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The result indicates that the mAP improved when the RO image numbers increased from 200 to 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' 2) Influence of hyperparameters: Besides the data combi- nation ratio, we also studied the influence of pasting number T and occlusion threshold t used in the SR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We set both the RO and SR image numbers to 800 and observed the performance of detectors trained with data sets collected using different T and t values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Although we previously used a different t value for the “Blue Tube”, we did not differentiate the tubes here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Like the study on different data combination ratios, this study also comprised two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the first part, we fixed t to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1 and increased T from 10 to 40 with a step length of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The upper section of Table IV shows the precision changes under the parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The ring ng0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='94 (d) a b cARXIV VERSION, 2022 10 TABLE III: Influence of data combination ratio AP Data Set Names Blue Purple White Purple Ring mAP RO800+SR200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 RO800+SR400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 RO800+SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='981 RO800+SR1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 RO200+SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='971 RO400+SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 RO800+SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='981 RO1600+SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='983 Note 1 Largest AP and mAP values are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note 2 We used the following hyper-parameter setting tblue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4 & tothers = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15, T = 30 to collect the SB data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The values were the same as the experiments in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' results exhibited a significant increase from 10 to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' However, an even larger T had little influence on the recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the second part, we set T to be 30 and varied t from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='20 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='80 with a step length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The lower section of Table IV shows the precision changes under the parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results exhibited a clear precision increase on the “Blue Tube”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We speculate that the reason was that the “Blue Tube” was shorter and vulnerable to occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' A larger t helped provide more occlusion cases in the training data set, leading to a higher detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results also indicated that the precision of the ”White Tube” and ”Purple Ring Tube” irregularly changed as the t increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' They were taller and did not suffer from occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Adding occlusions for them caused unexpected errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' For a complete observation, we recommend interested readers to compare with the third row of the table’s upper section to catch the changes starting from t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The T value of the upper section’s third row was the same as the rows in the lower section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' TABLE IV: Influence of parameters used for synthesis AP Params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' (T, t) Blue Purple White Purple Ring mAP (10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 (30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='80) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='988 Largest AP and mAP values are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Further Analysis on Synthetic Data We also studied the influence of cap variation and combina- tion ratio on synthetic data sets (the data sets collected using the SR, SB, or SR+SB methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The goal was to understand the best performance we could reach with synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' First, we fixed the number of images collected by the SR and SB methods to 800, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' We changed the number of cap region pictures (equals to the number of observation images multiplied by two) used for synthesis from 400 to 3200 in a 2-fold ratio to study the influence of cap variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The previsions YOLOv5x detectors using the changing data sets are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The results showed that the 400 row had competitive precision compared to the 1600 or 3200 rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The number was enough to support a satisfying detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The cap variations were thus considered to have a low influence on learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Second, we fix the number of cap region pictures to 3200 and change the number of images collected using the SR and SB methods, respectively, to study the influence of the combination ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Like the ablation study in Section V-B1, we divided the experiment here into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the first part, we set the number of images collected by the SR method to 800 and varied the number of images collected by the SB method from 200 to 1600 in a 2-fold ratio to understand the importance of the SB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The upper section of Table VI shows the precision of detectors trained using the varied data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The number of SB images did not appear to be positively correlated with the final detector’s precision, although the largest mAP was observed when the number of SB images was 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' In the second part, we fixed the number of images collected by the SB method to 800 and varied the number of images collected by the SR method to understand the importance of the SR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The lower section of Table VI shows the precision of detectors trained using the varied data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The result indicated that the mAP improved as the SR image number increased to 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' There was no significant difference when the image number increased from 800 to 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' TABLE V: The influence of #caps to synthesis AP #Caps Blue Purple White Purple Ring mAP 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='979 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='982 3200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 TABLE VI: Influence of the SR and SB ratio AP Data Set Names Blue Purple White Purple Ring mAP SB200 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='967 SB400 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 SB600 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 SB800 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 SB1600 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='957 SB800 + SR200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='955 SB800 + SR400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='942 SB800 + SR600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='945 SB800 + SR800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 SB800 + SR1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='980 Note 1 Largest AP and mAP values are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note 2 We used the following hyper-parameter setting tblue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='4 & tothers = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15, T = 30 to collect the SB data sets, and used the following hyper-parameter setting T = 30, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='15 (same for all tubes) to collect the SR data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The values were the same as the experiments in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' Note 2 We used 3200 segmented cap region pictures for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' ARXIV VERSION, 2022 11 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' CONCLUSIONS In this paper, we proposed an integrated robot observation and data synthesis framework for data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The pro- posed framework can significantly reduce the human effort in data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' It required only a single process and was a low-cost way to produce the combined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The experimental result showed that combined observation and synthetic images led to comparable performance to manual data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQfevxv/content/2301.01441v1.pdf'} +page_content=' The ablation studies provided a good guide on optimizing data configurations and parameter settings for training detectors using the combined data.' metadata={'source': 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/dev/null +++ b/0tFQT4oBgHgl3EQfDzUz/content/tmp_files/2301.13235v1.pdf.txt @@ -0,0 +1,3485 @@ +Joint calibration to SPX and VIX options with +signature-based models +Christa Cuchiero∗ +Guido Gazzani† +Janka M¨oller‡ +Sara Svaluto-Ferro§ +February 1, 2023 +Abstract +We consider a stochastic volatility model where the dynamics of the volatility are +described by linear functions of the (time extended) signature of a primary underlying +process, which is supposed to be some multidimensional continuous semimartingale. +Under the additional assumption that this primary process is of polynomial type, we +obtain closed form expressions for the VIX squared, exploiting the fact that the trun- +cated signature of a polynomial process is again a polynomial process. Adding to such +a primary process the Brownian motion driving the stock price, allows then to express +both the log-price and the VIX squared as linear functions of the signature of the corre- +sponding augmented process. This feature can then be efficiently used for pricing and +calibration purposes. Indeed, as the signature samples can be easily precomputed, the +calibration task can be split into an offline sampling and a standard optimization. For +both the SPX and VIX options we obtain highly accurate calibration results, showing +that this model class allows to solve the joint calibration problem without adding jumps +or rough volatility. +Keywords: signature methods, calibration of financial models, affine and polynomial pro- +cesses, S&P 500/VIX joint calibration +MSC (2020) Classification: 91B70, 62P05, 65C20. +Contents +1 +Introduction +2 +1.1 +State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Signature: definition and properties +6 +∗Vienna University, Department of Statistics and Operations Research, Data Science Uni Vienna, Kolin- +gasse 14-16 1, A-1090 Wien, Austria, christa.cuchiero@univie.ac.at +†Vienna University, Department of Statistics and Operations Research, Kolingasse 14-16 1, A-1090 Wien, +Austria, guido.gazzani@univie.ac.at +‡Vienna University, Department of Statistics and Operations Research, Kolingasse 14-16 1, A-1090 Wien, +Austria, janka.moeller@univie.ac.at +§University +of +Verona, +Department +of +Economics, +Via +Cantarane +24, +37129 +Verona, +Italy, +sara.svalutoferro@univr.it. +The first three authors gratefully acknowledge financial support through grant Y 1235 and grant I 3852 of +the Austrian Science Fund. All authors acknowledge financial support through the OEAD WTZ project FR +02/2022. +1 +arXiv:2301.13235v1 [q-fin.MF] 30 Jan 2023 + +3 +The model +10 +4 +Expected signature of polynomial diffusion processes +12 +5 +VIX options with signatures +16 +5.1 +Explicit formulas for the VIX . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +5.2 +Options on VIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +5.3 +Variance reduction for pricing VIX options +. . . . . . . . . . . . . . . . . . +21 +5.4 +Calibration to VIX options +. . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +5.4.1 +Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5.5 +The case of time-varying parameters . . . . . . . . . . . . . . . . . . . . . . +27 +6 +SPX as a signature-based model +28 +6.1 +Exploiting the affine nature of the signature: Fourier pricing of SPX and VIX +options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +6.2 +The case of time-varying parameters . . . . . . . . . . . . . . . . . . . . . . +34 +7 +Joint calibration of SPX and VIX options +35 +7.1 +Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +7.1.1 +First approach +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +7.1.2 +Second approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +A Numerical results for the Brownian motion case +42 +1 +Introduction +The joint calibration to SPX1 and VIX options is a problem that has gained a lot of attention +in quantitative finance since several years. It is sometimes still regarded as the holy grail of +volatility modeling, even though significant progress has been made recently (see below in +the literature overview). +One main reason for the increased interest is that the VIX index is no longer only used +as indicator of volatility, but rather as important underlying for many derivatives. In fact, +futures and options written on it are extensively used to hedge the volatility exposure of +option portfolios, see e.g. Rhoads (2011). Suppose that an investor has a long position on +the S&P 500 index. Although she might believe it has long-term prospects, it would be +desirable to reduce her exposure to short-term volatility. Buying VIX derivatives with the +belief that volatility is going to increase she might balance out these positions, while she +was wrong, the losses to her VIX position could be mitigated by gains to the existing trade. +Historically speaking, in March 2004, futures on VIX, which trade on the forward 30-days +realized volatility of the S&P 500, were launched on the Chicago Board Options Exchange +(CBOE). Two years later, in February 2006, also trading in European options written on +the VIX index started and has progressively increased since then. We address the reader +to the official website of CBOE2 for details on how the VIX is computed and traded in +practice, but just remark the following in view of expiration dates. +1The SPX is a theoretical index, in the sense that it is pegged to the value of the S&P 500 without +being based on a held portfolio of stock shares. +In the present paper we shall use SPX and S&P 500 +interchangeably. +2www.cboe.com/tradable products/vix/ +2 + +Remark 1.1. Note that for a given type of option just certain maturities are available. +This in particular implies that on different days options with different times to maturity +are traded. For example, options which expire exactly in one year are not available on +a daily basis. Concerning our concrete applications, typically, VIX options expire on the +Wednesday 30 days (or closest to 30 days) prior to the third Friday of the next calendar +month. On the other hand a monthly SPX option typically expires on the third Friday of +every month. +The steadily increasing liquidity on the market of VIX index products has marked the +need to jointly calibrate to both, the prices of options on the underlying S&P 500 index +and to prices of VIX derivatives. +From a data perspective the challenge, especially for short maturities, is to reconcile the +large negative skew of SPX options’ implied volatilities with relatively lower implied volatil- +ities arising from the VIX options. This has been discussed rigorously in Guyon (2020a), +where the author shows a necessary condition to solve the joint calibration problem. As +observed in the paper this translates, in the context of models with continuous trajectories, +to a volatility process with high mean-reversion speed and large negative correlation to the +S&P 500 index. In addition to that a high vol-of-vol is desirable in order to reproduce the +aforementioned negative skew of at-the-money SPX options, but this can yield too high +implied volatilities for the VIX options. +Inspired by Perez Arribas et al. (2020) and Cuchiero et al. (2022a), we consider here a +new type of stochastic volatility model for the discounted price process S = (St)t≥0 with +continuous trajectories. It is given by +dSt(ℓ) = St(ℓ)σS +t (ℓ)dBt, +for an initial condition S0 ∈ R+, a standard Brownian motion B, and a volatility process +σS satisfying +σS +t (ℓ) := ℓ(�Xt), +(1.1) +where ℓ is a linear map of the signature �Xt of a process � +X. +Specifically, the main in- +gredient in this modeling framework is a d-dimensional continuous semimartingale X = +(X1 +t , . . . , Xd +t )t≥0, which we call primary process. In most cases we then consider its aug- +mentation with time t denoted ( � +Xt)t≥0 = (t, X1 +t , . . . , Xd +t )t≥0. By modeling σS via (1.1) we +assume that the signature of � +X, denoted by �X (and rigorously introduced in Section 2), +serves as a linear regression basis for the volatility process, while the parameters of the +linear map ℓ have to be learned from (option price) data. Note that the parameters of X +are prespecified beforehand and can thus be seen – in analogy to machine learning terminol- +ogy – as hyperparameters (that of course can be optimized over some training or validation +set). As outlined below this is one of the crucial features that allows for the split of the +calibration task into precomputable samples and parameters to optimize. +Let us now highlight the implications of this modeling framework and the novelty of the +present work. +• The modeling framework can be seen as universal in the class of continuous non-rough +stochastic volatility models, which is a consequence of the universal approximation +result stated in Theorem 2.6. +3 + +• It is not only universal in an approximate sense, but truly nests several classical mod- +els (see Remark 3.4) and for instance also the ‘quintic Ornstein-Uhlenbeck volatility +model’, recently proposed by Abi Jaber et al. (2022b), which – with an additional +input curve – is shown to fit SPX and VIX smiles well. +• By choosing the parameters of ℓ appropriately, the modeling framework incorporates +both, purely Markovian (in (S, X)) and path-dependent models. +• Up to our knowledge, it is the first signature-based model that is employed for pricing +and calibration of VIX options as well as joint calibration, together with SPX options. +• We illustrate that the joint calibration problem can be solved in this framework with- +out jumps and rough volatility (compare also Guyon and Lekeufack (2022); Rømer +(2022); Abi Jaber et al. (2022b)). +• By using time-varying parameters we can go beyond short maturities both for SPX +and VIX options (as classically tackled in the literature) and achieve a joint calibration +also for longer maturities. +In order to achieve the highly accurate calibration results, illustrated in Section 5.4 and +Section 7, we exploit the following mathematical and numerical properties. +• Defining Z := (X, B), then not only σS(ℓ) but also the log-price log(S(ℓ)) can be +expressed as a linear function of the signature of �Z. The computational benefit is +immediate, since no (Euler) simulation scheme is needed to sample from the marginals +of the price process. In terms of the parameters ℓ, log(S(ℓ)) is the sum of a quadratic +function and a linear one, see Proposition 6.5. +• If X is additionally a polynomial process (see Cuchiero et al. (2012); Filipovi´c and +Larsson (2016)), then the VIX under our model can be computed analytically via +matrix exponentials. Indeed, in this case the forward variance can be represented by a +quadratic form in the parameters ℓ and the corresponding matrix can be computed by +polynomial technology, i.e. via matrix exponentials, see Theorem 5.1. This tractability +property is a consequence of the fact that the truncated signature of a polynomial +process is again a polynomial process (see Section 4). +• We can apply a Monte Carlo approach (potentially with variance reduction) for option +pricing and calibration, since the signature samples of �Z can be computed offline. +Indeed, due to the representations of VIX and log(S(ℓ)) described above, the same +samples can be used for every linear map ℓ. Therefore, the calibration task can be +split into an offline sampling and a standard optimization, as no simulation is needed +during the latter. +• Alternatively, a Fourier pricing approach for both VIX and SPX options can be used. +Indeed, by building on the fact that the signature of �Z is an affine process (with values +in the extended tensor algebra) as proved in Cuchiero et al. (2023), its Fourier-Laplace +transform can be computed by solving an (extended tensor algebra valued) Riccati +equation, which in turn can be used for Fourier-pricing as outlined in Section 6.1. +The remainder of the paper is organized as follows. Section 1.1 gives a review over the dif- +ferent contributions in the literature concerning the joint calibration problem. In Section 2 +4 + +we introduce the signature in the context of continuous semimartingales, its main properties +as well as notation used throughout the paper. Section 3 is dedicated to the introduction +of our signature-based model and the connections to classical and also recent stochastic +volatility models in the literature. Section 4 is then devoted to the discussion and proof +of the matrix exponential formula for the (conditional) expected signature of polynomial +processes. This result is at the core of Section 5, where we derive a tractable formula for +the VIX, needed for pricing VIX options and VIX futures. Building on these formulas, our +calibration results to VIX options are presented in Section 5.4.1. In Section 6, we then +prove, similarly as for the VIX, a tractable expression for S. Additionally, we exploit in +Section 6.1 the affine nature of the signature process (as proved in Cuchiero et al. (2023)), +to obtain a Fourier pricing approach within our modeling choice for both VIX and SPX op- +tions. We finally present the numerical results of the joint calibration problem in Section 7, +both in the case of constant parameters and with time-varying parameters, where the latter +are introduced in Section 5.5 and Section 6.2. +The data used in Section 5.4.1 and Section 7.1 were purchased from OptionMetrics3. +Python codes used to produce the numerical results of the present work are available from +the authors upon request. +1.1 +State of the art +This section is primarily dedicated to a literature review on the joint calibration problem +and secondly, to a brief overview on signature methods in finance. +First attempts to solve the joint calibration problem appear in Gatheral (2008), with +a double constant elasticity of variance model (CEV), which despite being rather flexible +cannot fit accurately the implied volatilities of SPX and VIX options jointly. Later on, the +belief that jumps in the SPX (or additionally also in the volatility) are necessary has lead +to the following contributions, Sepp (2012); Papanicolaou and Sircar (2014); Baldeaux and +Badran (2014); Kokholm and Stisen (2015); Pacati et al. (2018) and more recently Grzelak +(2022) with a new perspective on randomization. +Continuous stochastic volatility models based on Markovian semimartingales have also +been employed to solve the joint calibration problem. For instance, in Fouque and Saporito +(2018) a Heston model with stochastic vol-of-vol has been calibrated, however only for +maturities above 4 months where VIX options are less liquid. More recently, Rømer (2022) +considered a model where the volatility is driven by two Ornstein-Uhlenbeck (OU) processes +using a non-standard transformation function. The well-working choice of two OU-processes +illustrated there has been an inspiration for our concrete numerical implementations. We +also point out that the (non-rough) model introduced in Abi Jaber et al. (2022a,b), where +the volatility is described by a polynomial of order five in one single OU-process, falls +(apart from the additional input curve) into this class of continuous Markovian models +and is a particular instance of our framework. Let us also refer to the paper by Guyon +and Mustapha (2022), where a neural SDE model has been successfully jointly calibrated. +Within the class of continuous, however not necessarily Markovian models, Guyon and +Lekeufack (2022) conduct an empirical and statistical analysis as well as a joint calibration +for a family of models where the volatility depends on the paths of the asset. These models +can be turned into Markovian ones by using exponential kernels instead of general ones. +Two further distinct and rather new lines of research are worth being mentioned as well: +first, martingale optimal transport and second rough volatility. +3https://optionmetrics.com/ +5 + +The martingale optimal transport approach (see e.g. Guyon and Henry-Labordere (2013)) +is used to calibrate discrete-time models as proposed in Guyon (2020b, 2021). These models +are closely related to Schr¨odinger bridge problems, where the idea is to calibrate only the +drift of the volatility while keeping the volatility of volatility unchanged, see e.g. Henry- +Labordere (2019) or Guo et al. (2020, 2021) as well as the references therein regarding an +optimal transport approach. Although the calibration within that setting is accurate, it is +also computationally rather expensive and not amenable to calibrate to several maturities +jointly. These computational challenges have been tackled recently in Guyon and Bourgey +(2022) both in discrete and continuous time extending the contributions of Guyon (2020b, +2021). +In the area of rough volatility modeling, initiated by the seminal paper of Gatheral et al. +(2018), the main idea is to replace the standard Brownian motion in the volatility process +by a fractional Brownian motion. Even though the roughness of the trajectories found in +Gatheral et al. (2018), can also be explained by the estimation procedure as discussed e.g. in +Cont and Das (2023), the non-Markovianity given by the fractional Brownian motion with +Hurst parameter H < 0.5, manages to reproduce many stylized facts arising in financial +data. Several classical models have been enhanced with rougher noise, but for simplicity +we mention those employed in the SPX/VIX calibration. One example is the quadratic +rough Heston model introduced in Gatheral et al. (2020), which was in turn calibrated in +Rosenbaum and Zhang (2021) by relying on neural networks approaches, also exploited in +e.g. Bayer et al. (2019); Cuchiero et al. (2020). For a large class of rough volatility models +Jacquier et al. (2021) give new insights on the joint calibration problem by providing small- +time formulas of the at-the-money implied volatility, skew and curvature for both European +options on SPX and VIX options. In Rømer (2022) an exhaustive study of the flexibility +of different rough volatility models to joint SPX/VIX calibration is carried out, including +the rough Bergomi, the rough Heston and an extended rough Bergomi model. Some of +these, for instance the rough Heston model, have an affine structure i.e., can be embedded +in the class of affine Volterra processes, considered in Abi Jaber et al. (2019); Cuchiero +and Teichmann (2019, 2020). In particular they allow for Fourier pricing after solving the +related Riccati equations. This underlying structure is the building block of an extension +with jumps investigated in Bondi et al. (2022a) and recently employed in the context of the +joint calibration in Bondi et al. (2022b), where a rough Heston model with Hawkes-type +jumps with Fourier pricing is employed. +Concerning our framework, signature-based methods provide a generic non-parametric +way to extract characteristic features (linearly) and path-dependency from data, which is +essential in (machine) learning and calibration tasks in finance. This explains why these +techniques become more and more popular in mathematical finance, see e.g., Buehler et al. +(2020); Kalsi et al. (2020); Perez Arribas et al. (2020); Lyons et al. (2020); Ni et al. (2021); +Bayer et al. (2021); Min and Hu (2021); Akyildirim et al. (2022); Cuchiero et al. (2022a); +Cuchiero and M¨oller (2023); Cuchiero et al. (2022c) and the references therein. +2 +Signature: definition and properties +We start by introducing basic notions related to the definition of the signature of an Rd- +valued continuous semimartingale. This is similar as in Cuchiero et al. (2022a) or Bayer +et al. (2021), but to keep the paper self-contained we recall the essential definitions and +properties. +6 + +For each n ∈ N0 we define recursively the n-fold tensor product of Rd, +(Rd)⊗0 := R, +(Rd)⊗n := Rd ⊗ · · · ⊗ Rd +� +�� +� +n +. +For d ∈ N, we define the extended tensor algebra on Rd as +T((Rd)) := {a := (a0, . . . , an, . . . ) : an ∈ (Rd)⊗n}. +Similarly we introduce the truncated tensor algebra of order n ∈ N +T (n)(Rd) := {a ∈ T((Rd)) : am = 0, ∀m > n}, +and the tensor algebra T(Rd) := � +n∈N T (n)(Rd). Note that T (n)(Rd) has dimension dn := +(dn+1 − 1)/(d − 1). +For each a, b ∈ T((Rd)) and λ ∈ R we set +a + b := (a0 + b0, . . . , an + bn, . . . ), +λ · a := (λa0, . . . , λan, . . . ), +a ⊗ b := (c0, . . . , cn, . . . ), +where cn := �n +k=0 ak ⊗ bn−k. Observe that (T((Rd)), +, ·, ⊗) is a real non-commutative +algebra. +For a multi-index I := (i1, . . . , in) we set |I| := n. We also consider the empty index +I := ∅ and set |I| := 0. If n ≥ 1 or n ≥ 2 we set I′ := (i1, . . . , in−1), and I′′ := (i1, . . . , in−2), +respectively. We also use the notation +{I : |I| = n} := {1, . . . , d}n, +omitting the parameter d whenever this does not introduce ambiguity. Observe that multi- +indices can be identified with words, as it is done for instance in Lyons et al. (2020). +Next, for each |I| ≥ 1 we set +eI := ei1 ⊗ · · · ⊗ ein. +Observe that the set {eI : |I| = n} is an orthonormal basis of (Rd)⊗n. Denoting by e∅ the +basis element corresponding to (Rd)⊗0, each element of a ∈ T((Rd)) can thus be written as +a = +� +|I|≥0 +aIeI, +for some aI ∈ R. +Note that if an ∈ (Rd)⊗n we use non-bold notation whereas for the +components aI ∈ R we write them bold. Finally, for each a ∈ T(Rd) and each b ∈ T((Rd)) +we set +⟨a, b⟩ := +� +|I|≥0 +⟨aI, bI⟩. +Observe in particular that bI = ⟨eI, b⟩. +In the present work it will be useful to enumerate the elements of the truncated tensor +algebra. To this extent we introduce the isomorphism vec : T (n)(Rd) → Rdn and an injective +labeling function L : {I : |I| ≤ n} −→ {1, . . . , dn}, such that +vec(u) := +� +|I|≤n +eL (I)uI. +(2.1) +7 + +Finally, we denote by ( · )(1) : T((Rd)) → (T((Rd)))d and ( · )(2) : T((Rd)) → (T((Rd)))d×d +the shifts given by +u(1) := +� � +|I|≥0 +uIeI′ +� +(1{i|I|=1}, . . . , 1{i|I|=d})⊤, +u(2) := +� � +|I|≥0 +uIeI′′ +� +� +� +� +1{i|I|−1=i|I|=1} +· · · +1{i|I|−1=1,i|I|=d} +... +· · · +... +1{i|I|−1=d,i|I|=1} +· · · +1{i|I|−1=i|I|=d} +� +� +� . +(2.2) +Observe in particular that +⟨a, u(1)⟩ = +� +|I|≥0 +� +aIu(I1), . . . , aIu(Id) +�⊤ +and +⟨a, u(2)⟩ = +� +� +� +aIu(I11) +· · · +aIu(I1d) +... +· · · +... +aIu(Id1) +· · · +aIu(Idd) +� +� +� +for each a ∈ T(Rd). +Throughout the paper we fix a filtered probability space (Ω, F, (Ft)t≥0, Q) on which we +consider the stochastic processes to be defined. We are now ready to introduce the signature +of an Rd-valued continuous semimartingale. +Definition 2.1. Let X be a continuous Rd-valued semimartingale with d ≥ 1. The signature +of X is the T((Rd))-valued process (s, t) �→ Xs,t whose components are recursively defined +as +⟨e∅, Xs,t⟩ := 1, +⟨eI, Xs,t⟩ := +� t +s +⟨eI′, Xs,r⟩ ◦ dXin +r , +for each I = (i1, . . . , in) , I′ = (i1, . . . , in−1) and 0 ≤ s ≤ t, where ◦ denotes the Stratonovich +integral. Its projection Xn on T (n)(Rd) is given by +Xn +s,t = +� +|I|≤n +⟨eI, Xs,t⟩eI +and is called signature of X truncated at level n. If s = 0, we use the notation Xt and Xn +t , +respectively. +Observe that the signature of X and the signature of X − c coincide for each c ∈ R. +Moreover, with an equivalent notation we can write +Xt = +� +1, +� t +0 +1 ◦ dX1 +s , . . . , +� t +0 +1 ◦ dXd +s , +� t +0 +� � s +0 +1 ◦ dX1 +r +� +◦ dX1 +s , +� t +0 +� � s +0 +1 ◦ dX1 +r +� +◦ dX2 +s , . . . , +� t +0 +� � s +0 +1 ◦ dXd +r +� +◦ dXd +s , . . . +� +. +A well-known and extremely useful property of the signature is that every polynomial +function in the signature has a linear representation. For the precise statement we first +need to introduce the following concept (see also Definition 2.4 in Lyons et al. (2020) or +Section 2.2. in Bayer et al. (2021)). +8 + +Definition 2.2. For every two multi-indices I and J the shuffle product is defined recur- +sively as +eI � eJ := (eI′ � eJ) ⊗ ei|I| + (eI � eJ′) ⊗ ej|J|, +with eI � e∅ := e∅ � eI = eI. It extends to a, b ∈ T(Rd) as +a � b = +� +|I|,|J|≥0 +aIbJ(eI � eJ). +Observe that (T(Rd), +, �) is a commutative algebra, which in particular means that +the shuffle product is associative and commutative. +The proof of the paths version of the next result for can be found for instance in Ree +(1958) or Lyons et al. (2007). +Proposition 2.3 (Shuffle property). Let X be a continuous Rd-valued semimartingale and +I, J be two multi-indices. Then for each 0 ≤ s ≤ t +⟨eI, Xs,t⟩⟨eJ, Xs,t⟩ = ⟨eI � eJ, Xs,t⟩. +(2.3) +Proof. The result follows by induction using the chain rule for Stratonovich integrals. +We recall now an important property of the signature. +The result is known in the +rough paths literature (see for instance Boedihardjo et al. (2016)), but can also be proved +directly in the simpler situation of a continuous semimartingale that contains time as strictly +monotone component. +Lemma 2.4 (Uniqueness of the signature). Let X and Y be two continuous Rd-valued +semimartingales with X0 = Y0 = 0. Set � +Xt := (t, Xt), �Yt := (t, Yt) and let �X and �Y be the +corresponding signature processes. Then �XT = �YT if and only if Xt = Yt for each t ∈ [0, T]. +Proof. See Cuchiero et al. (2022a). +In order to combine the value of the signature on different time intervals Chen’s identity +going back to Chen (1957, 1977) turns out to be fundamental. +Lemma 2.5 (Chen’s identity). Let X be an Rd-valued semimartingale. Then +Xs,t = Xs,u ⊗ Xu,t +(2.4) +for each 0 ≤ s ≤ u ≤ t. This can equivalently be written as +⟨eI, Xs,t⟩ = +� +eI1⊗eI2=eI +⟨eI1, Xs,u⟩⟨eI2, Xu,t⟩, +for each multi-index I. +Proof. See Cuchiero et al. (2022a) for a direct proof using the definition of Stratonovich +integrals. +Let us recall also the universal approximation theorem of linear functions of the signature +in the context of continuous semimartingales as stated in Cuchiero et al. (2022a). We refer +to Cuchiero and M¨oller (2023), in the same context, for the situation involving not just the +approximation of a functional up to a fixed final value T, but a uniform approximation on +the whole time interval [0, T]. +9 + +Theorem 2.6 (Universal approximation theorem). Define the set +S(2) := {(�X2 +t )t∈[0,T](ω): ω ∈ Ω} +and consider a generic distance dS(2) on the set of trajectories given by S(2), with respect to +which the map from S(2) to R given by +ˆx2 �→ ⟨eI, S(|I|)(ˆx2)t⟩ +is continuous for each multi-index I and every t ∈ [0, T]. +Let K be a compact subset of S(2) and consider a continuous map f : K → R.4 Then +for every ε > 0 there exists some ℓ ∈ T(Rd) such that +sup +(�X2 +t )t∈[0,T ]∈K +|f((�X2 +t )t∈[0,T]) − ⟨ℓ, �XT ⟩|< ε, +almost surely. +Proof. See Theorem 2.12 and Remark 2.13 in Cuchiero et al. (2022a) for details concerning +the proof and the choice of the metric dS(2). +For the related concept of stochastic Taylor expansions and functional expansions we +refer the reader to Section 2.3 in Cuchiero et al. (2022a) and to Dupire and Tissot-Daguette +(2022), respectively. +Finally, we introduce the concept of polynomial diffusion process which will play a key +role for the computation of conditional expected signatures. Here we denote by √ · the +matrix square root. +Definition 2.7. Suppose that an Rd-valued process X = (Xt)t≥0 is a weak solution of +dXt = b(Xt)dt + +� +a(Xt)dWt, +X0 = x0 +(2.5) +for some d-dimensional Brownian motion W and some maps a : Rd → Sd ++ and b : Rd → Rd +such that aij is a polynomial of degree at most 2 and bj is a polynomial of degree at most +1 for each i, j ∈ {1, . . . , d}. Then we call X polynomial (diffusion) process. +3 +The model +We introduce now the model (St)t≥0 for the discounted dynamics of the S&P 500 index +already outlined in the introduction. Its dynamics under a risk-neutral probability measure +Q are given by +dSt = StσS +t dBt, +(3.1) +where S0 ∈ R+, σS = (σS +t )t≥0 is the volatility process to be specified and B = (Bt)t≥0 is a +one-dimensional Brownian motion, correlated with σS. We define additionally the instan- +taneous variance via Vt := (σS +t )2 for every t ≥ 0. Our modeling choice is to parametrize +the volatility process σS as a linear function of the time-extended signature of a primary +underlying process X, namely +σS +t (ℓ) := ℓ∅ + +� +0<|I|≤n +ℓI⟨eI, �Xt⟩, +(3.2) +where +4Compactness and continuity are defined with respect to dS(2). +10 + +• (Xt)t≥0 is a d-dimensional continuous semimartingale with positive quadratic variation +and (�Xt)t≥0 denotes the signature of its time extension ( � +Xt)t≥0 given by � +Xt := (t, Xt); +• ℓ := {ℓI ∈ R : |I| ≤ n} denotes the collection of parameters of the model, i.e., +ℓ ∈ R(d+1)n. +Furthermore, we denote by (Zt)t≥0 the process given by Zt = (Xt, Bt), by ( �Zt)t≥0 its +time extension, and by (�Zt)t≥0 the signature of ( �Zt)t≥0. The correlation matrix process +between the components of Z is denoted by +ρij = +[Zi, Zj] +� +[Zi] +� +[Zj] +∈ [−1, 1], +for all i, j = 1, . . . , d + 1, where [ · , · ] denotes the quadratic variation. +Observe that ρ +encodes in particular the correlation between X and B. In order to simplify the notation +we will drop the dependence on ℓ for the processes S = (St)t≥0 and (σS +t )t≥0 as in (3.1), +whenever this does not cause any confusion. +Remark 3.1. As an alternative definition for the volatility process (σS +t )t≥0 one can set +σS +t (ℓ) := ℓ∅ + +� +0<|I|≤n +ℓI⟨eI, �Xt−ε,t⟩, +for some fixed ε > 0. In this case the value of the volatility process σS at time t does not +depend on the whole trajectory of the primary process X, but just on its evolution from +t − ε to t. For an economically reasonable choice for ε the lags used in Section 3.4 of Guyon +and Lekeufack (2022) can be adapted to the current setting. +Remark 3.2 (Interest rates and dividends). In the model given by (3.1) we describe the +discounted prices and construct the VIX from them, in line with the definition of the CBOE +for the computation of the VIX. However, contingent claims are often expressed in terms +of undiscounted prices. If the dynamics of the discounted price process are given by (3.1), +the undiscounted one fulfills +d ˜St = (r − q) ˜Stdt + ˜StσS +t (ℓ)dBt, +where here r, q ∈ R denote the interest rate and the dividend, respectively. +Therefore +˜St(ℓ) = e(r−q)tSt(ℓ) and the price of a call option on the S&P 500 index under our model, +reads +C(T, K) = E +� +e−rT ( ˜ST (ℓ) − K)+� += E[e−rT (e(r−q)T ST (ℓ) − K)+] +where T > 0 denotes the maturity time and K ∈ R the undiscounted strike price. +Possible choices for the primary process X can be tractable processes such as an Ornstein- +Uhlenbeck (OU) process or a Brownian motion. For the proposed analysis we indeed need +that the corresponding conditional expected signature can be computed easily. Consider +the following assumption. +Assumption 3.3. � +X = (t, Xt)t≥0 is a polynomial diffusion process in the sense of Defini- +tion 2.7. +11 + +It is worth mentioning that the pool of primary processes that satisfy Assumption 3.3 is +rather wide, including for example correlated Brownian motions, geometric Brownian mo- +tions, OU processes, Cox-Ingersoll-Ross (CIR) processes, Jacobi processes, and all continu- +ous affine processes. If Assumption 3.3 is in force then �Xn is a finite-dimensional polynomial +process in sense of Filipovi´c and Larsson (2016) and Cuchiero et al. (2012). Hence, the (con- +ditional) expected signature of � +X can be found by solving a finite-dimensional ODE, i.e., +can be written in terms of a matrix exponential. For further details see Section 4 below. +Remark 3.4. We illustrate here that several classical and also recently considered stochas- +tic volatility models are nested within our modeling choice (3.2) under Assumption 3.3. +• Suppose that (Xt)t≥0 is a 1-dimensional OU process and let the order of the signature +be n = 1, with ℓ∅ = ℓ(0) = 0 and ℓ(1) ̸= 0. Then the process S = (St)t≥0 coincides +with the Stein-Stein model, as introduced in Stein and Stein (1991). +• Suppose that (Xt)t≥0 is a 1-dimensional geometric Brownian motion without drift +and let the order of the signature be n = 1, with ℓ∅ = ℓ(0) = 0 and ℓ(1) ̸= 0. Then +the process S = (St)t≥0 coincides with the SABR model, as introduced in initially in +Hagan et al. (2002) with β = 1. +• Suppose that (Xt)t≥0 is a 1-dimensional OU process and let the order of the signature +be n = 5, with ℓ∅, ℓ(1), ℓ(1,1,1), ℓ(1,1,1,1,1) non-zero and ℓI = 0 otherwise. Then the +process S = (St)t≥0 coincides with the model considered in Abi Jaber et al. (2022a,b) +with an exponential kernel (a part from the deterministic input curve considered +there additionally). If we allow for (Xt)t≥0 not to be a semimartingale and we do not +consider the time augmentation, we can also include fractional kernels and therefore +the whole class of Gaussian polynomial volatility models introduced in Abi Jaber et al. +(2022a) within our framework. +4 +Expected signature of polynomial diffusion processes +Let (Yt)t≥0 be a polynomial diffusion process in sense of Definition 2.7 whose dynamics are +given by +dYt = b(Yt)dt + σ(Yt)dWt, +Y0 = y0, +(4.1) +where σ(Yt) denotes the matrix square root of a(Yt). Recall that in this case the components +of a : Rd → Sd ++ are polynomials of degree at most 2, the components of b : Rd → Rd are +polynomials of degree at most 1, and W = (Wt)t≥0 is a d-dimensional Brownian motion. +Denote then by Y the corresponding signature. +We now explain how to employ the polynomial technology to compute the conditional +expected signature of (Yt)t≥0. Several representations of related quantities in particular for +the Brownian case can be found in the literature, see for instance Fawcett (2003), Lyons +and Victoir (2004), Lyons and Ni (2015), Boedihardjo et al. (2021), Rossi Ferrucci and Cass +(2022). Our approach follows Cuchiero et al. (2023) and is based on the classical theory +of polynomial processes (see Cuchiero et al. (2012) and Filipovi´c and Larsson (2016)). +Even though results for the corresponding infinite dimensional stochastic processes (see for +instance Cuchiero and Svaluto-Ferro (2021); Cuchiero et al. (2021b)) are needed in the case +of general signature SDEs considered in Cuchiero et al. (2023), the polynomial property of +(Yt)t≥0 here permits to stay in the finite dimensional setting. +12 + +Lemma 4.1. Let (Yt)t≥0 be the polynomial process given by (4.1) and b and a be the +corresponding drift and diffusion coefficients. Then +bj(y) = bc +j + +d +� +k=1 +bk +j yk +and +aij(y) = ac +ij + +d +� +k=1 +ak +ijyk + +d +� +k,h=1 +akh +ij ykyh, +for some bc +j,bk +j ,ac +ij, ak +ij, akh +ij = ahk +ij ∈ R. Moreover, +bj(Yt) = ⟨bj, Y1 +t ⟩ +and +aij(Yt) = ⟨aij, Y2 +t ⟩ +for +bj = +� +bc +j + +d +� +k=1 +bk +j Y k +0 +� +e∅ + +d +� +k=1 +bk +j ek +and +aij = +� +ac +ij + +d +� +k=1 +ak +ijY k +0 + +d +� +k,h=1 +akh +ij Y k +0 Y h +0 +� +e∅ + +d +� +k=1 +� +ak +ij + 2 +d +� +h=1 +akh +ij Y h +0 +� +ek + +d +� +k,h=1 +akh +ij ek � eh. +Observe that the upper index on Y k +0 and Y h +0 refers to Y ’s components and not to powers. +Proof. The first part follows by the observation that by definition of polynomial processes +b and a are polynomials of degree at most 1 and 2, respectively. For the second part it +then suffices to note that ⟨e∅, Y1 +t ⟩ = ⟨e∅, Y2 +t ⟩ = 1, ⟨ek, Y1 +t ⟩ = ⟨ek, Y2 +t ⟩ = (Y k +t − Y k +0 ), and +⟨ek � eh, Y2 +t ⟩ = (Y k +t − Y k +0 )(Y h +t − Y h +0 ). +Lemma 4.2. Let (Yt)t≥0 be the polynomial process given by (4.1) and let b and a as +in Lemma 4.1. The truncated signature (Yn +t )t≥0 is a polynomial process in the sense of +Definition 2.7 and for each |I| ≤ n it holds that +⟨eI, Yn +t ⟩ = +� t +0 +⟨LeI, Yn +s ⟩ds + +� t +0 +⟨eI′, Yn +s ⟩σi|I|(Ys)dWs, +where the operator L : T((Rd)) → T((Rd)) satisfies L(T (n)(Rd)) ⊆ T (n)(Rd) and is given by +LeI = eI′ � bi|I| + 1 +2eI′′ � ai|I|−1i|I|. +(4.2) +Proof. Let σj(Yt) denote the j-th row of σ(Yt). By definition of the signature, Stratonovich +integral and by the shuffle property we can compute +⟨eI, Yt⟩ = +� t +0 +⟨eI′, Ys⟩ ◦ d⟨ei|I|, Ys⟩ += +� t +0 +⟨eI′, Ys⟩d⟨ei|I|, Ys⟩ + 1 +2 +� t +0 +⟨eI′′, Ys⟩d[⟨ei|I|−1, Ys⟩, ⟨ei|I|, Ys⟩]s += +� t +0 +⟨eI′, Ys⟩⟨bi|I|, Ys⟩ds + +� t +0 +⟨eI′, Ys⟩σi|I|(Ys)dWs ++ 1 +2 +� t +0 +⟨eI′′, Ys⟩⟨ai|I|−1, Ys⟩ds += +� t +0 +⟨eI′ � bi|I|, Ys⟩ds + +� t +0 +⟨eI′, Ys⟩σi|I|(Ys)dWs + 1 +2 +� t +0 +⟨eI′′ � ai|I|−1i|I|, Ys⟩ds += +� t +0 +⟨LeI, Ys⟩ds + +� t +0 +⟨eI′, Ys⟩σi|I|(Ys)dWs, +13 + +for each |I| ≥ 0. Since |I � J| = |I| + |J| it holds that L(T (n)(Rd)) ⊆ T (n)(Rd). For |I| ≤ n +we thus get that the corresponding drift’s components are linear maps in Yn. Similarly, +since aij = � +|I|,|J|≤1 λIJ +ij eI � eJ for some λIJ +ij ∈ R and for |I| ≤ n we can compute +⟨eI′, Ys⟩σi|I|(Yt) +� +⟨eJ′, Ys⟩σi|J|(Yt) +�⊤ = ⟨eI′, Ys⟩⟨eJ′, Ys⟩⟨ai|I|j|J|, Ys⟩ += +� +|H1|,|H2|≤1 +λH1H2 +i|I|j|J|⟨eI′ � eH1, Ys⟩⟨eJ′ � eH2, Ys⟩, +we also have that the components of the corresponding diffusion matrix are polynomials of +degree 2 in Yn. Lemma 2.2 in Filipovi´c and Larsson (2016) yields the polynomial property. +Since the linear operator L maps the finite dimensional vector space T (n)(Rd) to itself, +it admits a matrix representation. +Definition 4.3. We call the operator L defined in (4.2) dual operator corresponding to Y. +For each |I| ≤ n set then ηIJ ∈ R such that +LeI = +� +|J|≤n +ηIJeJ, +and fix a labelling injective function L : {I : |I| ≤ n} → {1, . . . , dn} as introduced before +(2.1). We then call the matrix G ∈ Rdn×dn given by +GL (I)L (J) := ηIJ, +(4.3) +the dn-dimensional matrix representative of L. +Observe that using the notation of (2.1), for each u ∈ T (n)(Rd) the matrix representative +G of L satisfies +vec(Lu) = Gvec(u). +Theorem 4.4. Let (Yt)t≥0 be the polynomial process given by (4.1), (Ft)t≥0 be the filtration +generated by (Yt)t≥0 and let G be the dn-dimensional matrix representative of the dual +operator corresponding to Y. Then for each T, t ≥ 0 and each |I| ≤ n it holds +E[vec(Yn +T+t)|FT ] = etG⊤vec(Yn +T ), +or equivalently, +E[⟨eI, Yn +T+t⟩|FT ] = +� +|J|≤n +(etG⊤)L (I)L (J)⟨eJ, Yn +T ⟩, +where e( · ) denotes the matrix exponential. +Proof. By Lemma 4.2 we know that vec(Yn) is a polynomial process and Theorem 3.1 in +Filipovi´c and Larsson (2016) for polynomials of degree 1 yields the claim. +Example 4.5. For the present paper a crucial role is played by the polynomial process given +by time, a d-dimensional OU process, and a Brownian motion. Specifically, we consider the +process �Zt := ( � +Xt, Bt) where B is a Brownian motion and � +Xt = (t, Xt) with +dXj +t = κj(θj − Xj +t )dt + +� +a(Xt)dWt, +X0 = x0, +14 + +for aij(Xt) = σiσjρij, and W being a d-dimensional Brownian motion. We denote by ρj(d+1) +the correlation between Xj and B. Setting κd+1 := 0 and σd+1 := 1 we can see that �Z +satisfies (4.1) in d + 2 dimensions for +bj( �Zt) = 1{j=0} + κj(θj − �Zj +t )1{j̸=0} +and +aij( �Zt) = σiσjρij1{i,j̸=0}. +The corresponding b and a are given by bj = e∅(1{j=0} + κj(θj − �Zj +0)1{j̸=0}) − ejκj1{j̸=0} +and aij = e∅σiσjρij1{i,j̸=0} and we thus get +LeI = eI′(1{i|I|=0} + κi|I|(θi|I| − �Z +i|I| +0 )1{i|I|̸=0}) − (eI′ � ei|I|)κi|I|1{i|I|̸=0} ++ 1 +2eI′′σi|I|−1σi|I|ρi|I|−1i|I|1{i|I|−1,i|I|̸=0}. +An application of L to the first basis elements yields the following results: +• L(e1) = e∅κ1(θ1 − X1 +0) − e1κ1; +• L(eI ⊗ e0) = eI � b0 + eI′ � ai|I|0 = eI; +• L(e0 ⊗ e1 ⊗ e2) = e0 ⊗ e1κ2(θ2 − X2 +0) − (e0 ⊗ e1) � e2κ2 + 1 +2e0σ1σ2ρ12. +Letting (Ft)t≥0 be the filtration generated by ( �Zt)t≥0 by Theorem 4.4 we can conclude that +E[vec(�Zn +T+t)|FT ] = etG⊤vec(�Zn +T ), +(4.4) +or equivalently, +E[⟨eI, �Zn +T+t⟩|FT ] = +� +|J|≤n +(etG⊤)L (I)L (J)⟨eJ, �Zn +T ⟩, +(4.5) +where G denotes the (d + 2)n-dimensional matrix representative of L. In order to work +with the VIX it will be convenient to restrict our attention to the signature components of +(�Zt)t≥0 not involving B. The following remark will be useful. +Remark 4.6. Observe that given a subset E ⊆ {0, . . . , d + 1}, setting IE := {I : ij ∈ E} it +holds L(IE) ⊆ IE. This in particular implies that +LeI = +� +I∈IE +ηIJeJ +for each I ∈ IE. Choosing E = {0, . . . , d}, letting LE : IE → {1, . . . , (d+1)n} be a labelling +injective function, and setting GE +LE(I)LE(J) := ηIJ we can see that (4.5) reduces to +E[⟨eI, �Xn +T+t⟩|FT ] = +� +|J|≤n +(et(GE)⊤)LE(I)LE(J)⟨eJ, �Xn +T ⟩. +To simplify the notation we often drop the E from GE whenever this does not introduce +any confusion. +Remark 4.7. Let (Yt)t≥0 be a polynomial process and let Y−1 be defined via e∅ = Y−1 +s ⊗Ys, +i.e. ⟨e∅, Y−1 +s ⟩ = 1 and +� +eI1⊗eI2=eI +⟨eI1, Y−1 +s ⟩⟨eI2, Ys⟩ = 0, +15 + +for each |I| > 0. Observe that it can be defined recursively on |I| and each component of +Y−1 +s +corresponds to a linear combination of components of Ys of the same length or shorter. +Since by Chen’s identity (see Lemma 2.5) we have Ys ⊗ Ys,t = Yt, for each s ≤ u ≤ t and +|I| ≤ n we then get +E[⟨eI, Ys,t⟩|Fu] = E[⟨eI, Y−1 +s +⊗ Yt⟩|Fu] = +� +eI1⊗eI2=eI +⟨eI1, Y−1 +s ⟩E[⟨eI2, Yt⟩|Fu] += +� +eI1⊗eI2=eI +⟨eI1, Y−1 +s ⟩vec(eI2)⊤e(t−u)G⊤vec(Yn +u), +where G denotes the dn-dimensional matrix representative of the dual operator of Y. +5 +VIX options with signatures +In this section we discuss the implication on pricing VIX options under the model (3.1)- +(3.2) when Assumption 3.3 is in force. The implications of these on the log-price will be +investigated in Section 6. +The CBOE Volatility Index, known by its ticker symbol VIX, is a popular measure +of the market’s expected volatility of the S&P 500 index, calculated and published by +the Chicago Board Options Exchange (CBOE). The current VIX index value quotes the +expected annualized change in the S&P 500 index over the following 30 days, based on +options-based theory and current options-market data, more precisely +VIXT := +� +E +� +− 2 +∆ log +�ST+∆ +ST +� +|FT +� +, +(5.1) +where ∆ = 30 days and ST denotes the price process at time T > 0. With the term VIX +options we here usually refer to either put or calls written on VIX. In the present work we +will take into account without loss of generality only call options. +5.1 +Explicit formulas for the VIX +This section is dedicated to one of the main implication of our modeling framework, namely +an explicit formula for the VIX expression (5.1) for S following (3.1)-(3.2) under Assump- +tion 3.3. In particular we show in the next theorem that the computation of the VIX squared +reduces to a quadratic form in the parameters ℓ. The entries of the corresponding positive +semidefinite matrix can be computed by polynomial technology, i.e. by matrix exponential +as proved in Section 4. +Theorem 5.1. Let S = (St)t≥0 be a price process described by +dSt = StσS +t dBt, +where σS = (σS +t )t≥0 denotes the volatility process, B = (Bt)t≥0 a one-dimensional Brownian +motion. +(i) Assume that V = (σS)2 satisfies +E +�� T +0 +Vsds +� +< ∞. +(5.2) +16 + +Then the VIX index at T > 0 is given by +VIXT = +� +1 +∆E +�� T+∆ +T +Vtdt|FT +� +. +(5.3) +(ii) Assume that σS and X satisfy (3.2) and Assumption 3.3, respectively. Fix an injective +labeling function L : {I : |I| ≤ n} → {1, . . . , (d+1)2n+1} and let G be the (d+1)(2n+1)- +dimensional matrix representative of the dual operator corresponding to �X. Then (5.2) +is satisfied and +VIXT (ℓ) = +� +1 +∆ℓ⊤Q(T, ∆)ℓ, +(5.4) +where +QL (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)vec(�X2n+1 +T +), +(5.5) +and Id ∈ R(d+1)2n+1×(d+1)2n+1 denotes the identity matrix. More explicitly without the +vectorisation this reads +QL (I)L (J)(T, ∆) = +� +eK=(eI�eJ)⊗e0 +� +|H|≤2n+1 +(e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩. +Proof. Part (i) follows directly from an application of Itˆo’s formula. Indeed, for any t, T ≥ 0 +log(St) = log(S0) − 1 +2 +� t +0 +Vsds + +� t +0 +σS +s dBs, +and hence +− 2 +∆ log +�ST+∆ +ST +� += − 2 +∆(log(ST+∆) − log(ST )) = 1 +∆ +� T+∆ +T +Vsds − 2 +∆ +� T+∆ +T +σS +s dBs, +where the integral with respect to the Brownian motion B = (Bt)t≥0 vanishes once we take +the risk neutral conditional expectation, due to (5.2). This yields the alternative expression +for VIX2 +T and thus for VIXT . +For part (ii), observe that +Vt(ℓ) = +� � +|I|≤n +ℓI⟨eI, �Xt⟩ +�2 += +� +|I|,|J|≤n +ℓIℓJ⟨eI � eJ, �Xt⟩. +Since continuous polynomials processes have finite moments of every degree and (5.2) is +satisfied due to Lemma 4.2. The expression for Vt(ℓ) yields then +VIX2 +T (ℓ) = 1 +∆ +� +|I|,|J|≤n +ℓIℓJE +�� T+∆ +T +⟨eI � eI, �Xt⟩dt|FT +� += 1 +∆ℓ⊤Q(T, ∆)ℓ, +17 + +where for each T > 0 the matrix Q is given by +QL (I)L (J)(T, ∆) : = E +�� T+∆ +T +⟨eI � eJ, �Xt⟩dt|FT +� += E +�� T+∆ +0 +⟨eI � eJ, �Xt⟩dt − +� T +0 +⟨eI � eJ, �Xt⟩dt|FT +� += E +� +⟨(eI � eJ) ⊗ e0, �XT+∆⟩ − ⟨(eI � eJ) ⊗ e0, �XT ⟩|FT +� += E +� +⟨(eI � eJ) ⊗ e0, �XT+∆⟩|FT +� +− ⟨(eI � eJ) ⊗ e0, �XT ⟩. +By Theorem 4.4 we can rewrite the matrix Q as +QL (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤e∆G⊤vec(�X2n+1 +T +) +− vec((eI � eJ) ⊗ e0)⊤vec(�X2n+1 +T +) += vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)vec(�X2n+1 +T +), +and the claim follows. +Remark 5.2. Consider now the model described in Remark 3.1 and set for simplicity +ε ≥ ∆. Then the results of Theorem 5.1(ii) still hold however with +QL (I)L (J)(T, ∆) = +� +eI1⊗eI2=eI�eJ +� T+∆ +T +⟨eI1, �X−1 +t−ε⟩vec(eI2)⊤e(t−T)G⊤vec(�XT )dt, +where G denotes the (d + 1)2n+1-dimensional matrix representative of the dual operator +corresponding to �X. To adapt the proof we just need to note that for each t ∈ [T, T + ∆] +Remark 4.7 yields +E[⟨eI � eJ, �Xt−ε,t⟩|FT ] = +� +eI1⊗eI2=eI�eJ +⟨eI1, �X−1 +t−ε⟩vec(eI2)⊤e(t−T)G⊤vec(�XT ). +Note that since the integration’s variable t appears twice in this expression the time integral +cannot be incorporated in the signature. +Remark 5.3. Observe that accounting for the scaling factor of 100, conventionally intro- +duced by CBOE, the VIX index squared can equivalently be redefined (see e.g., Rosenbaum +and Zhang (2021); Rømer (2022)) as +VIX2 +T := 1002 +∆ E +�� T+∆ +T +Vtdt|FT +� +, +(5.6) +where T, t > 0 and ∆ = +1 +12, i.e., approximately 30 days. Notice that since the expressions +(5.3) and (5.6) differ only by a scaling factor, all the theoretical results of the present +work hold true disregarding this scaling. For sake of simplicity we will always use (5.3). We +address the reader to Chapter 11 in Gatheral (2011) for further details about the conventions +of CBOE and its link with (5.1). +We observe that the expression (5.5) is computationally appealing as we can unpack +the computation in three parts: compute the coordinate vector vec((eI � eJ) ⊗ e0), which +depends just on d > 0 and n > 0, calculate the matrix exponential of G⊤ which depends on +the choice of the primary process X, and finally sample �X2n+1 +T +which is the only part that +depends on the chosen maturity time T. +18 + +Remark 5.4. In general the computation of G ∈ R(d+1)2n+1×(d+1)2n+1, even if done only +once, can be costly. For this reason it can sometimes be interesting to avoid the last time +integral and to consider the following equivalent expression of the matrix Q, for |I|, |J| ≤ n: +QL (I)L (J)(T, ∆) = +� � T+∆ +T +vec(eI � eJ)⊤e(t−T)G⊤dt +� +vec(�X2n +T ), +(5.7) +where now G ∈ R(d+1)2n×(d+1)2n and where we use the fact that we can interchange the +conditional expectation with the time-integral by dominated convergence. As G is singular, +this time integral has to be computed numerically, in general. We propose here two possible +methods that can be used in order to compute it efficiently. +(i) Approximation of the time integral: e.g., via the trapezoidal rule also applied for +VIX2 in Bourgey and De Marco (2021). Hence if we consider the shuffled coordinates +vec(eI�eJ) of the exponential matrix we can use the symmetry of the shuffle to reduce +the number of integrals to be solved from ((d + 1)2n)2 to (d+1)n((d+1)n+1) +2 +· (d + 1)2n, +instead of (d2n)2. Observe that for our integral the error of such an approximation is +given by +Err(N) = − ∆2 +12N2 G⊤(eG⊤∆ − I) + O(N−3), +as N → +∞. +As a further dimension reduction one can exploit the polynomial +nature of �Xn to obtain a matrix representation of its second order moments. Without +entering into details, the matrix G would then be the matrix corresponding to the +linear operator acting on coefficients of polynomials of degree 2 in �Xn. Its dimension +would thus be (d+1)n((d+1)n+1) +2 +. +(ii) Approximation of the matrix exponential: we can avoid to approximate the +integral by approximating the matrix exponential. Assuming that +lim +N→+∞(G⊤∆)N = 0, +(5.8) +this can for instance be done via its Taylor expansion: +� T+∆ +T +etG⊤dt = ∆ +� +I + G⊤∆ +2! ++ · · · + (G⊤∆)N +N + 1! + O((G⊤∆)N+1) +� +. +Observe that (5.8) holds true whenever the spectral radius, i.e., the maximal eigen- +value in absolute value, of the matrix G⊤∆ is less than 1 (see for instance Theorem 1.5 +in Quarteroni et al. (2010)). This requirement suggests that for numerical purposes +the parameters of the primary underlying process have to be chosen accordingly. +An interesting example is given by the case where X is a d-dimensional correlated +Brownian motion, as considered for instance in Cuchiero et al. (2022a). In this case +the process has no linear drift and the corresponding matrix G is nilpotent, meaning +that Gn = 0, for each n big enough. +In general, this Taylor approach permits to avoid a numerical integration and produces +an accurate approximation, allocating as few memory as possible. +Remark 5.5. A further step in the direction of a fast evaluation of VIXT (ℓ) can be taken +by noticing that the matrix Q in (5.5) admits a Cholesky decomposition. Indeed since Q is +19 + +positive semidefinite and symmetric by the shuffle property, we know that there exists an +upper triangular matrix UT ∈ R(d+1)n×(d+1)n, with possible zero elements on the diagonal, +such that +Q(T, ∆) = UT U ⊤ +T , +where for sake of simplicity we drop the dependence on ∆ of UT . Hence the evaluation of +the VIXT (ℓ) reduces to +VIXT (ℓ) = +� +1 +∆ℓ⊤UT U ⊤ +T ℓ = +1 +√ +∆ +� +(U ⊤ +T ℓ)2 = +1 +√ +∆ +∥U ⊤ +T ℓ∥, +where here ∥ · ∥ denotes the Euclidean norm. We stress the fact that the Cholesky decom- +position can be carried out offline, and the computational benefit is immediate if several +samples of the signature are considered. +In the following remark we discuss a possible dimension reduction technique from which +one can benefit computationally. We follow the approach of Cuchiero et al. (2022b, 2021a), +where by employing the Johnson-Lindenstrauss Lemma a random projection of the signature +is considered, to which we refer as a randomized signature. A first way to use this tool is +the following. +Remark 5.6. Let d< ∈ N be the dimension of the space to which we would like to project +the signature of order n > 0, such that d< ≪ (d + 1)n. Consider A = (αij) ∈ Rd<×(d+1)n, +such that αij ∼ N(0, 1/d<). Then a possible way to employ the randomised signature is to +parametrize the volatility process as follows, +σS +t (ℓ) := ˜ℓ⊤A · vec(�Xn +t ) +where with ˜ℓ = ℓ · A⊤ ∈ Rd< we denote the randomised parameters. Due to the linearity of +integral and conditional expectation in (5.3) this modeling choice is equivalent to consider +the randomised matrix �Q ∈ Rd<×d< given by +�QL (I)L (J)(T, ∆) := AQL (I)L (J)(T, ∆)A⊤, +which leads to the following representation of VIXT (ℓ): +VIXT (ℓ) = +� +1 +∆ +˜ℓ⊤ ˜Q(T, ∆)˜ℓ. +Observe that even if this procedure does not reduce the number iterated integrals to be +computed offline, it reduces the number of parameters to calibrate, yielding in general to a +faster evaluation of VIXT (ℓ). +5.2 +Options on VIX +We here briefly describe some important aspect of options on VIX. First of all, note that +VIX options are written on VIX futures. The price process of a VIX future contract with +maturity T > 0, is given by +Ft(T) := E [VIXT |Ft] . +(5.9) +This has the following implications: +• The price of a VIX option depends on the maturity of the corresponding VIX future. +20 + +• In calibration procedures one should therefore not normalize, i.e. not use VIXt := +VIXt +E[VIXT ], as E[VIXT ] = 1. +• When calibrating to VIX options, we stress that we additionally calibrate to VIX +futures’ prices, see Section 5.4. This is important since future prices under the cali- +brated model are employed to compute its implied volatility surface. Including VIX +futures in the calibration leads to a consistent model, both for VIX options and VIX +futures, see e.g. Pacati et al. (2018); Guo et al. (2020); Guyon (2020a, 2021). Using +market prices of the VIX futures to invert the implied volatility surface could lead to +inconsistencies if one would like to price further derivatives with the calibrated model. +In this respect, let us here also comment on the computation of implied volatilities for +VIX call options. Two equivalent approaches can be used to compute implied volatilities +for a given maturity T > 0: +• use F0(T) in the so-called Black formula for options on futures (see Section 2.2 of +Papanicolaou and Sircar (2014)). +• Consider e−rT F0(T), with r > 0 the interest rate, in the Black-Scholes formula. +Recall that the Black formula coincides with Black-Scholes’ one when choosing the underly- +ing in the Black-Scholes formula to be e−r(T−t)Ft(T). This implies that the two approaches +are equivalent, i.e. lead to the same implied volatilities. In Section 5.4 we will only consider +futures’ prices at time t = 0 and maturity time T > 0, hence for sake of simplicity we use +the notation F(T) instead of F0(T). +5.3 +Variance reduction for pricing VIX options +We here discuss variance reduction techniques (see e.g. Glasserman (2004)) that can speed +up the calibration in the subsequently applied Monte Carlo approach further. The key idea +is to introduce a control variate, namely an easy to evaluate random variable Φcv such that +given T > 0 and K > 0, +E[Φcv] = 0, +Var +� +(VIXT (ℓ) − K)+ − Φcv� +< Var +� +(VIXT (ℓ) − K)+� +. +A well-working example of control variates used for pricing and calibrating neural SDE +models can be found in Cuchiero et al. (2020); Gierjatowicz et al. (2020), where Φcv is +constructed from hedging strategies. +In the following we describe two possible choices of control variates, which consist of +polynomials on VIX futures. We stress the fact that these can be seen as linear functions +of the signature of the primary process � +X, hence they belong to the class of sig-payoffs, see +Lyons et al. (2020); Perez Arribas et al. (2020) and Section 4.2.2 in Cuchiero et al. (2022a). +• The first example is to employ the VIX squared as main ingredient, see for instance +Bourgey and De Marco (2021); Guerreiro and Guerra (2022) for a similar choice within +a rough Bergomi model for pricing VIX options. This is particularly easy to treat in +our set up, as for any given maturity T > 0 we have +E[VIX2 +T (ℓ)] = 1 +∆ℓ⊤Qcv(T, ∆)ℓ, +21 + +with Qcv(T, ∆) := E[QL (I)L (J)(T, ∆)]. By Theorem 5.1 and Theorem 4.4 we indeed +have +Qcv +L (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id) E[vec(�X2n+1 +T +)] += vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)eTG⊤vec(�X2n+1 +0 +) += vec((eI � eJ) ⊗ e0)⊤(e(T+∆)G⊤ − eTG⊤)vec(�X2n+1 +0 +) +where G denotes the (d+1)2n+1-dimensional matrix representative of the dual operator +corresponding to �X and vec(�X2n+1 +0 +) = e∅ ∈ R(d+1)2n+1. +Observe that Qcv can again be computed offline similarly to the matrix Q. Thus to +compute the expectation of VIX2 +T (ℓ) we only have to evaluate the previous quadratic +form. To apply this now for pricing a call option with maturity T > 0 and strike +K > 0, we set +Φcv(ℓ, T, K) : = cT,K(∆VIX2 +T (ℓ) − ℓ⊤Qcv(T, ∆)ℓ), += cT,K(ℓ⊤(Q(T, ∆) − Qcv(T, ∆))ℓ), +where the constant cT,K maximizing the variance reduction is given by: +c∗ +T,K = Cov((VIXT (ℓ) − K)+, ℓ⊤Q(T, ∆)ℓ) +Var(ℓ⊤Q(T, ∆)ℓ) +. +Notice that also in this case both Q and Qcv satisfy the condition for applying the +Cholesky decomposition, leading to a faster evaluation of the control variate as dis- +cussed in Remark 5.5. Note that the Cholesky decomposition cannot be applied to +Q − Qcv, as this is in general an indefinite matrix. +• As a second example we consider a generic polynomial in VIX2 as control variate by +defining +Y cv +m (ℓ, T, K) = +m +� +i=0 +αi(T, K)(VIX2 +T (ℓ))i +(5.10) +where αi(T, K) are chosen to approximate the payoff (VIXT −K)+ with strike price K +for some m ≥ 1. The corresponding control-variate is then defined as Φcv(ℓ, T, K) := +cT,K (Y cv +m (ℓ, T, K) − E[Y cv +m (ℓ, T, K)]). Regarding the computational effort, let us re- +mark the following. +(i) VIX2 +T is computed anyway for every realisation and is hence already available, +therefore the computation of Y cv +m (ℓ, T, K) is not expensive. +(ii) It is possible to calculate E[Y cv +m (ℓ, T, K)] analytically relying on the moment +formula, see Theorem 4.4. +(iii) The choice of cT,K ∈ R is important and the optimal one, i.e., the one leading +the highest variance reduction, is given by the following expression +c∗ +T,K = Cov((VIXT (ℓ) − K)+, Y cv +m (ℓ, T, K)) +Var(Y cv +m (ℓ, T, K)) +, +see for instance Section 4.1.1 in Glasserman (2004). +We stress the fact that for m = 1 the two control variates introduced coincide. +22 + +5.4 +Calibration to VIX options +In this section we focus on the calibration to VIX options only. Let T be a set of maturities +and K a collection of strikes. Consider the model given by (3.1) and (3.2) and assume that +Assumption 3.3 is in force. +Using Monte Carlo compute an approximation of option and futures’ prices with NMC > +0 samples, i.e. +πmodel +VIX (ℓ, T, K) ≈ e−rT +NMC +NMC +� +i=1 +(VIXT (ℓ, ωi)−K)+, +F model +VIX (ℓ, T) ≈ +1 +NMC +NMC +� +i=1 +VIXT (ℓ, ωi), +(5.11) +where +VIXT (ℓ, ω) = +� +1 +∆ℓ⊤Q(T, ∆)(ω)ℓ = +1 +√ +∆ +∥U ⊤ +T (ω)ℓ∥. +It is crucial to note that in this framework a Monte Carlo approach is tractable since +for every ℓ the same samples can be used. This means that we do not need to carry out +any simulation during the optimization task. Indeed, the matrix Q can be simulated offline +while only the products with ℓ ∈ R(d+1)n enter in the calibration step. +Observe that an auxiliary randomization can be employed in every optimisation step as +discussed in Remark 5.6. Moreover, if we want to use control variates to reduce the variance +of the Monte Carlo estimator as described in the previous section, we would consider +πmodel +VIX (ℓ, T, K) ≈ e−rT +NV R +NV R +� +i=1 +(VIXT (ℓ, ωi) − K)+ − Φcv(ℓ, T, K)(ωi). +Due to the variance reduction the number of samples needed is NV R ≪ NMC and Φcv is as +in Section 5.3 +The calibration to VIX call options and the corresponding futures on T and K consists +in minimizing the functional +LVIX(ℓ) := +� +T∈T ,K∈K +L +� +πmodel +VIX (ℓ, T, K), πb,a +VIX(T, K), σb,a +VIX(T, K), F model +VIX (ℓ, T), F mkt +VIX(T) +� +, +(5.12) +where L denotes a real-valued loss function, F mkt +VIX(T) the market’s futures’ prices and +πb,a +VIX(T, K) := {πmkt,b +VIX (T, K), πmkt,a +VIX (T, K)}, +σb,a +VIX(T, K) := {σmkt,b +VIX (T, K), σmkt,a +VIX (T, K)}, +the market’s option bid/ask prices πmkt,b +VIX (T, K), πmkt,a +VIX (T, K), and bid/ask implied volatili- +ties σmkt,b +VIX (T, K), σmkt,a +VIX (T, K), respectively. We will specify the choice of the function L in +Section 5.4.1 and Section 7.1. +Remark 5.7 (Initial guess search). Since within our model choice we are given a quadratic +function in ℓ to be minimized, a stochastic optimization with an initial guess is employed. +In order to achieve faster convergence we consider an hyperparameter search to choose the +starting parameters. The steps are outlined as follows. +• Find the magnitude of the coefficients returning Monte Carlo prices of the VIX options +close to the one observable on the market. To this extent we sample Nℓ > 0 times +parameters ℓ ∈ Ji = [−10−i, 10−i](d+1)n, for i = 1, . . . , m with m > 0. +23 + +• Select J∗ ∈ (Ji)m +i=1 such that +J∗ ∈ argmini: ℓ∈Ji LVIX(ℓ). +• Choose the initial guess to be +ℓinitial ∈ argminℓ∈J∗ LVIX(ℓ). +5.4.1 +Numerical results +In the present section we report the results of the calibration to VIX options only. Here +we consider call options written on the VIX on the trading day 02/06/2021, the same as in +Guyon and Lekeufack (2022). We stress that for such recent dates the bid-ask spreads for +VIX options are rather tight with respect to older dated options as considered for instance +in Gatheral et al. (2020); Bondi et al. (2022b). The maturities are reported in the following +table with the corresponding range of strikes (in percentage) with respect to the market’s +futures prices. +T1 = 0.0383 +T2 = 0.0767 +T3 = 0.1342 +T4 = 0.2108 +T5 = 0.2875 +T6 = 0.3833 +(90%,250%) +(90%,250%) +(80%,310%) +(80%,300%) +(75%,395%) +(80%,405%) +We underline that the shortest maturity considered is 14 days. Regarding our modeling +choice we fix d = 2, n = 3, which means to calibrate 40 parameters. For X we choose a +2-dimensional Ornstein-Uhlenbeck processes, see Example 4.5, with the following (hyper- +parameter) configuration: +κ = (0.1, 25)⊤, +θ = (0.1, 4)⊤, +σ = (0.7, 10)⊤, +ρ = +� +� +1 +−0.577 +0.3 +· +1 +−0.6 +· +· +1 +� +� , +where the last column of ρ are the correlations with the Brownian motion B driving the price +process S. The motivation of this parameters choice is to mimic a rough or strong mean- +reverting model as suggested in Rogers (2023); Rømer (2022). We refer to Appendix A for +numerical results where we use only a correlated 2-dimensional Brownian motion as primary +process, which yields significantly worse results. Before stating the loss function L that we +employed in the calibration task, let us make the following remark. +Remark 5.8. Let f : R+ × R+ → R+ be the call pricing functional in the Black-Scholes +model, depending on the volatility σBS and the spot price ξ, i.e., f : (σBS, ξ) �→ f(σBS, ξ). +By Taylor expansion in an appropriate neighbourhood of (σmkt, ξmkt) we obtain +f(σBS, ξ) ≈f(σmkt, ξmkt) + ∂f +∂σ(σmkt, ξmkt)(σBS − σmkt) + ∂f +∂ξ (σmkt, ξmkt)(ξ − ξmkt), +which equivalently gives +(σBS − σmkt) ≈ +1 +∂f +∂σ(σmkt, ξmkt) +� +f(σBS, ξ) − f(σmkt, ξmkt) +� +− +∂f +∂ξ (σmkt, ξmkt) +∂f +∂σ(σmkt, ξmkt) +(ξ − ξmkt), +(5.13) +where we recognize for the derivatives with respect to σ and ξ, the Greeks Vega and Delta, +respectively. +24 + +Motivated by Remark 5.8 we propose, for a fixed maturity and strike price, the following +loss-function for β ∈ {0, 1} +Lβ(π, πmkt,b,a, σmkt,b,a, F, F mkt) = +(5.14) +�� +β˜1{π /∈[πmkt,b,πmkt,a]} + (1 − β) +���π − (πmkt,a + πmkt,b)/2 +�� + +��δmkte−rT (F − F mkt) +�� +υmkt(σmkt,a − σmkt,b) +�2 +, +where +• υmkt and δmkt denote the Vega and Delta of the option under the Black-Scholes model +which depend on the maturity and on the strike price; +• F and F mkt denote futures with maturity T such that the variables ξ, ξmkt appearing +in Remark 5.8 are ξ = e−rT F and ξmkt = e−rT F mkt, respectively; +• ˜1{x/∈[yb,ya]} := s(yb − x) + s(x − ya) for s(x) := 1 +2 tanh(100x) + 1 +2 a smooth version of +the indicator function. +Remark 5.9. +(i) We observe that by Remark 5.8 minimizing L0 is equivalent to mini- +mizing an upper bound of the square of the right-hand side of (5.13) normalized by +the bid-ask spread of the implied volatilities. Note that we slightly abused notation, +since υmkt and δmkt of course depend on the strike and the maturity. +(ii) Note that as ℓ �→ VIXT (ℓ, ω) = +1 +√ +∆∥U ⊤ +T ℓ∥ is convex and the call payoff is convex +and increasing, the model option and future prices are convex in ℓ. If β = 0 and the +initialization of ℓ is such that both the model and future prices are higher than the +market ones, then we actually deal with a convex optimization problem. +(iii) If our aim does not consist in calibrating to the mid-price or mid-implied-volatility +precisely, but we merely want to be within the bid-ask spreads we can set β = 1 +For the next calibration result we minimize L1 as introduced above with NMC = 80000 +Monte Carlo samples for the previous maturities and strikes. +25 + +Figure 1: The red crosses denote the bid-ask spreads (of the implied volatilities) for each +maturity, while the azure dots denote the calibrated implied volatilities of the model. On +the x-axis we find the strikes and on the y-axis we find the maturities. +We observe that the calibrated VIX smiles fall systematically in the bid-ask corridor for all +the maturities considered. We report additionally in the next tables the relative absolute +error between the market future prices and the calibrated ones for each maturity, i.e., +εT := |F mkt(T) − F model +VIX (ℓ∗, T)| +F mkt(T) +, +(5.15) +where ℓ∗ ∈ R40 denotes the calibrated parameters and here F model +VIX (ℓ∗, T) stands for the +calibrated future model price. In Figure 2 we can find an illustration of the calibrated and +the market futures’ term structure. +T1 = 0.0383 +T2 = 0.0767 +T3 = 0.1342 +εT1 = 7.0 × 10−6 +εT2 = 2.1 × 10−3 +εT3 = 1.3 × 10−5 +T4 = 0.2108 +T5 = 0.2875 +T6 = 0.3833 +εT4 = 1.5 × 10−4 +εT5 = 1.9 × 10−6 +εT6 = 1.3 × 10−6 +26 + +Implied Volatilities ViX 02-06-2021 +2.50 +2.25 +2.00 ++ ++ +1.75 ++ ++ +IV +1.50 +T +1.25 +1.00 ++ +H +0.75 +0.05 +0.10 +0.15 +0.20 +90 +80 +0.25 +70 +60 +Maturities +0.30 +50 +Strikes +40 +0.35 +30 +20 +0.40Figure 2: The blue circles denote the calibrated future prices and the red crosses the future +prices on the market, in between a linear interpolation is reported. +5.5 +The case of time-varying parameters +We now consider the case of maturity dependent parameters as for instance employed in +Gierjatowicz et al. (2020); Cuchiero et al. (2022a). +Since it will be important later on +to distinguish maturities of options written on the VIX index from maturities of options +written on the SPX index, we introduce the two sets T VIX and T SPX. +Let us fix here +T VIX = {T1, . . . , TN}, where Ti < Ti+1 for any in i = 1, . . . , N −1 and denote by ℓ(Ti) ∈ Rdn +the parameters depending on the maturity Ti > 0. We set T0 = 0 and TN+1 = +∞. Then, +we consider for any t ≥ 0 the volatility process to be +σS +t (ℓ) = +N +� +i=0 +� +|I|≤n +ℓI(Ti)1[Ti,Ti+1)(t)⟨eI, �Xt⟩. +(5.16) +Therefore the variance process reads as follows, +Vt(ℓ) = +N +� +i=0 +� +|J|,|I|≤n +ℓI(Ti)ℓJ(Ti)1[Ti,Ti+1)(t)⟨eI � eJ, �Xt⟩. +(5.17) +Assumption 5.10. Assume that for a set of maturities T VIX it holds that |Ti − Tj| ≥ ∆ +for all i ̸= j. +Proposition 5.11. Let T VIX be a set of maturities on the VIX index and let Q(T, τ) be +the matrix as defined in (5.5) (here for general τ > 0 instead of ∆). Then, under (5.17) the +VIX squared at time Ti ∈ T VIX is given by +VIX2 +Ti(ℓ) = 1 +∆ +� N +� +j=i +ℓ(Tj)⊤ (Q(Ti, (Tj+1 − Ti) ∧ ∆) − Q(Ti, (Tj − Ti) ∧ ∆)) ℓ(Tj) +� +. +Note that, if Ti+1 − Ti > ∆ (which is in particular holds under Assumption 5.10) then, +VIX2 +Ti(ℓ) = 1 +∆ℓ(Ti)⊤Q(Ti, ∆)ℓ(Ti). +27 + +Futures' Term Structure +Calibrated +Market +X +23 +22 +21 +20 +19 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +TProof. By the definition of the VIX, it holds that +VIX2 +Ti(ℓ) = 1 +∆E +�� Ti+∆ +Ti +N +� +j=i +� +|J|,|I|≤n +ℓI(Tj)ℓJ(Tj)1[Tj,Tj+1)(t)⟨eI � eJ, �Xt⟩dt +����FTi +� += 1 +∆ +N +� +j=i +� +|J|,|I|≤n +ℓI(Tj)ℓJ(Tj)E +�� Tj+1∧(Ti+∆) +Tj∧(Ti+∆) +⟨eI � eJ, �Xt⟩dt +����FTi +� += 1 +∆ +N +� +j=i +� +|J|,|I|≤n +ℓI(Tj)ℓJ(Tj) +� +E +�� Tj+1∧(Ti+∆) +Ti +⟨eI � eJ, �Xt⟩dt +����FTi +� +− E +�� Tj∧(Ti+∆) +Ti +⟨eI � eJ, �Xt⟩dt +����FTi +�� +and hence the first statement follows by the definition of Q in (5.5). +Notice that also in the case of Proposition 5.11, Remark 5.5 applies. +6 +SPX as a signature-based model +The goal of this section is to express the discounted price of the SPX, modeled via (3.1)-(3.2) +dSt(ℓ) = St(ℓ)σS +t (ℓ)dBt, +in terms of the signature of ( �Zt)t≥0 = (t, Xt, Bt)t≥0, allowing again to precompute its +samples and use the same ones for every ℓ. This is in the same spirit as in Cuchiero et al. +(2022a), even though there the asset price was directly modeled as linear function of the +signature of some primary process. +Recall that by (3.2) σS is parametrized as follows +σS +t (ℓ) := ℓ∅ + +� +0<|I|≤n +ℓI⟨eI, �Xt⟩, +where � +Xt = (t, Xt) with X a d-dimensional continuous semimartingale. Before addressing a +more tractable expression for S, that allows to avoid (Euler) simulation schemes, we recall +the following well-known integrability result. +Lemma 6.1. Assume that E[S0] < ∞. +Then, the process (St)t≥0 is a (non-negative) +supermartingale and in particular E[St] < ∞ for each t ≥ 0. +Proof. Note that St = S0E +�� · +0 σS +s dBs +� +t for all t ≥ 0. Moreover ( +� t +0 σS +s dBs)t≥0 is a local +martingale and hence, by the properties of the stochastic exponential, St is a non-negative +local martingale. It follows from Fatou’s Lemma that non-negative local martingales are +supermartingales. +In the following we suppose without loss of generality that S0 = 1. +Remark 6.2. Recall that if Novikov’s condition is satisfied, then a stochastic exponential +of the form St = E +�� · +0 σS +s dBs +� +t for t ∈ [0, T] is a true martingale. For σS +s as in (3.2), such +condition reads +E +� +exp +�1 +2 +� T +0 +Vt(ℓ)dt +�� +< +∞. +28 + +Observe that +E +� +exp +�1 +2 +� T +0 +Vt(ℓ)dt +�� += E +� +exp +�1 +2 +� +|I|,|J|≤n +ℓIℓJ +� T +0 +⟨eI � eJ, �Xt⟩dt +�� += E +� +exp +�1 +2ℓ⊤Q0(T)ℓ +�� +, +(6.1) +where for L : {I : |I| ≤ n} → {1, . . . , (d + 1)n}, +Q0 +L (I)L (J)(T) := ⟨(eI � eJ) ⊗ e0, �XT ⟩. +We point out that the previous condition is not necessarily satisfied for all ℓ ∈ R(d+1)n. +Indeed, let us consider X to be a one-dimensional Brownian motion and choose ℓ such that +the only non trivial component is the last one, i.e., +ℓI := +� +c ∈ R, +if I = (1, . . . , 1), |I| = n, +0, +otherwise. +(6.2) +Then, (6.1) translates into +E +� +exp +�c2 +2 +� T +0 +2n! +n!n! +X2n +t +2n! dt +�� += E +� +exp +� +c2 +2(n!)2 +� T +0 +X2n +t dt +�� +, +which is not finite in general, e.g. if n = 2, c = +√ +2(n!) then by Jensen’s inequality it follows +E +� +exp +�� T +0 +X4 +t dt +�� +≥ E +� 1 +T +� T +0 +eTX4 +t dt +� += 1 +T +� T +0 +E[eTX4 +t ]dt = +∞. +Remark 6.3. Let us underline the fact that, even if the process St(ℓ) is a non-negative +local martingale, we can still price options via risk neutral expectations without introducing +arbitrage. The reason is that, although E[ST (ℓ)] < S0, it is impossible to take a short +position on S(ℓ) and a long position on Vt := E[ST (ℓ)|Ft] for all t ≤ T because of credit +constraints, therefore, the condition of No Free Lunch with Vanishing Risk (NFLVR) as +introduced in Delbaen and Schachermayer (1994) is not violated. We address the reader to +Kardaras et al. (2015) for further details. +The key idea is to rewrite (3.1) as a type of signature-based model in sense of Cuchiero +et al. (2022a) including B = (Bt)t≥0 as part of the primary process. This is possible since Itˆo +integrals with respect to primary process’ components can be rewritten as linear functions +of the signature of the primary process itself. To do so, we introduce Assumption 6.4 on +Z = (X, B) in order to describe the correlation structure between B and X. +Assumption 6.4. For all i, j ∈ {1, . . . , d + 1} it holds +d[Zi, Zj]t = +� +|J|≤m +aJ +ij⟨eJ, �Zt⟩dt +for some m ∈ N, aJ +ij ∈ R and where �Zt = (t, Zt). +Notice that this assumption is in particular satisfied if X is a polynomial process as +in Assumption 3.3, correlated with B. As shown in the next proposition it allows for the +aforementioned tractable representation of S(ℓ). +29 + +Proposition 6.5. Let S = (St)t≥0 satisfy (3.1) with S0 = 1, and σS = (σS +t )t≥0 satisfy +(3.2). Suppose additionally that Z = (X, B) satisfies Assumption 6.4. Then, +log(St(ℓ)) = −1 +2ℓ⊤Q0(t)ℓ + +� +|I|≤n +ℓI⟨˜eB +I , �Zt⟩, +(6.3) +where +˜eB +∅ := ed+1, +˜eB +I := eI ⊗ ed+1 − +� +|J|≤m +aJ +i|I|(d+1) +2 +(eI′ � eJ) ⊗ e0, +for each |I| > 0, and the components of the matrix Q0(t) ∈ R(d+1)n×(d+1)n are given by +Q0 +L (I)L (J)(t) = ⟨(eI � eJ) ⊗ e0, �Xt⟩, +for a labeling function L : {I : |I| ≤ n} → {1, . . . , (d + 1)n}. +Proof. Under our assumptions we can compute +log(St(ℓ)) = −1 +2 +� t +0 +Vs(ℓ)ds + +� t +0 +σS +s (ℓ)dBs += −1 +2 +� +|I|,|J|≤n +ℓIℓJ +� t +0 +⟨eI � eJ, �Xs⟩ds + +� +|I|≤n +ℓI +� t +0 +⟨eI, �Xs⟩dBs +(∗) += −1 +2 +� +|I|,|J|≤n +ℓIℓJ⟨(eI � eJ) ⊗ e0, �Xt⟩ + +� +|I|≤n +ℓI⟨˜eB +I , �Zt⟩ += −1 +2ℓ⊤Q0(t)ℓ + +� +|I|≤n +ℓI⟨˜eB +I , �Zt⟩, +where for (∗) we used that +� t +0⟨eI, �Xs⟩dBs = ⟨˜eB +I , �Zt⟩ by Lemma 3.10 in Cuchiero et al. +(2022a). +Remark 6.6. Consider again the model described in Remark 3.1. +Then the results of +Proposition 6.5 still hold with +Q0 +L (I)L (J)(t) := +� t +0 +⟨eI � eJ, �Xs−ε,s⟩ds, +and +� t +0⟨eI, �Xs−ε,s⟩dBs instead of ⟨˜eB +I , �Zt⟩. Since the proof follows closely the proof of the +original result, we omit it. +Remark 6.7. +• Observe that since the matrix (⟨eI �eJ, �Xt⟩)|I|,|J|≤n is positive semidef- +inite, by monotonicity of the time integral on [0, t] for some t > 0, we also have +ℓ⊤Q0(t)ℓ ≥ 0, +for all ℓ ∈ R(d+1)n. This means that for any t > 0, we can rewrite the log-price as +log(St) = −1 +2∥(U 0 +t )⊤ℓ∥2+ +� +|I|≤n +ℓI⟨˜eB +I , �Zt⟩, +where U 0 +t is the upper-triangular matrix of the Cholesky decomposition of Q0(t). +30 + +• Notice that the log-price model in (6.3), it is not exactly a signature-based model +in the sense of Cuchiero et al. (2022a), as here it is given by a linear part in the +parameters ℓ and an additional quadratic part. It can also be rewritten as +d log(St) = −1 +2ℓ⊤ ˜Q(t)ℓdt + ℓ⊤vec(�Xn +t )dBt, +where ˜Q is given by +˜QL (I)L (J)(t) := ⟨eI � eJ, �Xt⟩. +• In order to sample the log-price at maturity, consistently with the VIX, we follow +the following road map. We simulate �Z and compute ⟨˜eB +I , �Z⟩ for each I as specified +above. Next, we drop from the samples of �Z the terms where B appears, i.e. the +components corresponding to indices containing the letter d + 1. The result coincides +with a sampling of �X and is then used to work with both Q and Q0. +This is equivalent to sampling �X for the variance process and to compute an additional +Itˆo integral as in (3.1). +In the following corollary we state the form of ˜eB +I when X is d-dimensional OU-process. +We omit the proof for sake of brevity. +Corollary 6.8. Let X be a d-dimensional OU-process as in Example 4.5 driven by a d- +dimensional Brownian motion with correlation matrix ρ. Then Assumption 6.4 is satisfied +and ˜eB +I is given by +˜eB +I = eI ⊗ ed+1 − 1 +21{i|I|̸=0}(σi|I|ρi|I|d+1)eI′ ⊗ e0, +for any multi-index I ̸= ∅. +Remark 6.9 (Variance reduction for pricing SPX options). Observe that a possible control +variate for reducing the variance of the Monte Carlo estimator for pricing SPX options is +the value at maturity of the log-price process. This means, +Φcv(ℓ, T, K) : = cT,K +� +log(ST (ℓ)) + 1 +2ℓ⊤Q0,cv(T)ℓ +� +, +where, using that the linear part (in ℓ) of log(ST (ℓ)) vanishes under the risk-neutral expec- +tation, we have +Q0,cv +L (I)L (J)(T) = vec((eI � eJ) ⊗ e0)⊤eTG⊤vec(�X2n+1 +0 +), +for G ∈ R(d+1)2n+1×(d+1)2n+1 denoting the (d + 1)2n+1-dimensional matrix representative of +the dual operator corresponding to �X. We choose the optimal c∗ +T,K ∈ R as +c∗ +T,K = Cov((ST (ℓ) − K)+, log(ST (ℓ))) +Var(log(ST (ℓ))) +. +31 + +6.1 +Exploiting the affine nature of the signature: Fourier pricing of SPX +and VIX options +This section is dedicated to outline how the linear parametrizations of the log-price and the +volatility process in �Z can be used for Fourier pricing. Consider again the model given by +(3.1)-(3.2) and let (Zt)t≥0 denote the primary process given by Zt = (Xt, Bt) introduced +after (3.2). Suppose for simplicity that +dZj +t = κj(θj − Zj +t )dt + σjdW j +t , +Zj +0 = 0, +for each j = 1, . . . , d + 1, where W denotes a (d + 1)-dimensional Brownian motion with +W d+1 = B. All parameters κj, θj, σj are in R with κd+1 = θd+1 = 0 and σd+1 = 1 so that +Zd+1 = W d+1 = B. Note that we do not account for correlations. +We illustrate now how to apply the results of Cuchiero et al. (2023) in the present +setting. Since ( �Zt)t≥0 is a polynomial process, by Lemma 4.1 there are b ∈ (T((Rd+2)))d+2 +and a ∈ (T((Rd+2)))(d+2)×(d+2) such that +d �Zj +t = ⟨bj, �Zt⟩dt + +� +⟨ajj, �Zt⟩dW j +t , +where bj = κjθje∅ − κjej and ajj = (σj)2e∅, using that (with a small abuse of notation) +κ0θ0 := 1, κj := 0 and σ0 := 0. Using the notation of (2.2) consider then the Riccati +operator R given by +R(u) = b⊤ � u(1) + 1 +2 Tr(a � +� +u(2) + (u(1))⊤ � u(1)� +) += +d+1 +� +j=0 +� +|I|≥0 +� +κjθju(Ij)eI + κju(Ij)ej � eI + 1 +2(σj)2(u(Ijj)eI + u2 +(Ij)eI � eI) +� +. +In this equation Tr : R(d+2)×(d+2) → R denotes the trace operator and is applied component +wise to the elements of (T((Rd+2)))(d+2)×(d+2). By Theorem 4.23 in Cuchiero et al. (2023), +we expect that +E[exp(⟨u, �ZT ⟩)] = exp(ψ(T)∅), +where ψ is a solution5 of the extended tensor algebra valued Riccati equation +∂tψ(t) = R(ψ(t)), +ψ(0) = u. +Choosing u as +u(ℓ) := −1 +2(ℓ � ℓ) ⊗ e0 + ˜ℓ +where ˜ℓ := � +|I|≤n ℓI˜eB +I , by Proposition 6.5 we get +log(St(ℓ)) = ⟨u(ℓ), �Zt⟩. +The representation of the Fourier-Laplace transform described above can then be used for +Fourier pricing. We dedicate the remaining part of this section to illustrate how this can +be done. +5We refer to Cuchiero et al. (2023) for the appropriate solution concept. +32 + +From Fourier analysis we know that for K > 0 and C < 0 it holds +(K − ey)+ = 1 +2π +� +R +e(iλ+C)y +K−C+1−iλ +(iλ + C)(iλ + C − 1)dλ. +This in particular implies that +E[(K − ST (ℓ))+] = 1 +2π +� +R +E[e(iλ+C) log(ST (ℓ))] +K−C+1−iλ +(iλ + C)(iλ + C − 1)dλ += 1 +2π +� +R +E[e⟨uλ,�ZT ⟩] +K−C+1−iλ +(iλ + C)(iλ + C − 1)dλ += 1 +2π +� +R +eψλ(T)∅ +K−C+1−iλ +(iλ + C)(iλ + C − 1)dλ, +where uλ := (iλ+C)u(ℓ) and ψλ is a solution of the Riccati equation with initial condition +ψλ(0) = uλ. +Let us now consider the case of VIX options where Fourier pricing can be applied by +computing the Fourier-Laplace transform of VIX squared, see also Sepp (2008); Papani- +colaou and Sircar (2014); Cao et al. (2020); Bondi et al. (2022b) and references therein +for a Fourier-based approach to pricing VIX options. +Fix a labelling injective function +L : {I : |I| ≤ n} → {1, . . . , (d + 1)(2n+1)} as introduced before (2.1) and recall that by +Theorem 5.1(ii) it holds +VIX2 +T (ℓ) = 1 +∆ℓ⊤Q(T, ∆)ℓ +for +QL (I)L (J)(T, ∆) = +� +eK=(eI�eJ)⊗e0 +� +|H|≤2n+1 +(e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩. +where G denotes the (d + 1)(2n+1)-dimensional matrix representative of the dual operator +corresponding to �X. Setting for |I|, |J| ≤ n +I(∆)u := +� +|K|,|H|≤2n+1 +(e∆G⊤ − Id)L (K)L (H)uKeH +we can write +VIX2 +T (ℓ) = 1 +∆ +� +|I|,|J|≤n +ℓIℓJ +� +eK=(eI�eJ)⊗e0 +� +|H|≤2n+1 +(e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩ += 1 +∆ +� +|K|,|H|≤2n+1 +(e∆G⊤ − Id)L (K)L (H)⟨eK, (ℓ � ℓ) ⊗ e0⟩⟨eH, �Xt⟩ += 1 +∆⟨I(∆)((ℓ � ℓ) ⊗ e0), �XT ⟩. +Also in this case since +1 +√ +2π +� +R eiλy(K − +� +|y|)+dy = +� +2 +π FS(K√ +|λ|) +|λ|3/2 +is integrable for FS(u) = +� u +0 sin(z2)dz, Fourier analysis yields +(K − √y)+ = 1 +π +� +R +e−iλy FS(K +� +|λ|) +|λ|3/2 +dλ, +33 + +for each y ≥ 0. This in particular implies that +E[(K − VIXT (ℓ))+] = 1 +π +� +R +E[e−iλVIX2 +T (ℓ)]FS(K +� +|λ|) +|λ|3/2 +dλ = 1 +π +� +R +eψλ(T)∅ FS(K +� +|λ|) +|λ|3/2 +dλ, +where ψλ is a solution of the Riccati equation with initial condition ψλ(0) = −iλv(ℓ) for +v(ℓ) := 1 +∆I(∆)((ℓ � ℓ) ⊗ e0). +Analogous formulations in terms of the error function are also possible, see for instance +Cao et al. (2020); Bondi et al. (2022b). In the same spirit one can also obtain a represen- +tation of future prices. We here do not provide an implementation of this Fourier pricing +approach but numerical experiments can be found in Cuchiero et al. (2023). +6.2 +The case of time-varying parameters +Analogously to Section 5.5, we now further enhance Proposition 6.5 by allowing the param- +eters ℓ to depend on the maturity. +Proposition 6.10. Let S = (St)t≥0 satisfy (3.1) with S0 = 1, and (σS +t )t≥0 satisfy (5.16) +for a set of maturities T VIX = {T1, . . . , TN}. Recall that in this case V = (Vt)t≥0 satisfy +Vt(ℓ) = +N +� +i=0 +� +|J|,|I|≤n +ℓI(Ti)ℓJ(Ti)1[Ti,Ti+1)(t)⟨eI � eJ, �Xt⟩. +Let Zt = (Xt, Bt) for all t ≥ 0, where X = (Xt)t≥0 is a d-dimensional continuous semi- +martingale. Then, under Assumptions 6.4 we get the following recursion for the log-price +process +log(St(ℓ 0 and a strike price K > 0 +reads in our model as follows +e−rT ( ˜ST (ℓ) − K)+ = e−rT +� +exp +� +(r − q)T − 1 +2ℓ⊤Q0(t)ℓ + +� +|I|≤n +ℓI⟨˜eB +I , �ZT ⟩ +� +− K +�+ +, +where ˜S denotes the undiscounted process as discussed in Remark 3.2 and r, q > 0 the +interest rate and the dividends, respectively. Recall also that the call option payoff written +on the VIX is given by +e−rT (VIXT (ℓ) − K)+ = e−rT +�� +1 +∆ℓ⊤Q(T, ∆)ℓ − K +�+ += e−rT +� 1 +√ +∆ +∥U ⊤ +T ℓ∥−K +�+ +, +where UT denotes the upper-triangular matrix of the Cholesky decomposition of the sym- +metric positive semidefinite matrix Q(T, ∆). +7.1 +Numerical results +Before presenting our numerical results, let us discuss two different ways of approaching the +joint calibration problem that can be found in the recent literature. +(i) The first approach consists in choosing for instance the first maturity of SPX and VIX +to coincide (or differ up to two days, see Remark 1.1), i.e., T SPX +1 += T VIX +1 +and then for +j ≥ 2, T SPX +j += T VIX +j−1 + ∆, see for instance Guyon (2021); Guo et al. (2021); Guyon +and Lekeufack (2022). +(ii) The second approach is to consider T SPX = T VIX, i.e., to choose the same (or close +together, see Remark 1.1) maturities both for SPX and VIX options. This perspective +has been adopted for instance by Gatheral et al. (2018); Rosenbaum and Zhang (2021); +Grzelak (2022); Bondi et al. (2022b). +36 + +0 +T1 +T2 +T1 + ∆ +T2 + ∆ +0 +T1 +T2 +T3 T1 + ∆ +T2 + ∆ +T3 + ∆ +Figure 3: The blue lines denote the time interval where the dynamics of the variance process +influence the SPX option up to the maturity time. +For instance the shortest blue line +denotes the time interval where the dynamics of the variance process enter up to maturity +T1. Similarly the red lines denote the corresponding ones for the VIX, as for instance the +variance process enters here in the time integral on [T1, T1 + ∆], see (5.3). On the upper +graph a representation of the joint calibration approach (i) is given where we notice that +the maturities of the VIX are chosen so that there is a maximal overlap with the ones of +the SPX. On the lower graph a representation of approach (ii) is given where the maturities +T = {T1, T2, T3} are considered. We observe that there is less overlap between the maturities +of the SPX and VIX than in approach (i). +Both approaches deal with the joint modeling of SPX and VIX options and in order to +be consistent with both viewpoints taken in the literature, we show how our signature-based +model solves the joint calibration within both settings. For this reason we split the rest of +the section in two subsection and discuss them separately. +7.1.1 +First approach +Here we consider call options for both indices on the trading day 02/06/2021, as in Guyon +and Lekeufack (2022). Maturities are reported in the following tables with the corresponding +range of strikes (in percentage) with respect to the spot and the market’s futures prices. +T VIX +1 += 0.0383 +T VIX +2 += 0.0767 +(90%,220%) +(90%,220%) +T SPX +1 += 0.0383 +T SPX +2 += 0.1205 +T SPX +3 += 0.1588 +(92%,105%) +(70%,105%) +(80%,120%) +We stress that the shortest maturity considered is of 14 days for both SPX and VIX, then the +second and third maturity of the SPX are 44 days and 58 days, respectively, and the second +one for the VIX is 28 days. Moreover, we consider a high moneyness level (up to 220%) +for VIX options, usually rather difficult to fit. Regarding our modeling choice we fix d = 3, +37 + +n = 3 and choose the primary process X to be a three dimensional Ornstein-Uhlenbeck +process (see Example 4.5) with parameters +κ = (0.1, 25, 10)⊤, +θ = (0.1, 4, 0.08)⊤, +σ = (0.7, 10, 5)⊤, +ρ = +� +� +� +� +1 +0.213 +−0.576 +0.329 +· +1 +−0.044 +−0.549 +· +· +1 +−0.539 +· +· +· +1 +� +� +� +� , +X0 = (1, 0.08, 2)⊤. +Note that with this configuration we need to calibrate 85 parameters, i.e., ℓ ∈ R85. Con- +cerning the calibration task, we solve (7.1) with λ = 0.35 with NMC = 80000 Monte Carlo +samples for the previous maturities and strikes. Furthermore we specify the loss function +L as in (5.14) for β = 1 both for SPX and VIX. +Figure 4: +In blue the calibrated implied volatilities smiles from top-left at maturities +T SPX +1 +, T VIX +1 +, T SPX +2 +, T SPX +3 +, T VIX +2 +. In red the corresponding bid-ask spreads. In the graphs +of the VIX smiles the red dashed line indicates the market future price at maturity and the +blue dashed line the calibrated one. +We report also the relative absolute error between the market future prices and the +calibrated ones as defined in (5.15): +T VIX +1 += 0.0383 +T VIX +2 += 0.0767 +εT VIX +1 += 9.8 · 10−6 +εT VIX +2 += 6.6 · 10−8 +38 + +SPX T=0.0383 +VIX T=0.0383 +SPX T=0.1205 +2.4 ++ +Calibrated +× ++ ++ +Calibrated +0.5 +0.250 + Bid/Ask +2.2 + Bid/Ask ++ +0.225 +++ +2.0 +0.4 - ++o+ +o+ ++o+ +1.8 +0.200 - +:+ ++ +≥ 1.6 - +≥ 0.3 +1.4 +0.150 +事 +* +Calibrated +0.2 - +1.2 +中 +Bid/Ask +0.125 ++ ++ ++$ +Market future +- +1.0 ++$+ +0.100 +--- Model future +丰串 ++i +0.1 +20 +30 +40 +45 +0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 +0.92 +0.94 +0.96 +0.98 +1.00 +1.02 +1.04 +25 +35 +Moneyness +Strike +MoneynessSPX T=0.1588 +VIXT=0.0767 ++ ++ +Calibrated +0.35 +1.8 ++ +Bid/Ask ++ ++ ++ +0.30 +1.6- ++ +0.25 +丰 +≥ +>1.4 ++o+ +丰 +++干 +丰 +丰 +0.20 - +++ ++ +1.2 - +Calibrated ++ ++ +Bid/Ask ++ +0.15 ++ +1.0- ++ + Market future + Model future ++ +0.8 +0.9 +1.0 +1.1 +1.2 +20 +25 +30 +35 +40 +45 +Moneyness +StrikeSimulation of time-series of SPX and VIX +Let ℓ⋆ ∈ R85 be the calibrated parameters +already used for Figure 4. We then fix T = 60 days the longest considered maturity for the +SPX and sample a trajectory for (Vt(ℓ⋆))t∈[0,T], (VIXt(ℓ⋆))t∈[0,T], (St(ℓ⋆))t∈[0,T]. Precisely, +we sample 12 grid points per day, i.e. we consider a 2 hours sampling per calendar day, for +a total of N = 720 grid points. The results of this simulation are reported in Figure 5. +Figure 5: On the top: one realization of the calibrated model S(ℓ⋆) for the SPX (in blue) and +the corresponding calibrated VIX (in red). On the bottom: the corresponding realization +of the calibrated variance process V (ℓ⋆). +Observe that even though ℓ∗ was only calibrated to option prices, the trajectories pro- +duced by the model are economically reasonable and also in line with several stylized facts, +such as negative correlation between SPX and VIX or volatility clustering. To obtain the +dynamics under the physical measure, these trajectories could still be adjusted by an appro- +priate market price of risk, but the quantities which are invariant under equivalent measure +changes like the volatility of volatility or the correlation stay the same. +The case of time-varying parameters +Next, we consider again the case of time-varying +parameters as introduced in Section 5.5 and Section 6.2 for VIX and SPX, respectively. +Although the joint calibration is mainly considered for short-dated options in the literature +as VIX options are then more liquid, it is even more challenging to provide an accurate +39 + +28 +1.005 +26 +1.000 +24 +0.995 +22 +0.990 ++ )*XI^ +20 +0.985 +18 +0.980 +16 +0.975 +14 +0.970 +0 +10 +20 +30 +40 +50 +60 +Days +0.06 +0.05 +0.04 +0.03 +0.02 +0.01 +M +M +0.00 +0 +10 +20 +30 +40 +50 +60 +Daysfit for both, short and long maturities. Allowing the parameters ℓ of our model to depend +on time, in particular on the maturities, we are able to calibrate additionally to longer +maturities than the ones considered in Figure 4. We consider for the choice of the primary +process the same configuration as we used for Figure 4. The procedure of our time-varying +calibration routine is as follows: +1. Calibrate jointly T SPX +1 +, T VIX +1 +and T SPX +2 +. +2. Use the parameters from the calibration of T SPX +j +and T VIX +j−1 to fit jointly the maturities +T SPX +j+1 and T VIX +j +for j = 2, . . . , J. +We consider J = 4, where the last maturity for the SPX is 170 days, and the last maturity +for the VIX is 77 days. For the first two maturities of the SPX and the first of the VIX we +consider the same moneyness ranges as in Figure 4, hence we specify here only the ranges +for the longer maturities: +T VIX +2 += 0.1342 +T VIX +3 += 0.2875 +T VIX +4 += 0.3833 +(90%,330%) +(78%,395%) +(80%,405%) +T SPX +3 += 0.2163 +T SPX +4 += 0.3696 +T SPX +5 += 0.4654 +(75%,125%) +(60%,135%) +(50%,145%) +We observe that for this choice of maturities Assumption 5.10 is satisfied. Hence the second +expression for the time-varying VIX is used from Proposition (5.11). On the other hand +in order to compute the price of the SPX options in the time-varying case we use the +representation of the log-price provided in Proposition 6.10. In (7.1), we employ λ = 0.25 +for each calibration within the rolling procedure and we consider always as loss function +Lβ as introduced in (5.14) for β = 0. It is worth mentioning that the initial parameter +search discussed in Remark 5.7, has been employed for calibrating jointly T SPX +1 +, T VIX +1 +and +T SPX +2 +, whereas for the next slices we have considered the previously calibrated parameters +as starting point of the optimization. +40 + +Figure 6: On the left-hand side: SPX smiles, in blue the calibrated implied volatilities and +in red the bid-ask spreads. On the right-hand side: VIX smiles, in blue the calibrated +implied volatilities and in red the bid-ask spreads. +Finally we report the absolute relative error on the VIX futures’ prices: +T VIX +1 += 0.0383 +T VIX +2 += 0.1205 +T VIX +3 += 0.1588 +T VIX +4 += 0.2108 +εT VIX +1 += 1.2 · 10−11 +εT VIX +2 += 2.3 · 10−5 +εT VIX +3 += 3.9 · 10−5 +εT VIX +4 += 2.9 · 10−6 +7.1.2 +Second approach +Let us now consider the second approach described at the beginning of Section 7.1. Specif- +ically, we consider a unique set of maturities for both SPX and VIX on the trading day of +02/06/2021. For this study, we do not consider time-varying parameters. In the following +table we report the moneyness ranges for SPX options in the second row and on the last +row the ones for VIX options: +T1 = 0.0192 +T2 = 0.0383 +T3 = 0.0575 +T4 = 0.0767 +(97%,104%) +(92%,105%) +(90%,110%) +(85%,110%) +(90%,200%) +(90%,220%) +(90%,290%) +(90%,290%) +We consider λ = 0.5 and as loss function L we employ (5.14) with β = 1 for VIX options +and the same (without futures) for SPX options. +41 + +Implied Volatilities SPX 02-06-2021 +Implied Volatilities VIX 02-06-2021 +2.25 +0.5 +2.00 +0.4 +1.75 +Calibrated +IV +IV +1.50 +Bid/Ask +0.3 +1.25 +0.2 +1.00 +0.75 +0.1 +1.3 +90 +0.40 +1.2 +80 +0.4 +0.35 +1.1 +70 +0.30 +1.0 +60 +0.3 +0.25 +0.9 +50 +0.20 +Maty +Strikes +40 +0.2 +0.8 +0.15 +30 +0.7 +0.1 +0.10 +20 +0.6 +0.05Figure 7: On the left-hand side: SPX smiles, in blue the calibrated implied volatilities and +in red the bid-ask spreads. On the right-hand side: VIX smiles, in blue the calibrated +implied volatilities and in red the bid-ask spreads. +We additionally report the relative absolute error of the calibrated VIX futures: +T VIX +1 += 0.0192 +T VIX +2 += 0.0383 +T VIX +3 += 0.0575 +T VIX +4 += 0.0767 +εT VIX +1 += 6.2 · 10−3 +εT VIX +2 += 1.2 · 10−5 +εT VIX +3 += 1.3 · 10−2 +εT VIX +4 += 1.6 · 10−3 +A +Numerical results for the Brownian motion case +This appendix is dedicated to the calibration to VIX options only, similarly as in Sec- +tion 5.4.1, however with the primary process (Xt)t≥0 being simply correlated Brownian +motions (similarly as in Cuchiero et al. (2022a)) instead of OU-processes. +To be precise, we here model given by (3.1)-(3.2), where (Xt)t≥0 is 2-dimensional Brow- +nian motion. The correlation matrix of Z = (X, B) is specified, as in Section 5.4.1, namely +by +ρ = +� +� +1 +−0.577 +0.3 +· +1 +−0.6 +· +· +1 +� +� . +For the other parameters we consider a truncation’s level n = 3, we sample NMC = 80000 +trajectories for Monte Carlo pricing, and we minimize the loss function (5.14) with β = 1 +to fit the same data-set as in Section 5.4.1. +42 + +Implied Volatilities SPX 02-06-2021 +Implied Volatilities VIX 02-06-2021 +2.75 +0.35 +2.50 +0.30 +2.25 ++ +++ +0.25 +2.00 +Calibrated +IV +IV 1.75 +Bid/Ask ++ +0.20 + +1.50 +0.15 +H +1.25 +1.00 +0.10 + +0.75 +1.10 +60 ++ +1.05 +50 +0.07 +0.07 +1.00 +0.06 +0.06 +40 +0.05 +/ness +Maturities +0.95 +30 + Strikes +0.04 +0.04 +0.90 +0.03 +0.03 +20 +0.02 +0.85 +0.02Figure 8: The red crosses denote the bid-ask spreads (of the implied volatilities) for each +maturity, while the azure dots denote the calibrated implied volatilities of the model. +We observe that with this specification the model is neither able to calibrate to all future +market prices (see Figure 9 below) nor to fit the market implied volatilites accurately. One +can indeed see that the model implied volatilities often do not lie within the bid-ask spreads, +in particular for high strikes and short maturities. +T1 = 0.0383 +T2 = 0.0767 +T3 = 0.1342 +εT1 = 2.1 × 10−5 +εT2 = 2.7 × 10−2 +εT3 = 2.1 × 10−7 +T4 = 0.2108 +T5 = 0.2875 +T6 = 0.3833 +εT4 = 1.6 × 10−6 +εT5 = 6.8 × 10−3 +εT6 = 2.1 × 10−3 +Figure 9: The blue circles denote the calibrated future prices and the red crosses the market +futures prices, in between a linear interpolation is applied. +43 + +Implied Volatilities ViX 02-06-2021 +2.50 +2.25 ++ +2.00 +1.75 +IV +1.50 +1.25 +1.00 +0.75 +0.05 +0.10 +0.15 +0.20 +90 +80 +0.25 +70 +60 +0.30 +Maturities +50 +Strikes +40 +0.35 +30 +20 +0.40Futures' Term Structure +Calibrated +Market +X +23 +22 +21 +20 +19 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +TReferences +E. Abi Jaber, M. Larsson, and S. Pulido. Affine Volterra processes. Annals of Applied +Probability, 29(5):3155–3200, 2019. +E. Abi Jaber, C. Illand, and S. 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The review of financial studies, 4(4):727–752, 1991. +48 + diff --git a/0tFQT4oBgHgl3EQfDzUz/content/tmp_files/load_file.txt b/0tFQT4oBgHgl3EQfDzUz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c125b101f8e4b50be2dd7790a737d85e03a48dd5 --- /dev/null +++ b/0tFQT4oBgHgl3EQfDzUz/content/tmp_files/load_file.txt @@ -0,0 +1,2309 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf,len=2308 +page_content='Joint calibration to SPX and VIX options with signature-based models Christa Cuchiero∗ Guido Gazzani† Janka M¨oller‡ Sara Svaluto-Ferro§ February 1, 2023 Abstract We consider a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process, which is supposed to be some multidimensional continuous semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the VIX squared, exploiting the fact that the trun- cated signature of a polynomial process is again a polynomial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the VIX squared as linear functions of the signature of the corre- sponding augmented process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This feature can then be efficiently used for pricing and calibration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Keywords: signature methods, calibration of financial models, affine and polynomial pro- cesses, S&P 500/VIX joint calibration MSC (2020) Classification: 91B70, 62P05, 65C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 State of the art .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5 2 Signature: definition and properties 6 ∗Vienna University, Department of Statistics and Operations Research, Data Science Uni Vienna, Kolin- gasse 14-16 1, A-1090 Wien, Austria, christa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='cuchiero@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='at †Vienna University, Department of Statistics and Operations Research, Kolingasse 14-16 1, A-1090 Wien, Austria, guido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='gazzani@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='at ‡Vienna University, Department of Statistics and Operations Research, Kolingasse 14-16 1, A-1090 Wien, Austria, janka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='moeller@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='at §University of Verona, Department of Economics, Via Cantarane 24, 37129 Verona, Italy, sara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='svalutoferro@univr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The first three authors gratefully acknowledge financial support through grant Y 1235 and grant I 3852 of the Austrian Science Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' All authors acknowledge financial support through the OEAD WTZ project FR 02/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='13235v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='MF] 30 Jan 2023 3 The model 10 4 Expected signature of polynomial diffusion processes 12 5 VIX options with signatures 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 Explicit formulas for the VIX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 First approach .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 37 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 Second approach .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 41 A Numerical results for the Brownian motion case 42 1 Introduction The joint calibration to SPX1 and VIX options is a problem that has gained a lot of attention in quantitative finance since several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It is sometimes still regarded as the holy grail of volatility modeling, even though significant progress has been made recently (see below in the literature overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' One main reason for the increased interest is that the VIX index is no longer only used as indicator of volatility, but rather as important underlying for many derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In fact, futures and options written on it are extensively used to hedge the volatility exposure of option portfolios, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Rhoads (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose that an investor has a long position on the S&P 500 index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Although she might believe it has long-term prospects, it would be desirable to reduce her exposure to short-term volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Buying VIX derivatives with the belief that volatility is going to increase she might balance out these positions, while she was wrong, the losses to her VIX position could be mitigated by gains to the existing trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Historically speaking, in March 2004, futures on VIX, which trade on the forward 30-days realized volatility of the S&P 500, were launched on the Chicago Board Options Exchange (CBOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Two years later, in February 2006, also trading in European options written on the VIX index started and has progressively increased since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We address the reader to the official website of CBOE2 for details on how the VIX is computed and traded in practice, but just remark the following in view of expiration dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 1The SPX is a theoretical index, in the sense that it is pegged to the value of the S&P 500 without being based on a held portfolio of stock shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the present paper we shall use SPX and S&P 500 interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 2www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='cboe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='com/tradable products/vix/ 2 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that for a given type of option just certain maturities are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This in particular implies that on different days options with different times to maturity are traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For example, options which expire exactly in one year are not available on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Concerning our concrete applications, typically, VIX options expire on the Wednesday 30 days (or closest to 30 days) prior to the third Friday of the next calendar month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' On the other hand a monthly SPX option typically expires on the third Friday of every month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The steadily increasing liquidity on the market of VIX index products has marked the need to jointly calibrate to both, the prices of options on the underlying S&P 500 index and to prices of VIX derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' From a data perspective the challenge, especially for short maturities, is to reconcile the large negative skew of SPX options’ implied volatilities with relatively lower implied volatil- ities arising from the VIX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This has been discussed rigorously in Guyon (2020a), where the author shows a necessary condition to solve the joint calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As observed in the paper this translates, in the context of models with continuous trajectories, to a volatility process with high mean-reversion speed and large negative correlation to the S&P 500 index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In addition to that a high vol-of-vol is desirable in order to reproduce the aforementioned negative skew of at-the-money SPX options, but this can yield too high implied volatilities for the VIX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Inspired by Perez Arribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020) and Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a), we consider here a new type of stochastic volatility model for the discounted price process S = (St)t≥0 with continuous trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It is given by dSt(ℓ) = St(ℓ)σS t (ℓ)dBt, for an initial condition S0 ∈ R+, a standard Brownian motion B, and a volatility process σS satisfying σS t (ℓ) := ℓ(�Xt), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) where ℓ is a linear map of the signature �Xt of a process � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Specifically, the main in- gredient in this modeling framework is a d-dimensional continuous semimartingale X = (X1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , Xd t )t≥0, which we call primary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In most cases we then consider its aug- mentation with time t denoted ( � Xt)t≥0 = (t, X1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , Xd t )t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By modeling σS via (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) we assume that the signature of � X, denoted by �X (and rigorously introduced in Section 2), serves as a linear regression basis for the volatility process, while the parameters of the linear map ℓ have to be learned from (option price) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that the parameters of X are prespecified beforehand and can thus be seen – in analogy to machine learning terminol- ogy – as hyperparameters (that of course can be optimized over some training or validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As outlined below this is one of the crucial features that allows for the split of the calibration task into precomputable samples and parameters to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us now highlight the implications of this modeling framework and the novelty of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The modeling framework can be seen as universal in the class of continuous non-rough stochastic volatility models, which is a consequence of the universal approximation result stated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 3 It is not only universal in an approximate sense, but truly nests several classical mod- els (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4) and for instance also the ‘quintic Ornstein-Uhlenbeck volatility model’, recently proposed by Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b), which – with an additional input curve – is shown to fit SPX and VIX smiles well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By choosing the parameters of ℓ appropriately, the modeling framework incorporates both, purely Markovian (in (S, X)) and path-dependent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Up to our knowledge, it is the first signature-based model that is employed for pricing and calibration of VIX options as well as joint calibration, together with SPX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We illustrate that the joint calibration problem can be solved in this framework with- out jumps and rough volatility (compare also Guyon and Lekeufack (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Rømer (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By using time-varying parameters we can go beyond short maturities both for SPX and VIX options (as classically tackled in the literature) and achieve a joint calibration also for longer maturities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to achieve the highly accurate calibration results, illustrated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 and Section 7, we exploit the following mathematical and numerical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Defining Z := (X, B), then not only σS(ℓ) but also the log-price log(S(ℓ)) can be expressed as a linear function of the signature of �Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The computational benefit is immediate, since no (Euler) simulation scheme is needed to sample from the marginals of the price process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In terms of the parameters ℓ, log(S(ℓ)) is the sum of a quadratic function and a linear one, see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If X is additionally a polynomial process (see Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Filipovi´c and Larsson (2016)), then the VIX under our model can be computed analytically via matrix exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, in this case the forward variance can be represented by a quadratic form in the parameters ℓ and the corresponding matrix can be computed by polynomial technology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' via matrix exponentials, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This tractability property is a consequence of the fact that the truncated signature of a polynomial process is again a polynomial process (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We can apply a Monte Carlo approach (potentially with variance reduction) for option pricing and calibration, since the signature samples of �Z can be computed offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, due to the representations of VIX and log(S(ℓ)) described above, the same samples can be used for every linear map ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Therefore, the calibration task can be split into an offline sampling and a standard optimization, as no simulation is needed during the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Alternatively, a Fourier pricing approach for both VIX and SPX options can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, by building on the fact that the signature of �Z is an affine process (with values in the extended tensor algebra) as proved in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023), its Fourier-Laplace transform can be computed by solving an (extended tensor algebra valued) Riccati equation, which in turn can be used for Fourier-pricing as outlined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 gives a review over the dif- ferent contributions in the literature concerning the joint calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In Section 2 4 we introduce the signature in the context of continuous semimartingales, its main properties as well as notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Section 3 is dedicated to the introduction of our signature-based model and the connections to classical and also recent stochastic volatility models in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Section 4 is then devoted to the discussion and proof of the matrix exponential formula for the (conditional) expected signature of polynomial processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This result is at the core of Section 5, where we derive a tractable formula for the VIX, needed for pricing VIX options and VIX futures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Building on these formulas, our calibration results to VIX options are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In Section 6, we then prove, similarly as for the VIX, a tractable expression for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Additionally, we exploit in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 the affine nature of the signature process (as proved in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023)), to obtain a Fourier pricing approach within our modeling choice for both VIX and SPX op- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We finally present the numerical results of the joint calibration problem in Section 7, both in the case of constant parameters and with time-varying parameters, where the latter are introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 and Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The data used in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 were purchased from OptionMetrics3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Python codes used to produce the numerical results of the present work are available from the authors upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 State of the art This section is primarily dedicated to a literature review on the joint calibration problem and secondly, to a brief overview on signature methods in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' First attempts to solve the joint calibration problem appear in Gatheral (2008), with a double constant elasticity of variance model (CEV), which despite being rather flexible cannot fit accurately the implied volatilities of SPX and VIX options jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Later on, the belief that jumps in the SPX (or additionally also in the volatility) are necessary has lead to the following contributions, Sepp (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Papanicolaou and Sircar (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Baldeaux and Badran (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Kokholm and Stisen (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Pacati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2018) and more recently Grzelak (2022) with a new perspective on randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Continuous stochastic volatility models based on Markovian semimartingales have also been employed to solve the joint calibration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For instance, in Fouque and Saporito (2018) a Heston model with stochastic vol-of-vol has been calibrated, however only for maturities above 4 months where VIX options are less liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' More recently, Rømer (2022) considered a model where the volatility is driven by two Ornstein-Uhlenbeck (OU) processes using a non-standard transformation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The well-working choice of two OU-processes illustrated there has been an inspiration for our concrete numerical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We also point out that the (non-rough) model introduced in Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a,b), where the volatility is described by a polynomial of order five in one single OU-process, falls (apart from the additional input curve) into this class of continuous Markovian models and is a particular instance of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us also refer to the paper by Guyon and Mustapha (2022), where a neural SDE model has been successfully jointly calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Within the class of continuous, however not necessarily Markovian models, Guyon and Lekeufack (2022) conduct an empirical and statistical analysis as well as a joint calibration for a family of models where the volatility depends on the paths of the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' These models can be turned into Markovian ones by using exponential kernels instead of general ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Two further distinct and rather new lines of research are worth being mentioned as well: first, martingale optimal transport and second rough volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 3https://optionmetrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='com/ 5 The martingale optimal transport approach (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Guyon and Henry-Labordere (2013)) is used to calibrate discrete-time models as proposed in Guyon (2020b, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' These models are closely related to Schr¨odinger bridge problems, where the idea is to calibrate only the drift of the volatility while keeping the volatility of volatility unchanged, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Henry- Labordere (2019) or Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020, 2021) as well as the references therein regarding an optimal transport approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Although the calibration within that setting is accurate, it is also computationally rather expensive and not amenable to calibrate to several maturities jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' These computational challenges have been tackled recently in Guyon and Bourgey (2022) both in discrete and continuous time extending the contributions of Guyon (2020b, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the area of rough volatility modeling, initiated by the seminal paper of Gatheral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2018), the main idea is to replace the standard Brownian motion in the volatility process by a fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Even though the roughness of the trajectories found in Gatheral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2018), can also be explained by the estimation procedure as discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' in Cont and Das (2023), the non-Markovianity given by the fractional Brownian motion with Hurst parameter H < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5, manages to reproduce many stylized facts arising in financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Several classical models have been enhanced with rougher noise, but for simplicity we mention those employed in the SPX/VIX calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' One example is the quadratic rough Heston model introduced in Gatheral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020), which was in turn calibrated in Rosenbaum and Zhang (2021) by relying on neural networks approaches, also exploited in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Bayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For a large class of rough volatility models Jacquier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021) give new insights on the joint calibration problem by providing small- time formulas of the at-the-money implied volatility, skew and curvature for both European options on SPX and VIX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In Rømer (2022) an exhaustive study of the flexibility of different rough volatility models to joint SPX/VIX calibration is carried out, including the rough Bergomi, the rough Heston and an extended rough Bergomi model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Some of these, for instance the rough Heston model, have an affine structure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', can be embedded in the class of affine Volterra processes, considered in Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero and Teichmann (2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In particular they allow for Fourier pricing after solving the related Riccati equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This underlying structure is the building block of an extension with jumps investigated in Bondi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) and recently employed in the context of the joint calibration in Bondi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b), where a rough Heston model with Hawkes-type jumps with Fourier pricing is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Concerning our framework, signature-based methods provide a generic non-parametric way to extract characteristic features (linearly) and path-dependency from data, which is essential in (machine) learning and calibration tasks in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This explains why these techniques become more and more popular in mathematical finance, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', Buehler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Kalsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Perez Arribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Bayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Min and Hu (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Akyildirim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero and M¨oller (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022c) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 2 Signature: definition and properties We start by introducing basic notions related to the definition of the signature of an Rd- valued continuous semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is similar as in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) or Bayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021), but to keep the paper self-contained we recall the essential definitions and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 6 For each n ∈ N0 we define recursively the n-fold tensor product of Rd, (Rd)⊗0 := R, (Rd)⊗n := Rd ⊗ · · · ⊗ Rd � �� � n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For d ∈ N, we define the extended tensor algebra on Rd as T((Rd)) := {a := (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ) : an ∈ (Rd)⊗n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Similarly we introduce the truncated tensor algebra of order n ∈ N T (n)(Rd) := {a ∈ T((Rd)) : am = 0, ∀m > n}, and the tensor algebra T(Rd) := � n∈N T (n)(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that T (n)(Rd) has dimension dn := (dn+1 − 1)/(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For each a, b ∈ T((Rd)) and λ ∈ R we set a + b := (a0 + b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , an + bn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ), λ · a := (λa0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , λan, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ), a ⊗ b := (c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , cn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ), where cn := �n k=0 ak ⊗ bn−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that (T((Rd)), +, ·, ⊗) is a real non-commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For a multi-index I := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , in) we set |I| := n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We also consider the empty index I := ∅ and set |I| := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If n ≥ 1 or n ≥ 2 we set I′ := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , in−1), and I′′ := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , in−2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We also use the notation {I : |I| = n} := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d}n, omitting the parameter d whenever this does not introduce ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that multi- indices can be identified with words, as it is done for instance in Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Next, for each |I| ≥ 1 we set eI := ei1 ⊗ · · · ⊗ ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that the set {eI : |I| = n} is an orthonormal basis of (Rd)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Denoting by e∅ the basis element corresponding to (Rd)⊗0, each element of a ∈ T((Rd)) can thus be written as a = � |I|≥0 aIeI, for some aI ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that if an ∈ (Rd)⊗n we use non-bold notation whereas for the components aI ∈ R we write them bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Finally, for each a ∈ T(Rd) and each b ∈ T((Rd)) we set ⟨a, b⟩ := � |I|≥0 ⟨aI, bI⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe in particular that bI = ⟨eI, b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the present work it will be useful to enumerate the elements of the truncated tensor algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To this extent we introduce the isomorphism vec : T (n)(Rd) → Rdn and an injective labeling function L : {I : |I| ≤ n} −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , dn}, such that vec(u) := � |I|≤n eL (I)uI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) 7 Finally, we denote by ( · )(1) : T((Rd)) → (T((Rd)))d and ( · )(2) : T((Rd)) → (T((Rd)))d×d the shifts given by u(1) := � � |I|≥0 uIeI′ � (1{i|I|=1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , 1{i|I|=d})⊤, u(2) := � � |I|≥0 uIeI′′ � � � � 1{i|I|−1=i|I|=1} · · 1{i|I|−1=1,i|I|=d} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 1{i|I|−1=d,i|I|=1} · · 1{i|I|−1=i|I|=d} � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) Observe in particular that ⟨a, u(1)⟩ = � |I|≥0 � aIu(I1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , aIu(Id) �⊤ and ⟨a, u(2)⟩ = � � � aIu(I11) · · aIu(I1d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' aIu(Id1) · · aIu(Idd) � � � for each a ∈ T(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Throughout the paper we fix a filtered probability space (Ω, F, (Ft)t≥0, Q) on which we consider the stochastic processes to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We are now ready to introduce the signature of an Rd-valued continuous semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let X be a continuous Rd-valued semimartingale with d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The signature of X is the T((Rd))-valued process (s, t) �→ Xs,t whose components are recursively defined as ⟨e∅, Xs,t⟩ := 1, ⟨eI, Xs,t⟩ := � t s ⟨eI′, Xs,r⟩ ◦ dXin r , for each I = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , in) , I′ = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , in−1) and 0 ≤ s ≤ t, where ◦ denotes the Stratonovich integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Its projection Xn on T (n)(Rd) is given by Xn s,t = � |I|≤n ⟨eI, Xs,t⟩eI and is called signature of X truncated at level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If s = 0, we use the notation Xt and Xn t , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that the signature of X and the signature of X − c coincide for each c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Moreover, with an equivalent notation we can write Xt = � 1, � t 0 1 ◦ dX1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , � t 0 1 ◦ dXd s , � t 0 � � s 0 1 ◦ dX1 r � dX1 s , � t 0 � � s 0 1 ◦ dX1 r � dX2 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , � t 0 � � s 0 1 ◦ dXd r � dXd s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' A well-known and extremely useful property of the signature is that every polynomial function in the signature has a linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For the precise statement we first need to introduce the following concept (see also Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 in Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020) or Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' in Bayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 8 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For every two multi-indices I and J the shuffle product is defined recur- sively as eI � eJ := (eI′ � eJ) ⊗ ei|I| + (eI � eJ′) ⊗ ej|J|, with eI � e∅ := e∅ � eI = eI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It extends to a, b ∈ T(Rd) as a � b = � |I|,|J|≥0 aIbJ(eI � eJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that (T(Rd), +, �) is a commutative algebra, which in particular means that the shuffle product is associative and commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The proof of the paths version of the next result for can be found for instance in Ree (1958) or Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 (Shuffle property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let X be a continuous Rd-valued semimartingale and I, J be two multi-indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then for each 0 ≤ s ≤ t ⟨eI, Xs,t⟩⟨eJ, Xs,t⟩ = ⟨eI � eJ, Xs,t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The result follows by induction using the chain rule for Stratonovich integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We recall now an important property of the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The result is known in the rough paths literature (see for instance Boedihardjo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2016)), but can also be proved directly in the simpler situation of a continuous semimartingale that contains time as strictly monotone component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 (Uniqueness of the signature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let X and Y be two continuous Rd-valued semimartingales with X0 = Y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Set � Xt := (t, Xt), �Yt := (t, Yt) and let �X and �Y be the corresponding signature processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then �XT = �YT if and only if Xt = Yt for each t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' See Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to combine the value of the signature on different time intervals Chen’s identity going back to Chen (1957, 1977) turns out to be fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 (Chen’s identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let X be an Rd-valued semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then Xs,t = Xs,u ⊗ Xu,t (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4) for each 0 ≤ s ≤ u ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This can equivalently be written as ⟨eI, Xs,t⟩ = � eI1⊗eI2=eI ⟨eI1, Xs,u⟩⟨eI2, Xu,t⟩, for each multi-index I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' See Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) for a direct proof using the definition of Stratonovich integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us recall also the universal approximation theorem of linear functions of the signature in the context of continuous semimartingales as stated in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We refer to Cuchiero and M¨oller (2023), in the same context, for the situation involving not just the approximation of a functional up to a fixed final value T, but a uniform approximation on the whole time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 9 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6 (Universal approximation theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Define the set S(2) := {(�X2 t )t∈[0,T](ω): ω ∈ Ω} and consider a generic distance dS(2) on the set of trajectories given by S(2), with respect to which the map from S(2) to R given by ˆx2 �→ ⟨eI, S(|I|)(ˆx2)t⟩ is continuous for each multi-index I and every t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let K be a compact subset of S(2) and consider a continuous map f : K → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 Then for every ε > 0 there exists some ℓ ∈ T(Rd) such that sup (�X2 t )t∈[0,T ]∈K |f((�X2 t )t∈[0,T]) − ⟨ℓ, �XT ⟩|< ε, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='12 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='13 in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) for details concerning the proof and the choice of the metric dS(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For the related concept of stochastic Taylor expansions and functional expansions we refer the reader to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) and to Dupire and Tissot-Daguette (2022), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Finally, we introduce the concept of polynomial diffusion process which will play a key role for the computation of conditional expected signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Here we denote by √ · the matrix square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose that an Rd-valued process X = (Xt)t≥0 is a weak solution of dXt = b(Xt)dt + � a(Xt)dWt, X0 = x0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) for some d-dimensional Brownian motion W and some maps a : Rd → Sd + and b : Rd → Rd such that aij is a polynomial of degree at most 2 and bj is a polynomial of degree at most 1 for each i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then we call X polynomial (diffusion) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 3 The model We introduce now the model (St)t≥0 for the discounted dynamics of the S&P 500 index already outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Its dynamics under a risk-neutral probability measure Q are given by dSt = StσS t dBt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) where S0 ∈ R+, σS = (σS t )t≥0 is the volatility process to be specified and B = (Bt)t≥0 is a one-dimensional Brownian motion, correlated with σS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We define additionally the instan- taneous variance via Vt := (σS t )2 for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Our modeling choice is to parametrize the volatility process σS as a linear function of the time-extended signature of a primary underlying process X, namely σS t (ℓ) := ℓ∅ + � 0<|I|≤n ℓI⟨eI, �Xt⟩, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) where 4Compactness and continuity are defined with respect to dS(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 10 (Xt)t≥0 is a d-dimensional continuous semimartingale with positive quadratic variation and (�Xt)t≥0 denotes the signature of its time extension ( � Xt)t≥0 given by � Xt := (t, Xt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ℓ := {ℓI ∈ R : |I| ≤ n} denotes the collection of parameters of the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', ℓ ∈ R(d+1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Furthermore, we denote by (Zt)t≥0 the process given by Zt = (Xt, Bt), by ( �Zt)t≥0 its time extension, and by (�Zt)t≥0 the signature of ( �Zt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The correlation matrix process between the components of Z is denoted by ρij = [Zi, Zj] � [Zi] � [Zj] ∈ [−1, 1], for all i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d + 1, where [ · , · ] denotes the quadratic variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that ρ encodes in particular the correlation between X and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to simplify the notation we will drop the dependence on ℓ for the processes S = (St)t≥0 and (σS t )t≥0 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1), whenever this does not cause any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As an alternative definition for the volatility process (σS t )t≥0 one can set σS t (ℓ) := ℓ∅ + � 0<|I|≤n ℓI⟨eI, �Xt−ε,t⟩, for some fixed ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In this case the value of the volatility process σS at time t does not depend on the whole trajectory of the primary process X, but just on its evolution from t − ε to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For an economically reasonable choice for ε the lags used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 of Guyon and Lekeufack (2022) can be adapted to the current setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 (Interest rates and dividends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the model given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) we describe the discounted prices and construct the VIX from them, in line with the definition of the CBOE for the computation of the VIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' However, contingent claims are often expressed in terms of undiscounted prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If the dynamics of the discounted price process are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1), the undiscounted one fulfills d ˜St = (r − q) ˜Stdt + ˜StσS t (ℓ)dBt, where here r, q ∈ R denote the interest rate and the dividend, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Therefore ˜St(ℓ) = e(r−q)tSt(ℓ) and the price of a call option on the S&P 500 index under our model, reads C(T, K) = E � e−rT ( ˜ST (ℓ) − K)+� = E[e−rT (e(r−q)T ST (ℓ) − K)+] where T > 0 denotes the maturity time and K ∈ R the undiscounted strike price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Possible choices for the primary process X can be tractable processes such as an Ornstein- Uhlenbeck (OU) process or a Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For the proposed analysis we indeed need that the corresponding conditional expected signature can be computed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' � X = (t, Xt)t≥0 is a polynomial diffusion process in the sense of Defini- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 11 It is worth mentioning that the pool of primary processes that satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 is rather wide, including for example correlated Brownian motions, geometric Brownian mo- tions, OU processes, Cox-Ingersoll-Ross (CIR) processes, Jacobi processes, and all continu- ous affine processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 is in force then �Xn is a finite-dimensional polynomial process in sense of Filipovi´c and Larsson (2016) and Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Hence, the (con- ditional) expected signature of � X can be found by solving a finite-dimensional ODE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', can be written in terms of a matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For further details see Section 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We illustrate here that several classical and also recently considered stochas- tic volatility models are nested within our modeling choice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose that (Xt)t≥0 is a 1-dimensional OU process and let the order of the signature be n = 1, with ℓ∅ = ℓ(0) = 0 and ℓ(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then the process S = (St)t≥0 coincides with the Stein-Stein model, as introduced in Stein and Stein (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose that (Xt)t≥0 is a 1-dimensional geometric Brownian motion without drift and let the order of the signature be n = 1, with ℓ∅ = ℓ(0) = 0 and ℓ(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then the process S = (St)t≥0 coincides with the SABR model, as introduced in initially in Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2002) with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose that (Xt)t≥0 is a 1-dimensional OU process and let the order of the signature be n = 5, with ℓ∅, ℓ(1), ℓ(1,1,1), ℓ(1,1,1,1,1) non-zero and ℓI = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then the process S = (St)t≥0 coincides with the model considered in Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a,b) with an exponential kernel (a part from the deterministic input curve considered there additionally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If we allow for (Xt)t≥0 not to be a semimartingale and we do not consider the time augmentation, we can also include fractional kernels and therefore the whole class of Gaussian polynomial volatility models introduced in Abi Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) within our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 4 Expected signature of polynomial diffusion processes Let (Yt)t≥0 be a polynomial diffusion process in sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7 whose dynamics are given by dYt = b(Yt)dt + σ(Yt)dWt, Y0 = y0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) where σ(Yt) denotes the matrix square root of a(Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Recall that in this case the components of a : Rd → Sd + are polynomials of degree at most 2, the components of b : Rd → Rd are polynomials of degree at most 1, and W = (Wt)t≥0 is a d-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Denote then by Y the corresponding signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We now explain how to employ the polynomial technology to compute the conditional expected signature of (Yt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Several representations of related quantities in particular for the Brownian case can be found in the literature, see for instance Fawcett (2003), Lyons and Victoir (2004), Lyons and Ni (2015), Boedihardjo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021), Rossi Ferrucci and Cass (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Our approach follows Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023) and is based on the classical theory of polynomial processes (see Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2012) and Filipovi´c and Larsson (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Even though results for the corresponding infinite dimensional stochastic processes (see for instance Cuchiero and Svaluto-Ferro (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2021b)) are needed in the case of general signature SDEs considered in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023), the polynomial property of (Yt)t≥0 here permits to stay in the finite dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 12 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let (Yt)t≥0 be the polynomial process given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) and b and a be the corresponding drift and diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then bj(y) = bc j + d � k=1 bk j yk and aij(y) = ac ij + d � k=1 ak ijyk + d � k,h=1 akh ij ykyh, for some bc j,bk j ,ac ij, ak ij, akh ij = ahk ij ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Moreover, bj(Yt) = ⟨bj, Y1 t ⟩ and aij(Yt) = ⟨aij, Y2 t ⟩ for bj = � bc j + d � k=1 bk j Y k 0 � e∅ + d � k=1 bk j ek and aij = � ac ij + d � k=1 ak ijY k 0 + d � k,h=1 akh ij Y k 0 Y h 0 � e∅ + d � k=1 � ak ij + 2 d � h=1 akh ij Y h 0 � ek + d � k,h=1 akh ij ek � eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that the upper index on Y k 0 and Y h 0 refers to Y ’s components and not to powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The first part follows by the observation that by definition of polynomial processes b and a are polynomials of degree at most 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For the second part it then suffices to note that ⟨e∅, Y1 t ⟩ = ⟨e∅, Y2 t ⟩ = 1, ⟨ek, Y1 t ⟩ = ⟨ek, Y2 t ⟩ = (Y k t − Y k 0 ), and ⟨ek � eh, Y2 t ⟩ = (Y k t − Y k 0 )(Y h t − Y h 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let (Yt)t≥0 be the polynomial process given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) and let b and a as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The truncated signature (Yn t )t≥0 is a polynomial process in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7 and for each |I| ≤ n it holds that ⟨eI, Yn t ⟩ = � t 0 ⟨LeI, Yn s ⟩ds + � t 0 ⟨eI′, Yn s ⟩σi|I|(Ys)dWs, where the operator L : T((Rd)) → T((Rd)) satisfies L(T (n)(Rd)) ⊆ T (n)(Rd) and is given by LeI = eI′ � bi|I| + 1 2eI′′ � ai|I|−1i|I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let σj(Yt) denote the j-th row of σ(Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By definition of the signature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Stratonovich integral and by the shuffle property we can compute ⟨eI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Yt⟩ = � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ ◦ d⟨ei|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ = � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩d⟨ei|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ + 1 2 � t 0 ⟨eI′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩d[⟨ei|I|−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ⟨ei|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩]s = � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩⟨bi|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ds + � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩σi|I|(Ys)dWs + 1 2 � t 0 ⟨eI′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩⟨ai|I|−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ds = � t 0 ⟨eI′ � bi|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ds + � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩σi|I|(Ys)dWs + 1 2 � t 0 ⟨eI′′ � ai|I|−1i|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ds = � t 0 ⟨LeI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩ds + � t 0 ⟨eI′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Ys⟩σi|I|(Ys)dWs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 13 for each |I| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since |I � J| = |I| + |J| it holds that L(T (n)(Rd)) ⊆ T (n)(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For |I| ≤ n we thus get that the corresponding drift’s components are linear maps in Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Similarly, since aij = � |I|,|J|≤1 λIJ ij eI � eJ for some λIJ ij ∈ R and for |I| ≤ n we can compute ⟨eI′, Ys⟩σi|I|(Yt) � ⟨eJ′, Ys⟩σi|J|(Yt) �⊤ = ⟨eI′, Ys⟩⟨eJ′, Ys⟩⟨ai|I|j|J|, Ys⟩ = � |H1|,|H2|≤1 λH1H2 i|I|j|J|⟨eI′ � eH1, Ys⟩⟨eJ′ � eH2, Ys⟩, we also have that the components of the corresponding diffusion matrix are polynomials of degree 2 in Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 in Filipovi´c and Larsson (2016) yields the polynomial property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since the linear operator L maps the finite dimensional vector space T (n)(Rd) to itself, it admits a matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We call the operator L defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) dual operator corresponding to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For each |I| ≤ n set then ηIJ ∈ R such that LeI = � |J|≤n ηIJeJ, and fix a labelling injective function L : {I : |I| ≤ n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , dn} as introduced before (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We then call the matrix G ∈ Rdn×dn given by GL (I)L (J) := ηIJ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) the dn-dimensional matrix representative of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that using the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1), for each u ∈ T (n)(Rd) the matrix representative G of L satisfies vec(Lu) = Gvec(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let (Yt)t≥0 be the polynomial process given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1), (Ft)t≥0 be the filtration generated by (Yt)t≥0 and let G be the dn-dimensional matrix representative of the dual operator corresponding to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then for each T, t ≥ 0 and each |I| ≤ n it holds E[vec(Yn T+t)|FT ] = etG⊤vec(Yn T ), or equivalently, E[⟨eI, Yn T+t⟩|FT ] = � |J|≤n (etG⊤)L (I)L (J)⟨eJ, Yn T ⟩, where e( · ) denotes the matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 we know that vec(Yn) is a polynomial process and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 in Filipovi´c and Larsson (2016) for polynomials of degree 1 yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For the present paper a crucial role is played by the polynomial process given by time, a d-dimensional OU process, and a Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Specifically, we consider the process �Zt := ( � Xt, Bt) where B is a Brownian motion and � Xt = (t, Xt) with dXj t = κj(θj − Xj t )dt + � a(Xt)dWt, X0 = x0, 14 for aij(Xt) = σiσjρij, and W being a d-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We denote by ρj(d+1) the correlation between Xj and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Setting κd+1 := 0 and σd+1 := 1 we can see that �Z satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) in d + 2 dimensions for bj( �Zt) = 1{j=0} + κj(θj − �Zj t )1{j̸=0} and aij( �Zt) = σiσjρij1{i,j̸=0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The corresponding b and a are given by bj = e∅(1{j=0} + κj(θj − �Zj 0)1{j̸=0}) − ejκj1{j̸=0} and aij = e∅σiσjρij1{i,j̸=0} and we thus get LeI = eI′(1{i|I|=0} + κi|I|(θi|I| − �Z i|I| 0 )1{i|I|̸=0}) − (eI′ � ei|I|)κi|I|1{i|I|̸=0} + 1 2eI′′σi|I|−1σi|I|ρi|I|−1i|I|1{i|I|−1,i|I|̸=0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' An application of L to the first basis elements yields the following results: L(e1) = e∅κ1(θ1 − X1 0) − e1κ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' L(eI ⊗ e0) = eI � b0 + eI′ � ai|I|0 = eI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' L(e0 ⊗ e1 ⊗ e2) = e0 ⊗ e1κ2(θ2 − X2 0) − (e0 ⊗ e1) � e2κ2 + 1 2e0σ1σ2ρ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Letting (Ft)t≥0 be the filtration generated by ( �Zt)t≥0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 we can conclude that E[vec(�Zn T+t)|FT ] = etG⊤vec(�Zn T ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4) or equivalently, E[⟨eI, �Zn T+t⟩|FT ] = � |J|≤n (etG⊤)L (I)L (J)⟨eJ, �Zn T ⟩, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) where G denotes the (d + 2)n-dimensional matrix representative of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to work with the VIX it will be convenient to restrict our attention to the signature components of (�Zt)t≥0 not involving B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The following remark will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that given a subset E ⊆ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d + 1}, setting IE := {I : ij ∈ E} it holds L(IE) ⊆ IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This in particular implies that LeI = � I∈IE ηIJeJ for each I ∈ IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Choosing E = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d}, letting LE : IE → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , (d+1)n} be a labelling injective function, and setting GE LE(I)LE(J) := ηIJ we can see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) reduces to E[⟨eI, �Xn T+t⟩|FT ] = � |J|≤n (et(GE)⊤)LE(I)LE(J)⟨eJ, �Xn T ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To simplify the notation we often drop the E from GE whenever this does not introduce any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let (Yt)t≥0 be a polynomial process and let Y−1 be defined via e∅ = Y−1 s ⊗Ys, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ⟨e∅, Y−1 s ⟩ = 1 and � eI1⊗eI2=eI ⟨eI1, Y−1 s ⟩⟨eI2, Ys⟩ = 0, 15 for each |I| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that it can be defined recursively on |I| and each component of Y−1 s corresponds to a linear combination of components of Ys of the same length or shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since by Chen’s identity (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) we have Ys ⊗ Ys,t = Yt, for each s ≤ u ≤ t and |I| ≤ n we then get E[⟨eI, Ys,t⟩|Fu] = E[⟨eI, Y−1 s ⊗ Yt⟩|Fu] = � eI1⊗eI2=eI ⟨eI1, Y−1 s ⟩E[⟨eI2, Yt⟩|Fu] = � eI1⊗eI2=eI ⟨eI1, Y−1 s ⟩vec(eI2)⊤e(t−u)G⊤vec(Yn u), where G denotes the dn-dimensional matrix representative of the dual operator of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5 VIX options with signatures In this section we discuss the implication on pricing VIX options under the model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) when Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 is in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The implications of these on the log-price will be investigated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The CBOE Volatility Index, known by its ticker symbol VIX, is a popular measure of the market’s expected volatility of the S&P 500 index, calculated and published by the Chicago Board Options Exchange (CBOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The current VIX index value quotes the expected annualized change in the S&P 500 index over the following 30 days, based on options-based theory and current options-market data, more precisely VIXT := � E � − 2 ∆ log �ST+∆ ST � |FT � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) where ∆ = 30 days and ST denotes the price process at time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' With the term VIX options we here usually refer to either put or calls written on VIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the present work we will take into account without loss of generality only call options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 Explicit formulas for the VIX This section is dedicated to one of the main implication of our modeling framework, namely an explicit formula for the VIX expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) for S following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) under Assump- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In particular we show in the next theorem that the computation of the VIX squared reduces to a quadratic form in the parameters ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The entries of the corresponding positive semidefinite matrix can be computed by polynomial technology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' by matrix exponential as proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let S = (St)t≥0 be a price process described by dSt = StσS t dBt, where σS = (σS t )t≥0 denotes the volatility process, B = (Bt)t≥0 a one-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (i) Assume that V = (σS)2 satisfies E �� T 0 Vsds � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) 16 Then the VIX index at T > 0 is given by VIXT = � 1 ∆E �� T+∆ T Vtdt|FT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) (ii) Assume that σS and X satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Fix an injective labeling function L : {I : |I| ≤ n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , (d+1)2n+1} and let G be the (d+1)(2n+1)- dimensional matrix representative of the dual operator corresponding to �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) is satisfied and VIXT (ℓ) = � 1 ∆ℓ⊤Q(T, ∆)ℓ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4) where QL (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)vec(�X2n+1 T ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) and Id ∈ R(d+1)2n+1×(d+1)2n+1 denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' More explicitly without the vectorisation this reads QL (I)L (J)(T, ∆) = � eK=(eI�eJ)⊗e0 � |H|≤2n+1 (e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Part (i) follows directly from an application of Itˆo’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, for any t, T ≥ 0 log(St) = log(S0) − 1 2 � t 0 Vsds + � t 0 σS s dBs, and hence − 2 ∆ log �ST+∆ ST � = − 2 ∆(log(ST+∆) − log(ST )) = 1 ∆ � T+∆ T Vsds − 2 ∆ � T+∆ T σS s dBs, where the integral with respect to the Brownian motion B = (Bt)t≥0 vanishes once we take the risk neutral conditional expectation, due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This yields the alternative expression for VIX2 T and thus for VIXT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For part (ii), observe that Vt(ℓ) = � � |I|≤n ℓI⟨eI, �Xt⟩ �2 = � |I|,|J|≤n ℓIℓJ⟨eI � eJ, �Xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since continuous polynomials processes have finite moments of every degree and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) is satisfied due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The expression for Vt(ℓ) yields then VIX2 T (ℓ) = 1 ∆ � |I|,|J|≤n ℓIℓJE �� T+∆ T ⟨eI � eI, �Xt⟩dt|FT � = 1 ∆ℓ⊤Q(T, ∆)ℓ, 17 where for each T > 0 the matrix Q is given by QL (I)L (J)(T, ∆) : = E �� T+∆ T ⟨eI � eJ, �Xt⟩dt|FT � = E �� T+∆ 0 ⟨eI � eJ, �Xt⟩dt − � T 0 ⟨eI � eJ, �Xt⟩dt|FT � = E � ⟨(eI � eJ) ⊗ e0, �XT+∆⟩ − ⟨(eI � eJ) ⊗ e0, �XT ⟩|FT � = E � ⟨(eI � eJ) ⊗ e0, �XT+∆⟩|FT � − ⟨(eI � eJ) ⊗ e0, �XT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 we can rewrite the matrix Q as QL (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤e∆G⊤vec(�X2n+1 T ) − vec((eI � eJ) ⊗ e0)⊤vec(�X2n+1 T ) = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)vec(�X2n+1 T ), and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider now the model described in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 and set for simplicity ε ≥ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then the results of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1(ii) still hold however with QL (I)L (J)(T, ∆) = � eI1⊗eI2=eI�eJ � T+∆ T ⟨eI1, �X−1 t−ε⟩vec(eI2)⊤e(t−T)G⊤vec(�XT )dt, where G denotes the (d + 1)2n+1-dimensional matrix representative of the dual operator corresponding to �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To adapt the proof we just need to note that for each t ∈ [T, T + ∆] Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7 yields E[⟨eI � eJ, �Xt−ε,t⟩|FT ] = � eI1⊗eI2=eI�eJ ⟨eI1, �X−1 t−ε⟩vec(eI2)⊤e(t−T)G⊤vec(�XT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that since the integration’s variable t appears twice in this expression the time integral cannot be incorporated in the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that accounting for the scaling factor of 100, conventionally intro- duced by CBOE, the VIX index squared can equivalently be redefined (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', Rosenbaum and Zhang (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Rømer (2022)) as VIX2 T := 1002 ∆ E �� T+∆ T Vtdt|FT � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6) where T, t > 0 and ∆ = 1 12, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', approximately 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Notice that since the expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6) differ only by a scaling factor, all the theoretical results of the present work hold true disregarding this scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For sake of simplicity we will always use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We address the reader to Chapter 11 in Gatheral (2011) for further details about the conventions of CBOE and its link with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We observe that the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) is computationally appealing as we can unpack the computation in three parts: compute the coordinate vector vec((eI � eJ) ⊗ e0), which depends just on d > 0 and n > 0, calculate the matrix exponential of G⊤ which depends on the choice of the primary process X, and finally sample �X2n+1 T which is the only part that depends on the chosen maturity time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 18 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In general the computation of G ∈ R(d+1)2n+1×(d+1)2n+1, even if done only once, can be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For this reason it can sometimes be interesting to avoid the last time integral and to consider the following equivalent expression of the matrix Q, for |I|, |J| ≤ n: QL (I)L (J)(T, ∆) = � � T+∆ T vec(eI � eJ)⊤e(t−T)G⊤dt � vec(�X2n T ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7) where now G ∈ R(d+1)2n×(d+1)2n and where we use the fact that we can interchange the conditional expectation with the time-integral by dominated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As G is singular, this time integral has to be computed numerically, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We propose here two possible methods that can be used in order to compute it efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (i) Approximation of the time integral: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', via the trapezoidal rule also applied for VIX2 in Bourgey and De Marco (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Hence if we consider the shuffled coordinates vec(eI�eJ) of the exponential matrix we can use the symmetry of the shuffle to reduce the number of integrals to be solved from ((d + 1)2n)2 to (d+1)n((d+1)n+1) 2 (d + 1)2n, instead of (d2n)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that for our integral the error of such an approximation is given by Err(N) = − ∆2 12N2 G⊤(eG⊤∆ − I) + O(N−3), as N → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As a further dimension reduction one can exploit the polynomial nature of �Xn to obtain a matrix representation of its second order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Without entering into details, the matrix G would then be the matrix corresponding to the linear operator acting on coefficients of polynomials of degree 2 in �Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Its dimension would thus be (d+1)n((d+1)n+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (ii) Approximation of the matrix exponential: we can avoid to approximate the integral by approximating the matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Assuming that lim N→+∞(G⊤∆)N = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8) this can for instance be done via its Taylor expansion: � T+∆ T etG⊤dt = ∆ � I + G⊤∆ 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' + · · · + (G⊤∆)N N + 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' + O((G⊤∆)N+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8) holds true whenever the spectral radius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', the maximal eigen- value in absolute value, of the matrix G⊤∆ is less than 1 (see for instance Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 in Quarteroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This requirement suggests that for numerical purposes the parameters of the primary underlying process have to be chosen accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' An interesting example is given by the case where X is a d-dimensional correlated Brownian motion, as considered for instance in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In this case the process has no linear drift and the corresponding matrix G is nilpotent, meaning that Gn = 0, for each n big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In general, this Taylor approach permits to avoid a numerical integration and produces an accurate approximation, allocating as few memory as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' A further step in the direction of a fast evaluation of VIXT (ℓ) can be taken by noticing that the matrix Q in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) admits a Cholesky decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed since Q is 19 positive semidefinite and symmetric by the shuffle property, we know that there exists an upper triangular matrix UT ∈ R(d+1)n×(d+1)n, with possible zero elements on the diagonal, such that Q(T, ∆) = UT U ⊤ T , where for sake of simplicity we drop the dependence on ∆ of UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Hence the evaluation of the VIXT (ℓ) reduces to VIXT (ℓ) = � 1 ∆ℓ⊤UT U ⊤ T ℓ = 1 √ ∆ � (U ⊤ T ℓ)2 = 1 √ ∆ ∥U ⊤ T ℓ∥, where here ∥ · ∥ denotes the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We stress the fact that the Cholesky decom- position can be carried out offline, and the computational benefit is immediate if several samples of the signature are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the following remark we discuss a possible dimension reduction technique from which one can benefit computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We follow the approach of Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b, 2021a), where by employing the Johnson-Lindenstrauss Lemma a random projection of the signature is considered, to which we refer as a randomized signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' A first way to use this tool is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let d< ∈ N be the dimension of the space to which we would like to project the signature of order n > 0, such that d< ≪ (d + 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider A = (αij) ∈ Rd<×(d+1)n, such that αij ∼ N(0, 1/d<).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then a possible way to employ the randomised signature is to parametrize the volatility process as follows, σS t (ℓ) := ˜ℓ⊤A · vec(�Xn t ) where with ˜ℓ = ℓ · A⊤ ∈ Rd< we denote the randomised parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Due to the linearity of integral and conditional expectation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) this modeling choice is equivalent to consider the randomised matrix �Q ∈ Rd<×d< given by �QL (I)L (J)(T, ∆) := AQL (I)L (J)(T, ∆)A⊤, which leads to the following representation of VIXT (ℓ): VIXT (ℓ) = � 1 ∆ ˜ℓ⊤ ˜Q(T, ∆)˜ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that even if this procedure does not reduce the number iterated integrals to be computed offline, it reduces the number of parameters to calibrate, yielding in general to a faster evaluation of VIXT (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 Options on VIX We here briefly describe some important aspect of options on VIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' First of all, note that VIX options are written on VIX futures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The price process of a VIX future contract with maturity T > 0, is given by Ft(T) := E [VIXT |Ft] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='9) This has the following implications: The price of a VIX option depends on the maturity of the corresponding VIX future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 20 In calibration procedures one should therefore not normalize, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' not use VIXt := VIXt E[VIXT ], as E[VIXT ] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' When calibrating to VIX options, we stress that we additionally calibrate to VIX futures’ prices, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is important since future prices under the cali- brated model are employed to compute its implied volatility surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Including VIX futures in the calibration leads to a consistent model, both for VIX options and VIX futures, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Pacati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Guyon (2020a, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Using market prices of the VIX futures to invert the implied volatility surface could lead to inconsistencies if one would like to price further derivatives with the calibrated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In this respect, let us here also comment on the computation of implied volatilities for VIX call options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Two equivalent approaches can be used to compute implied volatilities for a given maturity T > 0: use F0(T) in the so-called Black formula for options on futures (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 of Papanicolaou and Sircar (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider e−rT F0(T), with r > 0 the interest rate, in the Black-Scholes formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Recall that the Black formula coincides with Black-Scholes’ one when choosing the underly- ing in the Black-Scholes formula to be e−r(T−t)Ft(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This implies that the two approaches are equivalent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' lead to the same implied volatilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 we will only consider futures’ prices at time t = 0 and maturity time T > 0, hence for sake of simplicity we use the notation F(T) instead of F0(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 Variance reduction for pricing VIX options We here discuss variance reduction techniques (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Glasserman (2004)) that can speed up the calibration in the subsequently applied Monte Carlo approach further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The key idea is to introduce a control variate, namely an easy to evaluate random variable Φcv such that given T > 0 and K > 0, E[Φcv] = 0, Var � (VIXT (ℓ) − K)+ − Φcv� < Var � (VIXT (ℓ) − K)+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' A well-working example of control variates used for pricing and calibrating neural SDE models can be found in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Gierjatowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020), where Φcv is constructed from hedging strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the following we describe two possible choices of control variates, which consist of polynomials on VIX futures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We stress the fact that these can be seen as linear functions of the signature of the primary process � X, hence they belong to the class of sig-payoffs, see Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Perez Arribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020) and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The first example is to employ the VIX squared as main ingredient, see for instance Bourgey and De Marco (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Guerreiro and Guerra (2022) for a similar choice within a rough Bergomi model for pricing VIX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is particularly easy to treat in our set up, as for any given maturity T > 0 we have E[VIX2 T (ℓ)] = 1 ∆ℓ⊤Qcv(T, ∆)ℓ, 21 with Qcv(T, ∆) := E[QL (I)L (J)(T, ∆)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 we indeed have Qcv L (I)L (J)(T, ∆) = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id) E[vec(�X2n+1 T )] = vec((eI � eJ) ⊗ e0)⊤(e∆G⊤ − Id)eTG⊤vec(�X2n+1 0 ) = vec((eI � eJ) ⊗ e0)⊤(e(T+∆)G⊤ − eTG⊤)vec(�X2n+1 0 ) where G denotes the (d+1)2n+1-dimensional matrix representative of the dual operator corresponding to �X and vec(�X2n+1 0 ) = e∅ ∈ R(d+1)2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that Qcv can again be computed offline similarly to the matrix Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Thus to compute the expectation of VIX2 T (ℓ) we only have to evaluate the previous quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To apply this now for pricing a call option with maturity T > 0 and strike K > 0, we set Φcv(ℓ, T, K) : = cT,K(∆VIX2 T (ℓ) − ℓ⊤Qcv(T, ∆)ℓ), = cT,K(ℓ⊤(Q(T, ∆) − Qcv(T, ∆))ℓ), where the constant cT,K maximizing the variance reduction is given by: c∗ T,K = Cov((VIXT (ℓ) − K)+, ℓ⊤Q(T, ∆)ℓ) Var(ℓ⊤Q(T, ∆)ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Notice that also in this case both Q and Qcv satisfy the condition for applying the Cholesky decomposition, leading to a faster evaluation of the control variate as dis- cussed in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that the Cholesky decomposition cannot be applied to Q − Qcv, as this is in general an indefinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As a second example we consider a generic polynomial in VIX2 as control variate by defining Y cv m (ℓ, T, K) = m � i=0 αi(T, K)(VIX2 T (ℓ))i (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10) where αi(T, K) are chosen to approximate the payoff (VIXT −K)+ with strike price K for some m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The corresponding control-variate is then defined as Φcv(ℓ, T, K) := cT,K (Y cv m (ℓ, T, K) − E[Y cv m (ℓ, T, K)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Regarding the computational effort, let us re- mark the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (i) VIX2 T is computed anyway for every realisation and is hence already available, therefore the computation of Y cv m (ℓ, T, K) is not expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (ii) It is possible to calculate E[Y cv m (ℓ, T, K)] analytically relying on the moment formula, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (iii) The choice of cT,K ∈ R is important and the optimal one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', the one leading the highest variance reduction, is given by the following expression c∗ T,K = Cov((VIXT (ℓ) − K)+, Y cv m (ℓ, T, K)) Var(Y cv m (ℓ, T, K)) , see for instance Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 in Glasserman (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We stress the fact that for m = 1 the two control variates introduced coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 Calibration to VIX options In this section we focus on the calibration to VIX options only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let T be a set of maturities and K a collection of strikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider the model given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) and assume that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 is in force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Using Monte Carlo compute an approximation of option and futures’ prices with NMC > 0 samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' πmodel VIX (ℓ, T, K) ≈ e−rT NMC NMC � i=1 (VIXT (ℓ, ωi)−K)+, F model VIX (ℓ, T) ≈ 1 NMC NMC � i=1 VIXT (ℓ, ωi), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='11) where VIXT (ℓ, ω) = � 1 ∆ℓ⊤Q(T, ∆)(ω)ℓ = 1 √ ∆ ∥U ⊤ T (ω)ℓ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It is crucial to note that in this framework a Monte Carlo approach is tractable since for every ℓ the same samples can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This means that we do not need to carry out any simulation during the optimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, the matrix Q can be simulated offline while only the products with ℓ ∈ R(d+1)n enter in the calibration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that an auxiliary randomization can be employed in every optimisation step as discussed in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Moreover, if we want to use control variates to reduce the variance of the Monte Carlo estimator as described in the previous section, we would consider πmodel VIX (ℓ, T, K) ≈ e−rT NV R NV R � i=1 (VIXT (ℓ, ωi) − K)+ − Φcv(ℓ, T, K)(ωi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Due to the variance reduction the number of samples needed is NV R ≪ NMC and Φcv is as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 The calibration to VIX call options and the corresponding futures on T and K consists in minimizing the functional LVIX(ℓ) := � T∈T ,K∈K L � πmodel VIX (ℓ, T, K), πb,a VIX(T, K), σb,a VIX(T, K), F model VIX (ℓ, T), F mkt VIX(T) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='12) where L denotes a real-valued loss function, F mkt VIX(T) the market’s futures’ prices and πb,a VIX(T, K) := {πmkt,b VIX (T, K), πmkt,a VIX (T, K)}, σb,a VIX(T, K) := {σmkt,b VIX (T, K), σmkt,a VIX (T, K)}, the market’s option bid/ask prices πmkt,b VIX (T, K), πmkt,a VIX (T, K), and bid/ask implied volatili- ties σmkt,b VIX (T, K), σmkt,a VIX (T, K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We will specify the choice of the function L in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7 (Initial guess search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since within our model choice we are given a quadratic function in ℓ to be minimized, a stochastic optimization with an initial guess is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to achieve faster convergence we consider an hyperparameter search to choose the starting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The steps are outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Find the magnitude of the coefficients returning Monte Carlo prices of the VIX options close to the one observable on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To this extent we sample Nℓ > 0 times parameters ℓ ∈ Ji = [−10−i, 10−i](d+1)n, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , m with m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 23 Select J∗ ∈ (Ji)m i=1 such that J∗ ∈ argmini: ℓ∈Ji LVIX(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Choose the initial guess to be ℓinitial ∈ argminℓ∈J∗ LVIX(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 Numerical results In the present section we report the results of the calibration to VIX options only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Here we consider call options written on the VIX on the trading day 02/06/2021, the same as in Guyon and Lekeufack (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We stress that for such recent dates the bid-ask spreads for VIX options are rather tight with respect to older dated options as considered for instance in Gatheral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Bondi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The maturities are reported in the following table with the corresponding range of strikes (in percentage) with respect to the market’s futures prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='0383 T2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='0767 T3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1342 T4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2108 T5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2875 T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3833 (90%,250%) (90%,250%) (80%,310%) (80%,300%) (75%,395%) (80%,405%) We underline that the shortest maturity considered is 14 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Regarding our modeling choice we fix d = 2, n = 3, which means to calibrate 40 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For X we choose a 2-dimensional Ornstein-Uhlenbeck processes, see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5, with the following (hyper- parameter) configuration: κ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1, 25)⊤, θ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1, 4)⊤, σ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7, 10)⊤, ρ = � � 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6 1 � � , where the last column of ρ are the correlations with the Brownian motion B driving the price process S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The motivation of this parameters choice is to mimic a rough or strong mean- reverting model as suggested in Rogers (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Rømer (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We refer to Appendix A for numerical results where we use only a correlated 2-dimensional Brownian motion as primary process, which yields significantly worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Before stating the loss function L that we employed in the calibration task, let us make the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let f : R+ × R+ → R+ be the call pricing functional in the Black-Scholes model, depending on the volatility σBS and the spot price ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', f : (σBS, ξ) �→ f(σBS, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By Taylor expansion in an appropriate neighbourhood of (σmkt, ξmkt) we obtain f(σBS, ξ) ≈f(σmkt, ξmkt) + ∂f ∂σ(σmkt, ξmkt)(σBS − σmkt) + ∂f ∂ξ (σmkt, ξmkt)(ξ − ξmkt), which equivalently gives (σBS − σmkt) ≈ 1 ∂f ∂σ(σmkt, ξmkt) � f(σBS, ξ) − f(σmkt, ξmkt) � − ∂f ∂ξ (σmkt, ξmkt) ∂f ∂σ(σmkt, ξmkt) (ξ − ξmkt), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='13) where we recognize for the derivatives with respect to σ and ξ, the Greeks Vega and Delta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 24 Motivated by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8 we propose, for a fixed maturity and strike price, the following loss-function for β ∈ {0, 1} Lβ(π, πmkt,b,a, σmkt,b,a, F, F mkt) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='14) �� β˜1{π /∈[πmkt,b,πmkt,a]} + (1 − β) ���π − (πmkt,a + πmkt,b)/2 �� + ��δmkte−rT (F − F mkt) �� υmkt(σmkt,a − σmkt,b) �2 , where υmkt and δmkt denote the Vega and Delta of the option under the Black-Scholes model which depend on the maturity and on the strike price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' F and F mkt denote futures with maturity T such that the variables ξ, ξmkt appearing in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8 are ξ = e−rT F and ξmkt = e−rT F mkt, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' ˜1{x/∈[yb,ya]} := s(yb − x) + s(x − ya) for s(x) := 1 2 tanh(100x) + 1 2 a smooth version of the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (i) We observe that by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8 minimizing L0 is equivalent to mini- mizing an upper bound of the square of the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='13) normalized by the bid-ask spread of the implied volatilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that we slightly abused notation, since υmkt and δmkt of course depend on the strike and the maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (ii) Note that as ℓ �→ VIXT (ℓ, ω) = 1 √ ∆∥U ⊤ T ℓ∥ is convex and the call payoff is convex and increasing, the model option and future prices are convex in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' If β = 0 and the initialization of ℓ is such that both the model and future prices are higher than the market ones, then we actually deal with a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (iii) If our aim does not consist in calibrating to the mid-price or mid-implied-volatility precisely, but we merely want to be within the bid-ask spreads we can set β = 1 For the next calibration result we minimize L1 as introduced above with NMC = 80000 Monte Carlo samples for the previous maturities and strikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 25 Figure 1: The red crosses denote the bid-ask spreads (of the implied volatilities) for each maturity, while the azure dots denote the calibrated implied volatilities of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' On the x-axis we find the strikes and on the y-axis we find the maturities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We observe that the calibrated VIX smiles fall systematically in the bid-ask corridor for all the maturities considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We report additionally in the next tables the relative absolute error between the market future prices and the calibrated ones for each maturity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', εT := |F mkt(T) − F model VIX (ℓ∗, T)| F mkt(T) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='15) where ℓ∗ ∈ R40 denotes the calibrated parameters and here F model VIX (ℓ∗, T) stands for the calibrated future model price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In Figure 2 we can find an illustration of the calibrated and the market futures’ term structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='0383 T2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='0767 T3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1342 εT1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='0 × 10−6 εT2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 × 10−3 εT3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 × 10−5 T4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2108 T5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2875 T6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3833 εT4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 × 10−4 εT5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='9 × 10−6 εT6 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3 × 10−6 26 Implied Volatilities ViX 02-06-2021 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='00 + + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='75 + + IV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='50 T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='00 + H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='20 90 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='25 70 60 Maturities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='30 50 Strikes 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='35 30 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='40Figure 2: The blue circles denote the calibrated future prices and the red crosses the future prices on the market, in between a linear interpolation is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 The case of time-varying parameters We now consider the case of maturity dependent parameters as for instance employed in Gierjatowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since it will be important later on to distinguish maturities of options written on the VIX index from maturities of options written on the SPX index, we introduce the two sets T VIX and T SPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us fix here T VIX = {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , TN}, where Ti < Ti+1 for any in i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , N −1 and denote by ℓ(Ti) ∈ Rdn the parameters depending on the maturity Ti > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We set T0 = 0 and TN+1 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then, we consider for any t ≥ 0 the volatility process to be σS t (ℓ) = N � i=0 � |I|≤n ℓI(Ti)1[Ti,Ti+1)(t)⟨eI, �Xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='16) Therefore the variance process reads as follows, Vt(ℓ) = N � i=0 � |J|,|I|≤n ℓI(Ti)ℓJ(Ti)1[Ti,Ti+1)(t)⟨eI � eJ, �Xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='17) Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Assume that for a set of maturities T VIX it holds that |Ti − Tj| ≥ ∆ for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let T VIX be a set of maturities on the VIX index and let Q(T, τ) be the matrix as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5) (here for general τ > 0 instead of ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then, under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='17) the VIX squared at time Ti ∈ T VIX is given by VIX2 Ti(ℓ) = 1 ∆ � N � j=i ℓ(Tj)⊤ (Q(Ti, (Tj+1 − Ti) ∧ ∆) − Q(Ti, (Tj − Ti) ∧ ∆)) ℓ(Tj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that, if Ti+1 − Ti > ∆ (which is in particular holds under Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10) then, VIX2 Ti(ℓ) = 1 ∆ℓ(Ti)⊤Q(Ti, ∆)ℓ(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=" 27 Futures' Term Structure Calibrated Market X 23 22 21 20 19 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='40 TProof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By the definition of the VIX, it holds that VIX2 Ti(ℓ) = 1 ∆E �� Ti+∆ Ti N � j=i � |J|,|I|≤n ℓI(Tj)ℓJ(Tj)1[Tj,Tj+1)(t)⟨eI � eJ, �Xt⟩dt ����FTi � = 1 ∆ N � j=i � |J|,|I|≤n ℓI(Tj)ℓJ(Tj)E �� Tj+1∧(Ti+∆) Tj∧(Ti+∆) ⟨eI � eJ, �Xt⟩dt ����FTi � = 1 ∆ N � j=i � |J|,|I|≤n ℓI(Tj)ℓJ(Tj) � E �� Tj+1∧(Ti+∆) Ti ⟨eI � eJ, �Xt⟩dt ����FTi � − E �� Tj∧(Ti+∆) Ti ⟨eI � eJ, �Xt⟩dt ����FTi �� and hence the first statement follows by the definition of Q in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Notice that also in the case of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='11, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 6 SPX as a signature-based model The goal of this section is to express the discounted price of the SPX, modeled via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) dSt(ℓ) = St(ℓ)σS t (ℓ)dBt, in terms of the signature of ( �Zt)t≥0 = (t, Xt, Bt)t≥0, allowing again to precompute its samples and use the same ones for every ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is in the same spirit as in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a), even though there the asset price was directly modeled as linear function of the signature of some primary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Recall that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) σS is parametrized as follows σS t (ℓ) := ℓ∅ + � 0<|I|≤n ℓI⟨eI, �Xt⟩, where � Xt = (t, Xt) with X a d-dimensional continuous semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Before addressing a more tractable expression for S, that allows to avoid (Euler) simulation schemes, we recall the following well-known integrability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Assume that E[S0] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then, the process (St)t≥0 is a (non-negative) supermartingale and in particular E[St] < ∞ for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that St = S0E �� · 0 σS s dBs � t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Moreover ( � t 0 σS s dBs)t≥0 is a local martingale and hence, by the properties of the stochastic exponential, St is a non-negative local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It follows from Fatou’s Lemma that non-negative local martingales are supermartingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the following we suppose without loss of generality that S0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Recall that if Novikov’s condition is satisfied, then a stochastic exponential of the form St = E �� · 0 σS s dBs � t for t ∈ [0, T] is a true martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For σS s as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2), such condition reads E � exp �1 2 � T 0 Vt(ℓ)dt �� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 28 Observe that E � exp �1 2 � T 0 Vt(ℓ)dt �� = E � exp �1 2 � |I|,|J|≤n ℓIℓJ � T 0 ⟨eI � eJ, �Xt⟩dt �� = E � exp �1 2ℓ⊤Q0(T)ℓ �� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) where for L : {I : |I| ≤ n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , (d + 1)n}, Q0 L (I)L (J)(T) := ⟨(eI � eJ) ⊗ e0, �XT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We point out that the previous condition is not necessarily satisfied for all ℓ ∈ R(d+1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Indeed, let us consider X to be a one-dimensional Brownian motion and choose ℓ such that the only non trivial component is the last one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=', ℓI := � c ∈ R, if I = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , 1), |I| = n, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) Then, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) translates into E � exp �c2 2 � T 0 2n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' X2n t 2n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' dt �� = E � exp � c2 2(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' )2 � T 0 X2n t dt �� , which is not finite in general, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' if n = 2, c = √ 2(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=') then by Jensen’s inequality it follows E � exp �� T 0 X4 t dt �� ≥ E � 1 T � T 0 eTX4 t dt � = 1 T � T 0 E[eTX4 t ]dt = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us underline the fact that, even if the process St(ℓ) is a non-negative local martingale, we can still price options via risk neutral expectations without introducing arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The reason is that, although E[ST (ℓ)] < S0, it is impossible to take a short position on S(ℓ) and a long position on Vt := E[ST (ℓ)|Ft] for all t ≤ T because of credit constraints, therefore, the condition of No Free Lunch with Vanishing Risk (NFLVR) as introduced in Delbaen and Schachermayer (1994) is not violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We address the reader to Kardaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2015) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The key idea is to rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) as a type of signature-based model in sense of Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a) including B = (Bt)t≥0 as part of the primary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is possible since Itˆo integrals with respect to primary process’ components can be rewritten as linear functions of the signature of the primary process itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' To do so, we introduce Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 on Z = (X, B) in order to describe the correlation structure between B and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' For all i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d + 1} it holds d[Zi, Zj]t = � |J|≤m aJ ij⟨eJ, �Zt⟩dt for some m ∈ N, aJ ij ∈ R and where �Zt = (t, Zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Notice that this assumption is in particular satisfied if X is a polynomial process as in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3, correlated with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' As shown in the next proposition it allows for the aforementioned tractable representation of S(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 29 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let S = (St)t≥0 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) with S0 = 1, and σS = (σS t )t≥0 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose additionally that Z = (X, B) satisfies Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then, log(St(ℓ)) = −1 2ℓ⊤Q0(t)ℓ + � |I|≤n ℓI⟨˜eB I , �Zt⟩, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3) where ˜eB ∅ := ed+1, ˜eB I := eI ⊗ ed+1 − � |J|≤m aJ i|I|(d+1) 2 (eI′ � eJ) ⊗ e0, for each |I| > 0, and the components of the matrix Q0(t) ∈ R(d+1)n×(d+1)n are given by Q0 L (I)L (J)(t) = ⟨(eI � eJ) ⊗ e0, �Xt⟩, for a labeling function L : {I : |I| ≤ n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , (d + 1)n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Under our assumptions we can compute log(St(ℓ)) = −1 2 � t 0 Vs(ℓ)ds + � t 0 σS s (ℓ)dBs = −1 2 � |I|,|J|≤n ℓIℓJ � t 0 ⟨eI � eJ, �Xs⟩ds + � |I|≤n ℓI � t 0 ⟨eI, �Xs⟩dBs (∗) = −1 2 � |I|,|J|≤n ℓIℓJ⟨(eI � eJ) ⊗ e0, �Xt⟩ + � |I|≤n ℓI⟨˜eB I , �Zt⟩ = −1 2ℓ⊤Q0(t)ℓ + � |I|≤n ℓI⟨˜eB I , �Zt⟩, where for (∗) we used that � t 0⟨eI, �Xs⟩dBs = ⟨˜eB I , �Zt⟩ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10 in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider again the model described in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then the results of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 still hold with Q0 L (I)L (J)(t) := � t 0 ⟨eI � eJ, �Xs−ε,s⟩ds, and � t 0⟨eI, �Xs−ε,s⟩dBs instead of ⟨˜eB I , �Zt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since the proof follows closely the proof of the original result, we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that since the matrix (⟨eI �eJ, �Xt⟩)|I|,|J|≤n is positive semidef- inite, by monotonicity of the time integral on [0, t] for some t > 0, we also have ℓ⊤Q0(t)ℓ ≥ 0, for all ℓ ∈ R(d+1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This means that for any t > 0, we can rewrite the log-price as log(St) = −1 2∥(U 0 t )⊤ℓ∥2+ � |I|≤n ℓI⟨˜eB I , �Zt⟩, where U 0 t is the upper-triangular matrix of the Cholesky decomposition of Q0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 30 Notice that the log-price model in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='3), it is not exactly a signature-based model in the sense of Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022a), as here it is given by a linear part in the parameters ℓ and an additional quadratic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' It can also be rewritten as d log(St) = −1 2ℓ⊤ ˜Q(t)ℓdt + ℓ⊤vec(�Xn t )dBt, where ˜Q is given by ˜QL (I)L (J)(t) := ⟨eI � eJ, �Xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In order to sample the log-price at maturity, consistently with the VIX, we follow the following road map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We simulate �Z and compute ⟨˜eB I , �Z⟩ for each I as specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Next, we drop from the samples of �Z the terms where B appears, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' the components corresponding to indices containing the letter d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The result coincides with a sampling of �X and is then used to work with both Q and Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This is equivalent to sampling �X for the variance process and to compute an additional Itˆo integral as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the following corollary we state the form of ˜eB I when X is d-dimensional OU-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We omit the proof for sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let X be a d-dimensional OU-process as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 driven by a d- dimensional Brownian motion with correlation matrix ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 is satisfied and ˜eB I is given by ˜eB I = eI ⊗ ed+1 − 1 21{i|I|̸=0}(σi|I|ρi|I|d+1)eI′ ⊗ e0, for any multi-index I ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='9 (Variance reduction for pricing SPX options).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Observe that a possible control variate for reducing the variance of the Monte Carlo estimator for pricing SPX options is the value at maturity of the log-price process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This means, Φcv(ℓ, T, K) : = cT,K � log(ST (ℓ)) + 1 2ℓ⊤Q0,cv(T)ℓ � , where, using that the linear part (in ℓ) of log(ST (ℓ)) vanishes under the risk-neutral expec- tation, we have Q0,cv L (I)L (J)(T) = vec((eI � eJ) ⊗ e0)⊤eTG⊤vec(�X2n+1 0 ), for G ∈ R(d+1)2n+1×(d+1)2n+1 denoting the (d + 1)2n+1-dimensional matrix representative of the dual operator corresponding to �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We choose the optimal c∗ T,K ∈ R as c∗ T,K = Cov((ST (ℓ) − K)+, log(ST (ℓ))) Var(log(ST (ℓ))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 Exploiting the affine nature of the signature: Fourier pricing of SPX and VIX options This section is dedicated to outline how the linear parametrizations of the log-price and the volatility process in �Z can be used for Fourier pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Consider again the model given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) and let (Zt)t≥0 denote the primary process given by Zt = (Xt, Bt) introduced after (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Suppose for simplicity that dZj t = κj(θj − Zj t )dt + σjdW j t , Zj 0 = 0, for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , d + 1, where W denotes a (d + 1)-dimensional Brownian motion with W d+1 = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' All parameters κj, θj, σj are in R with κd+1 = θd+1 = 0 and σd+1 = 1 so that Zd+1 = W d+1 = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Note that we do not account for correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We illustrate now how to apply the results of Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023) in the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Since ( �Zt)t≥0 is a polynomial process, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1 there are b ∈ (T((Rd+2)))d+2 and a ∈ (T((Rd+2)))(d+2)×(d+2) such that d �Zj t = ⟨bj, �Zt⟩dt + � ⟨ajj, �Zt⟩dW j t , where bj = κjθje∅ − κjej and ajj = (σj)2e∅, using that (with a small abuse of notation) κ0θ0 := 1, κj := 0 and σ0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Using the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2) consider then the Riccati operator R given by R(u) = b⊤ � u(1) + 1 2 Tr(a � � u(2) + (u(1))⊤ � u(1)� ) = d+1 � j=0 � |I|≥0 � κjθju(Ij)eI + κju(Ij)ej � eI + 1 2(σj)2(u(Ijj)eI + u2 (Ij)eI � eI) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In this equation Tr : R(d+2)×(d+2) → R denotes the trace operator and is applied component wise to the elements of (T((Rd+2)))(d+2)×(d+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='23 in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023), we expect that E[exp(⟨u, �ZT ⟩)] = exp(ψ(T)∅), where ψ is a solution5 of the extended tensor algebra valued Riccati equation ∂tψ(t) = R(ψ(t)), ψ(0) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Choosing u as u(ℓ) := −1 2(ℓ � ℓ) ⊗ e0 + ˜ℓ where ˜ℓ := � |I|≤n ℓI˜eB I , by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 we get log(St(ℓ)) = ⟨u(ℓ), �Zt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' The representation of the Fourier-Laplace transform described above can then be used for Fourier pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We dedicate the remaining part of this section to illustrate how this can be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 5We refer to Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023) for the appropriate solution concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 32 From Fourier analysis we know that for K > 0 and C < 0 it holds (K − ey)+ = 1 2π � R e(iλ+C)y K−C+1−iλ (iλ + C)(iλ + C − 1)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This in particular implies that E[(K − ST (ℓ))+] = 1 2π � R E[e(iλ+C) log(ST (ℓ))] K−C+1−iλ (iλ + C)(iλ + C − 1)dλ = 1 2π � R E[e⟨uλ,�ZT ⟩] K−C+1−iλ (iλ + C)(iλ + C − 1)dλ = 1 2π � R eψλ(T)∅ K−C+1−iλ (iλ + C)(iλ + C − 1)dλ, where uλ := (iλ+C)u(ℓ) and ψλ is a solution of the Riccati equation with initial condition ψλ(0) = uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let us now consider the case of VIX options where Fourier pricing can be applied by computing the Fourier-Laplace transform of VIX squared, see also Sepp (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Papani- colaou and Sircar (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Bondi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b) and references therein for a Fourier-based approach to pricing VIX options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Fix a labelling injective function L : {I : |I| ≤ n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , (d + 1)(2n+1)} as introduced before (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) and recall that by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1(ii) it holds VIX2 T (ℓ) = 1 ∆ℓ⊤Q(T, ∆)ℓ for QL (I)L (J)(T, ∆) = � eK=(eI�eJ)⊗e0 � |H|≤2n+1 (e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' where G denotes the (d + 1)(2n+1)-dimensional matrix representative of the dual operator corresponding to �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Setting for |I|, |J| ≤ n I(∆)u := � |K|,|H|≤2n+1 (e∆G⊤ − Id)L (K)L (H)uKeH we can write VIX2 T (ℓ) = 1 ∆ � |I|,|J|≤n ℓIℓJ � eK=(eI�eJ)⊗e0 � |H|≤2n+1 (e∆G⊤ − Id)L (K)L (H)⟨eH, �XT ⟩ = 1 ∆ � |K|,|H|≤2n+1 (e∆G⊤ − Id)L (K)L (H)⟨eK, (ℓ � ℓ) ⊗ e0⟩⟨eH, �Xt⟩ = 1 ∆⟨I(∆)((ℓ � ℓ) ⊗ e0), �XT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Also in this case since 1 √ 2π � R eiλy(K − � |y|)+dy = � 2 π FS(K√ |λ|) |λ|3/2 is integrable for FS(u) = � u 0 sin(z2)dz, Fourier analysis yields (K − √y)+ = 1 π � R e−iλy FS(K � |λ|) |λ|3/2 dλ, 33 for each y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' This in particular implies that E[(K − VIXT (ℓ))+] = 1 π � R E[e−iλVIX2 T (ℓ)]FS(K � |λ|) |λ|3/2 dλ = 1 π � R eψλ(T)∅ FS(K � |λ|) |λ|3/2 dλ, where ψλ is a solution of the Riccati equation with initial condition ψλ(0) = −iλv(ℓ) for v(ℓ) := 1 ∆I(∆)((ℓ � ℓ) ⊗ e0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Analogous formulations in terms of the error function are also possible, see for instance Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Bondi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' In the same spirit one can also obtain a represen- tation of future prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' We here do not provide an implementation of this Fourier pricing approach but numerical experiments can be found in Cuchiero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='2 The case of time-varying parameters Analogously to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5, we now further enhance Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='5 by allowing the param- eters ℓ to depend on the maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let S = (St)t≥0 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='1) with S0 = 1, and (σS t )t≥0 satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='16) for a set of maturities T VIX = {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' , TN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Recall that in this case V = (Vt)t≥0 satisfy Vt(ℓ) = N � i=0 � |J|,|I|≤n ℓI(Ti)ℓJ(Ti)1[Ti,Ti+1)(t)⟨eI � eJ, �Xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Let Zt = (Xt, Bt) for all t ≥ 0, where X = (Xt)t≥0 is a d-dimensional continuous semi- martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content=' Then, under Assumptions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFQT4oBgHgl3EQfDzUz/content/2301.13235v1.pdf'} +page_content='4 we get the following recursion for the log-price process log(St(ℓ 0 give densities +to idempotent states via the Fourier transform, p �→ h(·p)/h(p), but the converse does +not hold (see Section 4 and Corollary 6.3). However it is shown here that every group-like +projection in the bidual yields a Pal set, and thus an idempotent state, but as seen in +Proposition 2.20 a converse statement does not hold. In general, it can only be said that +idempotent states are associated with group-like projections in the multiplier algebra of +the dual discrete quantum group [8]. +The language of wave-function collapse will be used talk about idempotent states with +group-like density, and later illustrate the difference between Haar and non-Haar idem- +potents: +Definition 2.14. Let q ∈ C(G)∗∗ be a projection and ϕ ∈ G. If ωϕ(q) > 0, then ϕ +conditioned by q = 1 is given by: +�qϕ(g) := ωϕ(qgq) +ωϕ(q) +(g ∈ C(G)), +and ϕ → �qϕ is referred to as wave-function collapse. Furthermore, say that a subset +S ⊆ G is stable under wave-function collapse if for all projections q ∈ C(G)∗∗, +(7) +(ϕ ∈ S and ωϕ(q) > 0) =⇒ �qϕ ∈ S. +The following is well known in the algebraic setting ([15], Prop. 1.8), and a similar +proof is known to work in the finite quantum group setting ([8], Corollary 4.2). For the +benefit of the reader, the proof is reproduced in the current setting: +Proposition 2.15. If p ∈ C(G) is a continuous group-like projection such that h(p) > 0, +then �ph ∈ G is an idempotent state. +Proof. Let φ = �ph. The difference between ωh and h can be suppressed here as ωh |C(G) = h. +Let f ∈ O(G): +(φ ⋆ φ)(f) = +1 +h(p)2 +� +h(pf(1)p)h(pf(2)p) = +1 +h(p)2 +� +h(f(1)p)h(f(2)p) += +1 +h(p)2(h ⊗ h) (∆(f)(p ⊗ p)) = +1 +h(p)2(h ⊗ h) (∆(f)∆(p)(1G ⊗ p)) += +1 +h(p)2(h ⊗ h) (∆(fp)(1G ⊗ p)) = +1 +h(p)2h(fp)h(p) = h(pfp) +h(p) += φ(f), +where the traciality of the Haar state, p2 = p, and (h⊗ϕ)(∆(f)(1G⊗g)) = h(f)ϕ(g) ([24], +Remark 2.2.2 i.) were used. By norm-continuity this implies that �ph is idempotent. +□ + +14 +J.P. MCCARTHY +Note that it is not claimed that the support projection of �ph ∈ G is p. In the below +this is assumed, and a nice description of the quasi-subgroup follows +Proposition 2.16. Let φ = �pφh be an idempotent with continuous group-like support +projection pφ ∈ C(G). Then +Sφ = {ϕ ∈ G : ϕ(pφ) = 1}. +Proof. Suppose that ϕ(pφ) = 1. Similarly to the proof of Proposition 2.15, for f ∈ O(G): +(8) +(φ ⋆ ϕ)(f) = +1 +h(pφ)(h ⊗ ϕ)(∆(fpφ)(1G ⊗ pφ)) = h(fpφ) +h(pφ) ϕ(pφ) = φ(f), +and by weak*-continuity, φ ⋆ ϕ = φ. On the other hand, suppose that φ ⋆ ϕ = φ so that +ϕ ∈ Sφ, the quasi-subgroup generated by φ. Applying (8) at f = pφ, with the existence +of �ph implying h(pφ) > 0: +(φ ⋆ ϕ)(pφ) = h(pφ) +h(pφ)ϕ(pφ) = φ(pφ) = 1 =⇒ ϕ(pφ) = 1. +□ +Proposition 2.17. If states ϕ1, ϕ2 on C(G) are supported on a group-like projection +p ∈ C(G)∗∗, then so is ϕ1 ⋆ ϕ2. +Proof. The proof for the finite case ([16], Prop. 3.12) applies with some adjustments. Let +(pλ) ⊂ O(G) converge σ-weakly to p ∈ C(G)∗∗. As the extension of ∆ to ∆∗∗ is σ-weakly +continuous +lim +λ +� +∆(pλ) +� +(1 ⊗ p) = p ⊗ p +The product is separately continuous, and ωϕ1 ⊗ ωϕ2 is σ-weakly continuous. +=⇒ lim +λ (ωϕ1 ⊗ ωϕ2) +� +pλ +(0) ⊗ pλ +(1)p = (ωϕ1 ⊗ ωϕ2)(p ⊗ p) +=⇒ lim +λ +� +ωϕ1(pλ +(0))ωϕ2(pλ +(1)p) = 1 +Note that as ϕ2 is supported on p: +=⇒ lim +λ +� +ϕ1(pλ +(0))ϕ2(pλ +(1)) = 1 +=⇒ lim +λ (ϕ1 ⋆ ϕ2)(pλ) = 1 +=⇒ lim +λ ωϕ1⋆ϕ2(pλ) = ωϕ1⋆ϕ2(p) = 1. +□ +Proposition 2.18. Suppose p ∈ C(G)∗∗ is a group-like projection. Then: +{ϕ ∈ G : ωϕ(p) = 1}, +is a Pal set, and so there is an idempotent φ supported on p such that pφ ≤ p. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +15 +Proof. First {ϕ ∈ G : ωϕ(p) = 1} is non-empty because p is normal and as ∥p∥C(G)∗∗ = 1, +there exists a state ω on C(G)∗∗ such that ω(p) = 1 [19], whose restriction to C(G) +is a state in Sp. +Weak*-closure and convexity are straightforward, and closure under +convolution follows from Proposition 2.17. +□ +Note that p is not necessarily equal to the support projection of the idempotent state +in {ϕ ∈ G : ωϕ(p) = 1}; and in the below the idempotent state in {ϕ ∈ G : ωϕ(p) = 1} +is not necessarily equal to φ. +Theorem 2.19. Suppose that an idempotent state φ ∈ G has group-like support projection +p ∈ C(G)∗∗. Then the quasi-subgroup generated by φ: +Sφ ⊆ {ϕ ∈ G : ωϕ(p) = 1}. +Proof. Consider ϕ ∈ Sφ not supported on p. Then, where q = 1G−p, ωϕ(q) > 0. Consider +ωϕ(q · q) ∈ C(G)∗ and note by Cauchy–Schwarz: +0 ≤ ωϕ(q · q) ≤ ϕ. +Then by Lemma 1.4: +ωϕ(q · q) ⋆ φ = ωϕ(q1Gq)φ = ωϕ(q)φ, +and it follows that �qϕ ∈ Sφ. Note �qϕ(p) = 0. +Using similar notation and techniques to Proposition 2.17, apply the σ-weakly contin- +uous ω�qϕ ⊗ ωφ to both sides of ∆∗∗(1G ⊗ p) = p ⊗ p, using the fact that p is the support +of φ: +=⇒ lim +λ +�� +ω�qϕ(pλ +(0)) ⊗ ωφ(pλ +(1)p) +� += ω�qϕ(p) ⊗ ωφ(p) +=⇒ lim +λ +�� +�qϕ(pλ +(0)) ⊗ ωφ(pλ +(1)p) +� += 0 +=⇒ lim +λ +�� +�qϕ(pλ +(0)) ⊗ φ(pλ +(1)) +� += 0 +=⇒ lim +λ +� +(�qϕ ⋆ φ)(pλ) +� += 0 +=⇒ lim +λ +� +φ(pλ) +� += 0 +=⇒ ωφ(p) = 0, +a nonsense, so ωϕ(q) = 0, and so ωϕ(p) = 1. +□ +It is not the case that every idempotent state φ has group-like support projection +pφ ∈ C(G)∗∗. Nor does Theorem 2.19 hold more generally: +Corollary 2.20. Suppose G is non-coamenable. Then the support projection ph ∈ C(G)∗∗ +of the Haar state is not a group-like projection. Furthermore: +{ϕ ∈ G : ωϕ(ph) = 1} ⊊ Sh. + +16 +J.P. MCCARTHY +Proof. Assume that the support ph ∈ C(G)∗∗ is a group-like projection. As ωh(1G) = 1, +1G − ph > 0 strictly as G is at the universal level and G is assumed non-coamenable. +Therefore there exists a state ωϕ on C(G)∗∗ such that +ωϕ(1G − ph) = 1 =⇒ ωϕ(ph) = 0. +Restrict ωϕ to a state ϕ on C(G). By Theorem 2.19 it follows that ϕ is not invariant +under the Haar state, which is absurd as Sh = G. +□ +There is a group-like projection p such that +{ϕ ∈ G : ωϕ(p) = 1} = Sh; +the unit p = 1G. +Note there is a relationship between quantum subgroups and wave-function collapse: +Proposition 2.21. ([9], Th. 3.3) Let G be a compact quantum group and φ ∈ C(G)∗ an +idempotent state. Then φ is a Haar idempotent if and only if the null-space +Nφ = {f ∈ C(G) : φ(|f|2) = 0} +is a two-sided ideal. +Note in the below ωϕ0 is the extension of the state ϕ0 on C(H) to a state on C(H)∗∗. +Lemma 2.22. Suppose that H ⊆ G via π : C(G) → C(H). Then the extension of ϕ0 ◦ π +to a state on C(G)∗∗ is given by: ωϕ0 ◦ π∗∗. +Proof. Consider f ∈ C(G). The result follows from the σ-continuity of the maps involved, +and π∗∗ |C(G) = π. +□ +Note that part (i) of the below is restricted to Haar idempotents coming from Haar +states on universal versions. +Theorem 2.23. Suppose that φ is an idempotent state on C(G). +(i) If φ is a (universal) Haar idempotent, then Sφ is closed under wave-function col- +lapse. +(ii) If φ is a non-Haar idempotent with group-like projection support, then Sφ is not +closed under wave-function collapse. +Proof. +(i) Suppose φ is a (universal) Haar idempotent via π : C(G) → C(H). By +Vaes’s Remark 1.5, every element of Sφ is of the form ϕ0 ◦ π for a state ϕ0 on +C(H). Suppose ϕ undergoes wave-function collapse to �qϕ. Then, using Lemma +2.22 +ωϕ(q) > 0 =⇒ ωϕ0(π∗∗(q)) > 0 +(ωϕ0 ∈ S(C(H)∗∗)). + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +17 +Using Lemma 2.22 again, it can be shown that �qϕ = ψ ◦ π, where: +ψ(g) = ωϕ0(π∗∗(q)gπ∗∗(q)) +ωϕ0(π∗∗(q)) +(g ∈ C(H), ωϕ0 ∈ S(C(H)∗∗)). +Thus, again by Vaes’s remark, ψ ◦ π and thus �qϕ ∈ Sφ, that is Sφ is closed under +wave-function collapse. +(ii) Suppose φ is a non-Haar idempotent with group-like support projection. By The- +orem 2.19 +Sφ ⊆ {ϕ ∈ G : ωϕ(pφ) = 1}. +As φ is a non-Haar idempotent, N∗∗ +φ += C(G)∗∗qφ is only a left ideal, and qφ +non-central. Suppose that for all uij ∈ C(G), uijqφuij ∈ N∗∗ +φ . Then uijqφuij = +uijqφuijqφ +=⇒ +uijqφuij = uijqφuijqφuij, so that uijqφuij is a projection. This +implies, because [uij, qφ]3 = 0 and [uij, qφ] is skew adjoint, that uijqφ = qφuij. +Therefore qφ is central and Nφ is an ideal. Therefore there exists uij such that +uijqφuij ̸∈ N∗∗ +φ : +ωφ(|uijqφuij|2) > 0. +By Cauchy–Schwarz: +0 < ωφ(|uijqφuij|2) ≤ ωφ(uijqφuij) ≤ ωφ(uij). +=⇒ � +uijφ(qφ) = ωφ(uijqφuij) +ωφ(uij) +> 0 =⇒ � +uijφ(pφ) < 1 =⇒ � +uijφ ̸∈ Sφ. +□ +3. Stabiliser quasi-subgroups +The analysis here is helped somewhat by defining the Birkhoff slice, a map Φ from the +state space of the algebra of continuous functions C(G) on a quantum permutation group +G to the doubly stochastic matrices: +Φ(ϕ) := (ϕ(uij))N +i,j=1. +Given a finite group G ⊆ SN and a partition P = B1 ⊔ · · · ⊔ Bk of {1, . . . , N}, the +P-stabiliser subgroup of G can be formed: +GP = {σ ∈ G : σ(Bp) = Bp, 1 ≤ p ≤ k}. +A P-stabiliser quasi-subgroup of G can also be defined. There are two, equivalent, defi- +nitions. The first definition uses the equivalence relation ∼P associated to the partition: +GP := {ϕ ∈ G: ϕ(uij) = 0 for all i ̸∼P j}. +Alternatively, consider the Birkhoff slice S(C(G)) → MN(C). By relabelling if necessary, +the blocks of a partition can be assumed to consist of consecutive labels. Define: +GP := {ϕ ∈ G : Φ(ϕ) is block diagonal with pattern P}, + +18 +J.P. MCCARTHY +that is: +ϕ ∈ GP ⇐⇒ Φ(ϕ) = + + +ΦB1(ϕ) +0 +· · · +0 +0 +ΦB2(ϕ) +· · · +0 +... +... +... +· · · +0 +0 +· · · +ΦBk(ϕ) + + , +where ΦBp(ϕ) = [ϕ(uij)]i,j∈Bp. +Theorem 3.1. For any partition P of {1, . . . , N}, GP is a quasi-subgroup. +Proof. That GP is convex, weak*-closed, and closed under convolution is straightforward +(using, for example that the Birkhoff slice is multiplicative Φ(ϕ1 ⋆ ϕ2) = Φ(ϕ1)Φ(ϕ2)). +The universal version gives ε ∈ GP so that GP is non-empty, and so a Pal set. +Suppose that φP is the associated idempotent. Therefore by Lemma 1.9: +φP(uij) = (φP ◦ S)(uij) = φP(uji). +For any fixed j ∈ {1, 2, . . . , N}, there exists i ∈ {1, 2, . . ., N} such that φP(uji) > 0. From +here: +φP(ujj) = (φP ⋆ φP)(ujj) = φP(uji)φP(uij) + +� +k̸=i +φP(ujk)φP(ukj) > 0. +To show that GP is equal to +SφP = {ϕ ∈ G : ϕ ⋆ φP = φP = φP ⋆ ϕ}, +suppose ϕ ∈ SφP, but ϕ ̸∈ GP. That implies there exists uij such that ϕ(uij) ̸= 0 with +i ̸∼P j. But this gives +φP(uij) = (ϕ ⋆ φP)(uij) = ϕ(uij)φP(ujj) + +� +k̸=j +ϕ(uik)φP(ukj) > 0, +a contradiction. +□ +For the partition j := {j} ⊔ ({1, 2, . . ., N}\{j}): +Gj = {ϕ ∈ G : ϕ(ujj) = 1}. +Note for any quantum permutation group G, and 1 ≤ j ≤ N, the diagonal element ujj is +a polynomial group-like projection: +∆(ujj)(1G ⊗ ujj) = +� N +� +k=1 +ujk ⊗ ukj +� +(1G ⊗ ujj) = ujj ⊗ ujj. +Using Proposition 2.16, it can be shown that the associated idempotent state is hj := � +ujjh, +that is: +hj(f) = h(ujjfujj) +h(ujj) +(f ∈ C(G)). + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +19 +The below is (almost) a special case of Theorem 2.23, but included as it uses different +proof techniques. +Theorem 3.2. The following are equivalent: +(i) hj is a Haar idempotent, +(ii) ujj is central, +(iii) Gj is stable under wave-function collapse. +Proof. (i) =⇒ (ii): assume hj is a Haar idempotent, say equal to hH◦π where π : C(G) → +C(H), uij �→ uH +ij, is the quotient map. Note that because hj(ujj) = hH(π(ujj)) = 1, and +hH is faithful on O(H), +1H = π(1G) = +N +� +m=1 +π(umj) = π(ujj), +so that π(ujj) = 1H is central in C(H). Assume that ujj is non-central. Then there exists +ukl ∈ C(G) such that |uklujj − ujjukl|2 > 0. Expanding: +ujjuklujj − ujjuklujjukl − uklujjuklujj + uklujjukl > 0. +Applying the Haar state, which is faithful on O(G), and using its traciality yields: +h(ujjuklujj) > h(ujjuklujjuklujj) +=⇒ hj(ukl) > hj(uklujjukl) +=⇒ hH(π(ukl)) > hH(π(uklujjukl)) = hH(π(ukl)π(ujj)π(ukl))) +=⇒ hH(π(ukl)) > hH(π(ukl)1Hπ(ukl))) = hH(π(ukl)), +an absurdity, and so ujj is central. +(ii) =⇒ (i): assume that ujj is central. +Nj := {f ∈ C(G) : hj(|f|2) = 0}. +If f ∈ Nj then h(ujjf ∗fujj) = 0 =⇒ fujj ∈ Nh, the null-space of the Haar state, so +that: +Nj = {f ∈ C(G) : fujj ∈ Nh}. +The rest of the argument is the same as ([8], Th. 4.5). +(ii) =⇒ (iii): assume that ujj is central. If ujj is central in C(G) then it is also central +in C(G)∗∗. Let ϕ ∈ Gj and q ∈ C(G)∗∗ such that ωϕ(q) > 0. Let pϕ ∈ C(G)∗∗ be the +support projection of ϕ. Note that +ωϕ(ujj) = ϕ(ujj) = 1 =⇒ pϕ ≤ ujj =⇒ pϕ = pϕujj. +Note +ωϕ(qujjq) = ωϕ(pϕqujjqpϕ) = ωϕ(pϕujjqqpϕ) = ωϕ(pϕqpϕ) = ωϕ(q). + +20 +J.P. MCCARTHY +It follows that: +�qϕ(ujj) = ωϕ(qujjq) +ωϕ(q) += 1 =⇒ �qϕ ∈ Gp. +(iii) +=⇒ (ii): assume now that ujj is non-central. Therefore there exists ukl ∈ C(G) +such that: +ujjukl ̸= uklujj. +Represent C(G) with the universal GNS representation πGNS(C(G)) ⊆ B(H). Denote +p := πGNS(ujj) and q := πGNS(ukl). +As pq ̸= qp, using Halmos two projections theory there exists a unit vector x ∈ ran p that +is orthogonal to both2 ran p ∩ ran q and ran p ∩ ker q. Define a state on C(G): +ϕ0(f) = ⟨x, πGNS(f)x⟩ +(f ∈ C(G)). +Note that: +ϕ0(ujj) = ⟨x, px⟩ = ⟨x, x⟩ = 1 =⇒ ϕ0 ∈ Gj. +Furthermore, together with x ∈ ran p +ϕ0(ukl) = ⟨x, qx⟩ = 1 =⇒ x ∈ ran q +ϕ0(ukl) = ⟨x, qx⟩ = 0 =⇒ x ∈ ker q +but x is orthogonal to both ran p ∩ ran q and ran p ∩ ker q so +0 < ⟨x, qx⟩ < 1 =⇒ 0 < ϕ0(ukl) < 1. +Now consider ϕ = � +uklϕ0: +ϕ(f) := ϕ0(uklfukl) +ϕ0(ukl) += ⟨qx, πGNS(f)qx⟩ +⟨qx, qx⟩ +(f ∈ C(G)). +In particular +ϕ(ujj) = ⟨qx, pqx⟩ +⟨qx, qx⟩ +Together with qx ∈ ran q: +ϕ(ujj) = 1 =⇒ qx ∈ ran p +ϕ(ujj) = 0 =⇒ qx ∈ ker p +By ([7], (6)), qx is orthogonal to ran p ∩ ran q and ker p ∩ ran q, and it follows that: +0 < ϕ(ujj) < 1, +that is, +ϕ0 ∈ Gj but � +uklϕ0 ̸∈ Gj. +□ +2in the notation of ([7],(1)), x ∈ M0 + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +21 +Consider, at the universal level: +(S+ +N)N := {ϕ ∈ S+ +N : ϕ(uNN) = 1}. +If H given by π : C(S+ +N) → C(H) is an isotropy subgroup in the sense that H ⊆ (S+ +N)N +and so π(uNN) = 1H, then H ⊆ S+ +N−1 by the universal property. In this way, where +πN−1 : C(S+ +N) → C(S+ +N−1) is the quotient +[u +S+ +N +ij ]N +i,j=1 → +� +uS+ +N−1 +0 +0 +1S+ +N−1 +� +, +the following is a maximal (set of states on an algebra of continuous functions on a) +quantum subgroup in the quasi-subgroup (S+ +N)N +(S+ +N−1)⊂S+ +N = {ϕ ◦ πN−1 : ϕ ∈ S+ +N−1}. +In the classical case, N ≤ 3, quasi-subgroups are subgroups, and so (S+ +N)N = (S+ +N−1)⊂S+ +N. +However, for N ≥ 4, the inclusion is proper. +Lemma 3.3. (Teo Banica) Consider a monomial of entries from the fundamental repre- +sentation u ∈ M4(C(S+ +4 )): +f = ui1j1 · · · uimjm. +Then f can only be zero for trivial reasons; i.e. if and only if there exists 2 ≤ n ≤ m such +that: +δin−1,in + δjn−1,jn = 1, +that is uin−1jn−1uinjn = 0. +Proof. With the notation from [5], namely c1, . . . , c4 ∈ SU2 being the Pauli matrices, and +x ∈ SU2 being a parameter, the Pauli representation of C(S+ +4 ) is: +π(uij) = Pcixcj, +the rank one projection on cixcj. Given unit norm ξ, Pξ(η) = ⟨η, ξ⟩ξ. By recurrence +Pξ1 · · ·Pξm(η) = ⟨η, ξm⟩⟨ξm, ξm−1⟩ · · ·⟨ξ2, ξ1⟩ξ1. +With η = ck, one of the Pauli matrices, therefore: +ui1j1 · · ·uimjm(ck) = Pci1xcj1 · · · Pcimxcjm(ck) += ⟨ck, cimxcjm⟩⟨cimxcjm, cim−1xcjm−1⟩ · · ·⟨ci2xcj2, ci1xcj1⟩ci1xck1. +Look at one of these inner products: +⟨cinxcjn, cin−1xcjn−1⟩ = tr(cinxcjn(cin−1xcjn−1)∗) += ± tr(cinxcjncjn−1x∗cin−1) += ± tr(cin−1cinxcjncjn−1x∗). + +22 +J.P. MCCARTHY +This vanishes for any x ∈ SU2 when one of cin−1cin or cjncjn−1 equals I2, and the other +does not, and so when +δin−1,in + δjn−1,jn = 1. +□ +Proposition 3.4. Let S+ +N be the quantum permutation group on N symbols with Haar +state h. Then, for any σ, τ ∈ SN: +h(ui1j1 · · · uinjn) = h(uσ(i1)τ(j1) · · · uσ(in)τ(jn)). +Proof. This is essentially ([17], Prop. 6.4), together with the fact that h is invariant. +□ +Corollary 3.5. Let S+ +N be the quantum permutation group on N ≥ 4 symbols. Then +|ui1j1ui2j2ui3j3|2 = 0. +for trivial reasons only. +Proof. Let 1 ≤ a, b, c, d, e, f ≤ 4 such that u +S+ +4 +ab u +S+ +4 +cd u +S+ +4 +ef +̸= 0. Using the quotient map +π4 : C(S+ +N) → C(S+ +4 ), u → diag(uS+ +4 , 1S+ +4 , . . . , 1S+ +4 ) +π4(|uabucduef|2) ̸= 0 =⇒ |uabucduef|2 ̸= 0. +Let σ(a) = i1, σ(c) = i2, σ(f) = i3 and similarly τ map b, d, f to j1, j2, j3. Proposition +3.4 gives +h(|ui1j1ui2j2ui3j3|2) = h(|uabucduef|2) ̸= 0 =⇒ |ui1j1ui2j2ui3j3|2 ̸= 0. +□ +Proposition 3.6. The inclusion (S+ +N−1)⊂S+ +N ⊂ (S+ +N)N is proper for N ≥ 4. +Proof. Note that for any (ϕ ◦ πN−1) ∈ (S+ +N−1)⊂S+ +N, +(ϕ ◦ πN−1)(u11u2Nu11) = ϕ(πN−1(u11u2Nu11)) = ϕ(πN−1(u11)πN−1(u2N)πN−1(u11)) = 0, +as πN−1(u2N) = 0. On the other hand, hN = � +uNNh, the idempotent in the stabiliser +quasi-subgroup (S+ +N)N, is not in (S+ +N−1)⊂S+ +N, because h faithful on O(S+ +N) implies +hN(u11u2Nu11) = h(uNNu11u2Nu11uNN) +h(uNN) += h(|u2Nu11uNN|2) +h(uNN) +> 0. +□ +Trying to do something for more complicated partitions of {1, . . . , N}, with an (ex- +plicit) idempotent state with a density with respect to ωh is in general more troublesome. +Consider for example: +Pi,j := ({1, . . . , N}\{i, j}) ⊔ {i} ⊔ {j}. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +23 +The obvious way to fix two points is to work with pi,j := uii ∧ ujj, an element of C(G)∗∗, +and given a quantum permutation ϕ ∈ G, define a subset of G by: +Gi,j := {ϕ ∈ G : ωϕ(pi,j) = 1}. +Note that Gi,j = Gi ∩ Gj. However the following is not in general well defined because +ωh(pi,j) is not necessarily strictly positive: +φi,j := ωh(pi,j · pi,j) +ωh(pi,j) +, +For example, consider the dual of the infinite dihedral group with the famous embedding +� +D∞ ⊂ S+ +4 . Working with alternating projection theory, and noting the Haar state on +C(� +D∞) is h(λ) = δλ,e, +ωh(p1,3) = lim +n→∞ h((u11u33)n) = lim +n→∞ +1 +4n = 0. +Proposition 3.7. The stabiliser quasi-subgroup (� +D∞)1,3 is the trivial group. +Proof. Let ϕ ∈ (� +D∞)1,3 so that ϕ(u11) = ϕ(u33) = 1. Then Φ(ϕ) = I4 and, as will be seen +later, by Proposition 4.1, ϕ is a character. There are four characters in � +D∞ and only the +counit has Birkhoff slice equal to the identity. +□ +By Proposition 4.3, p1,3 = pε, the support projection of the counit. +As C(� +D∞) is +coamenable, the Haar state is faithful and so ωh(pε) = 0 implies that pε ̸∈ C(� +D∞) (and +indeed p ∧ q ̸∈ C∗(p, q), the universal unital C∗-algebra generated by two projections). +Note that in general {ε} is a quantum subgroup of any quantum permutation group in +the sense that ε is a Haar idempotent via the quotient π : C(G) → C(e) to the trivial +group {e} ⊆ G: +[uij]N +i,j=1 → diag(1C, . . . , 1C). +4. Exotic quasi-subgroups of the quantum permutation group +A second reason for studying Pal sets and their generated quasi-subgroups is to pos- +tulate, or rather speculate, on, for some N ≥ 4, the existence of an exotic intermediate +quasi-subgroup: +SN ⊊ SN ⊊ S+ +N. +It is currently unknown whether or not there is a Haar idempotent giving an exotic +intermediate quantum subgroup SN ⊊ GN ⊊ S+ +N for some N ≥ 6. It is the case that +SN = S+ +N for N ≤ 3, and for N = 4 [4] and N = 5 [1] there is no such Haar idempotent. +Of course, if there is no exotic intermediate quasi-subgroup SN ⊊ SN ⊊ S+ +N then it is the +case that SN is a maximal quantum subgroup of S+ +N for all N, but of course this is stronger +than the non-existence of an exotic intermediate quantum subgroup. Indeed it is strictly + +24 +J.P. MCCARTHY +stronger in the sense that given a quantum permutation group G and its classical version +G ⊆ G (see below), the existence of a strictly intermediate quasi-subgroup G ⊊ S ⊊ G +does not imply a strictly intermediate quantum subgroup. For example, the finite dual +� +A5 has trivial classical version, and for any non-trivial subgroup H ⊂ A5 the non-Haar +idempotent 1H gives a strict intermediate quasi-subgroup: +{ε} ⊊ SH ⊊ � +A5. +However � +A5 has no non-trivial quantum subgroups because A5 is simple. +The idea for an example of an exotic intermediate quasi-subgroup would be to find a +Pal set given by some condition that is satisfied by the ‘elements of SN in S+ +N’ — and +some states non-zero on a commutator [f, g] ∈ C(S+ +N) — but not by the Haar state on +C(S+ +N). It will be seen that the ‘elements of SN in S+ +N’ correspond to the characters on +C(S+ +N). +4.1. The classical version of a quantum permutation group. The quotient of C(G) +by the commutator ideal is the algebra of functions on the characters on C(G). The +characters form a group G, with the group law given by the convolution: +ϕ1 ⋆ ϕ2 = (ϕ1 ⊗ ϕ2)∆, +the identity is the counit, and the inverse is the reverse ϕ−1 = ϕ ◦ S. +This section contains some general analysis for the support projections of characters on +algebras of continuous functions on quantum permutation groups. While passing to a von +Neumann algebra to talk about support projections, it will not be the conventional choice +of a von Neumann algebra associated to a compact quantum group. This conventional +choice is the algebra: +L∞(G) := Cr(G)′′. +As discussed previously, the current work is at the universal level, so instead consider the +bidual C(G)∗∗. +As before the Birkhoff slice aids the analysis. See [17] for more, where the following +proof is sketched. +Proposition 4.1. A state ϕ on C(G) is a character if and only if Φ(ϕ) is a permutation +matrix. +Proof. If ϕ is a character, +ϕ(uij) = ϕ(u2 +ij) = ϕ(uij)2 ⇒ ϕ(uij) = 0 or 1. +As it is doubly stochastic, it follows that Φ(ϕ) is a permutation matrix. Suppose now that +Φ(ϕ) = σ. Consider the GNS representation (Hσ, πσ, ξσ) associated to ϕ. By assumption +(9) +ϕ(uij) = ⟨ξσ, πσ(uij)(ξσ)⟩ = ⟨πσ(uij)(ξσ), πσ(uij)(ξσ)⟩ = ∥πσ(uij)(ξσ)∥2 = 0 or 1. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +25 +For f ∈ C(G), let (f (n))n≥1 ⊂ O(G) converge to f. For each f (n), (9) implies there exists +an ∈ C such that +πσ(f (n))(ξσ) = anξσ. +The representation πσ is norm continuous, and so πσ(f (n)) → πσ(f), and (πσ(f (n)))n≥1 is +Cauchy: +∥πσ(f (m)) − πσ(f (n))∥ → 0 +=⇒ |am − an|∥ξσ∥ → 0, +which implies that (an)n≥1 converges, to say af ∈ C. The norm convergence of f (n) → f +implies the strong convergence of πσ(f (n)) to πσ(f): +πσ(f)ξσ = lim +n→∞ +� +πσ(f (n))ξσ +� += lim +n→∞(anξσ) = afξσ. +Therefore +ϕ(gf) = ⟨ξσ, πσ(gf)ξσ⟩ = ⟨ξσ, πσ(g)πσ(f)(ξσ)⟩ += ⟨ξσ, πσ(g)afξσ⟩ = af⟨ξσ, πσ(g)ξσ⟩ = ϕ(g)ϕ(f). +□ +Define evσ : C(G) → C: +evσ(f) := πab(f)(σ) +(f ∈ C(G)). +This is a *-homomorphism, but in general evσ need not be non-zero. +Proposition 4.2. If ϕ is a state on C(G) such that Φ(ϕ) = σ, then ϕ = evσ. +Proof. Suppose that Φ(ϕ) = σ. We know that evσ is a *-homomorphism, and by Propo- +sition 4.1 so is ϕ. +As C(G) admits a character, πab is non-zero. +Furthermore, as *- +homomorphisms they are determined by their values on the generators: +ϕ(uij) = Φ(ϕ)ij = σij = δi,σ(j) = 1j→i(σ) = πab(uij)(σ) = evσ(uij). +□ +The classical version of G is therefore the finite group G ⊆ SN given by: +G := {evσ : σ ∈ SN, evσ ̸= 0}. +References to uij in the below are in the embedding: +C(G) ⊆ C(G)∗∗. +Note that the proof of (i) doesn’t use minimality to show that pσ is central: +Proposition 4.3. Associated to each character evσ on C(G) is a support projection +pσ ∈ C(G)∗∗ such that: +(i) pσ is a central projection in C(G)∗∗, and pσpτ = δσ,τpσ. +(ii) pσ = uσ(1),1 ∧ uσ(2),2 ∧ . . . ∧ uσ(N),N. + +26 +J.P. MCCARTHY +Proof. +(i) Note that +evσ(uσ(j),j) = 1 ⇒ ωσ(uσ(j),j) = 1 ⇒ pσ ≤ uσ(j),j, +while pσuij = 0 for i ̸= σ(j). Therefore pσ commutes with all of C(G) ⊆ C(G)∗∗ +and thus, via the Sherman–Takeda Theorem, pσ is in the commutant of C(G). +Everything in C(G)∗∗ commutes with the commutant of C(G). Any pair of per- +mutations σ ̸= τ are distinguished by some σ(j) ̸= τ(j), +pσpτ = pσuσ(j),juτ(j),jpτ = 0. +(ii) Let +qσ = uσ(1),1 ∧ uσ(2),2 ∧ . . . ∧ uσ(N),N. +Define +fσ := uσ(1),1 · · · uσ(N),N. +The sequence (f n +σ )n≥1 ⊂ C(G) converges σ-weakly to qσ. The extension ωσ of evσ +is a character implying that: +ωσ(qσ) = lim +n→∞ ωσ(f n +σ ) = 1 =⇒ pσ ≤ qσ. +Suppose r := qσ − pσ is non-zero. Then there exists a state ωr on C(G)∗∗ such +that ωr(r) = 1. Define a state ϕr on C(G) by: +ϕr(f) = ωr(rfr) +(f ∈ C(G)). +Then ϕr(uσ(j),j) = 1 =⇒ ϕx = evσ, by Proposition 4.2, with equal extensions ωr +and ωσ. However, in this case +ωσ(pσ) = ωr(pσ) = 0, +and this contradiction gives qσ = pσ. +□ +In the following, whenever evσ = 0, then so is pσ. Properties of the bidual summarised +in Section 1.4 are used. +Theorem 4.4. Where G ⊆ G is the classical version, define +pG := +� +σ∈G +pσ. +Then pG is a group-like projection in C(G)∗∗. In addition, pG is the support projection of +the Haar idempotent hC(G) ◦ πab. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +27 +Proof. Note pG is non-zero, as pεpG = pε. Consider pσ ̸= 0. Let (pλ +σ) ⊂ O(G) converge +σ-weakly to pσ ∈ C(G)∗∗. The extension of ∆ is σ-weakly continuous, and recall that pσ +is a meet of projections in O(G): +∆∗∗(pσ) = ∆∗∗(uσ(1),1 ∧ uσ(2),2 ∧ · · · ∧ uσ(N),N) += ∆(uσ(1),1) ∧ ∆(uσ(2),2) ∧ · · · ∧ ∆(uσ(N),N) += lim +n→∞ +�� +∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N) +�n� +. +Consider, for pτ ̸= 0 +∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N)(1G ⊗ pτ) += +� +N +� +k1,...,kN=1 +uσ(1),k1uσ(2),k2 · · · uσ(N),kN ⊗ uk1,1uk2,2 · · ·ukN,N +� +(1G ⊗ pτ) +Note pτ is central and +pτukj = +� +pτukj, +if k = τ(j) +0, +otherwise. , +and so +∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)(1G ⊗ pτ) += uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),Npτ += (uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)(1G ⊗ pτ) +Now +∆∗∗(pσ)(1G ⊗ pτ) = lim +n→∞ +� +∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N) +�n (1G ⊗ pτ) += lim +n→∞ +� +∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)n(1G ⊗ pτ) +� += lim +n→∞ +� +∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)n(1G ⊗ pτ)n� += lim +n→∞ +� +∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)(1G ⊗ pτ) +�n += lim +n→∞ +� +(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)(1G ⊗ pτ) +�n += lim +n→∞ +� +(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n(1G ⊗ pτ)n� += lim +n→∞ +� +(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n(1G ⊗ pτ) +� += lim +n→∞ +� +uσ(1),τ(1)uσ(2),τ(2) · · ·uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n� +(1G ⊗ pτ) += (pστ −1 ⊗ pτ)(1G ⊗ pτ) = pστ −1 ⊗ pτ. +Finally, sum ∆∗∗(pσ)(1G ⊗ pτ) over σ, τ ∈ G. + +28 +J.P. MCCARTHY +Note that C(G) = C(G)/Nab is finite dimensional, and so by (3): +C(G)∗∗ ∼= C(G) ⊕ N∗∗ +ab. +It follows that the support projection of hC(G) ◦ πab is pG. +□ +4.2. The (classically) random and truly quantum parts of a quantum permu- +tation. In the case of C(S+ +N), define pC := pSN and pQ := 1S+ +N − pC. In the rest of this +section the Gelfand–Birkhoff picture will be used: +ϕ ∈ S+ +N is a quantum permutation ⇐⇒ ϕ a state on C(S+ +N). +Definition 4.5. Let ϕ ∈ S+ +N be a quantum permutation. Say that ϕ +(i) is a (classically) random permutation if ωϕ(pQ) = 0, +(ii) is a genuinely quantum permutation if ωϕ(pQ) > 0, +(iii) is a mixed quantum permutation if 0 < ωϕ(pQ) < 1, +(iv) is a truly quantum permutation if ωϕ(pQ) = 1. +Random permutations are in bijection with probability measures ν ∈ Mp(SN): +ϕ random +⇐⇒ ϕ = ϕν where +ϕν(f) := +� +σ∈SN +πab(f)(σ)ν({σ}) +(f ∈ C(S+ +N)). +Theorem 4.6. Suppose hSN is the state on C(S+ +N) defined by hC(SN) ◦ πab. Then if +ϕ ⋆ hSN = hSN = hSN ⋆ ϕ, +ϕ is a random permutation. +Proof. This follows from Theorem 2.19. +□ +Lemma 4.7. Let ϕ, ρ be quantum permutations. The convolution operators ϕ → ρ ⋆ ϕ +and ϕ → ϕ ⋆ ρ are weak*-continuous +Proof. Follows from (ϕ ⋆ ρ)(f) = ϕ((IC(S+ +N) ⊗ ρ)∆(f)) = ρ((ϕ ⊗ IC(S+ +N))∆(f)). +□ +4.3. Exotic quasi-subgroups. +Theorem 4.8. Let ϕ ∈ S+ +N be genuinely quantum, ωϕ(pQ) > 0, and hSN ∈ S+ +N the Haar +idempotent hC(SN) ◦ πab. Form the idempotent φϕ from the weak*-limit of Ces`aro means +of ϕ, and then define an idempotent: +(10) +φ := w∗- lim +n→∞ +1 +n +n +� +k=1 +(hSN ⋆ φϕ)⋆k. +Then the quasi-subgroup generated satisfies: +SN ⊊ Sφ ⊆ S+ +N. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +29 +Proof. First let us show that SN ⊆ Sφ. +For any σ ∈ SN, and φn a Ces`aro mean of +(hSN ⋆ φϕ): +evσ ⋆φn = φn =⇒ w∗- lim +n→∞(evσ ⋆φn) = φ =⇒ evσ ⋆φ = φ =⇒ φ ⋆ evσ−1 = φ. +by Proposition 1.8. Similarly evσ−1 ⋆φn → φ which implies that φ ⋆ evσ = φ, and so +SN ⊆ Sφ +Now suppose for the sake of contradiction that φ is random. Then +φ ⋆ hSN = hSN = hSN ⋆ φ. +However for all Ces`aro means φn: +φn ⋆ ϕ = φn =⇒ φ ⋆ ϕ = φ =⇒ hSN ⋆ ϕ = hSN, +by left convolving both sides of φ ⋆ ϕ = φ with hSN. But Theorem 4.6 says in this case +that ϕ is random, a contradiction. +□ +If in fact for all genuinely quantum ϕ ∈ S+ +N it is the case that Sφ = S+ +N for φ given +by (10), then the maximality conjecture holds, and it is tenable to say that hSN and any +genuinely quantum permutation ϕ ∈ S+ +N generates S+ +N. +5. Convolution dynamics +This section will explore, with respect to pQ ∈ C(S+ +N)∗∗, the qualitative dynamics of +states on C(S+ +N) under convolution. Again, using the Gelfand–Birkhoff picture such states +will be referred to as quantum permutations. The results of this section are illustrated +qualitatively in a phase diagram, Figure 1. +5.1. The convolution of random and truly quantum permutations. +Lemma 5.1. Suppose p ∈ C(G)∗∗ is a group-like projection. Then, where q := 1G − p: +∆∗∗(q)(1G ⊗ p) = q ⊗ p. +Proof. Expand +∆∗∗(p + q)(1G ⊗ p) = (1G ⊗ p), +then multiply on the right with q ⊗ p. +□ +Proposition 5.2. Consider quantum permutations in S+ +N: +(i) The convolution of random permutations is random. +(ii) The convolution of a truly quantum permutation and a random permutation is +truly quantum. +(iii) The convolution of a truly quantum permutations can be random, mixed, or truly +quantum. +Proof. +(i) This is straightforward. + +30 +J.P. MCCARTHY +(ii) Let ϕ be truly quantum, and ϕν random with extension ων. Let (pλ +Q) ⊂ O(S+ +N) +converge σ-weakly to pQ. Using Lemma 5.1, mimic the proof of Theorem 2.19, +hitting both sides of +∆∗∗(pQ)(1S+ +N ⊗ pC) = pQ ⊗ pC, +with ωϕ ⊗ ων, to yield: +ωϕ⋆ϕν(pQ) = 1, +i.e. ϕ ⋆ ϕν is truly quantum. +(iii) It will be seen in Corollary 6.3 that the Haar state is truly quantum. Note that +for any N ≥ 4, the Kac–Paljutkin quantum group can be embedded G0 ⊂ S+ +N via +πG0. It can be shown that E11 ◦ πG0 is truly quantum, and (E11 ◦ πG0)⋆2 = ϕν is a +random permutation ([17], (4.6)). Let 0 ≤ c ≤ 1 and consider the truly quantum +permutation: +ϕ := +√ +1 − c (E11 ◦ πG0) + (1 − +√ +1 − c) h. +Then: +ϕ⋆2 = (1 − c)ϕν + c h =⇒ ϕ⋆2(pQ) = c. +□ +Corollary 5.3. If the convolution of two quantum permutations is a random permutation, +then either both are random, or both are truly quantum. +Proposition 5.4. A quantum permutation ϕ ∈ S+ +N can be written as a convex combination +of a random permutation and a truly quantum permutation. +Proof. If ϕ is random, or truly quantum, the result holds. +Assume ϕ is mixed. +The +projections pC, pQ ∈ C(S+ +N)∗∗ are central, and thus +ϕ = ωϕ(pC) � +pCϕ + ωϕ(pQ) � +pQϕ, +and � +pCϕ is random, while � +pQϕ is truly quantum. +□ +Definition 5.5. Let ϕ ∈ S+ +N be a quantum permutation. Define ϕC := � +pCϕ, the (classi- +cally) random part of ϕ, and ϕQ := � +pQϕ, the truly quantum part of ϕ. +Proposition 5.6. If ϕ ∈ S+ +N is a mixed quantum permutation with 0 < ωϕ(pQ) < 1, then +no finite convolution power ϕ⋆k is random, or truly quantum. +Proof. Let α := ωϕ(pQ) and write ϕ = (1 − α)ϕC + α ϕQ: +ϕ⋆k > (1 − α)kϕ⋆k +C +=⇒ ωϕ⋆k(pQ) ≤ 1 − (1 − α)k, +so no ϕ⋆k is truly quantum. In addition, ϕ⋆k = ϕ⋆ϕ⋆(k−1) cannot be random, by Corollary +5.3, because ϕ is neither random nor truly quantum. +□ +Definition 5.7. A quantum permutation ϕ ∈ S+ +N is called α-quantum if ωϕ(pQ) = α. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +31 +Proposition 5.8. If ϕ ∈ S+ +N is α-quantum and ρ ∈ S+ +N is β-quantum, then +α + β − 2αβ ≤ ωϕ⋆ρ(pQ) ≤ α + β − αβ. +Proof. Note that ϕ ⋆ ρ equals: +(1 − α)(1 − β)(ϕC ⋆ ρC) + β(1 − α)(ϕC ⋆ ρQ) + α(1 − β)(ϕQ ⋆ ρC) + αβ(ϕQ ⋆ ρQ). +Now apply Proposition 5.2. +□ +Definition 5.9. Where (ϕ, ρ) = (ϕ + ρ)/2 is the mean of two quantum permutations, a +quantum strictly 1-increasing pair of quantum permutations ϕ1, ϕ2 ∈ S+ +N are a pair such +that: +ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ). +A quantum strictly 2-increasing pair of quantum permutations are a pair such that: +ω(ϕ1⋆ϕ2)⋆2(pQ) > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ). +Inductively, a quantum strictly (n + 1)-increasing pair of quantum permutations are a +pair such that: +ω(ϕ1⋆ϕ2)⋆(2n)(pQ) > ω(ϕ1⋆ϕ2)⋆(2n−1)(pQ) > · · · > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ). +Proposition 5.10. Let ϕ1 ∈ S+ +N be an α-quantum permutation, and ϕ2 ∈ S+ +N a β- +quantum permutation. +(i) If (α, β) ̸= (0, 0), then if α = 1/4 or β < α/(4α −1), the pair (ϕ1, ϕ2) is quantum +strictly 1-increasing. +(ii) If (α, β) ̸= (0, 0), and β = α/(4α − 1), then: +ωϕ1⋆ϕ2(pQ) ≥ ω(ϕ1,ϕ2)(pQ). +Equality is possible, with e.g. quantum permutations coming from the Kac–Paljutkin +quantum group G0 ⊂ S+ +N. +(iii) If β > α/(4α − 1) then ωϕ1⋆ϕ2(pQ) can be less than, equal to, or greater than +ω(ϕ1,ϕ2)(pQ). +(iv) Let +(S+ +N × S+ +N)α,β := {(ϕ, ρ) : ωϕ(pQ) = α, ωρ(pQ) = β}. +Then +max{|ωϕ1⋆ϕ2(pQ) − ωϕ3⋆ϕ4(pQ)| : (ϕ1, ϕ2), (ϕ3, ϕ4) ∈ (S+ +N × S+ +N)α,β} = αβ. + +32 +J.P. MCCARTHY +Proof. For (i)-(iii) apply Proposition 5.8. For (iv), the maximum in Proposition 5.8 is +attained for +ϕ1 = (1 − α) hSN + α h +ϕ2 = (1 − β) hSN + β h +ϕ3 = (1 − α) hSN + α (E11 ◦ πG0) +ϕ4 = (1 − β) hSN + β (E11 ◦ πG0) +□ +Suppose that ϕ1 is α-quantum, and ϕ2 is β-quantum. The subset of S+ +N × S+ +N given +by condition (1) is called the QI-region. In this region the dynamics of the convolution +(ϕ1, ϕ2) → ϕ with respect to pQ cannot be too wild: +ωϕ1⋆ϕ2(pQ) ∈ +� +ω(ϕ1,ϕ2)(pQ), ω(ϕ1,ϕ2)(pQ) + αβ +� +. +Note that the width of this interval tends to zero for αβ → 0. +On the other hand, the region of S+ +N ×S+ +N given by (3) is called the QW-region, and the +dynamics can be more wild here. Given an arbitrary pair of quantum permutations in +this region, the convolution can be more, equal, or less quantum than the mean, and, as +αβ → 1, over the collection of (ϕ, ρ) ∈ QW the possible range of values of ωϕ⋆ρ(pQ) tends +to one. Tracing from QI towards QW, on the boundary ∂W (given by (2)) ‘conservation +of quantumness’, +ωϕ1⋆ϕ2(pQ) = ω(ϕ1,ϕ2)(pQ), +becomes possible for the first time. +Similarly, higher order regions can be defined: +(1) The region Q2I ⊆ QI given by β < (2α − 1)/(2α − 2) consists of quantum strictly +2-increasing pairs; +(2) The region Q3I ⊆ Q2I given by β < 1 − +√ +2/(1 − 2α) consists of quantum strictly +3-increasing pairs; +(3) The region Q 1 +2 W ⊆ QW given by β > (1 − 1/ +√ +2)/α consists of pairs of quantum +permutations (ϕ1, ϕ2) such that the pair (ϕ1 ⋆ ϕ2, ϕ1 ⋆ ϕ2) ̸∈ Q2I, etc. +5.2. The truly quantum part of an idempotent state. +Corollary 5.11. If φ ∈ S+ +N is an idempotent state, then +ωφ(pQ) ∈ {0} ∪ [1/2, 1]. +Proof. If φ is an idempotent state, +ωφ(pQ) = ωφ⋆φ(pQ). +The rest follows from Proposition 5.10. +□ + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +33 +Figure 1. The phase diagram for the convolution of α-quantum and β- +quantum permutations. +The phases are quantum increasing, QI, in the +bottom left, and quantum wild, QW, in the top right, with the bold line +∂W the boundary. +From the bottom left, Q3I ⊂ Q2I ⊂ QI, and then +touching ∂W on the diagonal, Q 1 +2W ⊂ QW. The region Q 1 +2W is such that +the convolution of states from this region cannot be too close to random: +indeed the convolution cannot fall inside Q2I. The line α = β represents +(ϕ, ϕ) → ϕ⋆2. The shading is proportional to αβ (see Proposition 5.10 (4)). +An idempotent on the boundary ∂W is the Haar idempotent hG0 associated with the +Kac–Paljutkin quantum group G0 ⊂ S+ +4 which satisfies ωhG0(pQ) = 1/2. +Example 5.12. Let G be a finite quantum group given by π : C(S+ +N) → C(G). Where +G ⊆ G is the classical version, the σ-weak extension π∗∗ to the biduals maps onto C(G), +and in particular π∗∗(pσ) ∈ C(G) is the support projection of +f �→ πab(π(f))(σ) +(f ∈ C(S+ +N)). +Let hG := hC(G) ◦ π with extension to the biduals ωG. From e.g. [13]: +ωG(pσ) = +1 +dim C(G) +(σ ∈ G). + +0.8 +0.6 +β +0.4- +02 +0: +0 +0.2 +0.4 +0.6 +0.834 +J.P. MCCARTHY +This implies that +(11) +ωG(pQ) = 1 − +|G| +dim C(G). +Let n ≥ 9, where Sn is generated by elements σ, τ of order two and three [18], and thus +there is an embedding � +Sn ⊂ S+ +5 given by Fourier type matrices uσ ∈ M2(C(� +Sn)) and +uτ ∈ M3(C(� +Sn)) ([2], Chapter 13): +u = +� +uσ +0 +0 +uτ +� +. +A finite dual �Γ ⊆ S+ +N has classical version with order equal to the number of one dimen- +sional representations of Γ (see [17] for more). Therefore the classical version of � +Sn is Z2 +and so, for n ≥ 9, the associated Haar idempotent: +(12) +ω� +Sn(pQ) = 1 − 2 +n!, +which tends to one for n → ∞. +This suggests the following study: consider +χN := {ωφ(pQ) : φ ∈ S+ +N, φ ⋆ φ = φ}. +It is the case that χN = {0} for N ≤ 3, and otherwise a non-singleton. By (12), 1 is a +limit point for χ5 ∩ [1/2, 1). Is there any other interesting behaviour: either at fixed N, +or asymptotically N → ∞? +It seems unlikely that there exists a finite exotic quantum permutation group SN ⊊ +GN ⊊ S+ +N for some N ≥ 6, but something can be said: +Proposition 5.13. An exotic finite quantum permutation group at order N satisfies: +dim C(G) ≥ 2N! +In particular, there is no exotic finite quantum group with dim C(G) < 1440. +Proof. This follows from (11) and Corollary 5.11, and the fact that any exotic quantum +permutation group SN ⊊ G ⊊ S+ +N must satisfy N ≥ 6. +□ +5.3. Periodicity. A periodicity in convolution powers of random permutations is possi- +ble. For example, suppose that G ⊆ SN and N ⊳ G is a normal subgroup. Consider the +probability ν uniform on the coset Ng. Then, where ϕν ∈ S+ +N is the associated state: +ϕν(f) = +� +σ∈SN +πab(f)(σ)ν({σ}) = +1 +|Ng| +� +τ∈N +πab(f)(τg) +(f ∈ C(S+ +N)), +the convolution powers (ϕ⋆k +ν )k≥0 are periodic, with period equal to the order of g. + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +35 +There can also be periodicity with respect to pQ. For example, ϕ := E11 ◦ πG0 is such +that +ϕ⋆k(pQ) = +� +0, +if k odd, +1, +if k odd. +Proposition 5.14. Suppose that ϕ ∈ S+ +N is truly quantum. If ϕ⋆k is random, then ϕ⋆(k+1) +is truly quantum. +Proof. Follows from Corollary 5.3. +□ +Corollary 5.15. Suppose that a truly quantum permutation ϕ has a random finite con- +volution power. Let k0 be the smallest such power. Then: +ωϕk(pQ) = +� +0, +if k +mod k0 = 0, +1, +otherwise. +Is there a quantum permutation with k0 > 2? This phenomenon suggests looking at +when the classical version of G is a normal quantum subgroup G⊳G. However, in general, +the classical periodicity associated with probability measures constant on cosets of N ⊳G +for G ⊆ SN does not extend to the quantum case. See [16], Section 4.3.1. +6. Integer fixed points quantum permutations +An example of an exotic intermediate quasi-subgroup would be nice: instead this section +presents a non-example. For a quantum permutation group G, consider the observable: +fix := +N +� +j=1 +ujj. +Note that σ(fix) ⊆ [0, N]. Consider a finite partition P of the spectrum into Borel subsets, +σ(fix) = +m +� +i=1 +Ei. +Borel functional calculus can be used to attach a (pairwise-distinct) label λi to each +Ei ⊆ σ(fix), and the number of fixed points of a quantum permutation ϕ can be measured +using fixP ∈ C(G)∗∗ given by: +fixP := +m +� +i=1 +λi 1Ei(fix). +Measurement is in the sense of algebraic quantum probability and the Gelfand–Birkhoff +picture: when a quantum permutation ϕ ∈ G is measured with a finite spectrum observ- +able f = � +λ∈σ(f) λ pλ in the bidual C(G)∗∗, the result is an element of σ(f), with f = λ +with probability ωϕ(pλ), and in that event there is wave-function collapse to �pλϕ. + +36 +J.P. MCCARTHY +Definition 6.1. A quantum permutation ϕ ∈ S+ +N has integer fixed points only if for all +Borel subsets E ⊆ σ(fix), +E ∩ {0, 1, . . . , N} = ∅ =⇒ ωϕ(1E(fix)) = 0. +Equivalently, if +ωϕ(1{0,1,...,N}(fix)) = 1. +Let F(G) ⊆ G be the set of quantum permutations with integer fixed points. +In the quotient πab : C(G) → C(G) to the classical version G ⊆ G, the number of fixed +points observable becomes a integer valued: +πab(fix) = fixG = +� +λ=0,1...,N +λ̸=N−1 +λ pλ, +with +pλ(σ) = +� +1, +if σ has λ fixed points, +0, +otherwise. +. +Therefore, random permutations ϕν ∈ S+ +N are elements of F(S+ +N). +There are plenty of concrete examples of genuinely quantum permutations with integer +fixed points: e.g. the quantum permutation ϕ := E11 ◦ πG0 has zero fixed points. So, +F(S+ +N) contains all the elements of SN in S+ +N, and also genuinely quantum permutations. +Proposition 6.2. For N ≥ 4, the Haar state on C(S+ +N) is not an element of F(S+ +N). In +fact: +ωh(1{x}(fix)) = 0 +(x ∈ [0, N]). +Proof. This follows from the fact that for N ≥ 4 the moments of fix with respect to +the Haar state are the Catalan numbers [3], and thus the corresponding measure is the +Marchenko-Pastur law of parameter one, which has no atoms: +ωh(1{x}(fix)) = +� +{x} +1 +2π +� +4 +t − 1 dt = 0. +□ +Corollary 6.3. For N ≥ 4, the Haar state on C(S+ +N) is truly quantum. +Proof. The Haar state h is genuinely quantum. Assume that h ∈ S+ +N is mixed: +ωh(pC) > 0 =⇒ ωh(pσ) > 0 +for some σ ∈ SN. Let qσ := 1S+ +N − pσ. Recalling that pσ is central: +ωh(f) = ωh(pσ) ( �pσh)(f) + ωh(qσ) ( �qσh)(f) +(f ∈ C(S+ +N)∗∗). + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +37 +Note that �pσh has a central minimal projection for support, which implies it is a character. +By Proposition 4.2, �pσh = evσ, which factors through the abelianisation πab: +evσ(f) = πab(f)(σ) +(f ∈ C(S+ +N)), +while the extension ωσ factors through π∗∗ +ab. Suppose that σ has λ ∈ {0, 1, . . . , N} fixed +points. Using Lemma 2.22, consider, where pλ = π∗∗ +ab(1{λ}(fix)), +ωσ(1{λ}(fix)) = pλ(σ) = 1, +=⇒ ωh(1{λ}(fix)) = ωh(pσ) ( �pσh)(1{λ}(fix)) + ωh(qσ) ( �qσh)(1{λ}(fix)) +≥ ωh(pσ) ωσ(1{λ}(fix)) = ωh(pσ) > 0, +contradicting Proposition 6.2. +□ +However, F(G) ⊆ G is in general not a Pal set: +Example 6.4. Let �S4 ⊂ S+ +5 by: +u = +� +u(12) +0 +0 +u(234) +� +. +Here u(12) ∈ M2(C( �S4)) and u(234) ∈ M3(C( �S4)) are Fourier-type magic unitaries associ- +ated with (12) and (234) ([2], Chapter 13). Consider the regular representation: +π : C( �S4) → B(C24). +Consider: +π(fix) = π(2e + (12) + (234) + (243)). +The spectrum contains λ± := (5 ± +√ +17)/2 (see [17]), but consider unit eigenvectors x2 +and x4 ∈ C24 of eigenvalues two and four that give quantum permutations: +ϕ2 = ⟨x2, π(·)x2⟩ and ϕ4 = ⟨x4, π(·)x4⟩, +with two and four fixed points. It can be shown that: +ϕ := 1 +2ϕ2 + 1 +2ϕ4 +is strict, that is |ϕ(σ)| = 1 for σ = e only, and therefore as the convolution in �S4 is +pointwise multiplication, +ϕ⋆k → δe, +which is the Haar state on C( �S4). The Haar state for finite quantum groups such as �S4 +is faithful, and so where pλ+ is the spectral projection associated with the eigenvalue λ+: +h� +S4(pλ+) > 0, +which implies that (ϕ⋆k)k≥0 does not converge to an element with integer fixed points, +and so F( �S4) is not a Pal set, and thus neither is F(S+ +N) for N ≥ 4. + +38 +J.P. MCCARTHY +Example 6.5. In the case of C(S+ +N) (N ≥ 4), the central algebra C(S+ +N)0 generated by the +characters of irreducible unitary representations is commutative [10], and generated by +fix, and so the central algebra C(S+ +N)0 ∼= C([0, N]), and the central states are given by +Radon probability measures. +The quantum permutation ‘uniform on quantum transpositions’, ϕtr from [10], is a +central state given by: +ϕtr(f) = f(N − 2) +(f ∈ C(S+ +N)0) +It has N − 2 fixed points (see [17]) but its convolution powers converge to the Haar state +h ∈ S+ +N, which is not in F(S+ +N) by Proposition 6.2. +Acknowledgement. Some of this work goes back to discussions with Teo Banica. Indeed +the proof of Lemma 3.3 is due to Teo. Thanks also to Matthew Daws for helping with +Section 1.4, Stefaan Vaes with Remark 1.5, and Ruy Exel with the argument in Theorem +2.23 (ii). +References +[1] T. Banica, Homogeneous quantum groups and their easiness level, Kyoto J. Math. 61, (2021) 1–30. +[2] T. Banica, Introduction to quantum groups, Springer Nature Switzerland, (2023), doi:10.1007/978- +3-031-23817-8. +[3] T. Banica and J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. +Jussieu 6 (2007), 381–414. +[4] T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626, (2009) +74–114. +[5] T. Banica and B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008), +942–961. +[6] E. B´edos, G. Murphy and L. Tuset, Co-amenability for compact quantum groups, J. Geom. Phys. +40 (2001) no. 2, 130–153. +[7] A. B¨ottcher, I.M. Spitkovsky, A gentle guide to the basics of two projections theory, Linear Algebra +Appl. 432 (6), (2010) 1412—1459. +[8] U. Franz and A. Skalski, On idempotent states on quantum groups, Journal of Algebra, 322, (2009), +no.5, 1774–1802. +[9] U. Franz, A. Skalski, and R. Tomatsu, Idempotent states on the compact quantum groups and their +classification on Uq(2), SUq(2), and SOq(3), Journal of Noncommutative Geometry 7 (2013), no.1, +221–254. +[10] A. Freslon, Cut-off phenomenon for random walks on free orthogonal free groups, Probab. Theory +Related Fields, 174 (2019), no 3–4, 731–760. +[11] I. Halperin, The product of projection operators. Acta Sci. Math. (Szeged) 23(1962), 96-–99. +[12] P. Kasprzak, and P.M. So�ltan, The Lattice of Idempotent States on a Locally Compact Quantum +Group, Publ. Res. Inst. Math. Sci., 56 (2020), 33–53. +[13] G.I. Kac and V.G. Paljutkin, Finite Group Rings, Trudy Moskov. Mat. Obˇsˇc. 15:224–261, 1966. +Translated in Trans. Moscow Math. Soc. (1967), 251–284, (1966). +[14] Y. Kawada, and K. Itˆo, On the probability distribution on a compact group. I, Proc. Phys.-Math. +Soc. Japan, 3 (1940), 22:977-988. . + +IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS +39 +[15] M. B. Landstand and A. Van Daele, Compact and discrete subgroups of algebraic quantum groups, +I (2007), available at arXiv:0702.458. +[16] J.P. McCarthy, The ergodic theorem for random walks on finite quantum groups, Communications +in Algebra, 49:9, (2021), 3850–3871, DOI:10.1080/00927872.2021.1908551 +[17] J.P. McCarthy, A state-space approach to quantum permutations, Exp. Math., 40(3), (2022), 628– +664. +[18] G.A. Miller,On the groups generated by 2 operators, Bull. Amer. Math. Soc. 7, (1901) 14-–32. +[19] G. J. Murphy, C∗-algebras and Operator Theory, Academic Press, Boston, (1990). +[20] A. Pal, A counterexample on idempotent states on a compact quantum group, Lett. Math. Phys., +37(1) (1996), 75–77. +[21] S. Sherman, The second adjoint of a C∗-algebra, Proceedings of the International Congress of Math- +ematicians (1): (1950) 470. +[22] Z. Takeda, Conjugate spaces of operator algebras Proceedings of the Japan Academy 30 (2) (1954) +90-–95. +[23] M. Takesaki, Theory of Operator Algebras I, Springer (1979). +[24] T. Timmermann, An Invitation to Quantum Groups and Duality, Eur. Math. Soc., (2008). +[25] S. +Vaes, +States +absorbed +by +a +Haar +idempotent +on +a +compact +quantum +group, +https://mathoverflow.net/q/438517, 14-01-2023. +[26] A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995), +3125-–3128. +[27] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211. +[28] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665. +[29] S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, +Invent. Math. 93 (1988), 35–76. +[30] S.L. Woronowicz, Compact quantum groups, in “Sym´etries quantiques” (Les Houches, 1995), North- +Holland, Amsterdam (1998), 845–884. +Department of Mathematics, Munster Technological University, Cork, Ireland. +jp.mccarthy@mtu.ie + diff --git a/2dFQT4oBgHgl3EQf2TZ3/content/tmp_files/load_file.txt b/2dFQT4oBgHgl3EQf2TZ3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd4f7e226ef567d23b67730a95ed54de450b44e7 --- /dev/null +++ b/2dFQT4oBgHgl3EQf2TZ3/content/tmp_files/load_file.txt @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf,len=1234 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='13423v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='OA] 31 Jan 2023 ANALYSIS FOR IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A minor adaptation of Van Daele’s proof yields an idempotent state in any non- empty weak∗-compact convolution-closed convex subset of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Contents Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Compact quantum groups 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Pal sets and quasi-subgroups 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Stabiliser quasi-subgroups 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Exotic quasi-subgroups of the quantum permutation group 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Convolution dynamics 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Integer fixed points quantum permutations 35 References 38 Introduction It is sometimes quipped that quantum groups are neither quantum nor groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' What- ever about compact quantum groups not being quantum, compact quantum groups are, of course, not in general classical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' On the other hand, compact Hausdorff groups are compact quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore, the classical theorems of the existence of the Haar measure, Peter–Weyl, Tannaka–Krein duality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=', can all be viewed as special cases of the quantum analogues proved by Woronowicz [29, 30], and thus naturally the theory of compact quantum groups has many commonalities with the theory of compact groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 46L30,46L67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' quantum permutations, idempotent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Not all classical theorems generalise so nicely: Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (Kawada–Itˆo Theorem, [14], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 3) Let G be a compact separable group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then a probability distribution on G is idempotent with respect to convolution if and only if it is the uniform distribution on a closed subgroup H ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quantum analogue of a closed subgroup, H ⊆ G, is given by a comultiplication- respecting surjective *-homomorphism π : C(G) → C(H), and the direct quantum ana- logue of the Kawada–Itˆo theorem would be that each state idempotent with respect to convolution is a Haar idempotent, that is a state on C(G) of the form hC(H) ◦ π (where hC(H) is the Haar state on C(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However in 1996 Pal discovered non-Haar idempotents in the Kac–Paljutkin quantum group [20], and thus the direct quantum analogue of the Kawada–Itˆo theorem is false (in fact there are counterexamples in the dual of S3, an even ‘smaller’ quantum group [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The null-spaces of Pal’s idempotent states are only one-sided ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Starting with [8], Franz, Skalski and coauthors undertook a general study of idempotent states on compact quantum groups, and, amongst other results, showed that the null-space being a one- sided rather than two-sided ideal is the only obstruction to an idempotent being Haar (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the case of quantum permutation groups, interpreting elements of the state space as quantum permutations, called the Gelfand–Birkhoff picture in [17], leads to the consideration of distinguished subsets of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In [17], using the fact that idempotent states in the case of finite quantum groups have group-like support ([8], Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2), subsets of the state space are associated to idempotent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The current work generalises this point of view: the subset associated to an idempotent state φ on the algebra of continuous functions on a quantum permutation group G is called a quasi- subgroup (after [12]), and given by the set of states absorbed by the idempotent: Sφ = {ϕ ∈ S(C(G)): ϕ ⋆ φ = φ = φ ⋆ ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Whenever a quasi-subgroup is given by a (universal) Haar idempotent, it is stable under wave-function collapse (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There is an obvious relationship between ideals and wave-function collapse: that all classical quasi-subgroups are subgroups is just another way of saying that there are no one-sided ideals in the commutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' An equivalence between Haar idempotent states and the stability of the associated quasi- subgroup under wave-function collapse is not proven here, but there is a partial result (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The other theme of the study of Franz, Skalski and coauthors is the relationship be- tween idempotent states and group-like projections, and culminates in a comprehensive statement about idempotent states being group-like projections in the multiplier algebra of the dual discrete quantum group [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This work contains no such comprehensive state- ment, but does extend the definition of continuous group-like projections p ∈ C(G) to IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 3 group-like projections p ∈ C(G)∗∗, the bidual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Idempotent states with group-like support projection are particularly well-behaved, however it is shown that in the non-coamenable case the support projection of the Haar state is not group-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The consideration of subsets of the state space leads directly to the key observation in this work that non-empty weak∗-compact convolution-closed convex subsets S of the state space, which are termed Pal sets, contain S-invariant idempotent states φS: ϕ ⋆ φS = φS = φS ⋆ ϕ (ϕ ∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This observation is via Van Daele’s proof of the existence of the Haar state [26] (os- tensibly for the apparently esoteric and pathological non-separable case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This observa- tion yields new examples of (generally) non-Haar idempotent states in the case of quan- tum permutation groups: namely from the stabiliser quasi-subgroups of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Pal sets, through their idempotent state, generate quasi-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider S3 ⊂ S+ 4 via C(S+ 4 ) → C(S+ 4 )/⟨u44 = 1⟩: this study yields the interesting example of an intermediate quasi-subgroup S3 ⊊ (S+ 4 )4 ⊊ S+ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where h is the Haar state on C(S+ 4 ), the (non-Haar) idempotent in (S+ 4 )4 is given by: φ(f) = h(u44fu44) h(u44) (f ∈ C(S+ 4 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quasi-subgroup shares many properties of the state space of C(S3), namely it is closed under convolution, closed under reverses ([17], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)), and contains an identity for the convolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' the counit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Moreover, if any quantum permutation ϕ ∈ (S+ 4 )4 is measured with u44 ∈ C(S+ 4 ) (in the sense of the Gelfand–Birkhoff picture), it gives one with probability one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' it fixes label four).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However, while it contains states non-zero on the commutator ideal of C(S+ 4 ), this isn’t a quantum permutation group on three labels because (S+ 4 )4 is not closed under wave-function collapse (the null-space of φ is one-sided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A famous open problem in the theory of quantum permutation groups is the maxi- mality conjecture: that the classical permutation group SN ⊆ S+ N is a maximal quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Following on from Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3 of [17], the current work considers the possibil- ity of an exotic intermediate quasi-subgroup strictly between the classical and quantum permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' An attack on the maximality conjecture via such methods is not a priori particularly promising, but some basic analysis of the support projections of the characters might be useful in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This analysis shows that the support projec- tion of the Haar idempotent associated with SN ⊂ S+ N is a group-like projection in the bidual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' One consequence of this is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8 which says that hSN and any “genuinely quantum permutation” generates a quasi-subgroup strictly bigger than SN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' an idem- potent state between hSN and the Haar state on C(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It isn’t hSN, but it could be (1) a non-Haar idempotent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' or, for some N ≥ 6, (2) the Haar idempotent from an exotic 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY quantum subgroup SN ⊊ GN ⊊ S+ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' or (3) the Haar state on C(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If it is always (3), a strictly stronger statement than the maximality conjecture, then the maximality conjecture holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using the Gelfand–Birkhoff picture, this particular analysis allows us to consider the (classically) random and truly quantum parts of a quantum permutation, and there are some basic rules governing the convolution of (classically) random quantum permutations and truly quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Some consequences of these are explored: for example, an idempotent state on C(S+ N) is either random, or “less than half” random (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Section 1 introduces compact quantum groups, and discusses Van Daele’s proof of the existence of the Haar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Key in this work is the restriction to universal algebras of continuous functions, and the reasons for this restriction are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A further restriction to quantum permutation groups is made, and finally some elementary properties of the bidual are summarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Section 2 introduces Pal sets, and asserts that they contain idempotent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Quasi-subgroups are defined to fix the non-injectivity of the association of a Pal set to its idempotent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The definition of a group-like projection is extended to include group-like projections in the bidual, and the interplay between such group-like projections and idempotent states is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Wave-function collapse is defined, and the question of stability of a quasi-subgroup under wave-function collapse studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In Section 3, stabiliser quasi-subgroups are defined, and it is shown that there is a strictly intermediate quasi-subgroup between S+ N−1 ⊂ S+ N and S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In Section 4, exotic quasi-subgroups of S+ N are considered (and by extension exotic quantum subgroups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Necessarily this section talks about the classical version of a quantum permutation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The support projections of characters are studied, and it is proved that the sum of these is a group-like projection in the bidual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the case of S+ N, this group-like projection is used to define the (classically) random and truly quantum parts of a quantum permutation, and it is proven that the Haar idempotent coming from SN ⊂ S+ N together with a quantum permutation with non-zero truly quantum part generates a non- classical quasi-subgroup in S+ N that is strictly bigger than SN (but possibly equal to S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In Section 5 the convolution of random and truly quantum permutations is considered, and as a corollary a number of quantitative and qualitative results around the random and truly quantum parts of convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In Section 6 there is a brief study of the number of fixed points of a quantum permutation, and it is shown that as a corollary of never having an integer number of fixed points, the Haar state is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Compact quantum groups 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition and the Haar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' An algebra of continuous functions on a (C∗-algebraic) compact quan- tum group G is a C∗-algebra C(G) with unit 1G together with a unital ∗-homomorphism ∆ : C(G) → C(G) ⊗ C(G) into the minimal tensor product that satisfies coassociativity and Baaj–Skandalis cancellation: ∆(C(G))(1G ⊗ C(G)) = ∆(C(G))(C(G) ⊗ 1G) = C(G) ⊗ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Woronowicz defined compact matrix quantum groups [28], and extended this definition to compact quantum groups [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In order to establish the existence of a Haar state, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 below, Woronowicz assumed that the algebra of functions was separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Shortly afterwards Van Daele removed this condition [26], and established the existence of a Haar state in the non-separable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quantum groups in the current work are compact matrix quantum groups, which are separable, however, a careful study of Van Daele’s proof suggests further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore, Van Daele’s proof will be teased out in some detail, and then adapted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that while Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4 are attributed here to Van Daele, it is pointed out by Van Daele that the techniques of their proofs were largely present in the work of Woronowicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define the convolution of states ϕ1, ϕ2 on C(G): ϕ1 ⋆ ϕ2 := (ϕ1 ⊗ ϕ2)∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 ([26, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The algebra of continuous functions C(G) on a compact quantum group admits a unique invariant state h, such that for all states ϕ on C(G): h ⋆ ϕ = h = ϕ ⋆ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3 ([26], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ be a state on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There exists a state φϕ on C(G) such that ϕ ⋆ φϕ = φϕ = ϕ ⋆ φϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define ϕn = 1 n(ϕ + ϕ⋆2 + · · · + ϕ⋆n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As the state space S(C(G)) is convex and closed under convolution, (ϕn)n≥1 ⊂ S(C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Via the weak*-compactness of the state space, Van Daele shows that φϕ, a weak*-limit point of (ϕn)n≥1, is ϕ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4 ([26], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ and φ be states on C(G) such that ϕ ⋆ φ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ρ ∈ C(G)∗ and 0 ≤ ρ ≤ ϕ, then also ρ ⋆ φ = ρ(1G)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where S(C(G)) is the state space of C(G), for each positive linear functional ω on C(G), define: Kω := {ϕ ∈ S(C(G)) : ω ⋆ ϕ = ω(1G)ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As per Van Daele, Kω is closed and thus compact with respect to the weak*-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is non-empty because ω can be normalised to a state �ω on C(G), and by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3, there exists φω ∈ K�ω and thus φω ∈ Kω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let φ ∈ Kω1+ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that both ω1, ω2 ≤ ω1 +ω2, and so by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4, φ ∈ Kω1 ∩Kω2 so that: Kω1+ω2 ⊂ Kω1 ∩ Kω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Assume that the intersection of the Kω over the positive linear functionals on C(G) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Thus, where the complement is with respect to S(C(G)): � ω pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Kc ω = S(C(G)), is an open cover of a compact set, and thus admits a finite subcover {Kc ωi : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , n} such that n� i=1 Kc ωi = S(C(G)) =⇒ n� i=1 Kωi = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ψ = �n i=1 ωi: the set Kψ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is also a subset of: n� i=1 Kωi = ∅, an absurdity, and so the intersection of all the Kω is non-empty, and thus there is a state h that is left-invariant for all positive linear functionals and thus for S(C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The universal and reduced versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A reference for this section is Timmer- mann [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A compact quantum group has a dense Hopf*-algebra of regular functions, O(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The algebra of regular functions has a minimal norm-completion, the reduced algebra of continuous functions, Cr(G), the image of the GNS representation associated to the Haar state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' and a maximal norm-completion, the universal algebra of continuous functions, Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The compact quantum group G is coamenable if O(G) has a unique norm-completion to an algebra of continuous functions on a compact quantum group, and so in particular Cr(G) ∼= Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The Haar state is faithful on O(G) and Cr(G), but Cr(G) does not in general admit a character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' On the other hand, Cu(G) does admit a character, but the Haar state is no longer faithful in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' After an abelianisation πab : C(G) → C(G)/Nab, and via Gelfand’s theorem, the algebra of continuous functions on the classical version of a compact quantum group is given by the algebra of continuous function on the set of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However, not every completion IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 7 Cα(G) of O(G) admits a classical version: in particular, when G is not coamenable the abelianisation of Cr(G) is zero, and Cr(G) admits no characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This work includes a study of the classical versions of quantum permutation groups G ⊆ S+ N, and working at the universal level ensures that talking about the classical version G ⊆ G makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quantum subgroup relation H ⊆ G will be given at the universal level: a quantum subgroup is given by a surjective *-homomorphism π : Cu(G) → Cu(H) that respects the comultiplication in the sense that: ∆Cu(H) ◦ π = (π ⊗ π) ◦ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Every such morphism of algebras of continuous function Cu(G) → Cu(H) restricts to a morphism on the level of regular functions O(G) → O(H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' and every morphism O(G) → O(H) extends to the level of universal algebras of continuous functions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Key in this work is the notion of a quasi-subgroup Sφ ⊆ S(Cu(G)), defined as the set of states ϕ that are absorbed by a given idempotent state φ on Cu(G): ϕ ⋆ φ = φ = φ ⋆ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If hH := hCα(H) ◦ π is a Haar idempotent associated with π : C(G) → Cα(H), it is the case that {ϕ ◦ π : ϕ ∈ S(Cα(H))} ⊆ ShH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As explained by Stefaan Vaes1 [25], in general this is not an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In particular the Haar state of Cr(G) in Cu(G), hr := hCr(G) ◦ πr, is in fact equal to the Haar state on Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Thus the quasi-subgroup generated by hr is the whole state space of Cu(G), but in the non-coamenable case there are states on Cu(G), such as the counit, that do not factor through πr, and thus in this case: {ϕ ◦ πr : ϕ ∈ S(Cr(G))} ⊊ Shr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Vaes goes on to prove that in the universal case of π : Cu(G) → Cu(H) that indeed: (1) {ϕ ◦ π : ϕ ∈ H} = ShH, and this is more satisfactory for a theory of quasi-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that Vaes’s observation yields Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6 as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There are issues related to the non-faithfulness of the Haar state on Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For example, suppose that π : Cr(G) → Cr(H) is a comultiplication-preserving quotient map and consider the Haar idempotent: φ := hCr(H) ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 1it is believed that (1) is not in the literature, however as its proof requires representation theory, not used in the current work, Vaes’s proof is omitted 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY As the Haar state is faithful on Cr(H), the null-space Nφ of φ coincides with ker π, and the support projection pφ ∈ Cr(G)∗∗ gives a nice direct sum structure to the bidual Cr(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For a non-coamenable compact quantum group H, and a quotient π : Cu(G) → Cu(H), the inclusion ker π ⊂ Nφ can be proper: Cu(G) → Cu(H) → Cu(H)/Nφ, with the final algebra of continuous functions isomorphic to Cr(H) ̸∼= Cu(H) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' From this point on, all algebras of continuous functions will be assumed universal, C(G) ∼= Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Careful readers can extract results which hold more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Quantum Permutation Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let C(X) be a C∗-algebra with unit 1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A (finite) partition of unity is a (finite) set of projections {pi}N i=1 ⊂ C(X) that sum to the identity: N � i=1 pi = 1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A matrix u ∈ MN(C(X)) is a magic unitary if the rows and columns are partitions of unity: N � k=1 uik = 1X = N � k=1 ukj (1 ≤ i, j ≤ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider the universal unital C∗-algebra: C(S+ N) := C∗(uij : u an N × N magic unitary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define (2) ∆(uij) = N � k=1 uik ⊗ ukj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using the universal property, Wang [27] shows that ∆ is a *-homomorphism, and S+ N is a compact quantum group, called the quantum permutation group on N symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note S+ N is not coamenable for N ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let G be a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A magic unitary u ∈ MN(C(G)) whose entries generate C(G) as a C∗-algebra, and such that ∆(uij) is given by (2), is called a magic fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A compact quantum group that admits such a magic fundamental representation is known as a quantum permutation group, and by the universal property G ⊆ S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 9 The relation G ⊆ S+ N yields a specific fundamental magic representation u ∈ MN(C(G)), and whether uij is a generator of C(G) or of C(S+ N) should be clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' From this point on, all quantum groups G will be assumed to be quantum permu- tations groups G ⊆ S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Again, careful readers can extract results which hold more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The antipode is given by: S(uij) = uji =⇒ S2(uij) = uij, that is quantum permutation groups are Kac, where the antipode is a bounded linear map satisfying S2 = IC(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ1, ϕ2 be states on C(G): (ϕ1 ⋆ ϕ2) ◦ S = (ϕ2 ◦ S) ⋆ (ϕ1 ◦ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where τ is the flip, f ⊗ g �→ g ⊗ f, in O(G): ∆ ◦ S = (S ⊗ S) ◦ τ ◦ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If f ∈ O(G), then using the antipodal property ((ϕ1 ⋆ ϕ2) ◦ S)(f) = ((ϕ2 ◦ S) ⋆ (ϕ1 ◦ S))(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The same holds for all f ∈ C(G) because the antipode is bounded, and the comultiplica- tion is a *-homomorphism, and thus both are norm-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='9 ([8], Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If a state φ on C(G) is idempotent, φ⋆φ = φ, then φ◦S = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The Bidual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the sequel the bidual C(X)∗∗ of a unital C∗-algebra C(X) will be utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Here some of its properties are summarised from Takesaki, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The bidual admits C(G)∗ as a predual, and so is a von Neumann algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' States ϕ on C(X) have extensions to states ωϕ on C(X)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where Nϕ = {f ∈ C(X) : ϕ(|f|2) = 0}, the σ-weak-closure is a σ-weakly-closed left ideal in a von Neumann algebra, and so of the form C(X)∗∗qϕ for some projection qϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The support projection of a state ϕ on C(X) is pϕ = 1X − qϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It has the property that: ϕ(f) = ωϕ(fpϕ) = ωϕ(pϕf) = ωϕ(pϕfpϕ) (f ∈ C(X)), and it is the smallest projection p ∈ C(X)∗∗ such that ωϕ(p) = 1 (if ωϕ(p) = 1 then ϕ is said to be supported on p, and pϕ ≤ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If N ⊆ C(X) is an ideal, then N∗∗ ⊆ C(X)∗∗ is σ-weakly closed, and so equal to C(X)∗∗q for a central projection q ∈ C(X)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, as C∗-algebras: (3) C(X)∗∗ ∼= (C(X)/N)∗∗ ⊕ N∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY The embedding C(X) ⊂ C(X)∗∗ is an isometry, so that C(X) is norm closed, and the norm closure of a norm dense *-subalgebra O(X) ⊆ C(X) in C(X)∗∗ is C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In addition, the σ-weak closures of O(X) and C(X) are both C(X)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A *-homomorphism T : C(X) → C(Y) extends to a σ-weakly continuous *-homomorphism: T ∗∗ : C(X)∗∗ → C(Y)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In particular, the extension of a character on C(X) is a character on C(X)∗∗, and thus the support projections of characters in C(X) are minimal projections in C(X)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The product on the bidual is separately σ-weakly continuous: � lim λ fλ � f = lim λ (fλf) (fλ, f ∈ C(X)∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Via the Sherman–Takeda Theorem [21, 22], projections p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , pN ∈ C(X) may be viewed as Hilbert space projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then (4) lim n→∞[(p1 · · · pN)n] = p1 ∧ · · · ∧ pN, strongly [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The powers of products of projections are in the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The strong and σ-strong coincide on the unit ball, and σ-strong convergence implies σ-weak convergence of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Finally, for any Borel set E ⊆ σ(f) of self-adjoint f ∈ C(X), the spectral projection 1E(f) ∈ C(X)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Pal sets and quasi-subgroups 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Pal sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The following notation/terminology is outlined in [17] and used hereafter: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Given a quantum permutation group G, the Gelfand–Birkhoff picture interprets elements of the state-space as quantum permutations, so that ϕ ∈ G means ϕ is a state on C(G), and a subset of the state space S(C(G)) can be denoted S ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A subset S ⊆ G is closed under convolution if ϕ, ρ ∈ S =⇒ ϕ ⋆ ρ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A subset S is closed under reverses if ϕ ∈ S =⇒ (ϕ ◦ S) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A subset S contains the identity if C(G) admits a counit ε, and ε ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that π : C(G) → C(H) gives a (closed) quantum subgroup H ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then the set: H⊆G := {ϕ ◦ π : ϕ ∈ H}, is closed under convolution, and closed under reverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There are subsets S ⊂ G that are closed under convolution, closed under reverses, and contain the identity that are not associated with quantum subgroups in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 11 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let Γ be a finite group with a non-normal subgroup Λ ⊂ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The state space of C(�Γ), denoted here �Γ, is the set of positive-definite functions on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define: (5) SΛ = {ϕ ∈ �Γ : ϕ(λ) = 1 for all λ ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The convolution for states on C(�Γ) is pointwise multiplication, therefore SΛ is closed under convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The reverse of ϕ ∈ �Γ is: (ϕ ◦ S)(γ) = ϕ(γ−1), and Λ is a group so SΛ is closed under reverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The identity, 1Γ ∈ SΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let G0 be the Kac–Paljutkin quantum group with algebra of functions C(G0) = Cf1 ⊕ Cf2 ⊕ Cf3 ⊕ Cf4 ⊕ M2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where f i is dual to fi, and Eij is dual to the matrix unit Eij in the M2(C) factor, the convex hulls co({f 1, f 4, E11}) and co({f 1, f 4, E22}) are closed under convolution, under reverses, and contain the identity, ε = f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let G be a quantum permutation group with uii ∈ C(G) non-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define a subset Gi ⊂ G by: Gi := {ϕ ∈ G : ϕ(uii) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This set is closed under convolution, and closed under reverses because S(uii) = uii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Finally ε ∈ Gi as ε(uij) = δi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' More in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A Pal set is a non-empty convex weak*-closed subset S ⊆ G that is closed under convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A Pal set S ⊆ G contains a unique S-invariant state, φS ∈ S, such that for all ϕ ∈ S: φS ⋆ ϕ = φS = ϕ ⋆ φS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This has exactly the same proof as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2, except rather than defining a Kω for each positive linear functional ω on C(G), they are defined only for each ω ∈ cone(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ The strength of the notion of a Pal set is that, as will be seen in Section 3, they can be easy to describe, and yield idempotent states with certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The problem with Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7 is that Pal sets are not in general sub-objects, not state-spaces of algebras of continuous functions on a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is possible to talk about compact quantum hypergroups in this setting [8, 9, 15], but this avenue will not be pursued here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore, the correspondence S → φS is not one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For example, the Pal set H⊆G yields the Haar idempotent hH ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The singleton {hH ◦ π} is a Pal set with the same idempotent hH ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Another such non-correspondence occurs for the Pal set of central states: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where: {uα ij : i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , dα, α ∈ Irr(G)} are matrix coefficients of mutually inequivalent irreducible unitary representations, a cen- tral state ϕ ∈ G is one such that for all α ∈ Irr(G) there exists ϕ(α) ∈ C such that: ϕ(uα ij) = ϕ(α)δi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The set of central states is a Pal set with idempotent state h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In [10], an S+ N analogue of the measure on SN constant on transpositions, a central state ϕtr on C(S+ N), is studied, and it is shown that the convolution powers (ϕ⋆k tr )k≥0 are a sequence of central states converging to the Haar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Quasi-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' To fix the non-injectivity of the association of a Pal set S with an idempotent φS, is to define a quasi-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This nomenclature of quasi-subgroup is inspired by Kasprzak and So�ltan [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Given an idempotent state φ ∈ G, the set: (6) Sφ := {ϕ ∈ G: ϕ ⋆ φ = φ = φ ⋆ ϕ} is a Pal set with idempotent state φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By associativity, Sφ is closed under convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Convexity is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For weak*-closure, let (ϕλ) ⊆ Sφ converge to ϕ ∈ G, and take f ∈ O(G): (ϕ ⋆ φ)(f) = � ϕ(f(1))φ(f(2)) = � � lim λ ϕλ(f(1)) � φ(f(2)) = lim λ � ϕλ(f(1))φ(f(2)) = lim λ ((ϕλ ⋆ φ)(f)) = lim λ φ(f) = φ(f) □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A quasi-subgroup is a subset of the state space of the form Sφ for an idempotent state φ on C(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' the quasi-subgroup generated by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quasi-subgroup Sφ is the largest Pal set with idempotent φ, and there is a one-to- one correspondence between quasi-subgroups and idempotent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Group-like projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Group-like projections (and their link with idempotent states) were first noted by Lanstad and Van Daele [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This definition can be extended to the bidual: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A group-like projection p ∈ C(G)∗∗ is a non-zero projection such that: ∆∗∗(p)(1G ⊗ p) = p ⊗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 13 In the finite case, there is a bijective correspondence between idempotent states and group-like projections: every idempotent state has group-like density with respect to the Haar state [8] (and this group-like density coincides with the support projection [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the compact case, continuous group-like projections p ∈ C(G) with h(p) > 0 give densities to idempotent states via the Fourier transform, p �→ h(·p)/h(p), but the converse does not hold (see Section 4 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However it is shown here that every group-like projection in the bidual yields a Pal set, and thus an idempotent state, but as seen in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='20 a converse statement does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In general, it can only be said that idempotent states are associated with group-like projections in the multiplier algebra of the dual discrete quantum group [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The language of wave-function collapse will be used talk about idempotent states with group-like density, and later illustrate the difference between Haar and non-Haar idem- potents: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let q ∈ C(G)∗∗ be a projection and ϕ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ωϕ(q) > 0, then ϕ conditioned by q = 1 is given by: �qϕ(g) := ωϕ(qgq) ωϕ(q) (g ∈ C(G)), and ϕ → �qϕ is referred to as wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore, say that a subset S ⊆ G is stable under wave-function collapse if for all projections q ∈ C(G)∗∗, (7) (ϕ ∈ S and ωϕ(q) > 0) =⇒ �qϕ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The following is well known in the algebraic setting ([15], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8), and a similar proof is known to work in the finite quantum group setting ([8], Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For the benefit of the reader, the proof is reproduced in the current setting: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If p ∈ C(G) is a continuous group-like projection such that h(p) > 0, then �ph ∈ G is an idempotent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let φ = �ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The difference between ωh and h can be suppressed here as ωh |C(G) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let f ∈ O(G): (φ ⋆ φ)(f) = 1 h(p)2 � h(pf(1)p)h(pf(2)p) = 1 h(p)2 � h(f(1)p)h(f(2)p) = 1 h(p)2(h ⊗ h) (∆(f)(p ⊗ p)) = 1 h(p)2(h ⊗ h) (∆(f)∆(p)(1G ⊗ p)) = 1 h(p)2(h ⊗ h) (∆(fp)(1G ⊗ p)) = 1 h(p)2h(fp)h(p) = h(pfp) h(p) = φ(f), where the traciality of the Haar state, p2 = p, and (h⊗ϕ)(∆(f)(1G⊗g)) = h(f)ϕ(g) ([24], Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=') were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By norm-continuity this implies that �ph is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Note that it is not claimed that the support projection of �ph ∈ G is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the below this is assumed, and a nice description of the quasi-subgroup follows Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let φ = �pφh be an idempotent with continuous group-like support projection pφ ∈ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then Sφ = {ϕ ∈ G : ϕ(pφ) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that ϕ(pφ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Similarly to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='15, for f ∈ O(G): (8) (φ ⋆ ϕ)(f) = 1 h(pφ)(h ⊗ ϕ)(∆(fpφ)(1G ⊗ pφ)) = h(fpφ) h(pφ) ϕ(pφ) = φ(f), and by weak*-continuity, φ ⋆ ϕ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' On the other hand, suppose that φ ⋆ ϕ = φ so that ϕ ∈ Sφ, the quasi-subgroup generated by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Applying (8) at f = pφ, with the existence of �ph implying h(pφ) > 0: (φ ⋆ ϕ)(pφ) = h(pφ) h(pφ)ϕ(pφ) = φ(pφ) = 1 =⇒ ϕ(pφ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If states ϕ1, ϕ2 on C(G) are supported on a group-like projection p ∈ C(G)∗∗, then so is ϕ1 ⋆ ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The proof for the finite case ([16], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='12) applies with some adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let (pλ) ⊂ O(G) converge σ-weakly to p ∈ C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As the extension of ∆ to ∆∗∗ is σ-weakly continuous lim λ � ∆(pλ) � (1 ⊗ p) = p ⊗ p The product is separately continuous, and ωϕ1 ⊗ ωϕ2 is σ-weakly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' =⇒ lim λ (ωϕ1 ⊗ ωϕ2) � pλ (0) ⊗ pλ (1)p = (ωϕ1 ⊗ ωϕ2)(p ⊗ p) =⇒ lim λ � ωϕ1(pλ (0))ωϕ2(pλ (1)p) = 1 Note that as ϕ2 is supported on p: =⇒ lim λ � ϕ1(pλ (0))ϕ2(pλ (1)) = 1 =⇒ lim λ (ϕ1 ⋆ ϕ2)(pλ) = 1 =⇒ lim λ ωϕ1⋆ϕ2(pλ) = ωϕ1⋆ϕ2(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose p ∈ C(G)∗∗ is a group-like projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then: {ϕ ∈ G : ωϕ(p) = 1}, is a Pal set, and so there is an idempotent φ supported on p such that pφ ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' First {ϕ ∈ G : ωϕ(p) = 1} is non-empty because p is normal and as ∥p∥C(G)∗∗ = 1, there exists a state ω on C(G)∗∗ such that ω(p) = 1 [19], whose restriction to C(G) is a state in Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Weak*-closure and convexity are straightforward, and closure under convolution follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Note that p is not necessarily equal to the support projection of the idempotent state in {ϕ ∈ G : ωϕ(p) = 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' and in the below the idempotent state in {ϕ ∈ G : ωϕ(p) = 1} is not necessarily equal to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that an idempotent state φ ∈ G has group-like support projection p ∈ C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then the quasi-subgroup generated by φ: Sφ ⊆ {ϕ ∈ G : ωϕ(p) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider ϕ ∈ Sφ not supported on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, where q = 1G−p, ωϕ(q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider ωϕ(q · q) ∈ C(G)∗ and note by Cauchy–Schwarz: 0 ≤ ωϕ(q · q) ≤ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4: ωϕ(q · q) ⋆ φ = ωϕ(q1Gq)φ = ωϕ(q)φ, and it follows that �qϕ ∈ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note �qϕ(p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using similar notation and techniques to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='17, apply the σ-weakly contin- uous ω�qϕ ⊗ ωφ to both sides of ∆∗∗(1G ⊗ p) = p ⊗ p, using the fact that p is the support of φ: =⇒ lim λ �� ω�qϕ(pλ (0)) ⊗ ωφ(pλ (1)p) � = ω�qϕ(p) ⊗ ωφ(p) =⇒ lim λ �� �qϕ(pλ (0)) ⊗ ωφ(pλ (1)p) � = 0 =⇒ lim λ �� �qϕ(pλ (0)) ⊗ φ(pλ (1)) � = 0 =⇒ lim λ � (�qϕ ⋆ φ)(pλ) � = 0 =⇒ lim λ � φ(pλ) � = 0 =⇒ ωφ(p) = 0, a nonsense, so ωϕ(q) = 0, and so ωϕ(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ It is not the case that every idempotent state φ has group-like support projection pφ ∈ C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Nor does Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19 hold more generally: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose G is non-coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then the support projection ph ∈ C(G)∗∗ of the Haar state is not a group-like projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore: {ϕ ∈ G : ωϕ(ph) = 1} ⊊ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Assume that the support ph ∈ C(G)∗∗ is a group-like projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As ωh(1G) = 1, 1G − ph > 0 strictly as G is at the universal level and G is assumed non-coamenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore there exists a state ωϕ on C(G)∗∗ such that ωϕ(1G − ph) = 1 =⇒ ωϕ(ph) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Restrict ωϕ to a state ϕ on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19 it follows that ϕ is not invariant under the Haar state, which is absurd as Sh = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ There is a group-like projection p such that {ϕ ∈ G : ωϕ(p) = 1} = Sh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' the unit p = 1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note there is a relationship between quantum subgroups and wave-function collapse: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' ([9], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3) Let G be a compact quantum group and φ ∈ C(G)∗ an idempotent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then φ is a Haar idempotent if and only if the null-space Nφ = {f ∈ C(G) : φ(|f|2) = 0} is a two-sided ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note in the below ωϕ0 is the extension of the state ϕ0 on C(H) to a state on C(H)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that H ⊆ G via π : C(G) → C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then the extension of ϕ0 ◦ π to a state on C(G)∗∗ is given by: ωϕ0 ◦ π∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider f ∈ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The result follows from the σ-continuity of the maps involved, and π∗∗ |C(G) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Note that part (i) of the below is restricted to Haar idempotents coming from Haar states on universal versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that φ is an idempotent state on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) If φ is a (universal) Haar idempotent, then Sφ is closed under wave-function col- lapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) If φ is a non-Haar idempotent with group-like projection support, then Sφ is not closed under wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) Suppose φ is a (universal) Haar idempotent via π : C(G) → C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By Vaes’s Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5, every element of Sφ is of the form ϕ0 ◦ π for a state ϕ0 on C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose ϕ undergoes wave-function collapse to �qϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='22 ωϕ(q) > 0 =⇒ ωϕ0(π∗∗(q)) > 0 (ωϕ0 ∈ S(C(H)∗∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 17 Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='22 again, it can be shown that �qϕ = ψ ◦ π, where: ψ(g) = ωϕ0(π∗∗(q)gπ∗∗(q)) ωϕ0(π∗∗(q)) (g ∈ C(H), ωϕ0 ∈ S(C(H)∗∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Thus, again by Vaes’s remark, ψ ◦ π and thus �qϕ ∈ Sφ, that is Sφ is closed under wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) Suppose φ is a non-Haar idempotent with group-like support projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19 Sφ ⊆ {ϕ ∈ G : ωϕ(pφ) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As φ is a non-Haar idempotent, N∗∗ φ = C(G)∗∗qφ is only a left ideal, and qφ non-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that for all uij ∈ C(G), uijqφuij ∈ N∗∗ φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then uijqφuij = uijqφuijqφ =⇒ uijqφuij = uijqφuijqφuij, so that uijqφuij is a projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This implies, because [uij, qφ]3 = 0 and [uij, qφ] is skew adjoint, that uijqφ = qφuij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore qφ is central and Nφ is an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore there exists uij such that uijqφuij ̸∈ N∗∗ φ : ωφ(|uijqφuij|2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By Cauchy–Schwarz: 0 < ωφ(|uijqφuij|2) ≤ ωφ(uijqφuij) ≤ ωφ(uij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' =⇒ � uijφ(qφ) = ωφ(uijqφuij) ωφ(uij) > 0 =⇒ � uijφ(pφ) < 1 =⇒ � uijφ ̸∈ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Stabiliser quasi-subgroups The analysis here is helped somewhat by defining the Birkhoff slice, a map Φ from the state space of the algebra of continuous functions C(G) on a quantum permutation group G to the doubly stochastic matrices: Φ(ϕ) := (ϕ(uij))N i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Given a finite group G ⊆ SN and a partition P = B1 ⊔ · · · ⊔ Bk of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N}, the P-stabiliser subgroup of G can be formed: GP = {σ ∈ G : σ(Bp) = Bp, 1 ≤ p ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A P-stabiliser quasi-subgroup of G can also be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There are two, equivalent, defi- nitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The first definition uses the equivalence relation ∼P associated to the partition: GP := {ϕ ∈ G: ϕ(uij) = 0 for all i ̸∼P j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Alternatively, consider the Birkhoff slice S(C(G)) → MN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By relabelling if necessary, the blocks of a partition can be assumed to consist of consecutive labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define: GP := {ϕ ∈ G : Φ(ϕ) is block diagonal with pattern P}, 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY that is: ϕ ∈ GP ⇐⇒ Φ(ϕ) = \uf8ee \uf8ef\uf8ef\uf8f0 ΦB1(ϕ) 0 · · 0 0 ΦB2(ϕ) · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' · · 0 0 · · ΦBk(ϕ) \uf8f9 \uf8fa\uf8fa\uf8fb , where ΦBp(ϕ) = [ϕ(uij)]i,j∈Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For any partition P of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N}, GP is a quasi-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' That GP is convex, weak*-closed, and closed under convolution is straightforward (using, for example that the Birkhoff slice is multiplicative Φ(ϕ1 ⋆ ϕ2) = Φ(ϕ1)Φ(ϕ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The universal version gives ε ∈ GP so that GP is non-empty, and so a Pal set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that φP is the associated idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='9: φP(uij) = (φP ◦ S)(uij) = φP(uji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For any fixed j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N}, there exists i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=', N} such that φP(uji) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' From here: φP(ujj) = (φP ⋆ φP)(ujj) = φP(uji)φP(uij) + � k̸=i φP(ujk)φP(ukj) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' To show that GP is equal to SφP = {ϕ ∈ G : ϕ ⋆ φP = φP = φP ⋆ ϕ}, suppose ϕ ∈ SφP, but ϕ ̸∈ GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' That implies there exists uij such that ϕ(uij) ̸= 0 with i ̸∼P j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' But this gives φP(uij) = (ϕ ⋆ φP)(uij) = ϕ(uij)φP(ujj) + � k̸=j ϕ(uik)φP(ukj) > 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ For the partition j := {j} ⊔ ({1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=', N}\\{j}): Gj = {ϕ ∈ G : ϕ(ujj) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note for any quantum permutation group G, and 1 ≤ j ≤ N, the diagonal element ujj is a polynomial group-like projection: ∆(ujj)(1G ⊗ ujj) = � N � k=1 ujk ⊗ ukj � (1G ⊗ ujj) = ujj ⊗ ujj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='16, it can be shown that the associated idempotent state is hj := � ujjh, that is: hj(f) = h(ujjfujj) h(ujj) (f ∈ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 19 The below is (almost) a special case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='23, but included as it uses different proof techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The following are equivalent: (i) hj is a Haar idempotent, (ii) ujj is central, (iii) Gj is stable under wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) =⇒ (ii): assume hj is a Haar idempotent, say equal to hH◦π where π : C(G) → C(H), uij �→ uH ij, is the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that because hj(ujj) = hH(π(ujj)) = 1, and hH is faithful on O(H), 1H = π(1G) = N � m=1 π(umj) = π(ujj), so that π(ujj) = 1H is central in C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Assume that ujj is non-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then there exists ukl ∈ C(G) such that |uklujj − ujjukl|2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Expanding: ujjuklujj − ujjuklujjukl − uklujjuklujj + uklujjukl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Applying the Haar state, which is faithful on O(G), and using its traciality yields: h(ujjuklujj) > h(ujjuklujjuklujj) =⇒ hj(ukl) > hj(uklujjukl) =⇒ hH(π(ukl)) > hH(π(uklujjukl)) = hH(π(ukl)π(ujj)π(ukl))) =⇒ hH(π(ukl)) > hH(π(ukl)1Hπ(ukl))) = hH(π(ukl)), an absurdity, and so ujj is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) =⇒ (i): assume that ujj is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Nj := {f ∈ C(G) : hj(|f|2) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If f ∈ Nj then h(ujjf ∗fujj) = 0 =⇒ fujj ∈ Nh, the null-space of the Haar state, so that: Nj = {f ∈ C(G) : fujj ∈ Nh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The rest of the argument is the same as ([8], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) =⇒ (iii): assume that ujj is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ujj is central in C(G) then it is also central in C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ ∈ Gj and q ∈ C(G)∗∗ such that ωϕ(q) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let pϕ ∈ C(G)∗∗ be the support projection of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that ωϕ(ujj) = ϕ(ujj) = 1 =⇒ pϕ ≤ ujj =⇒ pϕ = pϕujj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note ωϕ(qujjq) = ωϕ(pϕqujjqpϕ) = ωϕ(pϕujjqqpϕ) = ωϕ(pϕqpϕ) = ωϕ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY It follows that: �qϕ(ujj) = ωϕ(qujjq) ωϕ(q) = 1 =⇒ �qϕ ∈ Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (iii) =⇒ (ii): assume now that ujj is non-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore there exists ukl ∈ C(G) such that: ujjukl ̸= uklujj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Represent C(G) with the universal GNS representation πGNS(C(G)) ⊆ B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Denote p := πGNS(ujj) and q := πGNS(ukl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As pq ̸= qp, using Halmos two projections theory there exists a unit vector x ∈ ran p that is orthogonal to both2 ran p ∩ ran q and ran p ∩ ker q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define a state on C(G): ϕ0(f) = ⟨x, πGNS(f)x⟩ (f ∈ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that: ϕ0(ujj) = ⟨x, px⟩ = ⟨x, x⟩ = 1 =⇒ ϕ0 ∈ Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore, together with x ∈ ran p ϕ0(ukl) = ⟨x, qx⟩ = 1 =⇒ x ∈ ran q ϕ0(ukl) = ⟨x, qx⟩ = 0 =⇒ x ∈ ker q but x is orthogonal to both ran p ∩ ran q and ran p ∩ ker q so 0 < ⟨x, qx⟩ < 1 =⇒ 0 < ϕ0(ukl) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Now consider ϕ = � uklϕ0: ϕ(f) := ϕ0(uklfukl) ϕ0(ukl) = ⟨qx, πGNS(f)qx⟩ ⟨qx, qx⟩ (f ∈ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In particular ϕ(ujj) = ⟨qx, pqx⟩ ⟨qx, qx⟩ Together with qx ∈ ran q: ϕ(ujj) = 1 =⇒ qx ∈ ran p ϕ(ujj) = 0 =⇒ qx ∈ ker p By ([7], (6)), qx is orthogonal to ran p ∩ ran q and ker p ∩ ran q, and it follows that: 0 < ϕ(ujj) < 1, that is, ϕ0 ∈ Gj but � uklϕ0 ̸∈ Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 2in the notation of ([7],(1)), x ∈ M0 IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 21 Consider, at the universal level: (S+ N)N := {ϕ ∈ S+ N : ϕ(uNN) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If H given by π : C(S+ N) → C(H) is an isotropy subgroup in the sense that H ⊆ (S+ N)N and so π(uNN) = 1H, then H ⊆ S+ N−1 by the universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In this way, where πN−1 : C(S+ N) → C(S+ N−1) is the quotient [u S+ N ij ]N i,j=1 → � uS+ N−1 0 0 1S+ N−1 � , the following is a maximal (set of states on an algebra of continuous functions on a) quantum subgroup in the quasi-subgroup (S+ N)N (S+ N−1)⊂S+ N = {ϕ ◦ πN−1 : ϕ ∈ S+ N−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the classical case, N ≤ 3, quasi-subgroups are subgroups, and so (S+ N)N = (S+ N−1)⊂S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However, for N ≥ 4, the inclusion is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (Teo Banica) Consider a monomial of entries from the fundamental repre- sentation u ∈ M4(C(S+ 4 )): f = ui1j1 · · · uimjm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then f can only be zero for trivial reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' if and only if there exists 2 ≤ n ≤ m such that: δin−1,in + δjn−1,jn = 1, that is uin−1jn−1uinjn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' With the notation from [5], namely c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , c4 ∈ SU2 being the Pauli matrices, and x ∈ SU2 being a parameter, the Pauli representation of C(S+ 4 ) is: π(uij) = Pcixcj, the rank one projection on cixcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Given unit norm ξ, Pξ(η) = ⟨η, ξ⟩ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By recurrence Pξ1 · · ·Pξm(η) = ⟨η, ξm⟩⟨ξm, ξm−1⟩ · · ·⟨ξ2, ξ1⟩ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' With η = ck, one of the Pauli matrices, therefore: ui1j1 · · ·uimjm(ck) = Pci1xcj1 · · · Pcimxcjm(ck) = ⟨ck, cimxcjm⟩⟨cimxcjm, cim−1xcjm−1⟩ · · ·⟨ci2xcj2, ci1xcj1⟩ci1xck1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Look at one of these inner products: ⟨cinxcjn, cin−1xcjn−1⟩ = tr(cinxcjn(cin−1xcjn−1)∗) = ± tr(cinxcjncjn−1x∗cin−1) = ± tr(cin−1cinxcjncjn−1x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY This vanishes for any x ∈ SU2 when one of cin−1cin or cjncjn−1 equals I2, and the other does not, and so when δin−1,in + δjn−1,jn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let S+ N be the quantum permutation group on N symbols with Haar state h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, for any σ, τ ∈ SN: h(ui1j1 · · · uinjn) = h(uσ(i1)τ(j1) · · · uσ(in)τ(jn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This is essentially ([17], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4), together with the fact that h is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let S+ N be the quantum permutation group on N ≥ 4 symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then |ui1j1ui2j2ui3j3|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' for trivial reasons only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let 1 ≤ a, b, c, d, e, f ≤ 4 such that u S+ 4 ab u S+ 4 cd u S+ 4 ef ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using the quotient map π4 : C(S+ N) → C(S+ 4 ), u → diag(uS+ 4 , 1S+ 4 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , 1S+ 4 ) π4(|uabucduef|2) ̸= 0 =⇒ |uabucduef|2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let σ(a) = i1, σ(c) = i2, σ(f) = i3 and similarly τ map b, d, f to j1, j2, j3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4 gives h(|ui1j1ui2j2ui3j3|2) = h(|uabucduef|2) ̸= 0 =⇒ |ui1j1ui2j2ui3j3|2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The inclusion (S+ N−1)⊂S+ N ⊂ (S+ N)N is proper for N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that for any (ϕ ◦ πN−1) ∈ (S+ N−1)⊂S+ N, (ϕ ◦ πN−1)(u11u2Nu11) = ϕ(πN−1(u11u2Nu11)) = ϕ(πN−1(u11)πN−1(u2N)πN−1(u11)) = 0, as πN−1(u2N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' On the other hand, hN = � uNNh, the idempotent in the stabiliser quasi-subgroup (S+ N)N, is not in (S+ N−1)⊂S+ N, because h faithful on O(S+ N) implies hN(u11u2Nu11) = h(uNNu11u2Nu11uNN) h(uNN) = h(|u2Nu11uNN|2) h(uNN) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Trying to do something for more complicated partitions of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N}, with an (ex- plicit) idempotent state with a density with respect to ωh is in general more troublesome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider for example: Pi,j := ({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N}\\{i, j}) ⊔ {i} ⊔ {j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 23 The obvious way to fix two points is to work with pi,j := uii ∧ ujj, an element of C(G)∗∗, and given a quantum permutation ϕ ∈ G, define a subset of G by: Gi,j := {ϕ ∈ G : ωϕ(pi,j) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that Gi,j = Gi ∩ Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However the following is not in general well defined because ωh(pi,j) is not necessarily strictly positive: φi,j := ωh(pi,j · pi,j) ωh(pi,j) , For example, consider the dual of the infinite dihedral group with the famous embedding � D∞ ⊂ S+ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Working with alternating projection theory, and noting the Haar state on C(� D∞) is h(λ) = δλ,e, ωh(p1,3) = lim n→∞ h((u11u33)n) = lim n→∞ 1 4n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The stabiliser quasi-subgroup (� D∞)1,3 is the trivial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ ∈ (� D∞)1,3 so that ϕ(u11) = ϕ(u33) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then Φ(ϕ) = I4 and, as will be seen later, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1, ϕ is a character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There are four characters in � D∞ and only the counit has Birkhoff slice equal to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3, p1,3 = pε, the support projection of the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As C(� D∞) is coamenable, the Haar state is faithful and so ωh(pε) = 0 implies that pε ̸∈ C(� D∞) (and indeed p ∧ q ̸∈ C∗(p, q), the universal unital C∗-algebra generated by two projections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that in general {ε} is a quantum subgroup of any quantum permutation group in the sense that ε is a Haar idempotent via the quotient π : C(G) → C(e) to the trivial group {e} ⊆ G: [uij]N i,j=1 → diag(1C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Exotic quasi-subgroups of the quantum permutation group A second reason for studying Pal sets and their generated quasi-subgroups is to pos- tulate, or rather speculate, on, for some N ≥ 4, the existence of an exotic intermediate quasi-subgroup: SN ⊊ SN ⊊ S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is currently unknown whether or not there is a Haar idempotent giving an exotic intermediate quantum subgroup SN ⊊ GN ⊊ S+ N for some N ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is the case that SN = S+ N for N ≤ 3, and for N = 4 [4] and N = 5 [1] there is no such Haar idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Of course, if there is no exotic intermediate quasi-subgroup SN ⊊ SN ⊊ S+ N then it is the case that SN is a maximal quantum subgroup of S+ N for all N, but of course this is stronger than the non-existence of an exotic intermediate quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Indeed it is strictly 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY stronger in the sense that given a quantum permutation group G and its classical version G ⊆ G (see below), the existence of a strictly intermediate quasi-subgroup G ⊊ S ⊊ G does not imply a strictly intermediate quantum subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For example, the finite dual � A5 has trivial classical version, and for any non-trivial subgroup H ⊂ A5 the non-Haar idempotent 1H gives a strict intermediate quasi-subgroup: {ε} ⊊ SH ⊊ � A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However � A5 has no non-trivial quantum subgroups because A5 is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The idea for an example of an exotic intermediate quasi-subgroup would be to find a Pal set given by some condition that is satisfied by the ‘elements of SN in S+ N’ — and some states non-zero on a commutator [f, g] ∈ C(S+ N) — but not by the Haar state on C(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It will be seen that the ‘elements of SN in S+ N’ correspond to the characters on C(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The classical version of a quantum permutation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quotient of C(G) by the commutator ideal is the algebra of functions on the characters on C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The characters form a group G, with the group law given by the convolution: ϕ1 ⋆ ϕ2 = (ϕ1 ⊗ ϕ2)∆, the identity is the counit, and the inverse is the reverse ϕ−1 = ϕ ◦ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This section contains some general analysis for the support projections of characters on algebras of continuous functions on quantum permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' While passing to a von Neumann algebra to talk about support projections, it will not be the conventional choice of a von Neumann algebra associated to a compact quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This conventional choice is the algebra: L∞(G) := Cr(G)′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As discussed previously, the current work is at the universal level, so instead consider the bidual C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As before the Birkhoff slice aids the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' See [17] for more, where the following proof is sketched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A state ϕ on C(G) is a character if and only if Φ(ϕ) is a permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ is a character, ϕ(uij) = ϕ(u2 ij) = ϕ(uij)2 ⇒ ϕ(uij) = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As it is doubly stochastic, it follows that Φ(ϕ) is a permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose now that Φ(ϕ) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider the GNS representation (Hσ, πσ, ξσ) associated to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By assumption (9) ϕ(uij) = ⟨ξσ, πσ(uij)(ξσ)⟩ = ⟨πσ(uij)(ξσ), πσ(uij)(ξσ)⟩ = ∥πσ(uij)(ξσ)∥2 = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 25 For f ∈ C(G), let (f (n))n≥1 ⊂ O(G) converge to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For each f (n), (9) implies there exists an ∈ C such that πσ(f (n))(ξσ) = anξσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The representation πσ is norm continuous, and so πσ(f (n)) → πσ(f), and (πσ(f (n)))n≥1 is Cauchy: ∥πσ(f (m)) − πσ(f (n))∥ → 0 =⇒ |am − an|∥ξσ∥ → 0, which implies that (an)n≥1 converges, to say af ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The norm convergence of f (n) → f implies the strong convergence of πσ(f (n)) to πσ(f): πσ(f)ξσ = lim n→∞ � πσ(f (n))ξσ � = lim n→∞(anξσ) = afξσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore ϕ(gf) = ⟨ξσ, πσ(gf)ξσ⟩ = ⟨ξσ, πσ(g)πσ(f)(ξσ)⟩ = ⟨ξσ, πσ(g)afξσ⟩ = af⟨ξσ, πσ(g)ξσ⟩ = ϕ(g)ϕ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Define evσ : C(G) → C: evσ(f) := πab(f)(σ) (f ∈ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This is a *-homomorphism, but in general evσ need not be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ is a state on C(G) such that Φ(ϕ) = σ, then ϕ = evσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that Φ(ϕ) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' We know that evσ is a *-homomorphism, and by Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1 so is ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' As C(G) admits a character, πab is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Furthermore, as *- homomorphisms they are determined by their values on the generators: ϕ(uij) = Φ(ϕ)ij = σij = δi,σ(j) = 1j→i(σ) = πab(uij)(σ) = evσ(uij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ The classical version of G is therefore the finite group G ⊆ SN given by: G := {evσ : σ ∈ SN, evσ ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' References to uij in the below are in the embedding: C(G) ⊆ C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that the proof of (i) doesn’t use minimality to show that pσ is central: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Associated to each character evσ on C(G) is a support projection pσ ∈ C(G)∗∗ such that: (i) pσ is a central projection in C(G)∗∗, and pσpτ = δσ,τpσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) pσ = uσ(1),1 ∧ uσ(2),2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' ∧ uσ(N),N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) Note that evσ(uσ(j),j) = 1 ⇒ ωσ(uσ(j),j) = 1 ⇒ pσ ≤ uσ(j),j, while pσuij = 0 for i ̸= σ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore pσ commutes with all of C(G) ⊆ C(G)∗∗ and thus, via the Sherman–Takeda Theorem, pσ is in the commutant of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Everything in C(G)∗∗ commutes with the commutant of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Any pair of per- mutations σ ̸= τ are distinguished by some σ(j) ̸= τ(j), pσpτ = pσuσ(j),juτ(j),jpτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) Let qσ = uσ(1),1 ∧ uσ(2),2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' ∧ uσ(N),N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define fσ := uσ(1),1 · · · uσ(N),N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The sequence (f n σ )n≥1 ⊂ C(G) converges σ-weakly to qσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The extension ωσ of evσ is a character implying that: ωσ(qσ) = lim n→∞ ωσ(f n σ ) = 1 =⇒ pσ ≤ qσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose r := qσ − pσ is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then there exists a state ωr on C(G)∗∗ such that ωr(r) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define a state ϕr on C(G) by: ϕr(f) = ωr(rfr) (f ∈ C(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then ϕr(uσ(j),j) = 1 =⇒ ϕx = evσ, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2, with equal extensions ωr and ωσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However, in this case ωσ(pσ) = ωr(pσ) = 0, and this contradiction gives qσ = pσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ In the following, whenever evσ = 0, then so is pσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Properties of the bidual summarised in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where G ⊆ G is the classical version, define pG := � σ∈G pσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then pG is a group-like projection in C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In addition, pG is the support projection of the Haar idempotent hC(G) ◦ πab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note pG is non-zero, as pεpG = pε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider pσ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let (pλ σ) ⊂ O(G) converge σ-weakly to pσ ∈ C(G)∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The extension of ∆ is σ-weakly continuous, and recall that pσ is a meet of projections in O(G): ∆∗∗(pσ) = ∆∗∗(uσ(1),1 ∧ uσ(2),2 ∧ · · · ∧ uσ(N),N) = ∆(uσ(1),1) ∧ ∆(uσ(2),2) ∧ · · · ∧ ∆(uσ(N),N) = lim n→∞ �� ∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N) �n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider, for pτ ̸= 0 ∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N)(1G ⊗ pτ) = � N � k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=',kN=1 uσ(1),k1uσ(2),k2 · · · uσ(N),kN ⊗ uk1,1uk2,2 · · ·ukN,N � (1G ⊗ pτ) Note pτ is central and pτukj = � pτukj, if k = τ(j) 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' and so ∆(uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)∆(uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2) · · ·∆(uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)(1G ⊗ pτ) = uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · · uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='Npτ = (uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · · uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)(1G ⊗ pτ) Now ∆∗∗(pσ)(1G ⊗ pτ) = lim n→∞ � ∆(uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)∆(uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2) · · ·∆(uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N) �n (1G ⊗ pτ) = lim n→∞ � ∆(uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)∆(uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2) · · ·∆(uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)n(1G ⊗ pτ) � = lim n→∞ � ∆(uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)∆(uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2) · · ·∆(uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)n(1G ⊗ pτ)n� = lim n→∞ � ∆(uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1)∆(uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2) · · ·∆(uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)(1G ⊗ pτ) �n = lim n→∞ � (uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · · uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)(1G ⊗ pτ) �n = lim n→∞ � (uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · · uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)n(1G ⊗ pτ)n� = lim n→∞ � (uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · · uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)n(1G ⊗ pτ) � = lim n→∞ � uσ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(1)uσ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(2) · · ·uσ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='τ(N) ⊗ uτ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1uτ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 · · · uτ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='N)n� (1G ⊗ pτ) = (pστ −1 ⊗ pτ)(1G ⊗ pτ) = pστ −1 ⊗ pτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Finally, sum ∆∗∗(pσ)(1G ⊗ pτ) over σ, τ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Note that C(G) = C(G)/Nab is finite dimensional, and so by (3): C(G)∗∗ ∼= C(G) ⊕ N∗∗ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It follows that the support projection of hC(G) ◦ πab is pG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The (classically) random and truly quantum parts of a quantum permu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the case of C(S+ N), define pC := pSN and pQ := 1S+ N − pC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the rest of this section the Gelfand–Birkhoff picture will be used: ϕ ∈ S+ N is a quantum permutation ⇐⇒ ϕ a state on C(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ ∈ S+ N be a quantum permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Say that ϕ (i) is a (classically) random permutation if ωϕ(pQ) = 0, (ii) is a genuinely quantum permutation if ωϕ(pQ) > 0, (iii) is a mixed quantum permutation if 0 < ωϕ(pQ) < 1, (iv) is a truly quantum permutation if ωϕ(pQ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Random permutations are in bijection with probability measures ν ∈ Mp(SN): ϕ random ⇐⇒ ϕ = ϕν where ϕν(f) := � σ∈SN πab(f)(σ)ν({σ}) (f ∈ C(S+ N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose hSN is the state on C(S+ N) defined by hC(SN) ◦ πab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then if ϕ ⋆ hSN = hSN = hSN ⋆ ϕ, ϕ is a random permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ, ρ be quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The convolution operators ϕ → ρ ⋆ ϕ and ϕ → ϕ ⋆ ρ are weak*-continuous Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Follows from (ϕ ⋆ ρ)(f) = ϕ((IC(S+ N) ⊗ ρ)∆(f)) = ρ((ϕ ⊗ IC(S+ N))∆(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Exotic quasi-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ ∈ S+ N be genuinely quantum, ωϕ(pQ) > 0, and hSN ∈ S+ N the Haar idempotent hC(SN) ◦ πab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Form the idempotent φϕ from the weak*-limit of Ces`aro means of ϕ, and then define an idempotent: (10) φ := w∗- lim n→∞ 1 n n � k=1 (hSN ⋆ φϕ)⋆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then the quasi-subgroup generated satisfies: SN ⊊ Sφ ⊆ S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' First let us show that SN ⊆ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For any σ ∈ SN, and φn a Ces`aro mean of (hSN ⋆ φϕ): evσ ⋆φn = φn =⇒ w∗- lim n→∞(evσ ⋆φn) = φ =⇒ evσ ⋆φ = φ =⇒ φ ⋆ evσ−1 = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Similarly evσ−1 ⋆φn → φ which implies that φ ⋆ evσ = φ, and so SN ⊆ Sφ Now suppose for the sake of contradiction that φ is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then φ ⋆ hSN = hSN = hSN ⋆ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However for all Ces`aro means φn: φn ⋆ ϕ = φn =⇒ φ ⋆ ϕ = φ =⇒ hSN ⋆ ϕ = hSN, by left convolving both sides of φ ⋆ ϕ = φ with hSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' But Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6 says in this case that ϕ is random, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ If in fact for all genuinely quantum ϕ ∈ S+ N it is the case that Sφ = S+ N for φ given by (10), then the maximality conjecture holds, and it is tenable to say that hSN and any genuinely quantum permutation ϕ ∈ S+ N generates S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Convolution dynamics This section will explore, with respect to pQ ∈ C(S+ N)∗∗, the qualitative dynamics of states on C(S+ N) under convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Again, using the Gelfand–Birkhoff picture such states will be referred to as quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The results of this section are illustrated qualitatively in a phase diagram, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The convolution of random and truly quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose p ∈ C(G)∗∗ is a group-like projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, where q := 1G − p: ∆∗∗(q)(1G ⊗ p) = q ⊗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Expand ∆∗∗(p + q)(1G ⊗ p) = (1G ⊗ p), then multiply on the right with q ⊗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider quantum permutations in S+ N: (i) The convolution of random permutations is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) The convolution of a truly quantum permutation and a random permutation is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (iii) The convolution of a truly quantum permutations can be random, mixed, or truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) This is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY (ii) Let ϕ be truly quantum, and ϕν random with extension ων.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let (pλ Q) ⊂ O(S+ N) converge σ-weakly to pQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1, mimic the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='19, hitting both sides of ∆∗∗(pQ)(1S+ N ⊗ pC) = pQ ⊗ pC, with ωϕ ⊗ ων, to yield: ωϕ⋆ϕν(pQ) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' ϕ ⋆ ϕν is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (iii) It will be seen in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3 that the Haar state is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that for any N ≥ 4, the Kac–Paljutkin quantum group can be embedded G0 ⊂ S+ N via πG0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It can be shown that E11 ◦ πG0 is truly quantum, and (E11 ◦ πG0)⋆2 = ϕν is a random permutation ([17], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let 0 ≤ c ≤ 1 and consider the truly quantum permutation: ϕ := √ 1 − c (E11 ◦ πG0) + (1 − √ 1 − c) h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then: ϕ⋆2 = (1 − c)ϕν + c h =⇒ ϕ⋆2(pQ) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If the convolution of two quantum permutations is a random permutation, then either both are random, or both are truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A quantum permutation ϕ ∈ S+ N can be written as a convex combination of a random permutation and a truly quantum permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ is random, or truly quantum, the result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Assume ϕ is mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The projections pC, pQ ∈ C(S+ N)∗∗ are central, and thus ϕ = ωϕ(pC) � pCϕ + ωϕ(pQ) � pQϕ, and � pCϕ is random, while � pQϕ is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ ∈ S+ N be a quantum permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Define ϕC := � pCϕ, the (classi- cally) random part of ϕ, and ϕQ := � pQϕ, the truly quantum part of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ ∈ S+ N is a mixed quantum permutation with 0 < ωϕ(pQ) < 1, then no finite convolution power ϕ⋆k is random, or truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let α := ωϕ(pQ) and write ϕ = (1 − α)ϕC + α ϕQ: ϕ⋆k > (1 − α)kϕ⋆k C =⇒ ωϕ⋆k(pQ) ≤ 1 − (1 − α)k, so no ϕ⋆k is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In addition, ϕ⋆k = ϕ⋆ϕ⋆(k−1) cannot be random, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3, because ϕ is neither random nor truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A quantum permutation ϕ ∈ S+ N is called α-quantum if ωϕ(pQ) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 31 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ ∈ S+ N is α-quantum and ρ ∈ S+ N is β-quantum, then α + β − 2αβ ≤ ωϕ⋆ρ(pQ) ≤ α + β − αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that ϕ ⋆ ρ equals: (1 − α)(1 − β)(ϕC ⋆ ρC) + β(1 − α)(ϕC ⋆ ρQ) + α(1 − β)(ϕQ ⋆ ρC) + αβ(ϕQ ⋆ ρQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Now apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where (ϕ, ρ) = (ϕ + ρ)/2 is the mean of two quantum permutations, a quantum strictly 1-increasing pair of quantum permutations ϕ1, ϕ2 ∈ S+ N are a pair such that: ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A quantum strictly 2-increasing pair of quantum permutations are a pair such that: ω(ϕ1⋆ϕ2)⋆2(pQ) > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Inductively, a quantum strictly (n + 1)-increasing pair of quantum permutations are a pair such that: ω(ϕ1⋆ϕ2)⋆(2n)(pQ) > ω(ϕ1⋆ϕ2)⋆(2n−1)(pQ) > · · · > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let ϕ1 ∈ S+ N be an α-quantum permutation, and ϕ2 ∈ S+ N a β- quantum permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (i) If (α, β) ̸= (0, 0), then if α = 1/4 or β < α/(4α −1), the pair (ϕ1, ϕ2) is quantum strictly 1-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (ii) If (α, β) ̸= (0, 0), and β = α/(4α − 1), then: ωϕ1⋆ϕ2(pQ) ≥ ω(ϕ1,ϕ2)(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Equality is possible, with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' quantum permutations coming from the Kac–Paljutkin quantum group G0 ⊂ S+ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (iii) If β > α/(4α − 1) then ωϕ1⋆ϕ2(pQ) can be less than, equal to, or greater than ω(ϕ1,ϕ2)(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (iv) Let (S+ N × S+ N)α,β := {(ϕ, ρ) : ωϕ(pQ) = α, ωρ(pQ) = β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then max{|ωϕ1⋆ϕ2(pQ) − ωϕ3⋆ϕ4(pQ)| : (ϕ1, ϕ2), (ϕ3, ϕ4) ∈ (S+ N × S+ N)α,β} = αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 32 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For (i)-(iii) apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For (iv), the maximum in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8 is attained for ϕ1 = (1 − α) hSN + α h ϕ2 = (1 − β) hSN + β h ϕ3 = (1 − α) hSN + α (E11 ◦ πG0) ϕ4 = (1 − β) hSN + β (E11 ◦ πG0) □ Suppose that ϕ1 is α-quantum, and ϕ2 is β-quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The subset of S+ N × S+ N given by condition (1) is called the QI-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In this region the dynamics of the convolution (ϕ1, ϕ2) → ϕ with respect to pQ cannot be too wild: ωϕ1⋆ϕ2(pQ) ∈ � ω(ϕ1,ϕ2)(pQ), ω(ϕ1,ϕ2)(pQ) + αβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that the width of this interval tends to zero for αβ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' On the other hand, the region of S+ N ×S+ N given by (3) is called the QW-region, and the dynamics can be more wild here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Given an arbitrary pair of quantum permutations in this region, the convolution can be more, equal, or less quantum than the mean, and, as αβ → 1, over the collection of (ϕ, ρ) ∈ QW the possible range of values of ωϕ⋆ρ(pQ) tends to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Tracing from QI towards QW, on the boundary ∂W (given by (2)) ‘conservation of quantumness’, ωϕ1⋆ϕ2(pQ) = ω(ϕ1,ϕ2)(pQ), becomes possible for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Similarly, higher order regions can be defined: (1) The region Q2I ⊆ QI given by β < (2α − 1)/(2α − 2) consists of quantum strictly 2-increasing pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (2) The region Q3I ⊆ Q2I given by β < 1 − √ 2/(1 − 2α) consists of quantum strictly 3-increasing pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' (3) The region Q 1 2 W ⊆ QW given by β > (1 − 1/ √ 2)/α consists of pairs of quantum permutations (ϕ1, ϕ2) such that the pair (ϕ1 ⋆ ϕ2, ϕ1 ⋆ ϕ2) ̸∈ Q2I, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The truly quantum part of an idempotent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If φ ∈ S+ N is an idempotent state, then ωφ(pQ) ∈ {0} ∪ [1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If φ is an idempotent state, ωφ(pQ) = ωφ⋆φ(pQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The rest follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 33 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The phase diagram for the convolution of α-quantum and β- quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The phases are quantum increasing, QI, in the bottom left, and quantum wild, QW, in the top right, with the bold line ∂W the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' From the bottom left, Q3I ⊂ Q2I ⊂ QI, and then touching ∂W on the diagonal, Q 1 2W ⊂ QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The region Q 1 2W is such that the convolution of states from this region cannot be too close to random: indeed the convolution cannot fall inside Q2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The line α = β represents (ϕ, ϕ) → ϕ⋆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The shading is proportional to αβ (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='10 (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' An idempotent on the boundary ∂W is the Haar idempotent hG0 associated with the Kac–Paljutkin quantum group G0 ⊂ S+ 4 which satisfies ωhG0(pQ) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let G be a finite quantum group given by π : C(S+ N) → C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Where G ⊆ G is the classical version, the σ-weak extension π∗∗ to the biduals maps onto C(G), and in particular π∗∗(pσ) ∈ C(G) is the support projection of f �→ πab(π(f))(σ) (f ∈ C(S+ N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let hG := hC(G) ◦ π with extension to the biduals ωG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' From e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' [13]: ωG(pσ) = 1 dim C(G) (σ ∈ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4- 02 0: 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='834 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY This implies that (11) ωG(pQ) = 1 − |G| dim C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let n ≥ 9, where Sn is generated by elements σ, τ of order two and three [18], and thus there is an embedding � Sn ⊂ S+ 5 given by Fourier type matrices uσ ∈ M2(C(� Sn)) and uτ ∈ M3(C(� Sn)) ([2], Chapter 13): u = � uσ 0 0 uτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A finite dual �Γ ⊆ S+ N has classical version with order equal to the number of one dimen- sional representations of Γ (see [17] for more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore the classical version of � Sn is Z2 and so, for n ≥ 9, the associated Haar idempotent: (12) ω� Sn(pQ) = 1 − 2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=', which tends to one for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This suggests the following study: consider χN := {ωφ(pQ) : φ ∈ S+ N, φ ⋆ φ = φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It is the case that χN = {0} for N ≤ 3, and otherwise a non-singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By (12), 1 is a limit point for χ5 ∩ [1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Is there any other interesting behaviour: either at fixed N, or asymptotically N → ∞?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It seems unlikely that there exists a finite exotic quantum permutation group SN ⊊ GN ⊊ S+ N for some N ≥ 6, but something can be said: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' An exotic finite quantum permutation group at order N satisfies: dim C(G) ≥ 2N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In particular, there is no exotic finite quantum group with dim C(G) < 1440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This follows from (11) and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='11, and the fact that any exotic quantum permutation group SN ⊊ G ⊊ S+ N must satisfy N ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A periodicity in convolution powers of random permutations is possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For example, suppose that G ⊆ SN and N ⊳ G is a normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider the probability ν uniform on the coset Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then, where ϕν ∈ S+ N is the associated state: ϕν(f) = � σ∈SN πab(f)(σ)ν({σ}) = 1 |Ng| � τ∈N πab(f)(τg) (f ∈ C(S+ N)), the convolution powers (ϕ⋆k ν )k≥0 are periodic, with period equal to the order of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 35 There can also be periodicity with respect to pQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For example, ϕ := E11 ◦ πG0 is such that ϕ⋆k(pQ) = � 0, if k odd, 1, if k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that ϕ ∈ S+ N is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' If ϕ⋆k is random, then ϕ⋆(k+1) is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that a truly quantum permutation ϕ has a random finite con- volution power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let k0 be the smallest such power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Then: ωϕk(pQ) = � 0, if k mod k0 = 0, 1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Is there a quantum permutation with k0 > 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This phenomenon suggests looking at when the classical version of G is a normal quantum subgroup G⊳G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' However, in general, the classical periodicity associated with probability measures constant on cosets of N ⊳G for G ⊆ SN does not extend to the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' See [16], Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Integer fixed points quantum permutations An example of an exotic intermediate quasi-subgroup would be nice: instead this section presents a non-example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For a quantum permutation group G, consider the observable: fix := N � j=1 ujj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Note that σ(fix) ⊆ [0, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider a finite partition P of the spectrum into Borel subsets, σ(fix) = m � i=1 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Borel functional calculus can be used to attach a (pairwise-distinct) label λi to each Ei ⊆ σ(fix), and the number of fixed points of a quantum permutation ϕ can be measured using fixP ∈ C(G)∗∗ given by: fixP := m � i=1 λi 1Ei(fix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Measurement is in the sense of algebraic quantum probability and the Gelfand–Birkhoff picture: when a quantum permutation ϕ ∈ G is measured with a finite spectrum observ- able f = � λ∈σ(f) λ pλ in the bidual C(G)∗∗, the result is an element of σ(f), with f = λ with probability ωϕ(pλ), and in that event there is wave-function collapse to �pλϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 36 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' A quantum permutation ϕ ∈ S+ N has integer fixed points only if for all Borel subsets E ⊆ σ(fix), E ∩ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N} = ∅ =⇒ ωϕ(1E(fix)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Equivalently, if ωϕ(1{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=',N}(fix)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let F(G) ⊆ G be the set of quantum permutations with integer fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the quotient πab : C(G) → C(G) to the classical version G ⊆ G, the number of fixed points observable becomes a integer valued: πab(fix) = fixG = � λ=0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=',N λ̸=N−1 λ pλ, with pλ(σ) = � 1, if σ has λ fixed points, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Therefore, random permutations ϕν ∈ S+ N are elements of F(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' There are plenty of concrete examples of genuinely quantum permutations with integer fixed points: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' the quantum permutation ϕ := E11 ◦ πG0 has zero fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' So, F(S+ N) contains all the elements of SN in S+ N, and also genuinely quantum permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For N ≥ 4, the Haar state on C(S+ N) is not an element of F(S+ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In fact: ωh(1{x}(fix)) = 0 (x ∈ [0, N]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' This follows from the fact that for N ≥ 4 the moments of fix with respect to the Haar state are the Catalan numbers [3], and thus the corresponding measure is the Marchenko-Pastur law of parameter one, which has no atoms: ωh(1{x}(fix)) = � {x} 1 2π � 4 t − 1 dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' For N ≥ 4, the Haar state on C(S+ N) is truly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The Haar state h is genuinely quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Assume that h ∈ S+ N is mixed: ωh(pC) > 0 =⇒ ωh(pσ) > 0 for some σ ∈ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let qσ := 1S+ N − pσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Recalling that pσ is central: ωh(f) = ωh(pσ) ( �pσh)(f) + ωh(qσ) ( �qσh)(f) (f ∈ C(S+ N)∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS 37 Note that �pσh has a central minimal projection for support, which implies it is a character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2, �pσh = evσ, which factors through the abelianisation πab: evσ(f) = πab(f)(σ) (f ∈ C(S+ N)), while the extension ωσ factors through π∗∗ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Suppose that σ has λ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' , N} fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='22, consider, where pλ = π∗∗ ab(1{λ}(fix)), ωσ(1{λ}(fix)) = pλ(σ) = 1, =⇒ ωh(1{λ}(fix)) = ωh(pσ) ( �pσh)(1{λ}(fix)) + ωh(qσ) ( �qσh)(1{λ}(fix)) ≥ ωh(pσ) ωσ(1{λ}(fix)) = ωh(pσ) > 0, contradicting Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' □ However, F(G) ⊆ G is in general not a Pal set: Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Let �S4 ⊂ S+ 5 by: u = � u(12) 0 0 u(234) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Here u(12) ∈ M2(C( �S4)) and u(234) ∈ M3(C( �S4)) are Fourier-type magic unitaries associ- ated with (12) and (234) ([2], Chapter 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider the regular representation: π : C( �S4) → B(C24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Consider: π(fix) = π(2e + (12) + (234) + (243)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The spectrum contains λ± := (5 ± √ 17)/2 (see [17]), but consider unit eigenvectors x2 and x4 ∈ C24 of eigenvalues two and four that give quantum permutations: ϕ2 = ⟨x2, π(·)x2⟩ and ϕ4 = ⟨x4, π(·)x4⟩, with two and four fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' It can be shown that: ϕ := 1 2ϕ2 + 1 2ϕ4 is strict, that is |ϕ(σ)| = 1 for σ = e only, and therefore as the convolution in �S4 is pointwise multiplication, ϕ⋆k → δe, which is the Haar state on C( �S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The Haar state for finite quantum groups such as �S4 is faithful, and so where pλ+ is the spectral projection associated with the eigenvalue λ+: h� S4(pλ+) > 0, which implies that (ϕ⋆k)k≥0 does not converge to an element with integer fixed points, and so F( �S4) is not a Pal set, and thus neither is F(S+ N) for N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' 38 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' MCCARTHY Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' In the case of C(S+ N) (N ≥ 4), the central algebra C(S+ N)0 generated by the characters of irreducible unitary representations is commutative [10], and generated by fix, and so the central algebra C(S+ N)0 ∼= C([0, N]), and the central states are given by Radon probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' The quantum permutation ‘uniform on quantum transpositions’, ϕtr from [10], is a central state given by: ϕtr(f) = f(N − 2) (f ∈ C(S+ N)0) It has N − 2 fixed points (see [17]) but its convolution powers converge to the Haar state h ∈ S+ N, which is not in F(S+ N) by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Some of this work goes back to discussions with Teo Banica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Indeed the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='3 is due to Teo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Thanks also to Matthew Daws for helping with Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='4, Stefaan Vaes with Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='5, and Ruy Exel with the argument in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content='23 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Banica, Homogeneous quantum groups and their easiness level, Kyoto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFQT4oBgHgl3EQf2TZ3/content/2301.13423v1.pdf'} +page_content=' Math.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..e24b82c3d2cdc030b38bb5f0aeef7676b91a7cba --- /dev/null +++ b/59E4T4oBgHgl3EQf1g16/content/tmp_files/2301.05291v1.pdf.txt @@ -0,0 +1,3262 @@ + +1 + +Scalable synthesis and characterization of multilayer +γ-graphyne, new carbon crystals with a small direct +bandgap +Authors: Victor G. Desyatkin1‡, William B. Martin1‡, Ali E. Aliev2, Nathaniel E. Chapman1, +Alexandre F. Fonseca3, Douglas S. Galvão3, E. Roy Miller4, Kevin H. Stone5, Zhong Wang2, +Dante Zakhidov6, F. Ted Limpoco7, Sarah R. Almahdali1, Shane M. Parker4, Ray H. Baughman2, +and Valentin O. Rodionov1* +Affiliations: +1Department of Macromolecular Science and Engineering, Case Western Reserve University; +2100 Adelbert Road, Cleveland, OH 44106, USA. +2Alan G. MacDiarmid NanoTech Institute, University of Texas at Dallas; 800 West Campbell +Road, Richardson, TX 75080, USA. +3Applied Physics Department, Institute of Physics “Gleb Wataghin”, University of Campinas; +Campinas, SP, 13083-970, Brazil. +4Department of Chemistry, Case Western Reserve University; 10900 Euclid Ave, Cleveland, OH +44106, USA. + + +2 + +5Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory; 2575 +Sand Hill Road, Menlo Park, CA 94025, USA. +6Department of Materials Science and Engineering, Stanford University; 496 Lomita Mall, +Stanford, CA 94305, USA. +7Oxford Instruments Asylum Research; 6310 Hollister Ave, Santa Barbara, CA 93117, USA. + +Abstract: γ-Graphyne is the most symmetric sp2/sp1 allotrope of carbon, which can be viewed as +graphene uniformly expanded through insertion of two-carbon acetylenic units between all the +aromatic rings. To date, synthesis of bulk γ-graphyne has remained a challenge. We here report +the synthesis of multilayer γ-graphyne through crystallization-assisted irreversible cross-coupling +polymerization. Comprehensive characterization of this new carbon phase is described, including +synchrotron X-ray diffraction, electron diffraction, lateral force microscopy, Raman and infrared +spectroscopy, and cyclic voltammetry. Experiments indicate that γ-graphyne is a 0.48 eV bandgap +semiconductor, with a hexagonal a-axis spacing of 6.88 Å and an interlayer spacing of 3.48 Å, +which is consistent with theoretical predictions. The observed crystal structure has an aperiodic +sheet stacking. The material is thermally stable up to 240 °C but undergoes a transformation at +higher temperatures. While conventional 2D polymerizations and reticular chemistry rely on error +correction through reversibility, we demonstrate that a periodic covalent lattice can be synthesized +under purely kinetic control. The reported methodology is scalable and inspires extension to other +allotropes of the graphyne family. + + + + +3 + +Introduction: Over five hundred carbon phases have been theoretically predicted, but few have +been experimentally realized.1 Allotropes based on hexagonal lattices of sp2 hybridized carbons, +including few-layer graphene,2 nanotubes,3 and fullerenes,4 are synthetically accessible at scale. +These materials have been revolutionary for fundamental physics and materials science, and found +applications in post-silicon electronics, high-capacity batteries, organic solar cells, extreme- +strength composites, and other fields. + +Figure 1. Graphyne family allotropes of carbon. a, Graphynes-n (graphdiyne for n = 2). b, +12,12,12-Graphyne. c, 6,6,12-Graphyne. d, Graphyne (or γ-graphyne). +In contrast, advances in the synthesis of nonbenzenoid carbon allotropes have been limited. +Several sp2-based nanographenes with non-hexagonal rings,5, 6 as well as an extended +nonbenzenoid biphenylene network,7 were described. Materials containing sp1 hybridized linear +arrangements remain even more elusive. Linear polyyne chains -(C≡C)n- are unstable even for +moderate chain lengths unless they are stabilized with bulky end-capping groups8 or confined +inside carbon nanotubes.9 Graphynes, a family of hybrid lattices combining sp1 and sp2 carbons, +were first theoretically proposed by Baughman, Eckhardt, and Kertesz in 1987.10 These allotropes +can be formally viewed as graphenes that are expanded through insertion of acetylenic groups. +Graphynes are commonly divided into two types: graphynes-n in which aromatic rings are +separated by n acetylenic bonds (Fig. 1a), and x,y,z-graphynes featuring at least some non-aromatic +sp2 carbons (Fig. 1b and 1c). Theoretical studies of some graphyne lattices suggest unique +electronic and chemical properties,11 including an intrinsic band gap and electrochemical + +a +b +d +12-DBA +4 + +capacities that far exceed graphite.12 To date, few-layer graphyne-2 or graphdiyne (Fig. 1a, n = 2) +is the only graphyne family allotrope that has been synthesized (typically at sub-milligram scale) +and thoroughly characterized.13 +γ-Graphyne (Fig. 1d), the basic graphyne-n homolog (n = 1), is especially intriguing. Single- +layer γ-graphyne is predicted to be a semiconductor with a moderate band gap,14 ultrafast charge +carrier mobility comparable to that of graphene,15 high thermal conductivity,16 and exceptional +strength.17 Because of these properties, γ-graphyne could form the basis for the next generation +carbon-based devices. γ-Graphyne oligomers containing up to four dehydrobenzo[12]annulene +(12-DBA) repeat units have been prepared through multistep organic synthesis,18, 19 but synthesis +of the extended crystalline lattice remains challenging. +Pyrolytic and vapor-deposition methodologies commonly used for the controlled synthesis of +benzenoid carbons are unsuitable for sp1-based structures, as acetylenes readily convert to +graphene or amorphous carbon at elevated temperatures. While graphdiyne13 and graphdiyne- +graphene heterostructures20 have been synthesized via templated solution-phase 2D +polymerizations, a similar approach has not been previously attempted for γ-graphyne. Reported +graphdiyne syntheses rely on polymerization of highly energetic hexaethynylbenzene through +Glaser-Hay sp1-sp1 coupling, which can be conveniently localized at metal surfaces.21 A +comparable synthesis of γ-graphyne would require either coupling between sp1 and sp2 carbons, +or de novo formation of three acetylenic bonds per each 12-DBA repeat unit. The most common +and general methodology for sp1-sp2 C-C coupling is the Sonogashira reaction.22 The mechanism +of this reaction is thought to involve a homogeneous Pd0 catalytic cycle, like all other Pd-catalyzed +cross-couplings. Therefore, attempting to confine this chemistry to the surface of a template would +be challenging. Moreover, previously reported γ-graphyne oligomers are distorted from planarity, + + +5 + +due to steric hindrance introduced by the terminal functionalities.23 This inevitable distortion, as +well as the typically poor solubility of oligomers, limits stepwise extension of the lattice beyond +3-4 12-DBA units.18 +Results and Discussion: Here, we demonstrate that under appropriately adjusted Sonogashira +coupling conditions, an A3B3-type monomer, 1,3,5-tribromo-2,4,6-triethynylbenzene (TBTEB, +Fig. 2a), can be polymerized into extended γ-graphyne. The main idea that guided our thinking is +that an effective route to graphynes and similar rigid 2D polymers could proceed through reactions +that create multiple connections in a single step or through a series of kinetically coupled fast steps. +Such mechanism would bypass the kinetic dead-end of partially connected intermediates, as each +monomer unit will “click” into place. Furthermore, this polymerization would be self-correcting. +Defects in the growing lattice would be the most reactive sites due to local distortions and strain, +which could be relieved upon multi-site reaction with the monomer. Thus, our two primary aims +were to establish reaction conditions favoring exhaustive coupling of the multifunctional TBTEB, +and to find a way to template the formation of the desired 2D lattice instead of disordered +hyperbranched structures. +Several types of Suzuki-Miyaura, Kumada, and Negishi cross-couplings favor exhaustive +substitution in multifunctional substrates. This mode of reactivity has previously been exploited +for the syntheses of polyfunctional arenes24 and low-defect hyperbranched polyphenylenes,25 as +well as for pseudo-living chain polymerizations.26 In all these cases it is assumed that an +exceptionally reactive Pd species is formed after the initial catalytic cycle.24 The subsequent +coupling steps are catalyzed by this Pd species, proceed inside the solvent cage, and are diffusion- +controlled. We reasoned that we could extend this reaction mode to Sonogashira-type chemistry. +Furthermore, while the nature of the Cu-mediated catalytic cycle in Sonogashira coupling remains + + +6 + +largely unexplored, it is broadly understood to involve Cu acetylides.22 The latter can assume either +three-dimensional or low-dimensional polymeric forms27 or possibly associate with a metal +surface. We hypothesized that a Cu surface could template the γ-graphyne lattice, like it +presumably does in the reported syntheses of graphdiyne.13 +We synthesized TBTEB using a known method28 and screened its reactivity under a range of +conditions (Fig. 2a and Table S1, SI). In the control experiment in the absence of catalysts, TBTEB +decomposed in refluxing pyridine with a half-life of ~24 hours, yielding an amorphous +carbonaceous material (Fig. S19, SI). X-ray photoelectron spectroscopy (XPS) survey indicated +moderate loss of Br through spontaneous hydrodebromination (Fig. S12 and S14d, SI). A similar +featureless carbon was produced in the control experiment with just a Pd pre-catalyst and no source +of Cu (Fig. S18, SI). +Experiments performed in the presence of both Pd and Cu produced outcomes dependent on the +state of the metals. Pd(II) pre-catalysts, as well as PEPPSI-IPr, which we selected for its propensity +for multi-site coupling,24 yielded amorphous carbons broadly comparable to the control products. +However, for stoichiometric loading of Pd(PPh3)4 in the presence of Cu foil, we obtained a black +lustrous material (Fig. S15b, SI). Transmission electron microscopy (TEM) and scanning electron +microscopy (SEM) images of this material revealed flakes composed of stacks of flat sheets (Fig. +S17 and S21, SI). Selected area electron diffraction (SAED) experiments produced dotted ring +patterns (Fig. S17e-f, SI), and no Moiré fringes were observed in bright-field TEM, indicating sub- +micron crystalline domains with random orientation. +While the possibility of the layered flakes being a phase of γ-graphyne was intriguing, the data +were insufficient to make a structural assignment. XPS survey indicated that the product was +primarily carbonaceous, but contaminated by Pd, P, and C from the catalyst (Fig. S10, SI). Since + + +7 + +a multilayered material was obtained, we reasoned that the coupling reaction cannot be confined +to the surface of the foil and decided to investigate sources of Cu other than the metallic surface. +To our surprise and delight, reactions employing soluble CuI also yielded layered flakes (Fig. 2b +and S15a, S16, SI). The crystallinity of this material was significantly improved over the product +produced using Cu foil, with Moiré fringes observable in bright-field TEM (Fig. S16a, SI). Some +of the flakes had well-defined hexagonal shapes (Fig. 2b and S16d, SI). Micron-size hexagonal +prisms with terraces could be observed in SEM (Fig. 2c and S20d, SI). A similar hexagonal +morphology was previously reported for graphdiyne produced by interfacial synthesis.29 + +Figure 2. Two-dimensional polymerizations of TBTEB. a, Overview of selected reaction +conditions. b-c, Representative bright-field TEM and SEM of the carbon flakes obtained from +Pd(PPh3)4/CuI homogeneous reaction. Inset in b emphasizes the hexagonal shape of the layer. d, +High resolution XPS for the C1s region of the sample in b. +The higher crystallinity of the product obtained through the optimized homogeneous +Pd(PPh3)4/CuI protocol allowed for more efficient removal of contaminants. Survey XPS of this + +a +Pd(PPh, +Cuo foil, pyridine +Pd(PPhs)4 +110℃ +Br +B +small crystallline domains +Cul, pyridine +110 °C +Pd(PPhs)4 or no cat. +TBTEB +pyridine +110 °C +large crystallline domains +amorphous +d +3 +6 +Residuals +Fit +Intensity (a.u.) +C-H(sp) +C-C (sp) +C-C (sp2) +C-Br (sp2) +C=0 +*H- +0 +280 +285 +290 +100nm +2 μm +Binding Energy (eV) +8 + +product indicated a level of contamination with Pd and P that was below the detection limit of the +technique (Fig. S9, SI). We acquired high-resolution XPS data for the C1s region of this material +(Fig. 2d), as well as for three controls: the product of the Cu foil synthesis, the Pd-only reaction, +and thermal reaction products (Fig. S13, SI). The C1s peak can be deconvolved into five sub- +peaks, corresponding to C-H (terminal alkyne sp1),30 C≡C (internal alkyne sp1), C=C (aromatic +sp2), aromatic C-Br, and C=O carbons.31 The contribution of C=O is negligible for all samples, +indicating little to no oxidation under the reducing/anaerobic reaction conditions. Without the +contribution of the sp1 subpeak, none of the fits converge, which strongly supports the presence of +acetylenic bonds in all products. XPS indicates a 1:1 ratio of sp1 to sp2 carbons in the crystalline +material synthesized by the homogeneous Pd(PPh3)4/CuI protocol, which is consistent with +γ-graphyne. This ratio is much higher in the control samples (Fig. S13b-d, SI), due to extensive +side reactions and contamination with aromatic impurities. The π-π* “shake up” peak at 290 eV is +commonly observed in XPS of graphitic carbons and graphene, as well as small aromatic +molecules.31 Notably, this peak does not appear in the XPS of graphdiyne.20 The “shake up” feature +was negligible for the product of the homogeneous Cu protocol (Fig. 2d), strongly suggesting that +this material is not graphitic. The peak was prominent for the control products that were also +contaminated with P (Fig. S10, S11, and S13c-d, SI), indicating that it may be related to adsorbed +PPh3. +The initial structural identification of the crystalline carbon material was made via synchrotron +X-ray powder diffraction (PXRD) using a 0.728 Å wavelength (Fig. 3a). We observed a peak at +7.0° 2θ, which matches the predicted11 5.96 Å spacing between (1010) planes of γ-graphyne (Fig. +3a, left inset and Fig. 3d). The intense peak at 12.0° 2θ could be indexed as the (0003) plane, +corresponding to an interlayer distance of 3.48 Å (Fig. 3a, right inset). To index the other observed + + +9 + +peaks, we explored the possible crystal structures of γ-graphyne. While there is a single stable +crystallographic configuration for two graphene sheets, a variety of arrangements are possible for +bilayer γ-graphyne. Some of these bilayer stackings have been previously identified.32 To +systematically survey the possible structures, we used density functional theory (DFT) to analyze +the potential energy surface for a γ-graphyne bilayer. The computed surface (Fig. 3e) is a function +of the horizontal offset of the upper γ-graphyne layer relative to the lower layer with a fixed +interlayer distance of 3.35 Å. The chosen interlayer distance was based on first-pass optimization +of the bilayer geometry and is underestimated due to the difficulty of computationally modeling +van der Waals interactions.32 The calculations identified two types of local energy minima: one +where the upper layer aromatic rings overlay the 12-DBA rings of the lower layer (Fig. 3e, binding +site B), and the second one, where there is a fixed lateral distance between the centers of the upper +and lower layer aromatic rings (Fig. 3e, binding site A). The absolute energy minimum of the +former arrangement gives rise to a single crystal structure with AB mode of stacking corresponding +to the P63mc space group. However, at least six bilayer structures are possible for binding at site +A with energy minima that are nearly identical within the error of the calculations, giving rise to a +multitude of AB, ABC, or more complex arrangements for multiple sheets. To further explore this +complex energy map, which would be expensive for DFT calculations, we ran fully atomistic +reactive molecular dynamics (MD) simulations for 3-6 layers of γ-graphyne. In all cases, the +simulations converged to stacks of sheets bound exclusively at A sites (Fig. S34 and S35, SI). +However, for multiple sheets stacked through A sites, the energy barrier for transitioning between +different A site configurations is extremely small, since this energy difference depends upon the +van der Waals interactions between non-adjacent graphyne sheets. Modeling indicated that several +of the less-ordered stacking modes identified by the MD and DFT calculations produce PXRD + + +10 + +patterns that closely match our experimental diffraction patterns. Thus, we could index the peak at +14.5° 2θ as the (2112) plane. + +Figure 3. X-ray and electron diffraction patterns of γ-graphyne. a, Synchrotron PXRD pattern +(0.728 Å radiation) of γ-graphyne produced by the Pd(PPh3)4/CuI protocol. Left inset (red): peak +at 7.0° 2θ superimposed with the modeled (1010) peak for γ-graphyne with P3112 stacking. Right +inset (black): peak at 12.0° 2θ superimposed with the (0003) peak for the same model. The dashed +blue line is the (0002) peak center for graphite. b, Representative SAED (white) of the same +material overlaid with the simulated (red) c orientation diffraction pattern for a structure with no +systematic absences. c, SAED of the sample region in b, rotated by 45°. d, DFT-generated model +of a γ-graphyne sheet overlaid with crystal planes of interest. e, DFT-generated potential energy +surface for the stacking of two γ-graphyne sheets. The binding energy calculated for a single unit +cell (highlighted). +We further explored the structure and symmetry of the crystals using electron diffraction. The +spot SAED patterns of γ-graphyne produced with the Pd(PPh3)4/CuI protocol were exceptionally +well-defined, consistent for different regions of the sample, and independent of the size or presence + +a +b +0.344 +nm +1 +(0003) +0.5 +(2112) +(2110) +Intensity (a.u.) +0 +0 +↑(1010) +66.57 +7.5 +11.5 +12 +12.5 +(1070) +0.5 +5 +7.5 +10 +12.5 +15 +2 nm +20 (degrees) +14.0 +C +0.450 nm +d +e +10 +0.560 nm +Binding Energy (kcal/mol) +0.344 nm +5 +B +12.5 +Shift ( +ib +0 +> +11.0 +-5 +(1010) +-10 +·0.596 nm +9.5 +-5 +0 +5 +2 nm-1 +X Shift (A) +11 + +of a selected area aperture. The near-perfect uniformity of the patterns indicates that the material +consists of crystalline domains that are sufficiently large to span the entire illuminated region of +our typical imaging frame of 2×2 μm (Fig. 2b and Fig. S16, SI), indicating crystalline domain sizes +of at least 1-3 μm.33 The diffraction patterns observed from the flat areas of the sample had perfect +hexagonal symmetry (Fig. 3b). Using bond distances calculated by DFT and the interlayer distance +obtained from PXRD, we built models for several plausible stacking modes of γ-graphyne. These +models were used to simulate electron diffraction in the c, b, and intermediate crystal orientations +that lie ~45° to the (0001) pole (Fig. S23, SI). The simulated c orientation patterns exclusively +involve spacings in the basal plane (Fig. S22c and S22d, SI). The first- and second-order reflections +in the experimental diffraction pattern correspond to d-spacings of 5.96 Å and 3.44 Å, which +perfectly match the theoretically calculated spacings11 for the (1010) and (1120) plane sets of +γ-graphyne (Fig. 3b and 3d). Both of these distances are defined in part by the length of the +acetylenic bond. Some of the more symmetric space groups, such as Cmcm and R3m, are expected +to produce diffraction patterns with systematic absences. Since there were no such systematic +absences in the observed diffraction pattern, these space groups could be conclusively eliminated. +Additional SAED patterns were obtained for an alternate sample orientation. The initial position +of the stage was chosen to yield the most symmetric spot intensity distribution, which corresponds +to a beam normal to the basal plane and coincident with the a axis. Then the sample was rotated +around the b axis. As the sample rotation reached ~45°, diffraction patterns that involve the z +spacings began to appear. The experimental diffractograms in this orientation provided groups of +closely spaced spots (Fig. 3c), suggesting defects in the layer stacking. Such defects would not +appear in the c orientation diffractograms, since the stacking mode only affects reflections +involving z spacing (Fig. S24, SI). The symmetry of the patterns agrees with our simulations for + + +12 + +this intermediate orientation (Fig. S23e, 23f, 23h, SI). As no systematic absences were observed, +we can exclude some of the more symmetric space groups, most notably the P63mc space group. +The experimental diffraction patterns were most consistent with either one of the lower symmetry +stacking modes, such as P3112 (Fig. S22e, SI) or an aperiodic superlattice. It is important to note +that despite their multi-spot character, the observed diffractograms are not indicative of turbostratic +stacking, which would produce ring patterns. The shape-factor effect alters the geometry of the +diffracted beam,34 which introduces error into the determination of spot centers. Although this +prohibits precise measurement of interplanar spacings, the simulations indicate that the interlayer +spacing estimated from SAED agrees with our PXRD data (Fig. S23, SI). +We further probed the structure of γ-graphyne flakes by using lateral force microscopy (LFM). +LFM is a technique closely related to contact-mode atomic force microscopy (AFM). In LFM, the +scanning tip is rastered across the surface of the sample while maintaining contact, and the +torsional moment due to stick-slip friction is measured. LFM can achieve resolution approaching +that of scanning tunneling microscopy (STM),35 which is often the technique of choice for direct +imaging of atomic arrangements. However, STM cannot be usefully applied to our sample due to +the presence of absorbates that roughen its surface. In contrast, lattice-resolution LFM imaging +can often be performed even on rough substrates under ambient conditions.36 We prepared a +sample for imaging by ultrasonicating the γ-graphyne flakes in water and casting them on a freshly +cleaved surface of highly oriented pyrolytic graphite (HOPG). A lateral deflection map for a flat +region of one of the γ-graphyne flakes (Fig. 4a-b) was then recorded. In the same imaging session, +with the same tip, we then obtained a lateral deflection map of the underlying HOPG as a reference +(Fig. 4d). Fast Fourier transform (FFT) of LFM images of both the γ-graphyne flake and HOPG +showed hexagonal arrangements of high intensity spots, indicating hexagonal symmetry for both + + +13 + +lattices (Fig. S27b-c, SI). The ~2.4 Å periodicity observed for the HOPG lattice (Fig. 4e, Fig. S26 +and S27c, SI) agrees with the expected value.35 The lattice of the γ-graphyne flake is visibly less +dense (Fig. 4c). Its ~3.4 Å (one half of the a-axis spacing) periodicity agrees with the theoretically +predicted distance between the (1120) planes of γ-graphyne, which is also observed as a second- +order reflection in electron diffraction (Fig. 3b, 3d, and 4f). + +Figure 4. Scanning probe microscopy of a γ-graphyne flake. a, AFM topography map of a +γ-graphyne flake. b and d, Lateral deflection maps of a region of the γ-graphyne flake and the +HOPG substrate control in a. c and e, Plots of lateral response along the linear traces from b and +d. Grey bars mark the confidence interval for lattice constants. f, Fast Fourier Transform (FFT) of +the map from b. Circles highlight the periodicity for the (2110) planes of γ-graphyne from electron +diffraction (Fig. 3b). +The observed Raman spectra are consistent with expectations for γ-graphyne.37, 38 The distinctive +Y (A1g) band corresponding to the C≡C stretch of internal triple bonds appears at 2197 cm-1 (Fig. +5a). The G band, which corresponds to the E2g modes of the aromatic rings, is centered at +1565 cm-1, exhibiting the predicted softening37 relative to the G bands of common graphitic +materials (~1580 cm-1) due to additional resonant configurations from π-electron delocalization + +a +5 +b +2.5 +Lateral Response (mV) +C +Lateral Response (mV) +1 +Topography (nm) +4 +1.25 +0 +3 +0 +2 +-1.25 +-1 +0 +0.5 +1 +1.5 +um +nm +2 +-2.5 +Distance (nm) +0.5 +e +Lateral Response (mV) +0.5 +Lateral Response (mV) +0.25 +0 +.0 +-0.25 +-0.5 +0 +0.5 +1 +1.5 +1nm +2 +2 nm1 +-0.5 +Distance +(nm) +14 + +along the acetylenic linkages.39 A broad D band, corresponding to A1g breathing of aromatic rings, +is observed at ~1350 cm-1 and is expected to be sensitive to domain size, lattice defects, and the +excitation wavelength. A broad survey scan (100 – 3000 cm-1) showed no other Raman features +for the 405 nm excitation wavelength, most notably no C-H stretches (2800-3000 cm-1) or the Yʹ +band at ~1900 cm-1 characteristic of diacetylenes.29 In the spectra of the TBTEB monomer (Fig. +S32, SI), a band at 2114 cm-1, corresponding to the stretch of the terminal C≡C-H triple bonds, is +observed but not seen in the spectra of γ-graphyne, most likely due to the low ratio of internal to +terminal triple bond sites. + +Figure 5. Vibrational spectroscopy of γ-graphyne. a, Raman spectrum of γ-graphyne measured +using an excitation wavelength of 405 nm (4.3 mW). b, Raman spectra of γ-graphyne measured +using an excitation wavelength of 535 nm. Black trace: <0.3 mW for less than 10 s. Red trace: +<0.3 mW, after high-power irradiation at 7.5 mW for 30 s. c, Mid-IR spectrum of polycrystalline +γ-graphyne. Inset: deconvolution of the alkyne absorption peak at 2190 cm-1. +It is important to note that our attempts to obtain Raman spectra of γ-graphyne using 532 nm +excitation, for above 0.75 mW incident power using a 0.55 NA 50× objective, resulted in rapid +and irreversible transformation of the material. This transformation was marked by the bleaching +of the Y band, the increase in the intensity of the D band, and the shift of the G band to ~1590 cm-1 +(Fig. 5b). The resulting Raman signature is similar to that of disordered graphitic carbons.40 The + +b +2 +a +405 nm +532 nm +c +1 - +←1513 +G (E2g) +D (Ag) +←1588 +1565 +Envelope +1343 +1.2 +Y(Atg) +G (E2g) +C=C-Ar (int) +C=C-Ar(Br) (int) +2197 +0.75 +1592 +(a.u.) +(a.u.) +(a.u.) +0.75 +C=C-H (term) +1339 +Intensity ( +Intensity +Intensity +2000 +2400 +0.5 +2190 cm-1 +0.5- +B (A1) +Y (Aa) +2197 +0.25 +0.25- +MMW +0 +500 +1000 +1500 +2000 +2500 +500 +1000 +1500 +2000 +2500 +1000 1500 2000 2500 3000 3500 4000 +Raman Shift (cm-1) +Raman Shift (cm-1) +Wavenumber (cm-1) +15 + +transformation is not due to direct oxidation, since experiments conducted in air and under ~1×10-3 +mbar vacuum resulted in similar observations. γ-Graphyne retains its characteristic Raman +signature under prolonged high power 405 nm illumination (>4 mW), while rapidly transforming +under 532 nm laser light (Fig. S28c, SI), which suggests a photochemical process. Under stable +low power 532 nm illumination (<0.3 mW), the degree of transformation does not linearly +correlate with the exposure dose (Fig S28a-b, SI), suggesting thresholding photochemical +behavior. A plausible pathway of the transformation involves Masamune-Bergman +cycloaromatization, which has been demonstrated for a single 12-DBA subunit on a copper +surface.41 The observed splitting of the G band under 532 nm excitation, but not 405 nm excitation +(black trace, Fig. 5b) is not yet well understood, and could be due to partial transformation from +the 532 nm light even when measured at a relatively low excitation intensity. +The micro-FTIR spectra of polycrystalline flakes of γ-graphyne featured wide-band absorption +in the fingerprint region, as well as between 2800-3700 cm-1 (Fig. 5c). Absorption in these regions +is associated with vibrations of the aromatic rings of γ-graphyne, as well as contributions from the +C-H and O-H groups on the periphery of the sheets or belonging to adventitious small-molecule +adsorbates. The distorted line shape of these bands, as well as the change in line shape observed +for flakes of different thicknesses suggests significant contribution from resonant Mie scattering.42, +43 Thus, we could not identify specific vibration frequencies or their true intensities for low- and +high-frequency mid-IR regions. However, a distinctive peak corresponding to acetylenic bonds +centered at ~2190 cm-1 was observed in the mid-IR silent region (1700-2500 cm-1). Since the C≡C +stretch is IR-inactive for symmetric alkynes, an ideal infinite monolayer of γ-graphyne would not +possess this band. However, symmetry breaking due to stacking of graphyne sheets, as well as +finite and defective sheets, could activate this absorption, like for the appearance of the D band in + + +16 + +the Raman spectra of milled graphite.44 The IR absorption reveals a prominent low-frequency +shoulder (Fig. 5c, inset), due to the contribution of the terminal alkyne species at sheet edges (C≡C- +H stretch, ~2115 cm-1). + +Figure 6. Electronic properties of γ-graphyne. a, Determination of the optical bandgap from the +near-IR spectrum of polycrystalline γ-graphyne. b, Cyclic voltammetry of γ-graphyne powder on +glassy carbon. These are three-electrode measurements in 0.1 M n-Bu4N·PF6/acetonitrile +supporting electrolyte, with Pt counter electrode and an Ag/AgNO3 (0.1 M) reference electrode at +a scan rate of 50 mV/s. +The near-IR absorbance spectrum of γ-graphyne shows a strong onset of the fundamental +electronic edge at high frequencies (5000-7500 cm-1), which is manifested as a monotonic decrease +of absorbance with respect to frequency, and as an optical phonon fingerprint in the infrared +shoulder (Fig. 6a). The dependence of absorbance on photon energy within the electronic edge can +be used to estimate the electronic band gap, Eg, and determine the type of semiconductor. The +absorption coefficient α is defined as α = (1/x)×ln(1/T), where x is the sample thickness and T is +transmittance, the ratio of transmitted to incident light intensity. Within the semi-classical theory +of the optical absorption of crystalline direct band gap semiconductors, α(E) is expected to be 0 +for E < Eg, and proportional to (E – Eg)1/2 for E ≥ Eg.45, 46 Thus, Eg can be estimated by plotting α2 +versus E, and extrapolating the linear region of the curve to the energy axis (Fig. 6a). This α2 versus + +a +5 +b +4 +Y-graphyne on glassy carbon +Y-graphyne annealed at 600 °C on glassy carbon +4 +glassy carbon +F += 602 mV +oxidation +Current (μA) +3 +106 +0 +E +132 mV +reduction +-2 +1 +E_ = 0.48 ± 0.05 +4 +0 +9 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +-1 +-0.5 +0 +0.5 +1 +1.5 +Photon Energy (eV) +Potential (V vs. Ag/Agt) +17 + +E plot gives an optical band gap of Eg = 0.48 ± 0.05 eV for γ-graphyne, where the uncertainty +indicates the standard deviation for over 20 polycrystalline particles. +To investigate the redox properties of γ-graphyne, a working electrode was prepared by +ultrasonically dispersing 1 mg of γ-graphyne in 1 mL of a 1:1 v/v mixture of ethanol and water, +and then casting this suspension onto a glassy carbon electrode with subsequent air drying. Three- +electrode cyclic voltammetry (CV) measurements were then conducted in acetonitrile with n- +Bu4N·PF6 supporting electrolyte for potentials between -1.0 and +1.5 V vs. Ag/Ag+ (0.1 M). An +oxidation peak at 602 mV and a reduction peak at 132 mV were observed (Fig. 6b, blue trace), +corresponding to an electrochemical bandgap of 0.47 eV.47 This value agrees with the bandgap +determined by the optical absorption measurements. +To evaluate thermal stability, γ-graphyne particles were supported over holes in thin mica films +by single-layer carbon nanotube sheets and then heated to gradually increasing set points under +vacuum (1.3×10-3 mbar). After one hour at the set temperature, the sample assembly was cooled +down and transferred to the FTIR microscope for analysis. The same sample region was analyzed +for consistency. Heating up to 240 °C did not produce any major changes in FTIR spectra beyond +a slight decrease in the intensity of the 2800-3700 cm-1 band, likely due to the removal of +adventitious adsorbates (Fig. S30a, SI). Loss of the alkyne band at 2100-2200 cm-1 was observed +at just over 240 °C, suggesting an onset of a structural transformation. The material lost all the +alkyne absorbance after the one-hour thermal cycling processes reached 350 °C (Fig. S30b, SI). +The absorbance shoulder corresponding to terminal alkynes disappeared before the internal alkyne +components of the band. The loss of alkyne absorption was accompanied by the disappearance of +the Mie scattering bands in the fingerprint region and the region between 2800-3700 cm-1, +suggesting significant rearrangement of the microcrystalline structure. The material annealed to + + +18 + +600 °C lost the characteristic reduction and oxidation peaks in CV, indicating a major chemical +change (Fig. 6b, red trace). Despite the dramatic transformation of the IR signature, the optical +images of the thermally treated samples revealed no textural or geometric changes on heating up +to 450 °C (Fig. S30c, SI). Due to its sensitivity to the different alkyne bonds present in the sample, +the IR absorbance was found to be a useful marker for monitoring the structural changes of γ- +graphyne. +Our data indicates that the material we synthesized is multilayer γ-graphyne. Contrary to +expectations, we found that no external template is required for synthesizing highly crystalline +γ-graphyne. There is no experimental evidence even in our reactions performed with Cu foil that +any polymerization is happening on the surface. The lower crystallinity of the Cu foil products is +likely due to the reduced concentration of catalytic Cu species in solution. This results in slower +Sonogashira coupling and a higher extent of side reactions compared to the homogeneous +Pd(PPh3)4/CuI protocol. +We tried to understand why TBTEB preferentially polymerizes into multilayer γ-graphyne +flakes, as opposed to amorphous branched structures. As the polymerization appears to be self- +templating, we assumed the existence of an attractive supramolecular interaction between TBTEB +and the lattice of γ-graphyne. Self-assembly through solvophobically driven π-stacking has been +documented for several phenylene ethynylene oligomers and macrocycles structurally related to +graphynes.48 To explore the potential supramolecular interactions in our system, we computed by +DFT the structure and potential energy surface for a single TBTEB molecule bound to a +γ-graphyne monolayer. Our calculations predict that TBTEB would associate with the surface of +graphyne at two types of binding sites (around the aromatic rings and over 12-DBA rings) with a +binding energy in excess of 20 kcal/mol (Fig. S33, SI). The monomer species outcompetes toluene, + + +19 + +whose binding energy we estimate as ~11 kcal/mol. If every TBTEB species were to react while +bound over the underlying layer of γ-graphyne, one of the many local energy minimum stackings +or a mixture of the stackings could result. +Conclusions: To our knowledge, our synthesis of γ-graphyne is the first example of an ordered +covalent lattice formed spontaneously under purely kinetic control. Typical covalent organic +frameworks and metal-organic frameworks are held together through bonds that are reversible. +This reversibility is considered critical for continuous error correction during the +reaction/crystallization process.49 Conventional thinking predicts that irreversible polymerization +of an A3B3-type monomer, not employing a strict geometric constraint on reactivity, must yield +only disordered branched structures. However, since we routinely observed micron-scale +γ-graphyne crystallites, 2D polymerization assisted by crystallization must be presently kinetically +favored over random 3D growth. Furthermore, the initial nucleation of flat graphyne sheets appears +to be a highly probable event. The high fidelity of the resulting lattices indicates that the system is +capable of correcting errors despite the irreversibility of Sonogashira coupling. At a minimum, the +reaction must proceed comparably well at both lattice edges and internal defect sites. Since +“patching” a single internal defect requires forming six new chemical bonds, it is highly likely that +these bond-making steps are kinetically coupled. The apparent capability for error correction, as +well as the strong dependence of the product structure on the nature of the Pd pre-catalyst, strongly +corroborate our original hypothesis of a multi-site coupling mechanism. +Similar cross-coupling methodology could conceivably be applied to the synthesis of other +graphyne-family allotropes, as well as to theoretically proposed heteroatom-doped derivatives.50 +Performing the reaction at interfaces may provide access to extended few-layer or monolayer +sheets rather than microcrystalline powders. These extended sheets could form the basis of the first + + +20 + +γ-graphyne-based devices, especially since we observe a small, direct band gap. Further +exploration of the chemical and physical properties of γ-graphyne is under way in our laboratories. + +ASSOCIATED CONTENT +Materials and methods, experimental procedures, NMR and FT-IR spectra, crystallographic +information for possible stacking modes of γ-graphyne sheets, supplemental figures and +discussion referred to in the text, and computational data (.PDF) +Models of possible stacking modes of γ-graphyne sheets (.CIF) + + +AUTHOR INFORMATION +Corresponding Author +*Correspondence to: Prof. V.O. Rodionov. Email: vor2@case.edu +Author Contributions +‡These authors contributed equally. + +ACKNOWLEDGMENT +We are grateful to Prof. Jessica Bickel for discussions of scanning probe microscopy data. We +thank the US Department of Energy (R01AB123456) and the National Science Foundation (GFRP +Award 1451075 to WBM) for funding. RHB acknowledges support from Robert A. Welch +Foundation (grant AT-0029). DSG and AFF acknowledge support from São Paulo Research +Foundation (FAPESP, awards #2013/08293-7 and #2020/02044-9), National Council for + + +21 + +Scientific and Technological Development (CNPq), and the John David Rogers Computing Center +(CCJDR) at the Institute of Physics “Gleb Wataghin”, University of Campinas. We thank Oxford +Instruments Asylum Research for providing access to Cypher VRS AFM instrument. This work +made use of the High Performance Computing Resource in the Core Facility for Advanced +Research Computing at Case Western Reserve University. Raman spectroscopy was performed at +the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under +award ECCS-2026822. The use of the Stanford Synchrotron Radiation Lightsource, SLAC +National Accelerator Laboratory, was supported by the U.S. Department of Energy, Office of +Science, Office of Basic Energy Sciences, under Contract DE-AC02-76SF00515. + +REFERENCES +(1) Hoffmann, R.; Kabanov, A. 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Chapman,1 Alexandre F. +Fonseca,3 Douglas S. Galvão,3 Ericka Roy Miller,4 Kevin H. Stone,5 Zhong Wang,2 Dante +Zakhidov,6 F. Ted Limpoco,7 Sarah R. Almahdali,1 Shane M. Parker,4 Ray H. Baughman,2 and +Valentin O. Rodionov1* +‡These authors contributed equally. + +1Department of Macromolecular Science and Engineering, Case Western Reserve University; +2100 Adelbert Road, Cleveland, OH 44106, USA. +2Alan G. MacDiarmid NanoTech Institute, University of Texas at Dallas; 800 West Campbell +Road, Richardson, TX 75080, USA. +3Applied Physics Department, Institute of Physics “Gleb Wataghin”, University of Campinas; +Campinas, SP, 13083-970, Brazil. +4Department of Chemistry, Case Western Reserve University; 10900 Euclid Ave, Cleveland, OH +44106, USA. +5Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory; 2575 Sand +Hill Road, Menlo Park, CA 94025, USA. +6Department of Materials Science and Engineering, Stanford University; 496 Lomita Mall, +Stanford, CA 94305, USA. +7Oxford Instruments Asylum Research; 6310 Hollister Ave, Santa Barbara, CA 93117, USA. + +*Correspondence to: Prof. Valentin O. Rodionov. Email: vor2@case.edu + + + + +2 +Materials and Methods +Materials +All reagents and solvents were acquired from commercial suppliers (Acros Organics, Sigma- +Adrich, TCI Chemicals, Fisher Scientific, Oakwood Chemical and VWR International) and used +without further purification, unless otherwise noted. Tetrahydrofuran (THF) was distilled over +Na/benzophenone. Triethylamine (TEA) was distilled over CaH2. Anhydrous pyridine (Py) was +purchased from Acros in AcroSeal packaging and used without further purification. Copper foil, +1.0 mm thick Puratronic 99.999% (metals basis) with 25x50 mm lateral dimensions, was +purchased from Alfa Aesar and cut into 5x10 mm pieces. These pieces of were then sequentially +sonicated for 20 minutes in 3M HCl, water, ethanol, and acetone, dried under high vacuum at +ambient temperature, and immediately used. + +Synthetic Methods +Reactions were monitored by thin-layer chromatography (TLC) carried out on 0.25 mm +MilliporeSigma aluminum-backed silica gel plates (60F-254). Plates were visualized using 254 nm +UV light and basic potassium permanganate stain (1.5 g KMnO4, 0.5 g NaOH, and 10 g K2CO3 in +150 ml water; terminal alkynes stain yellow). Flash chromatography was performed on Luknova +SuperSepTM (230-400 mesh) silica gel. Reactions requiring anhydrous or air-free conditions were +performed under positive pressure of N2 or Ar using standard Schlenk line techniques. + +Nuclear Magnetic Resonance (NMR) Spectrometry +NMR spectra were recorded on a Bruker Avance III HD 500 spectrometer operating at 500.24 +(1H), 125.79 (13C), or 99.37 (29Si) MHz and equipped with Bruker Ascend 500 MHz US Narrow +Bore Magnet and Broadband Prodigy TCI CryoProbe. NMR spectra were referenced to TMS (1H, +13C, 29Si) or residual solvent peaks. Chemical shifts (δ) are reported in parts per million (ppm). + +Gas Chromatography – Mass Spectrometry +GC-MS analyses were performed on an Agilent 5977B GC/MSD instrument equipped with +an Agilent 7890B automatic liquid sampler. Before injection of the sample, the 10 μL syringe was + + +3 +cleaned with acetone and ethyl acetate (3x10 μL each). 1 μL of the sample was then automatically +injected into the instrument. The method used a 3-minute solvent delay. The oven was initially set +at 60 °C and held at this temperature for 2.25 minutes before increasing the temperature to 225 °C +at 35 °C/min rate. Data analysis was performed using Agilent MassHunter Qualitative Analysis +Navigator. + +Infrared Spectroscopy +Routine small molecule FTIR spectra were collected on an Agilent Cary 630 FTIR instrument +equipped with single-reflection germanium or diamond attenuated total reflectance (ATR) +modules. The instrument was calibrated before sampling against a newly cleaned (acetone) and +dried crystal surface. Solid samples were placed directly on the crystal and secured with a needle +press. 32 scans from 4000 to 550 cm-1 were recorded. A background was collected for each sample +(512 scans). +We attempted to obtain FTIR spectra of γ-graphyne using the routine spectrometer described +above. When using the diamond ATR module, we observed primarily artifacts related to the +diamond substrate itself (Fig. S29a), suggesting that the refractive index (RI) of γ-graphyne in +mid-IR is higher than that of diamond (n = 2.4). Some IR bands in the fingerprint region (600- +1800 cm-1) and at ~2100 cm-1 could be observed using a germanium ATR crystal (n = 4.0) (Fig. +S29b). However, this spectrum featured low intensity and unfavorable signal-to-noise ratio +because of the low penetration depth for ATR FTIR on germanium. Likewise, we could not obtain +satisfactory data using the KBr pellet technique due to the low absorbance. +Therefore, for all subsequent experiments we utilized a PerkinElmer Spotlight 200i FTIR +Microscopy System equipped with a Spectrum Two spectrometer capable of both transmittance and +reflectance measurements in the mid-IR to near-IR range (600-7800 cm-1). +The micro-FTIR spectra were collected at ambient conditions in reflection mode using an +8 cm-1 resolution, 50x50 μm2 aperture, and 100 scans. The typical polycrystalline particles chosen +were on average ~300×300 μm and 20 μm thick. Spectra was deconvolved using CasaXPS +software.1, 2 The peaks were modeled using a Gaussian/Lorentzian (SGL(10)) sum formula SGL(p) +where the mixing between the two peak types are determined by m = p/100. SGL(100) is pure +Lorentzian while SGL(0) is entirely Gaussian. A linear background was used. + + + +4 +Raman Spectroscopy +Raman spectra were obtained using a Horiba Labram HR Evolution confocal Raman +microscope with a 405 and 532 nm excitation laser source in reflection geometry. An Edwards T- +Station 85 turbo pump connected to a Janis ST-500 stage was used for vacuum control. Samples +were measured in vacuum at a pressure reading of 1-2×10-5 mbar to control for possible oxidation. +The vacuum monitoring was done with a pressure gauge situated next to the turbo pump. It is most +likely that the pressure in the Janis ST-500 stage was closer to 1×10-3 mbar. A 50x 0.55NA +Mitutoyo long working-distance objective (13 mm) was used to collect Raman spectra from +diffraction-limited spot sizes of ~620 nm (532 nm laser) and ~470 nm (405 nm laser). A low laser +power of less than ~0.3 mW for the 532 nm laser was used for ideal measuring conditions to reduce +beam damage and inhibit decomposition. Powers up to 3.0 mW were used to explore laser light +induced transformations. For the 405 nm laser, laser powers between 0.43 and 4.3 mW were used. +Multiple spots were measured on each sample and no differences were found in the Raman spectra. + +Melting Points +Melting points were determined with a Mettler Toledo MP50 Melting Point System. + +Preparation of Exfoliated Samples +Analyte dispersions in water (1 mg/mL) were prepared by ultrasonication using a Branson +SFX550 Sonifier. A 1/8" double-step microtip (Branson p/n 101-063-212) was used. Samples were +processed for 15 minutes at 50% amplitude. + +Scanning Electron Microscopy (SEM) +The SEM images of Fig. S20a-b and S21 were acquired on an FEI Apreo 2 SEM operating at +5kV and an FEI Inspect F-50 operating at 30kV, respectively. For these images, dried material was +added to carbon tape and mounted on SEM sample stands and sputtered with a thin layer of gold. +The SEM images of γ-graphyne in Fig. 2 (Main Text) and Fig. S20c-d were acquired by a +high resolution Zeiss Supra-40 system, using an in-lens detector and a 10 kV accelerating voltage. + + +5 +The samples for these images were not coated with gold. The small γ-graphyne particles were +grounded to the gold-covered silicon substrate using silver paste. + +Transmission Electron Microscopy (TEM) and Selected Area Electron Diffraction (SAED) +Exfoliated samples were analyzed by TEM. Prior to sample preparation, 200-Cu C-B grids +were plasma-treated for 30 seconds using an Emitech K100x glow discharger. 3 µL of sample +dispersion was added to the grid and allowed to absorb for 5 minutes before the excess solvent +was wicked. The grid was then transferred to a single-tilt sample holder and imaged on an FEI +Tecnai 20 TEM operating at 200 kV in low-dose mode. Images were recorded on a Tvips F416. +Data was collected using SerialEM software. Tilting was performed with the equipped alpha- +rotation goniometer. +SAED patterns were recorded on an FEI Tecnai 20 TEM using a 40 μm selected area aperture +and in the absence of the selected area aperture. The obtained patterns were calibrated against the +(111) planes of evaporated aluminum (plane spacing 0.2338 nm) on a 3 mm grid. The calibration +sample was purchased from Electron Microscopy Sciences (EMS p/n 80044). + +X-ray Photoelectron Spectroscopy (XPS) +Samples were spread onto double-sided copper tape for XPS analysis. Surveys and high- +resolution spectra were acquired on a PHI VersaProbe II Scanning XPS Microprobe using a +monochromatic Al X-ray at pressures of 10-10 to 10-7 Torr. The data was smoothed by using the +Savitzky-Golay method, with a smoothing width of five, and analyzed using CasaXPS software.1, +2 +A Tougaard background3 was applied to each peak before deconvolution. All peak fits used +generalized Voigt-like peak shapes, as this function is most appropriate for fitting asymmetric XPS +signals.4 CasaXPS provides a generalized Voigt function described as Lorentzian Finite: LF(α, β, +w, n, m), where the first three parameters (α, β, w) affect the Lorentzian line shape and its +asymmetry and the final two (n, m) change the width of the Gaussian function and the number of +times convolution with the Lorentzian component occurs.5 Symmetrical peak parameters for the +LF line shape were used: LF(1, 1, 255, 360, 6), values derived from default symmetric peak shape +settings for CasaXPS. All sub-peak widths were constrained to full width at half maximum + + +6 +(FWHM) of 1.6 eV or less. The subpeaks are located at 283.7 eV (terminal alkyne sp1),6 284.6 eV +(aromatic sp2), 285.3 eV (internal alkyne sp1), 286.9 eV (aromatic C-Br), and 288.5 eV (carbonyl +C=O).6, 7 All peaks were allowed a ± 0.2 eV padding to the peak position. + +Synchrotron Powder X-ray Diffraction +Powder diffraction data were collected at the Stanford Synchrotron Radiation Lightsource +(SSRL, SLAC National Accelerator Laboratory) beamline 2-1. Samples were loaded into thin- +walled glass capillaries of 0.5mm nominal diameter. The capillaries were spun during data +collection to improve powder averaging. Measurements were made using 0.7281 Å wavelength +X-rays. Diffracted X-rays were collected using a Pilatus 100K area detector mounted +approximately 700 mm from the sample on the 2-theta arm of the beamline diffractometer. Each +image covers an angular range of approximately 5 degrees. Images were collected 0.0625 degrees +apart, providing substantial overlap between images, and subsequently merged and integrated into +the final powder diffraction profile using custom software available at the beamline. + +Scanning Probe Microscopy +Scanning probe microscopy was performed on an Oxford Instruments Asylum Research +Cypher VRS AFM instrument equipped with an Olympus RC800PSA probe with a nominal spring +constant of 0.39 N/m. Images were taken at 5 nm scan size, 16 Hz line rates, and 256 x 256 pixels. +The analyte powder was suspended in HPLC grade water and cast on freshly cleaved highly +oriented pyrolytic graphite (HOPG). The sample flakes were located optically using the AFM +camera and the tip precisely positioned on the material using the scanner as inertial motors. Lateral +force microscopy (LFM) was the imaging mode used to obtained lattice resolution images. In this +mode, the tip is rastered orthogonal to the long axis of the cantilever. The torsional moment of the +cantilever due to stick-slip friction was registered as the lateral signal in the photodetector, which +is sensitive to the lattice corrugation of the surface. + + + + +7 +Computation and Modeling +Density Functional Theory (DFT) Calculations +All DFT calculations were performed using 2D periodic boundary conditions via the RIPER +module of TURBOMOLE/7.5.8-12 In all cases the PBE density functional13 with D3 dispersion +corrections14 and Becke-Johnson damping15 was used. All the calculated geometries were in broad +agreement with prior computational studies of γ-graphyne.16-18 +Potential energy surfaces of both a γ-graphyne bilayer and individual molecules/graphyne +supercells were computed as a function of the horizontal offset of the upper γ-graphyne layer or +molecule relative to the lower layer with a fixed interlayer distance of 3.35 Å. Binding energies +for each structure were calculated as adsorption energies (������������������������������������������������������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������������������������������������������������������������������������������). A +9x9 k-point grid was used for the γ-graphyne bilayer structures. The monomer/graphyne supercells +were built using a 3x3 γ-graphyne monolayer to ensure 1 nm spacing between adjacent periodic +images of monomers. Due to the resulting repetitiveness of these supercells, a coarser 3x3 k-point +grid was used throughout the potential energy surface scan calculations. The def2-SVP basis set 19 +was used throughout. +Local bilayer minima identified from the potential energy surfaces were refined by geometry +optimization with the def2-TZVP basis set20 and a 9x9 k-point grid. Similarly, selected binding +site supercell structures were optimized with def2-SVP and a 9x9 k-point grid. Final binding +energies for all structures were calculated with def2-TZVP as well, along with a finer 17x17 k- +point grid. + +Molecular Dynamics (MD) Simulations +The energy and dynamics of a few different stackings of γ-graphynes were investigated using +classical MD simulations. LAMMPS package21 was used with the ReaxFF reactive force field.22, +23 ReaxFF is a state-of-the-art potential previously applied to simulations of structural, mechanical, +and thermal properties of carbon nanostructures,24-28 including graphynes.29-32 +The following γ-graphyne stackings were used as initial structures: AA, AB1 and AB2, and +ABC (Fig. S35). AB1 and AB2 were previously proposed by Yun.33 AB1 and ABC structures +were investigated by Ducéré and Chauvin34 as the most probable stackings of γ-graphyne. +However, as we will see below, neither corresponds to the lowest energy stacking mode. + + +8 +All the structures were prepared with 3 and/or 6 layers and were first energy-minimized and +then either equilibrated at 300 K or quenched from 1000 to 1 K. The energy minimizations were +performed with periodic boundary conditions (PBC) imposed along all directions in space, +including the possibility to relax the size along the PBC directions. Combination of energy +minimization with free evolution algorithms as suggested by Sihn35 was used to ensure the lowest- +energy structure is obtained. This protocol was recently used by Kanegae and Fonseca36 to study +elastic properties of graphynes. Thermal equilibration of the structures was performed with PBC +by applying a Langevin thermostat to all atoms with damping factor of 1.0 fs and timestep of 0.025 +fs, for a total period of 500 ps (or 20 × 106 timesteps). Quenching of the structures was simulated +at the same conditions as thermal equilibration except that a Boltzmann distribution of velocities +corresponding to the initial temperature of 1000 K was initially attributed to the system, then the +temperature was allowed to decrease from 1000 to 1 K over a period of 500 ps. + +SAED and PXRD Simulations +The lattice parameters and bond lengths were obtained from DFT calculations (vide supra) +and a previously published computational study.16 SAED and PXRD simulations were performed +using the CrystalMaker software suite.37 A model of a single γ-graphyne sheet was built in +CrystalMaker using a hexagonal P6 lattice with parameters a and c set to 6.86 Å and 3.4 Å, +respectively. The asymmetric unit comprised four atoms placed at 0.208, 0.412, 0.589, 0.795 along +the hexagonal P6 x axis. The basic models corresponding to various sheet stacking modes were +constructed using Vesta.38 Models shown in Fig S22c and S22e-h were generated from DFT +simulations and were passed through Avogadro/spglib39 in an attempt to identify the space group +associated with each model. + + + + +9 +Small Molecule Synthesis +1,3,5-tribromo-2,4,6-triiodobenzene +Br +Br +Br +Br +Br +Br +I +I +I +KI, H5IO6 +H2SO4, rt, 72 h + +The synthesis procedure for 1,3,5-tribromo-2,4,6-triiodobenzene was adapted from the +literature.40 To concentrated H2SO4 (500 mL) at room temperature was added periodic acid (41.03 +g, 180 mmol) in small portions over 15 min. After dissolution of the periodic acid, crushed KI +(89.64 g, 540 mmol) was added in small portions at 0 °C over 1 h. To the resulting deep purple +solution at 0 °C was added 1,3,5-tribromobenzene (18.89 g, 60.0 mmol) in small portions over 25 +min. After the solution was stirred at room temperature for 72 h, the resulting thick mixture was +poured onto ice. The resulting precipitate was filtered and washed with H2O (5 × 200 mL) and +then MeOH (5 × 200 mL). The product was recrystallized twice from pyridine/EtOH 1:4 (1000 +mL) to yield a solid. The solid was dried at under high vacuum for 1 day to give 1,3,5-tribromo- +2,4,6-triiodobenzene 2 (28 g, 67%) as a pale-yellow solid. Mp > 300 °C (decomposition); FTIR +(neat) νmax = 1488, 1354, 1262, 1227, 1147, 1002, 858, 771, 739, 554, 508 cm-1. 13C NMR (126 +MHz, DMSO-d6) δ 138.61 (CBr), 108.23 (CI). EI-MS fragmentation: m/z 695.5, 693.5, 691.5, +689.5, 567.6, 566.6, 565.6, 564.6, 439.7, 437.6. UV/vis (CHCl3, C =6.874 × 10-5 M): λmax (ɛ) = +227 (5000), 248 (27700), 283 (5600 M–1cm–1). + +((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) +Br +Br +Br +I +I +I +Br +Br +Br +Si +Si +Si +Si +PdCl2(PPh3)2, CuI, PPh3 + +Et3N, THF, 80°C, 48 h + +1,3,5-tribromo-2,4,6-triiodobenzene 2 (346 mg, 0.5 mmol), [PdCl2(PPh3)2] (105 mg, 0.15 +mmol, 30 mol%), CuI (19 mg, 0.1 mmol, 20 mol%), Et3N (50 mL) and THF (40 mL) were added +to a dry three-necked flask. Ethynyltrimethylsilane (736.7 mg, 1.07 mL, 7.5 mmol) and Ph3P (52 +mg, 0.2 mmol, 40 mol%) were added to the mixture. The mixture was stirred at 80 °C for 48 h + + +10 +under argon. After the removal of solvent on a rotary evaporator, DCM (100 mL) was added to +the residue and filtered through Celite. The mixture was washed with water (20 mL) and NaCl(aq) +(20 mL), dried over anhydrous Na2SO4, and the solvent was removed under reduced pressure. The +residue was further purified by flash chromatography using n-hexane as the eluent to yield ((2,4,6- +tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) 3 as a white solid (175 mg, +0.29 mmol, yield: 58%). Rf (hexane) = 0.3. Mp = 110-111 °C; FTIR (neat) νmax = 2958, 2160, +1376, 1340, 1245, 1019, 834, 758, 708, 658, 633, 539 cm-1. 1H NMR (500 MHz, CDCl3): δ = 0.29 +ppm [s, 27H, Si(CH3)3]. 13C NMR (126 MHz, CDCl3): d 129.09 (CBr), 127.49 (C6C≡C), 106.79 +(C≡CSi), 101.83 (C6C≡C), –0.23 [Si(CH3)3] ppm. 29Si NMR (99 MHz, CDCl3) δ -15.87 ppm. EI- +MS fragmentation: m/z 603.9, 602, 601.9, 590.9, 589.9. 588.9, 588.9, 587.9, 586.9, 584.9. UV/vis +(CHCl3, C = 5.436 × 10-5 M): λmax (ɛ) = 260 (45700), 271 (44600), 289 (39600 M–1cm–1). + +1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB +Br +Br +Br +Si +Si +Si +Br +Br +Br +0°C, 15 min +TBAF, THF + +To a solution of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) +(151 mg, 0.25 mmol) in THF (15 mL) was added 0.55 mL TBAF (75% solution in water, 1.5 +mmol) and stirred at 0 °C for 15 min. The solution was then diluted with ethyl acetate and washed +with distilled water and dried with anhydrous Na2SO4. The solvent was removed on a rotary +evaporator. The residue was further purified by flash chromatography using n-hexane as the eluent +to give TBTEB as a white solid (84 mg, 0.216 mmol, yield: 87%). Rf (hexane) = 0.2. FTIR (neat) +νmax = 3275, 2922, 2112, 1519, 1368, 1336, 965, 736, 681, 634 cm-1. 1H NMR (500 MHz, CDCl3): +δ = 5.16 (s, 3H, (C≡CH) ppm. 13C NMR (126 MHz, CDCl3): δ 129.90 (CBr), 126.33 (C6C≡CH), +91.87 (C≡CH), 80.97 (C6C≡CH) ppm. EI-MS fragmentation: m/z 390.8, 389.9, 388.8, 387.8, +386.8, 385,8, 384,8, 383.8. UV/vis (CHCl3, C = 4.136 × 10-5 M): λmax (ɛ) = 212 (7100), 218 (8300), +260 (58100), 278 nm (33500 M–1cm–1). + + + + +11 +Synthesis of Carbon Materials +Table S1. Representative Polymerization Reactions of TBTEB.a +Entry +[Pd] / mol% +[Cu] / +mol% +Additive / +mol% +Solvent / Base +Temp, °C +Time, h +Product +1 +Pd(PPh3)4 / 100% +CuI / +8% +- +Pyridine +110 +72 +high +crystallinity +2 +Pd(PPh3)4 / 100% +Cu foil1 +- +Pyridine +110 +48 +low +crystallinity +3 +Pd(PPh3)4 / 100% +- +- +Pyridine +110 +72 +amorphous +4 +- +- +- +Pyridine +110 +72 +amorphous +5 +PdCl2(PPh3)2 / +30% +Cu/Si +Ph3P / 40% +THF/ Et3N +(44:56) +80 +48 +amorphous +6 +- +Cu/Si +- +Pyridine +110 +72 +amorphous +7 +- +Cu wire +- +Pyridine +110 +72 +amorphous +8 +Pd(PPh3)4 / 100% +Cu wire +- +Pyridine +110 +72 +low +crystallinity +9 +- +Cu +wire1 +- +Pyridine +100 +72 +amorphous +10 +Pd(PPh3)4 / 40% +- +- +Pyridine +100 +48 +amorphous +11 +Pd(PPh3)4 / 100% +Cu/Si +- +Pyridine +110 +72 +amorphous +12 +- +Cu foil1 +- +Pyridine +110 +72 +amorphous +13 +PdCl2(PPh3)2 / +60% +Cu/Si +Ph3P / 80% +THF/ Et3N (1:1) +80 +72 +amorphous +14 +- +Cu foil +- +Pyridine +70 +72 +No Material +1Copper foil was treated by sonicating in 3M HCl, water, ethanol, and acetone, sequentially, +for 20 minutes, dried under vacuum at rt and used immediately. +a All the reactions were performed under positive pressure of Ar (Schlenk line with a Hg +bubbler). + + + + +12 +General Synthetic Procedure for Carbon Materials +In a typical procedure, TBTEB, Pd(PPh3)4, and Cu were charged to a Schlenk tube under argon +atmosphere and solvent was added. The tube was sealed, and the contents degassed by three freeze- +pump-thaw cycles. The reaction mixture was stirred under argon atmosphere and heated for 72 +hours. The reaction mixture was concentrated on a rotary evaporator. The solid product was +washed with methanol, ethanol, isopropanol, toluene, hexanes, ethyl acetate and acetone. The +washing procedure involved dispersing the material in the corresponding solvent by gentle +sonication, followed by centrifugation. Conditions for selected experiments from Table S1 are +detailed below. +Table S1, Entry 1 +TBTEB (116 mg, 0.3 mmol), Pd(PPh3)4 (347 mg, 0.3 mmol) and CuI (4.6 mg, 0.024 mmol) +reacted in anhydrous pyridine (50 mL) using the general procedure. +Typical mass of the crude product after centrifugation and drying on low vacuum (1-2 Torr) +over 10 hours is ~90% of the monomer mass (104 mg for the scale above). After extensive drying +at high vacuum (10 mTorr) and/or heating to 100°C for 72 hours the mass decreased to ~60% of +the original monomer mass (68 mg for the scale above). TLC indicated monomer conversion is +quantitative. +Table S1, Entry 2 +TBTEB (116 mg, 0.3 mmol), Pd(PPh3)4 (347 mg, 0.3 mmol) and several pieces of copper foil +reacted in a mixture of anhydrous pyridine (50 mL) using the general procedure. +Table S1, Entry 3 +TBTEB (116 mg, 0.3 mmol) and Pd(PPh3)4 (347 mg, 0.3 mmol) reacted in pyridine (50 mL) +using the general procedure. No Cu was used. +Table S1, Entry 4 +TBTEB (116 mg, 0.3 mmol) refluxed in pyridine (50 mL). Neither Pd(PPh3)4 nor copper was +used. + + + + +13 +Spectroscopic and Imaging Data and Supplementary Discussion +-10 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 +130 +140 +150 +160 +170 +180 +190 +200 +210 +f1 (ppm) +-2E+07 +0 +2E+07 +4E+07 +6E+07 +8E+07 +1E+08 +1E+08 +1E+08 +2E+08 +2E+08 +2E+08 +2E+08 +2E+08 +3E+08 +3E+08 +3E+08 +3E+08 +39.02 +39.19 +39.35 +39.52 +39.69 +39.85 +40.02 +108.23 +138.61 +Br +Br +Br +I +I +I +a +b +a +b + +Fig. S1. 13C NMR spectrum of 1,3,5-tribromo-2,4,6-triiodobenzene in DMSO-d6. + + + + +14 +Br +Br +Br +Si +Si +Si +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.0 +f1 (ppm) +0 +5E+07 +1E+08 +2E+08 +2E+08 +2E+08 +3E+08 +4E+08 +4E+08 +0.29 +7.26 + +Fig. S2. 1H NMR spectrum of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris- +(trimethylsilane) in CDCl3. + + + + +15 + +Fig. S3. 13C NMR spectrum of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris- +(trimethylsilane) in CDCl3. + + + +-0.23 +-4E+08 +e +-4E+08 +Br +Br +-3E+08 +a +Br +d +-2E+08 +-2E+08 +-2E+08 +-1E+08 +a +C +b +-5E+07 +210 +200 +190 +180 +170 +160 +150 +140 +130 +120 +110 +100 +90 +80 +70 +60 + 50 +40 +30 +20 +10 +0 +-10 +f1 (ppm) +16 +Br +Br +Br +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.0 +f1 (ppm) +0.0 +5.0×107 +1.0×108 +1.5×108 +2.0×108 +2.5×108 +3.0×108 +3.5×108 +4.0×108 +4.5×108 +5.0×108 +2.50 DMSO-d6 +3.34 +5.15 + +Fig. S4. 1H NMR spectrum of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB in DMSO-d6. + + + + +17 +Br +Br +Br +-10 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 +130 +140 +150 +160 +170 +180 +190 +200 +210 +f1 (ppm) +-5.0×107 +0.0 +5.0×107 +1.0×108 +1.5×108 +2.0×108 +2.5×108 +3.0×108 +3.5×108 +4.0×108 +4.5×108 +5.0×108 +5.5×108 +6.0×108 +39.19 +39.35 +39.52 +39.52 DMSO-d6 +39.69 +39.85 +80.97 +91.85 +126.33 +129.89 +a +b +c +d +a +b +c +d + +Fig. S5. 13C NMR spectrum of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB in DMSO-d6. + + + + +18 + +Fig. S6. FTIR of 1,3,5-tribromo-2,4,6-triiodobenzene (ATR-FTIR on diamond). + + + +70 +60 - +w +50 - +(%) +40 - +30 - +20 - +Br +10 - +Br +Br +0 +4000 +3500 +3000 +2500 +2000 +1500 +1000 +500 +Wavenumber (cm-1) +19 + +Fig. S7. FTIR of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) +(ATR-FTIR on germanium). + + + +100 +90 +80 - +2901 V.(CH3) +70 - +2160 v(C三C) +2958 Vas(CH3) +-09 +50 - +T +1019 Si-Ar +40 +1340 (CH3) in TMS +Br +30 +633 v(C-Br) +1245 Si-Ar +20 - +Br +Br +10- +-Si +834 v(Si-CH3) +0 +4000 +3500 +3000 +2500 +2000 +1500 +1000 +500 +Wavenumber (cm-1) +20 + +Fig. S8. FTIR of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB (ATR-FTIR on germanium). + + + +90 +80 +70- +2113.4 +-09 +1C"C1: +(%) +50 - +T +40 - +3281.9 +C=CIH +30 - +Br +1338.1 +20 - +10 - +Br +Br +C +631.7 +Br +0 +4000 +3500 +3000 +2500 +2000 +1500 +1000 +500 +Wavenumber (cm-1) +21 + +Fig. S9. XPS survey corresponding to Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine). + + + +c +C 83.51% +0 4.62% +2.0 1 +Br 11.87% +Intensity (a.u.) +1.5 +0 +Br +1.0 +Br +Br +0.5 +0.0 +1000 +750 +500 +250 +0 +Binding Energy (eV) +22 + +Fig. S10. XPS survey corresponding to Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in +pyridine). + + + +c +2.0 C 84.49% +0 5.73% +Br 5.02% +1.5 +P 4.05% +Intensity (a.u.) +Pd 0.7% +1.0 +0 +Pd +0.5 +Br +Br +p +Br +0.0 +1000 +750 +500 +250 +0 +Binding Energy (eV) +23 + +Fig. S11. XPS survey corresponding to Table S1, Entry 3 (TBTEB and Pd(PPh3)4 in pyridine, no +Cu). + + + +c +C 82.80% +0 7.31% +2.0 JBr 4.79% +P 4.59% +Intensity (a.u.) +1.5 - +IPd 0.5% +0 +Pd +1.0 +0.5 +Br +Br +Br +0.0 +1000 +750 +500 +250 +0 +Binding Energy (eV) +24 + +Fig. S12. XPS survey corresponding to Table S1, Entry 4 (thermal decomposition of TBTEB in +refluxing pyridine). + +C +C 79.41% +N 5.38% +1.5 - +0 3.41% +Br 11.80% +Intensity (a.u.) +1.0 +N +0 +Br +Br +M +Br +0.5 +0.0 +1000 +750 +500 +250 +0 +Binding Energy (eV) +25 + +Fig. S13. High resolution XPS data for C1s peak regions. a, Table S1, Entry 1 (TBTEB and +Pd(PPh3)4/CuI in pyridine). b, Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine). c, +Table S1, Entry 3 (control experiment with TBTEB and Pd(PPh3)4 in pyridine, no Cu). d, Table +S1, Entry 4 (control experiment with no catalysts). + + +a +4 +d +Residuals +.5 +Fit +Intensity (a.u.) +C-H(sp) +Intensity (a.u.) +Intensity (a.u.) +3 +Intensity (a.u.) +4. +.2 +C-C (sp) +? +C-C (sp2) +3 +.2 +C-Br (sp2) +.2 +C=O +Tt-nt* +1 +1 +1 +0 +0 +0 +0 +280 +285 +290 +280 +284 +288 +292 +280 +284 +288 +292 +280 +284 +288 +292 +Binding Energy (eV) +Binding Energy (eV) +Binding Energy (eV) +Binding Energy (eV) +26 + +Fig. S14. High resolution Br 3d region XPS spectra of selected carbon material samples. a, Table +S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine). b, Table S1, Entry 2 (TBTEB and +Pd(PPh3)4/Cu foil in pyridine). c, Table S1, Entry 3 (control experiment with TBTEB and +Pd(PPh3)4 in pyridine, no Cu). d, Table S1, Entry 4 (control experiment with no catalysts). + + + +a +b +.5 +5 +d +.5 +Residuals +Fit +Intensity (a.u.) +.4 +Intensity (a.u.) +4 +Intensity (a.u.) +.4 +Intensity (a.u.) +2 +C-Br +3 +3 +3 +Br +2 +2 +2 +.1 +1. +.1 +1. +0 +0 +0 +0 +65 +70 +75 +65 +70 +75 +65 +70 +75 +65 +70 +75 +Binding Energy (eV) +Binding Energy (eV) +Binding Energy (eV) +Binding Energy (eV) +27 +High Resolution Br 3d XPS +High-resolution Br 3d XPS spectra of the carbon materials from Table S1, Entries 1-4 are +presented in Fig. S14. The Br signal manifests as a pair of peaks due to spin-orbit coupling, +yielding a Br3d3/2 peak at a higher binding energy and Br3d5/2 peak at a lower value. These coupled +peaks are well-defined in XPS, with a separation of 1.04 eV and intensity ratio of 0.671. +Prior to the reaction, bromine species constitute ~20% atomic percent (at%) of the monomer. +A decrease in this fraction is observed in all cases after the reaction. The residual Br 3d peak can +be deconvolved to two distinctive species: Br covalently bonded to aromatic carbon (71.4, 70.5 +eV),7, 41, 42 corresponding to partially unreacted sites, and weakly coordinated/anionic Br (67.5- +69.5 eV)42-46 trapped within the carbon matrix or on the edges or surfaces of γ-graphyne sheets. +After the reaction catalyzed by Pd(PPh3)4/CuI in pyridine (Table S1, Entry 1) XPS survey +indicates an apparent 12 at% residual Br (Fig. S9), consisting almost entirely of covalent C(Ar)- +Br (Fig. S14a). The structure of the fit includes contributions from at least two C(Ar)-Br species, +presumably mono- and di-substituted. As this material consists of micron-scale crystallites (Fig. +S16a-d), and the surface penetration depth for XPS in carbon materials is 10 nm or less, we +attribute this C(Ar)-Br signal to the unreacted edge groups. Because of the composition difference +between edge and bulk of the material, a survey XPS measurement likely results in a significant +overestimation of the Br content. It is important to note that the number of terminal alkyne C-H +groups estimated from fitting of the C1s XPS signal (Fig. 2d, Main Text and S13a) correlates well +with the apparent C(Ar)-Br content, as would be expected for complementary edge groups. +For the rest of the reactions surveyed (Table S1, Entries 2-4) the carbon products are +significantly contaminated with the weakly bound Br species (Fig. S14b-d). As these products are +significantly less crystalline than the product of the optimized Pd(PPh3)4/CuI protocol, small- +molecule impurities get trapped in disordered carbon matrices. This is also evidenced through +observation of Pd, P, and N contaminants derived from solvents and/or catalysts in the +corresponding survey XPS spectra (Figs. S9-S12). While the small molecule contamination +precludes quantitative interpretation of the results, important chemical insights can be obtained by +examining the speciation of Br in the products. It can be clearly seen that thermal decomposition +of TBTEB is accompanied by Br loss, likely through spontaneous hydrodebromination (Fig S14d). +Pd(PPh3)4 alone is capable of activating the C(Ar)-Br sites, accelerating the rate of +hydrodebromination (compare Fig. S14c and S14d). + + + +28 +Fig. S15. a, Photo of the carbon material corresponding to Table S1, Entry 1 (TBTEB and +Pd(PPh3)4/CuI in pyridine). b, Photo of the carbon material corresponding to Table S1, Entry 2 +(TBTEB and Pd(PPh3)4/Cu foil in refluxing pyridine). + + + +b +29 + +Fig. S16. a-d, Representative bright field TEM images of the carbon product corresponding to +Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine). + + + +a +b +100nm +100nm +C +d +100nm +100nm +30 + +Fig. S17. a-f, Representative bright field TEM images of the carbon product corresponding to +Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine). (e-f) Representative SAED ring +patterns for the same material. + +b +2 nm-1 +2 nm +31 + +Fig. S18. a-c, Representative bright field TEM images of the carbon product corresponding to +Table S1, Entry 3 (control experiment with TBTEB and Pd(PPh3)4 in pyridine, no Cu). + + + +a +b +100nm +100nm +100nm +32 + +Fig. S19. a-c, Representative bright field TEM images of the carbon product corresponding to +Table S1, Entry 4 (thermal decomposition of TBTEB in refluxing pyridine). + + + +b +100nm +100nm +33 + +Fig. S20. a-d, Representative SEM images of the carbon product corresponding to Table S1, Entry +1 (TBTEB and Pd(PPh3)4/CuI in pyridine). + + + +a +b +10μm +10 μm +C +d +10 μm +um +34 + +Fig. S21. a-b, Representative SEM images of the carbon product corresponding to Table S1, Entry +2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine). + + + +a +b +10 μm +10 μm +35 + +Fig. S22. Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED +patterns in the c orientation (sheets perpendicular to the incident beam, a-h, bottom). Interatomic +distances are obtained from DFT calculations. + + +a +AA +b +AB +c +AB +d +ABC +P6.mc +P6 +Cmcm +R3m +3 +2 nm +2 nml +2 nmr +2 nm +ABC +ABC +ABC +ABC +h +Q +P1 +pi +P3,12 +(P1 +2 nm +2 nmrl +2 nml +2 nml +36 + +Fig. S23. Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED +patterns for 45° rotation from the (0001) pole to a viewing direction (matching the experimental +diffraction pattern rotated 45°, a-h, bottom). Interatomic distances are obtained from DFT +calculations. + + + +a +AA +b +AB +c +AB +d +ABC +2 nml +2 nml +2 nm +2 nm +ABC +f +ABC +e +g +ABC +h +ABC +2 nml +2 nml +2 nm +2 nm +37 + +Fig. S24. Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED +patterns in the b orientation (sheets parallel to the incident beam, a-h, bottom). Interatomic +distances are obtained from DFT calculations. + + + +a +AA +b +AB +c +AB +d +ABC +2 nm +2 nm +2 nmr +2 nmrl +e +ABC +f +ABC +g +ABC +h +ABC +2 nml +2 nml +2 nmt +2 nml +38 +Table S2. Crystallographic information for the structure in Fig 22-24a. +AA Stacking +File: AA.cif +Hexagonal +P 6 +a = b = 6.8826 Å c = 3.4 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.205 +0 +0 +C2 +0.412 +0 +0 + + + + + +39 +Table S3. Crystallographic information for the structure in Fig 22-24b. +AB Stacking +File: AB_1_P_63_mmc.cif +Hexagonal +P 63 m c +a = b = 6.8826 Å c = 6.8 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.12833 +0.66667 0 +C2 +0.9213 +0.66667 0 + + + + + +40 +Table S4. Crystallographic information for structure in Fig 22-24c. +AB Stacking +File: AB_2_C_mcm.cif +Orthorhombic +C m c m +a = 11.86455 b = 6.8826 Å c = 6.8 Å +α = β = γ = 90° +Atom +X +Y +Z +C1 +0.60351 0.27153 +0.25 +C2 +0.7056 0.16932 +0.25 +C3 +0 0.08195 +0.25 +C4 +0 0.28643 +0.25 +C5 +0.39653 0.47845 +0.25 +C6 +0.79423 0.08036 +0.25 +C7 +0 0.66825 +0.25 +C8 +0 0.46396 +0.25 + + + + + +41 +Table S5. Crystallographic information for the structure in Fig 22-24d. +ABC Stacking +File: ABC_1_R_3m.cif +Trigonal +R 3 m +a = 6.8826 Å b = 6.8826 Å c = 10.2 +Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.205 +0 +0 +C2 +0.412 +0 +0 + + + + + + +42 +Table S6. Crystallographic information for the structure in Fig 22-24e. +ABC Stacking +File: ABC_2_P_31_1_2.cif +Trigonal +P31 1 2 +a = 6.8826 Å b = 6.8826 Å c = 10.2 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +-0.0461 0.25305 0.33416 +C2 +0.15811 0.25303 0.33675 +C3 +0.74697 0.45999 0.33221 +C4 +0.74719 0.66451 0.33033 +C5 +0.54 0.04594 0.33418 +C6 +0.33594 0.84148 0.33416 + + + + + + +43 +Table S7. Crystallographic information for the structure in Fig 22-24f. +ABC Stacking +File: ABC_3_P_1.cif +Triclinic +P 1 +a = 6.8826 Å b = 6.8826 Å c = 10.2 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.2082 0.00369 0.03346 +C2 +0.41232 0.00348 0.03615 +C3 +0.00125 0.21104 0.02974 +C4 +0.00128 0.41553 0.02911 +C5 +0.7941 +0.7971 0.03339 +C6 +0.58989 0.59289 +0.036 +C7 +0.79437 0.00396 0.03178 +C8 +0.58975 0.00353 0.03455 +C9 +0.00093 0.79713 0.03197 +C10 +0.00084 0.59282 0.03011 +C11 +0.20831 0.21084 0.03177 +C12 +0.41291 0.41502 0.03447 +C13 +0.20746 0.25244 0.36408 +C14 +0.41183 0.25267 +0.3614 +C15 +0.00061 0.45964 0.36408 +C16 +0.00071 0.66389 0.36511 +C17 +0.79344 0.04594 0.36413 +C18 +0.58888 0.84196 0.36166 +C19 +0.79355 0.25266 0.36279 +C20 +0.58943 0.25262 0.36005 +C21 +0.00029 0.04573 0.36571 +C22 +0.00025 0.84136 +0.3658 +C23 +0.2077 0.45976 0.36282 +C24 +0.41172 0.66394 0.36035 +C25 +0.5396 +-0.0693 +0.6984 +C26 +0.74397 +-0.0694 0.69948 +C27 +0.33301 0.13814 +0.6988 +C28 +0.33305 0.34252 0.69902 +C29 +0.12586 +0.7244 0.69882 +C30 +-0.0786 0.52011 0.69897 +C31 +0.12615 +-0.0688 0.69937 +C32 +-0.0782 +-0.0686 0.70003 +C33 +0.33238 0.72407 0.69829 +C34 +0.33218 0.51958 0.69867 +C35 +0.53995 0.13784 0.69835 +C36 +0.74438 0.34213 0.69858 + + + + + +45 +Table S8. Crystallographic information for the structure in Fig 22-24g. +ABC Stacking +File: ABC_4_P_1.cif +Triclinic +P 1 +a = 6.8826 Å b = 6.8826 Å c = 10.2 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.20667 0.00525 0.03487 +C2 +0.99656 0.04592 0.37042 +C3 +0.99663 0.84175 0.37282 +C4 +0.20381 0.46018 0.36719 +C5 +0.40783 0.66417 0.36718 +C6 +0.96727 0.26154 0.70381 +C7 +0.1718 0.26159 0.70446 +C8 +0.5854 0.25291 0.36175 +C9 +0.76012 0.46863 0.69988 +C10 +0.55331 0.05453 0.69973 +C11 +0.34904 0.85037 +0.697 +C12 +0.55351 0.26145 0.70105 +C13 +0.34928 0.26152 0.70302 +C14 +0.76017 +0.0547 0.70185 +C15 +0.75975 0.85009 0.69941 +C16 +0.75989 0.67277 0.69767 +C17 +0.78956 0.25282 0.36422 +C18 +0.58537 0.84163 0.36731 +C19 +0.78967 0.04595 0.36734 +C20 +0.41085 0.00506 0.03731 +C21 +0.99956 0.21238 0.03077 +C22 +0.99958 0.41685 0.02999 +C23 +0.79264 0.79833 0.03505 +C24 +0.58845 0.59413 0.03776 +C25 +0.79274 0.00519 0.03296 +C26 +0.58814 0.00481 0.03561 +C27 +0.99955 0.79858 0.03354 +C28 +0.99956 0.59434 0.03143 +C29 +0.20664 0.21233 0.03304 +C30 +0.41119 0.41657 0.03596 +C31 +0.20355 0.25287 0.36769 +C32 +0.40794 0.25318 0.36406 +C33 +0.99653 0.45992 0.36686 +C34 +0.99683 0.66426 0.37044 +C35 +0.96722 0.46863 0.70158 +C36 +0.17143 +0.6732 0.69867 + + + + + +46 +Table S9. Crystallographic information for the structure in Fig 22-24h. +ABC Stacking +File: ABC_5_P_1.cif +Triclinic +P 1 +a = 6.8826 Å b = 6.8826 Å c = 10.2 Å +α = β = 90°, γ = 120° +Atom +X +Y +Z +C1 +0.20729 0.00285 0.03076 +C2 +0.4116 0.00274 0.03013 +C3 +0.0004 0.21004 0.03198 +C4 +0.00035 0.41442 0.03159 +C5 +0.79356 0.79639 0.03191 +C6 +0.58913 0.59204 0.03168 +C7 +0.7936 0.00297 0.03192 +C8 +0.58918 0.00287 +0.0308 +C9 +0.0003 0.79627 0.03138 +C10 +0.00036 0.59192 0.03118 +C11 +0.20737 +0.21 0.03135 +C12 +0.41183 0.41439 0.03123 +C13 +0.53792 +0.6814 0.36681 +C14 +0.74207 0.68142 0.36949 +C15 +0.33055 0.88835 0.36378 +C16 +0.33035 0.09262 +0.3637 +C17 +0.12385 0.47443 0.36693 +C18 +-0.0802 0.27021 0.36939 +C19 +0.1238 0.68116 0.36551 +C20 +-0.0807 +0.6808 0.36811 +C21 +0.33076 0.47473 0.36565 +C22 +0.33079 0.27053 0.36458 +C23 +0.53784 0.88843 0.36541 +C24 +0.74236 0.09251 0.36793 +C25 +0.53617 +-0.0708 0.69795 +C26 +0.74074 +-0.0704 0.69524 +C27 +0.32902 0.13614 0.69763 +C28 +0.32888 0.34036 0.69964 +C29 +0.1223 0.72255 0.69796 +C30 +-0.0823 0.51833 0.69522 +C31 +0.12216 +-0.0709 0.69619 +C32 +-0.082 +-0.0708 0.69357 +C33 +0.32914 +0.7225 0.70012 +C34 +0.3292 0.51803 0.70085 +C35 +0.53628 0.13642 0.69616 +C36 +0.74048 0.34073 0.69353 + + +47 + +Fig. S25. LFM lattice images of the carbon product corresponding to Table S1, Entry 1 +(γ-graphyne) and reference HOPG substrate. a, Raw LFM image of a 10x10 nm flat area of a +γ-graphyne flake (Fig. 4a, Main Text). b, Low-pass filtered version of the LFM image in (a). c, +Raw LFM image of a 10x10 nm area of the reference HOPG substrate obtained several nm from +the edge of the sample flake. d, Low-pass filtered version of the LFM image in (c). + + + +a +2.5 +Y-graphyne +Lateral Response (mV) +1.25 +0 +-1.25 +-2.5 +1.0 +Lateral Response (mV) +0.5 +HOPG +0 +-0.5 +-1.0 +48 + +Fig. S26. HOPG substrate measurements used for the calibration of the LFM images. a, Lateral +deflection map of the HOPG surface, raw image. b, Plot of lateral response along the linear trace +from (a). c, FFT of the lateral response map from a. d, Low-pass filtered lateral deflection map +from (a). + + + +b +uV +500 μV +750 +Lateral Signal +0 +-500 +0 +2 nm +c +0.00 +750 +49 + +Fig. S27. Analysis of periodicity through FFT of scanning microscopy images. a, SAED of the +product corresponding to Table S1, Entry 1 (γ-graphyne). Reflections corresponding to the (2110) +crystallographic planes are highlighted. b, FFT of the AFM height image of the same material +(Fig. 4a, Main Text). The bright-spot pattern is a perfect match for the reflections highlighted in +a. c, FFT of the LFM image of HOPG substrate (Fig. S25b). The bright-spot pattern is consistent +with the expected periodicity for a graphene sheet (distance of ~2.1 Å between (1010) +crystallographic planes). The periodicity of HOPG standard is clearly different and from that of γ- +graphyne. + + + +a +b +C +2110 +2 nm-1 +2 nm-1 +2 nm-1 +50 + +Fig. S28. Multi-power and multi-wavelength Raman spectroscopy of γ-graphyne and structural +transformations under laser light. a, Spectra obtained over 10s (#1, black) and 500s (#2, red) +intervals with 532 nm excitation (0.3 mW incident power). The spectra were collected sequentially +(10s, then 500s) from the same sample region. b, Spectra of the sample region in a obtained over +10s at 3.0 mW (magenta, #3) and 0.3 mW (green, #4) incident power. The spectra were obtained +sequentially (0.3 mW, then 3.0 mW). The experiment was performed after the low-power +experiments in a. c, Sequentially obtained spectra of γ-graphyne with 532 nm excitation (10s, 0.3 +mW incident power, yellow, #1), 405 nm excitation (10s, 4.3 mW incident power, black, #2), and +532 nm excitation (10s, 0.3 mW incident power, red, #3). The spectra were acquired from the same +sample region. + + + +a +#1:532nm@0.3mW10s +b +#3:532nm@3.0mW10s +c +#1: 532 nm @ 0.3 mW +#2:532nm@0.3mW500s +#4:532nm@0.3mW10s +#2:405nm@4.3mW +#3:532nm@0.3mW +(a.u.) + Intensity (a.u.) +Normalized Intensity (a.u.) +Normalized Intensity ( +Normalized +] +] +1 +10001200 1400 1600 1800 2000 2200 2400 +1000 1200 1400 1600 1800 2000 2200 2400 +1000120014001600 180020002200 2400 +Raman Shift (cm-1) +Raman Shift (cm-1) +Raman Shift (cm-1) +51 +Discussion of the Multi-Power Raman Experiments +Figure S28 shows a series of measurements performed to probe whether the transformation of +γ-graphyne observed under laser light (Fig. 5b, Main Text) is photochemical or photothermal. +First, a spectrum was obtained over 10s with 532 nm excitation at 0.3 mW incident power +(Fig. S28a, black trace). Following this, a spectrum was collected for 500s from the same region +at the same power setting (Fig. S28a, red trace). This experiment delivered 50× the photon +exposure dose compared to the baseline 10s experiment. No change in the Raman signature was +observed. +Then, the incident power was increased to 3.0 mW, and a spectrum of the same sample region +was collected over 10s (Fig. S28b, magenta trace). This spectrum is dominated by the high +fluorescent background characteristic of disordered/polymeric carbons.47 Finally, a spectrum of +the same sample region is collected over 10s at 0.3 mW incident power (Fig. S28, green trace). +This spectrum is similar to the typical spectra of disordered graphitic materials. No γ-graphyne- +specific features are discernible, indicating a near-instant structural transformation during the high- +power exposure. The photon exposure dose delivered in the 3.0 mW high power experiment is 10× +compared to the baseline experiment, but only 20% that of the 500s low-power experiment (Fig. +S28a, black trace). Thus, the degree of transformation does not linearly correlate with the exposure +dose, indicating thresholding photochemical behavior or a photothermal process. +The experiment in Fig. S28c provides evidence against a purely photothermal process. In this +experiment, the 4.3 mW high power 405 nm light exposure does not induce a structural +transformation, with stable power 532 nm spectra taken before and after the 405 nm exposure +showing no changes. If the process was purely photothermal, then the 4.3 mW 405 nm exposure +should have induced transformation similar to the 3.0 mW 532 nm exposure (Fig S29b). Our +current hypothesis is that γ-graphyne is photochemically sensitive to 532 nm light above a certain +power threshold. However, we cannot conclusively eliminate a photothermal mechanism and +speculate that a combination of both photochemical and photothermal transformations is possible. + + + + +52 + +Fig. S29. ATR-FTIR spectroscopy of bulk γ-graphyne. a, diamond crystal, and b, Ge crystal. The +prominent feature at ~2100 cm-1 in spectrum a is a diamond crystal artifact and not the acetylenic +absorbance band of γ-graphyne. + + + +a +Intensity (a.u.) +0.5 +b +Intensity (a.u.) +0.5 +4000 +3800 +3600 +3400 +3200 +3000 +2800 +2600 +2400 +2200 +2000 +1800 +1600 +1400 +1200 +1000 +800 +Wavenumber (cm'1) +53 + +Fig. S30. Evaluation of thermal stability of γ-graphyne. a, FTIR spectra of a polycrystalline γ- +graphyne particle that was heated to increasing set-point temperatures under vacuum (1.3×10-3 +mbar). The sample was kept at every temperature set-point for one hour, and then cooled down to +room temperature for spectroscopic measurements. The indicated temperature is the set-point +temperature before the spectral measurement. The sample was suspended in a round hole (D ~200 +μm) fabricated in mica film (10x10x0.01mm), and fixed at the edges using bundles from a thin +forest-drawn carbon nanotube sheet. b, Intensity of the internal alkyne peak at 2188 cm-1 as a +function of the maximum setpoint temperature for successive annealing. c, Picture series for the +polycrystalline particle that was heated in vacuum from ambient temperature up to an upper set- +point temperature of 450°C, showing no noticeable changes in particle dimensions or texture. + + + +a +b +25℃ +320°℃ +2188 cm1 +150°℃ +340℃ +200°C +360°℃ +240°C +400°℃ +Absorbance (a.u.) +0.75 +280°C +450°℃ +Absorbance (a.u.) +0.75 +300°C +600°℃ +0.5 +0.5 +0.25 +0.25 +0 +0 +4000 +3000 +2000 +1000 +0 +100 +200 +300 +400 +500 +600 +Wavenumber (cm1) +Temperature (°C) +C +25°℃ +150℃ +200°℃ +240 °C +280°℃ +300 +μm 200 +100 +0 +100 +200 +300 +400 +μm +300°℃ +320°C +340°C +360°C +450°C +54 + +Fig. S31. Micro-FTIR spectra of γ-graphyne and TBTEB monomer. Top panel: absorbance +spectrum of γ-graphyne for a ~300x300 µm polycrystalline pellet 20 µm thick. Bottom panel: +absorbance spectrum of TBTEB. The spectra were collected in reflection mode using a 50x50 µm +aperture, 8 cm-1 resolution, and 100 scans. + + + +2.0 +-Graphyne +a +Absorbance, +1.8 +1.6 +1.4 +1.2 +1.0 +a.u. +2.5 +Precursor (TBTEB) +=( +3 +Absorbance, +2.0 +H +II +H +1.5 +1.0 +7000 +6000 +5000 +4000 +3000 +2000 +1000 +Wavenumber, cm-1 +55 + +Fig. S32. Raman spectra of TBTEB and biphenyl acetylene demonstrating the frequency shift of +terminal alkyne compared to internal alkyne. + + + +TBTEB +Biphenyl Acetylene +2222 cm-1 +0.75 +Intensity (a.u.) +0.5 +2114 cm-1 +0.25 +0 +2060 +2080 +2100 +2120 +2140 +2160 +2180 +2200 +2220 +2240 +2260 +2280 +Raman Shift (cm'1) +56 + +Fig. S33. DFT-generated potential energy surfaces for binding of TBTEB monomer to a γ- +graphyne monolayer. + + + +14.0 +5 +Binding Energy (kcal/mol) +12.5 +Y Shift (A) +11.0 +Br +-5 +9.5 +-10 +5 +0 +10 +X Shift (A) +57 + +Fig. S34. Upper views of the γ-graphyne stacking modes used for molecular dynamics (MD) +simulations. From top-left to bottom-right, the packing modes are: AA, AB1, AB2 and ABC. +Colors are used to highlight different layers. + + + + +58 + +Fig. S35. Final frames of MD quench simulations of AA (left), AB1 (middle) and ABC (right) +stacking structures of γ-graphynes with 6 layers. The top row shows all 6 layers. The middle and +bottom rows show pairs (or triples in the bottommost right structure) of layers to help visualize the +local mode of stacking. + + + + +59 +Discussion of MD Simulation Results +Table S10 shows the cohesion energies of the initial stacking structures (referred to as AA, +AB1, AB2, and ABC, Fig. S34) computed using the ReaxFF potential. ReaxFF predicts that the +lowest energy stacking mode is AB2. However, the differences in the formation energies are so +small that the structures might be considered energetically equivalent. Furthermore, it is likely that +these stacking modes are easily interconvertible. + +Table S10. ReaxFF energies for possible stacking modes of γ-graphyne. +Structure +Energy [eV/atom] +AA +-8.0010 +AB1 +-8.0018 +ABC +-8.0026 +AB2 +-8.0052 + +The dynamics of these stacking structures is interesting. Thermal equilibration or quenching +simulations starting from the AA, AB1 or ABC stacking structures end with structures where pairs +or triples of layers adopt one of the two smallest energy stackings, AB2 or ABC. Figure S35 shows +that under thermal fluctuations, the layers are relatively free to move and find out the best local +stacking. Panels in Fig. S35 confirm that the AB2 is the preferable stacking for γ-graphyne bilayer, +which agrees with DFT calculation results. In this stacking mode, the upper layer is bound at an +A-type site (Fig. 3e, Main Text). + + + + +60 +References +(1) CasaXPS v.2.3.24PR1.0; Casa Software Ltd.: Teignmouth, United Kingdom, 2020. +(2) Fairley, N.; Fernandez, V.; Richard‐Plouet, M.; Guillot-Deudon, C.; Walton, J.; Smith, E.; +Flahaut, D.; Greiner, M.; Biesinger, M.; Tougaard, S.; et al. Systematic and collaborative approach +to problem solving using X-ray photoelectron spectroscopy. Appl. Surf. Sci. Adv. 2021, 5. DOI: +10.1016/j.apsadv.2021.100112 +(3) Tougaard, S.; Jansson, C. Comparison of validity and consistency of methods for quantitative +XPS peak analysis. Surf. Interface Anal. 1993, 20 (13), 1013-1046. DOI: 10.1002/sia.740201302. +(4) Major, G. H.; Avval, T. G.; Patel, D. I.; Shah, D.; Roychowdhury, T.; Barlow, A. J.; Pigram, +P. J.; Greiner, M.; Fernandez, V.; Herrera‐Gomez, A.; et al. 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DOI: 10.1103/PhysRevB.64.075414 + + diff --git a/59E4T4oBgHgl3EQf1g16/content/tmp_files/load_file.txt b/59E4T4oBgHgl3EQf1g16/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d64123a4282f32373f38d528c973240a1892ec1 --- /dev/null +++ b/59E4T4oBgHgl3EQf1g16/content/tmp_files/load_file.txt @@ -0,0 +1,3315 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf,len=3314 +page_content='1 Scalable synthesis and characterization of multilayer γ-graphyne, new carbon crystals with a small direct bandgap Authors: Victor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Desyatkin1‡, William B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Martin1‡, Ali E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Aliev2, Nathaniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chapman1, Alexandre F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Fonseca3, Douglas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Galvão3, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Roy Miller4, Kevin H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Stone5, Zhong Wang2, Dante Zakhidov6, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Ted Limpoco7, Sarah R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Almahdali1, Shane M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Parker4, Ray H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Baughman2, and Valentin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rodionov1* Affiliations: 1Department of Macromolecular Science and Engineering, Case Western Reserve University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2100 Adelbert Road, Cleveland, OH 44106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2Alan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' MacDiarmid NanoTech Institute, University of Texas at Dallas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 800 West Campbell Road, Richardson, TX 75080, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3Applied Physics Department, Institute of Physics “Gleb Wataghin”, University of Campinas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Campinas, SP, 13083-970, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4Department of Chemistry, Case Western Reserve University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 10900 Euclid Ave, Cleveland, OH 44106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2 5Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2575 Sand Hill Road, Menlo Park, CA 94025, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6Department of Materials Science and Engineering, Stanford University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 496 Lomita Mall, Stanford, CA 94305, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 7Oxford Instruments Asylum Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6310 Hollister Ave, Santa Barbara, CA 93117, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Abstract: γ-Graphyne is the most symmetric sp2/sp1 allotrope of carbon, which can be viewed as graphene uniformly expanded through insertion of two-carbon acetylenic units between all the aromatic rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To date, synthesis of bulk γ-graphyne has remained a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We here report the synthesis of multilayer γ-graphyne through crystallization-assisted irreversible cross-coupling polymerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Comprehensive characterization of this new carbon phase is described, including synchrotron X-ray diffraction, electron diffraction, lateral force microscopy, Raman and infrared spectroscopy, and cyclic voltammetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Experiments indicate that γ-graphyne is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 eV bandgap semiconductor, with a hexagonal a-axis spacing of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='88 Å and an interlayer spacing of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 Å, which is consistent with theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The observed crystal structure has an aperiodic sheet stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The material is thermally stable up to 240 °C but undergoes a transformation at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' While conventional 2D polymerizations and reticular chemistry rely on error correction through reversibility, we demonstrate that a periodic covalent lattice can be synthesized under purely kinetic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The reported methodology is scalable and inspires extension to other allotropes of the graphyne family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3 Introduction: Over five hundred carbon phases have been theoretically predicted, but few have been experimentally realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 Allotropes based on hexagonal lattices of sp2 hybridized carbons, including few-layer graphene,2 nanotubes,3 and fullerenes,4 are synthetically accessible at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These materials have been revolutionary for fundamental physics and materials science, and found applications in post-silicon electronics, high-capacity batteries, organic solar cells, extreme- strength composites, and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Graphyne family allotropes of carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Graphynes-n (graphdiyne for n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, 12,12,12-Graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, 6,6,12-Graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, Graphyne (or γ-graphyne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In contrast, advances in the synthesis of nonbenzenoid carbon allotropes have been limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Several sp2-based nanographenes with non-hexagonal rings,5, 6 as well as an extended nonbenzenoid biphenylene network,7 were described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Materials containing sp1 hybridized linear arrangements remain even more elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Linear polyyne chains -(C≡C)n- are unstable even for moderate chain lengths unless they are stabilized with bulky end-capping groups8 or confined inside carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9 Graphynes, a family of hybrid lattices combining sp1 and sp2 carbons, were first theoretically proposed by Baughman, Eckhardt, and Kertesz in 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='10 These allotropes can be formally viewed as graphenes that are expanded through insertion of acetylenic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Graphynes are commonly divided into two types: graphynes-n in which aromatic rings are separated by n acetylenic bonds (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1a), and x,y,z-graphynes featuring at least some non-aromatic sp2 carbons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1b and 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Theoretical studies of some graphyne lattices suggest unique electronic and chemical properties,11 including an intrinsic band gap and electrochemical a b d 12 DBA 4 capacities that far exceed graphite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='12 To date, few-layer graphyne-2 or graphdiyne (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1a, n = 2) is the only graphyne family allotrope that has been synthesized (typically at sub-milligram scale) and thoroughly characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='13 γ-Graphyne (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1d), the basic graphyne-n homolog (n = 1), is especially intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Single- layer γ-graphyne is predicted to be a semiconductor with a moderate band gap,14 ultrafast charge carrier mobility comparable to that of graphene,15 high thermal conductivity,16 and exceptional strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='17 Because of these properties, γ-graphyne could form the basis for the next generation carbon-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' γ-Graphyne oligomers containing up to four dehydrobenzo[12]annulene (12-DBA) repeat units have been prepared through multistep organic synthesis,18, 19 but synthesis of the extended crystalline lattice remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Pyrolytic and vapor-deposition methodologies commonly used for the controlled synthesis of benzenoid carbons are unsuitable for sp1-based structures, as acetylenes readily convert to graphene or amorphous carbon at elevated temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' While graphdiyne13 and graphdiyne- graphene heterostructures20 have been synthesized via templated solution-phase 2D polymerizations, a similar approach has not been previously attempted for γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Reported graphdiyne syntheses rely on polymerization of highly energetic hexaethynylbenzene through Glaser-Hay sp1-sp1 coupling, which can be conveniently localized at metal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='21 A comparable synthesis of γ-graphyne would require either coupling between sp1 and sp2 carbons, or de novo formation of three acetylenic bonds per each 12-DBA repeat unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The most common and general methodology for sp1-sp2 C-C coupling is the Sonogashira reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='22 The mechanism of this reaction is thought to involve a homogeneous Pd0 catalytic cycle, like all other Pd-catalyzed cross-couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Therefore, attempting to confine this chemistry to the surface of a template would be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Moreover, previously reported γ-graphyne oligomers are distorted from planarity, 5 due to steric hindrance introduced by the terminal functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='23 This inevitable distortion, as well as the typically poor solubility of oligomers, limits stepwise extension of the lattice beyond 3-4 12-DBA units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='18 Results and Discussion: Here, we demonstrate that under appropriately adjusted Sonogashira coupling conditions, an A3B3-type monomer, 1,3,5-tribromo-2,4,6-triethynylbenzene (TBTEB, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2a), can be polymerized into extended γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The main idea that guided our thinking is that an effective route to graphynes and similar rigid 2D polymers could proceed through reactions that create multiple connections in a single step or through a series of kinetically coupled fast steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Such mechanism would bypass the kinetic dead-end of partially connected intermediates, as each monomer unit will “click” into place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Furthermore, this polymerization would be self-correcting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Defects in the growing lattice would be the most reactive sites due to local distortions and strain, which could be relieved upon multi-site reaction with the monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Thus, our two primary aims were to establish reaction conditions favoring exhaustive coupling of the multifunctional TBTEB, and to find a way to template the formation of the desired 2D lattice instead of disordered hyperbranched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Several types of Suzuki-Miyaura, Kumada, and Negishi cross-couplings favor exhaustive substitution in multifunctional substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This mode of reactivity has previously been exploited for the syntheses of polyfunctional arenes24 and low-defect hyperbranched polyphenylenes,25 as well as for pseudo-living chain polymerizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='26 In all these cases it is assumed that an exceptionally reactive Pd species is formed after the initial catalytic cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='24 The subsequent coupling steps are catalyzed by this Pd species, proceed inside the solvent cage, and are diffusion- controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We reasoned that we could extend this reaction mode to Sonogashira-type chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Furthermore, while the nature of the Cu-mediated catalytic cycle in Sonogashira coupling remains 6 largely unexplored, it is broadly understood to involve Cu acetylides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='22 The latter can assume either three-dimensional or low-dimensional polymeric forms27 or possibly associate with a metal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We hypothesized that a Cu surface could template the γ-graphyne lattice, like it presumably does in the reported syntheses of graphdiyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='13 We synthesized TBTEB using a known method28 and screened its reactivity under a range of conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2a and Table S1, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In the control experiment in the absence of catalysts, TBTEB decomposed in refluxing pyridine with a half-life of ~24 hours, yielding an amorphous carbonaceous material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S19, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' X-ray photoelectron spectroscopy (XPS) survey indicated moderate loss of Br through spontaneous hydrodebromination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S12 and S14d, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A similar featureless carbon was produced in the control experiment with just a Pd pre-catalyst and no source of Cu (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S18, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Experiments performed in the presence of both Pd and Cu produced outcomes dependent on the state of the metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Pd(II) pre-catalysts, as well as PEPPSI-IPr, which we selected for its propensity for multi-site coupling,24 yielded amorphous carbons broadly comparable to the control products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, for stoichiometric loading of Pd(PPh3)4 in the presence of Cu foil, we obtained a black lustrous material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S15b, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Transmission electron microscopy (TEM) and scanning electron microscopy (SEM) images of this material revealed flakes composed of stacks of flat sheets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S17 and S21, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Selected area electron diffraction (SAED) experiments produced dotted ring patterns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S17e-f, SI), and no Moiré fringes were observed in bright-field TEM, indicating sub- micron crystalline domains with random orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' While the possibility of the layered flakes being a phase of γ-graphyne was intriguing, the data were insufficient to make a structural assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS survey indicated that the product was primarily carbonaceous, but contaminated by Pd, P, and C from the catalyst (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S10, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Since 7 a multilayered material was obtained, we reasoned that the coupling reaction cannot be confined to the surface of the foil and decided to investigate sources of Cu other than the metallic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To our surprise and delight, reactions employing soluble CuI also yielded layered flakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2b and S15a, S16, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The crystallinity of this material was significantly improved over the product produced using Cu foil, with Moiré fringes observable in bright-field TEM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S16a, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Some of the flakes had well-defined hexagonal shapes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2b and S16d, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Micron-size hexagonal prisms with terraces could be observed in SEM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2c and S20d, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A similar hexagonal morphology was previously reported for graphdiyne produced by interfacial synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Two-dimensional polymerizations of TBTEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Overview of selected reaction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b-c, Representative bright-field TEM and SEM of the carbon flakes obtained from Pd(PPh3)4/CuI homogeneous reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Inset in b emphasizes the hexagonal shape of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, High resolution XPS for the C1s region of the sample in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The higher crystallinity of the product obtained through the optimized homogeneous Pd(PPh3)4/CuI protocol allowed for more efficient removal of contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Survey XPS of this a Pd(PPh, Cuo foil, pyridine Pd(PPhs)4 110℃ Br B small crystallline domains Cul, pyridine 110 °C Pd(PPhs)4 or no cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' TBTEB pyridine 110 °C large crystallline domains amorphous d 3 6 Residuals Fit Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') C H(sp) C C (sp) C C (sp2) C Br (sp2) C=0 H 0 280 285 290 100nm 2 μm Binding Energy (eV) 8 product indicated a level of contamination with Pd and P that was below the detection limit of the technique (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S9, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We acquired high-resolution XPS data for the C1s region of this material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2d), as well as for three controls: the product of the Cu foil synthesis, the Pd-only reaction, and thermal reaction products (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S13, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The C1s peak can be deconvolved into five sub- peaks, corresponding to C-H (terminal alkyne sp1),30 C≡C (internal alkyne sp1), C=C (aromatic sp2), aromatic C-Br, and C=O carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='31 The contribution of C=O is negligible for all samples, indicating little to no oxidation under the reducing/anaerobic reaction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Without the contribution of the sp1 subpeak, none of the fits converge, which strongly supports the presence of acetylenic bonds in all products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS indicates a 1:1 ratio of sp1 to sp2 carbons in the crystalline material synthesized by the homogeneous Pd(PPh3)4/CuI protocol, which is consistent with γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This ratio is much higher in the control samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S13b-d, SI), due to extensive side reactions and contamination with aromatic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The π-π* “shake up” peak at 290 eV is commonly observed in XPS of graphitic carbons and graphene, as well as small aromatic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='31 Notably, this peak does not appear in the XPS of graphdiyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='20 The “shake up” feature was negligible for the product of the homogeneous Cu protocol (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2d), strongly suggesting that this material is not graphitic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The peak was prominent for the control products that were also contaminated with P (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S10, S11, and S13c-d, SI), indicating that it may be related to adsorbed PPh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The initial structural identification of the crystalline carbon material was made via synchrotron X-ray powder diffraction (PXRD) using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='728 Å wavelength (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We observed a peak at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0° 2θ, which matches the predicted11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='96 Å spacing between (1010) planes of γ-graphyne (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3a, left inset and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The intense peak at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0° 2θ could be indexed as the (0003) plane, corresponding to an interlayer distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 Å (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3a, right inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To index the other observed 9 peaks, we explored the possible crystal structures of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' While there is a single stable crystallographic configuration for two graphene sheets, a variety of arrangements are possible for bilayer γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Some of these bilayer stackings have been previously identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='32 To systematically survey the possible structures, we used density functional theory (DFT) to analyze the potential energy surface for a γ-graphyne bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The computed surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3e) is a function of the horizontal offset of the upper γ-graphyne layer relative to the lower layer with a fixed interlayer distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='35 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The chosen interlayer distance was based on first-pass optimization of the bilayer geometry and is underestimated due to the difficulty of computationally modeling van der Waals interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='32 The calculations identified two types of local energy minima: one where the upper layer aromatic rings overlay the 12-DBA rings of the lower layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3e, binding site B), and the second one, where there is a fixed lateral distance between the centers of the upper and lower layer aromatic rings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3e, binding site A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The absolute energy minimum of the former arrangement gives rise to a single crystal structure with AB mode of stacking corresponding to the P63mc space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, at least six bilayer structures are possible for binding at site A with energy minima that are nearly identical within the error of the calculations, giving rise to a multitude of AB, ABC, or more complex arrangements for multiple sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To further explore this complex energy map, which would be expensive for DFT calculations, we ran fully atomistic reactive molecular dynamics (MD) simulations for 3-6 layers of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In all cases, the simulations converged to stacks of sheets bound exclusively at A sites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S34 and S35, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, for multiple sheets stacked through A sites, the energy barrier for transitioning between different A site configurations is extremely small, since this energy difference depends upon the van der Waals interactions between non-adjacent graphyne sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Modeling indicated that several of the less-ordered stacking modes identified by the MD and DFT calculations produce PXRD 10 patterns that closely match our experimental diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Thus, we could index the peak at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5° 2θ as the (2112) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' X-ray and electron diffraction patterns of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Synchrotron PXRD pattern (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='728 Å radiation) of γ-graphyne produced by the Pd(PPh3)4/CuI protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Left inset (red): peak at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0° 2θ superimposed with the modeled (1010) peak for γ-graphyne with P3112 stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Right inset (black): peak at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0° 2θ superimposed with the (0003) peak for the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The dashed blue line is the (0002) peak center for graphite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Representative SAED (white) of the same material overlaid with the simulated (red) c orientation diffraction pattern for a structure with no systematic absences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, SAED of the sample region in b, rotated by 45°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, DFT-generated model of a γ-graphyne sheet overlaid with crystal planes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' e, DFT-generated potential energy surface for the stacking of two γ-graphyne sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The binding energy calculated for a single unit cell (highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We further explored the structure and symmetry of the crystals using electron diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The spot SAED patterns of γ-graphyne produced with the Pd(PPh3)4/CuI protocol were exceptionally well-defined, consistent for different regions of the sample, and independent of the size or presence a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='344 nm 1 (0003) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 (2112) (2110) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0 0 ↑(1010) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 (1070) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 15 2 nm 20 (degrees) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='450 nm d e 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='560 nm Binding Energy (kcal/mol) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='344 nm 5 B 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Shift ( ib 0 > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 5 (1010) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='596 nm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 5 0 5 2 nm 1 X Shift (A) 11 of a selected area aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The near-perfect uniformity of the patterns indicates that the material consists of crystalline domains that are sufficiently large to span the entire illuminated region of our typical imaging frame of 2×2 μm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S16, SI), indicating crystalline domain sizes of at least 1-3 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33 The diffraction patterns observed from the flat areas of the sample had perfect hexagonal symmetry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Using bond distances calculated by DFT and the interlayer distance obtained from PXRD, we built models for several plausible stacking modes of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These models were used to simulate electron diffraction in the c, b, and intermediate crystal orientations that lie ~45° to the (0001) pole (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S23, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The simulated c orientation patterns exclusively involve spacings in the basal plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S22c and S22d, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The first- and second-order reflections in the experimental diffraction pattern correspond to d-spacings of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='96 Å and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='44 Å, which perfectly match the theoretically calculated spacings11 for the (1010) and (1120) plane sets of γ-graphyne (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3b and 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Both of these distances are defined in part by the length of the acetylenic bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Some of the more symmetric space groups, such as Cmcm and R3m, are expected to produce diffraction patterns with systematic absences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Since there were no such systematic absences in the observed diffraction pattern, these space groups could be conclusively eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Additional SAED patterns were obtained for an alternate sample orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The initial position of the stage was chosen to yield the most symmetric spot intensity distribution, which corresponds to a beam normal to the basal plane and coincident with the a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Then the sample was rotated around the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' As the sample rotation reached ~45°, diffraction patterns that involve the z spacings began to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The experimental diffractograms in this orientation provided groups of closely spaced spots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3c), suggesting defects in the layer stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Such defects would not appear in the c orientation diffractograms, since the stacking mode only affects reflections involving z spacing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S24, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The symmetry of the patterns agrees with our simulations for 12 this intermediate orientation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S23e, 23f, 23h, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' As no systematic absences were observed, we can exclude some of the more symmetric space groups, most notably the P63mc space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The experimental diffraction patterns were most consistent with either one of the lower symmetry stacking modes, such as P3112 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S22e, SI) or an aperiodic superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' It is important to note that despite their multi-spot character, the observed diffractograms are not indicative of turbostratic stacking, which would produce ring patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The shape-factor effect alters the geometry of the diffracted beam,34 which introduces error into the determination of spot centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Although this prohibits precise measurement of interplanar spacings, the simulations indicate that the interlayer spacing estimated from SAED agrees with our PXRD data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S23, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We further probed the structure of γ-graphyne flakes by using lateral force microscopy (LFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' LFM is a technique closely related to contact-mode atomic force microscopy (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In LFM, the scanning tip is rastered across the surface of the sample while maintaining contact, and the torsional moment due to stick-slip friction is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' LFM can achieve resolution approaching that of scanning tunneling microscopy (STM),35 which is often the technique of choice for direct imaging of atomic arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, STM cannot be usefully applied to our sample due to the presence of absorbates that roughen its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In contrast, lattice-resolution LFM imaging can often be performed even on rough substrates under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='36 We prepared a sample for imaging by ultrasonicating the γ-graphyne flakes in water and casting them on a freshly cleaved surface of highly oriented pyrolytic graphite (HOPG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A lateral deflection map for a flat region of one of the γ-graphyne flakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4a-b) was then recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In the same imaging session, with the same tip, we then obtained a lateral deflection map of the underlying HOPG as a reference (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Fast Fourier transform (FFT) of LFM images of both the γ-graphyne flake and HOPG showed hexagonal arrangements of high intensity spots, indicating hexagonal symmetry for both 13 lattices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S27b-c, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Å periodicity observed for the HOPG lattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4e, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S26 and S27c, SI) agrees with the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='35 The lattice of the γ-graphyne flake is visibly less dense (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Its ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Å (one half of the a-axis spacing) periodicity agrees with the theoretically predicted distance between the (1120) planes of γ-graphyne, which is also observed as a second- order reflection in electron diffraction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3b, 3d, and 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Scanning probe microscopy of a γ-graphyne flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, AFM topography map of a γ-graphyne flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b and d, Lateral deflection maps of a region of the γ-graphyne flake and the HOPG substrate control in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c and e, Plots of lateral response along the linear traces from b and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Grey bars mark the confidence interval for lattice constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' f, Fast Fourier Transform (FFT) of the map from b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Circles highlight the periodicity for the (2110) planes of γ-graphyne from electron diffraction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The observed Raman spectra are consistent with expectations for γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='37, 38 The distinctive Y (A1g) band corresponding to the C≡C stretch of internal triple bonds appears at 2197 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The G band, which corresponds to the E2g modes of the aromatic rings, is centered at 1565 cm-1, exhibiting the predicted softening37 relative to the G bands of common graphitic materials (~1580 cm-1) due to additional resonant configurations from π-electron delocalization a 5 b 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Lateral Response (mV) C Lateral Response (mV) 1 Topography (nm) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0 3 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 um nm 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Distance (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 e Lateral Response (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Lateral Response (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1nm 2 2 nm1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Distance (nm) 14 along the acetylenic linkages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='39 A broad D band, corresponding to A1g breathing of aromatic rings, is observed at ~1350 cm-1 and is expected to be sensitive to domain size, lattice defects, and the excitation wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A broad survey scan (100 – 3000 cm-1) showed no other Raman features for the 405 nm excitation wavelength, most notably no C-H stretches (2800-3000 cm-1) or the Yʹ band at ~1900 cm-1 characteristic of diacetylenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29 In the spectra of the TBTEB monomer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S32, SI), a band at 2114 cm-1, corresponding to the stretch of the terminal C≡C-H triple bonds, is observed but not seen in the spectra of γ-graphyne, most likely due to the low ratio of internal to terminal triple bond sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Vibrational spectroscopy of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Raman spectrum of γ-graphyne measured using an excitation wavelength of 405 nm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Raman spectra of γ-graphyne measured using an excitation wavelength of 535 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Black trace: <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW for less than 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Red trace: <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW, after high-power irradiation at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 mW for 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Mid-IR spectrum of polycrystalline γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Inset: deconvolution of the alkyne absorption peak at 2190 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' It is important to note that our attempts to obtain Raman spectra of γ-graphyne using 532 nm excitation, for above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 mW incident power using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='55 NA 50× objective, resulted in rapid and irreversible transformation of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This transformation was marked by the bleaching of the Y band, the increase in the intensity of the D band, and the shift of the G band to ~1590 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The resulting Raman signature is similar to that of disordered graphitic carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='40 The b 2 a 405 nm 532 nm c 1 - ←1513 G (E2g) D (Ag) ←1588 1565 Envelope 1343 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 Y(Atg) G (E2g) C=C-Ar (int) C=C-Ar(Br) (int) 2197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 1592 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 C=C-H (term) 1339 Intensity ( Intensity Intensity 2000 2400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 2190 cm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5- B (A1) Y (Aa) 2197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25- MMW 0 500 1000 1500 2000 2500 500 1000 1500 2000 2500 1000 1500 2000 2500 3000 3500 4000 Raman Shift (cm-1) Raman Shift (cm-1) Wavenumber (cm-1) 15 transformation is not due to direct oxidation, since experiments conducted in air and under ~1×10-3 mbar vacuum resulted in similar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' γ-Graphyne retains its characteristic Raman signature under prolonged high power 405 nm illumination (>4 mW), while rapidly transforming under 532 nm laser light (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28c, SI), which suggests a photochemical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Under stable low power 532 nm illumination (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW), the degree of transformation does not linearly correlate with the exposure dose (Fig S28a-b, SI), suggesting thresholding photochemical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A plausible pathway of the transformation involves Masamune-Bergman cycloaromatization, which has been demonstrated for a single 12-DBA subunit on a copper surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='41 The observed splitting of the G band under 532 nm excitation, but not 405 nm excitation (black trace, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5b) is not yet well understood, and could be due to partial transformation from the 532 nm light even when measured at a relatively low excitation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The micro-FTIR spectra of polycrystalline flakes of γ-graphyne featured wide-band absorption in the fingerprint region, as well as between 2800-3700 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Absorption in these regions is associated with vibrations of the aromatic rings of γ-graphyne, as well as contributions from the C-H and O-H groups on the periphery of the sheets or belonging to adventitious small-molecule adsorbates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The distorted line shape of these bands, as well as the change in line shape observed for flakes of different thicknesses suggests significant contribution from resonant Mie scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='42, 43 Thus, we could not identify specific vibration frequencies or their true intensities for low- and high-frequency mid-IR regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, a distinctive peak corresponding to acetylenic bonds centered at ~2190 cm-1 was observed in the mid-IR silent region (1700-2500 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Since the C≡C stretch is IR-inactive for symmetric alkynes, an ideal infinite monolayer of γ-graphyne would not possess this band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, symmetry breaking due to stacking of graphyne sheets, as well as finite and defective sheets, could activate this absorption, like for the appearance of the D band in 16 the Raman spectra of milled graphite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='44 The IR absorption reveals a prominent low-frequency shoulder (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5c, inset), due to the contribution of the terminal alkyne species at sheet edges (C≡C- H stretch, ~2115 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Electronic properties of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Determination of the optical bandgap from the near-IR spectrum of polycrystalline γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Cyclic voltammetry of γ-graphyne powder on glassy carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These are three-electrode measurements in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 M n-Bu4N·PF6/acetonitrile supporting electrolyte, with Pt counter electrode and an Ag/AgNO3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 M) reference electrode at a scan rate of 50 mV/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The near-IR absorbance spectrum of γ-graphyne shows a strong onset of the fundamental electronic edge at high frequencies (5000-7500 cm-1), which is manifested as a monotonic decrease of absorbance with respect to frequency, and as an optical phonon fingerprint in the infrared shoulder (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The dependence of absorbance on photon energy within the electronic edge can be used to estimate the electronic band gap, Eg, and determine the type of semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The absorption coefficient α is defined as α = (1/x)×ln(1/T), where x is the sample thickness and T is transmittance, the ratio of transmitted to incident light intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Within the semi-classical theory of the optical absorption of crystalline direct band gap semiconductors, α(E) is expected to be 0 for E < Eg, and proportional to (E – Eg)1/2 for E ≥ Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='45, 46 Thus, Eg can be estimated by plotting α2 versus E, and extrapolating the linear region of the curve to the energy axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This α2 versus a 5 b 4 Y-graphyne on glassy carbon Y-graphyne annealed at 600 °C on glassy carbon 4 glassy carbon F = 602 mV oxidation Current (μA) 3 106 0 E 132 mV reduction -2 1 E_ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='05 4 0 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9 -1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Photon Energy (eV) Potential (V vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Ag/Agt) 17 E plot gives an optical band gap of Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='05 eV for γ-graphyne, where the uncertainty indicates the standard deviation for over 20 polycrystalline particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To investigate the redox properties of γ-graphyne, a working electrode was prepared by ultrasonically dispersing 1 mg of γ-graphyne in 1 mL of a 1:1 v/v mixture of ethanol and water, and then casting this suspension onto a glassy carbon electrode with subsequent air drying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Three- electrode cyclic voltammetry (CV) measurements were then conducted in acetonitrile with n- Bu4N·PF6 supporting electrolyte for potentials between -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 V vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Ag/Ag+ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' An oxidation peak at 602 mV and a reduction peak at 132 mV were observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6b, blue trace), corresponding to an electrochemical bandgap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='47 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='47 This value agrees with the bandgap determined by the optical absorption measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To evaluate thermal stability, γ-graphyne particles were supported over holes in thin mica films by single-layer carbon nanotube sheets and then heated to gradually increasing set points under vacuum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3×10-3 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After one hour at the set temperature, the sample assembly was cooled down and transferred to the FTIR microscope for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The same sample region was analyzed for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Heating up to 240 °C did not produce any major changes in FTIR spectra beyond a slight decrease in the intensity of the 2800-3700 cm-1 band, likely due to the removal of adventitious adsorbates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S30a, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Loss of the alkyne band at 2100-2200 cm-1 was observed at just over 240 °C, suggesting an onset of a structural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The material lost all the alkyne absorbance after the one-hour thermal cycling processes reached 350 °C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S30b, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The absorbance shoulder corresponding to terminal alkynes disappeared before the internal alkyne components of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The loss of alkyne absorption was accompanied by the disappearance of the Mie scattering bands in the fingerprint region and the region between 2800-3700 cm-1, suggesting significant rearrangement of the microcrystalline structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The material annealed to 18 600 °C lost the characteristic reduction and oxidation peaks in CV, indicating a major chemical change (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6b, red trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Despite the dramatic transformation of the IR signature, the optical images of the thermally treated samples revealed no textural or geometric changes on heating up to 450 °C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S30c, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Due to its sensitivity to the different alkyne bonds present in the sample, the IR absorbance was found to be a useful marker for monitoring the structural changes of γ- graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Our data indicates that the material we synthesized is multilayer γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Contrary to expectations, we found that no external template is required for synthesizing highly crystalline γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' There is no experimental evidence even in our reactions performed with Cu foil that any polymerization is happening on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The lower crystallinity of the Cu foil products is likely due to the reduced concentration of catalytic Cu species in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This results in slower Sonogashira coupling and a higher extent of side reactions compared to the homogeneous Pd(PPh3)4/CuI protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We tried to understand why TBTEB preferentially polymerizes into multilayer γ-graphyne flakes, as opposed to amorphous branched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' As the polymerization appears to be self- templating, we assumed the existence of an attractive supramolecular interaction between TBTEB and the lattice of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Self-assembly through solvophobically driven π-stacking has been documented for several phenylene ethynylene oligomers and macrocycles structurally related to graphynes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 To explore the potential supramolecular interactions in our system, we computed by DFT the structure and potential energy surface for a single TBTEB molecule bound to a γ-graphyne monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Our calculations predict that TBTEB would associate with the surface of graphyne at two types of binding sites (around the aromatic rings and over 12-DBA rings) with a binding energy in excess of 20 kcal/mol (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S33, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The monomer species outcompetes toluene, 19 whose binding energy we estimate as ~11 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' If every TBTEB species were to react while bound over the underlying layer of γ-graphyne, one of the many local energy minimum stackings or a mixture of the stackings could result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Conclusions: To our knowledge, our synthesis of γ-graphyne is the first example of an ordered covalent lattice formed spontaneously under purely kinetic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Typical covalent organic frameworks and metal-organic frameworks are held together through bonds that are reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This reversibility is considered critical for continuous error correction during the reaction/crystallization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='49 Conventional thinking predicts that irreversible polymerization of an A3B3-type monomer, not employing a strict geometric constraint on reactivity, must yield only disordered branched structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, since we routinely observed micron-scale γ-graphyne crystallites, 2D polymerization assisted by crystallization must be presently kinetically favored over random 3D growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Furthermore, the initial nucleation of flat graphyne sheets appears to be a highly probable event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The high fidelity of the resulting lattices indicates that the system is capable of correcting errors despite the irreversibility of Sonogashira coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' At a minimum, the reaction must proceed comparably well at both lattice edges and internal defect sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Since “patching” a single internal defect requires forming six new chemical bonds, it is highly likely that these bond-making steps are kinetically coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The apparent capability for error correction, as well as the strong dependence of the product structure on the nature of the Pd pre-catalyst, strongly corroborate our original hypothesis of a multi-site coupling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Similar cross-coupling methodology could conceivably be applied to the synthesis of other graphyne-family allotropes, as well as to theoretically proposed heteroatom-doped derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='50 Performing the reaction at interfaces may provide access to extended few-layer or monolayer sheets rather than microcrystalline powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These extended sheets could form the basis of the first 20 γ-graphyne-based devices, especially since we observe a small, direct band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Further exploration of the chemical and physical properties of γ-graphyne is under way in our laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ASSOCIATED CONTENT Materials and methods, experimental procedures, NMR and FT-IR spectra, crystallographic information for possible stacking modes of γ-graphyne sheets, supplemental figures and discussion referred to in the text, and computational data (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='PDF) Models of possible stacking modes of γ-graphyne sheets (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='CIF) AUTHOR INFORMATION Corresponding Author *Correspondence to: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rodionov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Email: vor2@case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='edu Author Contributions ‡These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ACKNOWLEDGMENT We are grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Jessica Bickel for discussions of scanning probe microscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We thank the US Department of Energy (R01AB123456) and the National Science Foundation (GFRP Award 1451075 to WBM) for funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' RHB acknowledges support from Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Welch Foundation (grant AT-0029).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' DSG and AFF acknowledge support from São Paulo Research Foundation (FAPESP, awards #2013/08293-7 and #2020/02044-9), National Council for 21 Scientific and Technological Development (CNPq), and the John David Rogers Computing Center (CCJDR) at the Institute of Physics “Gleb Wataghin”, University of Campinas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We thank Oxford Instruments Asylum Research for providing access to Cypher VRS AFM instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This work made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Case Western Reserve University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Raman spectroscopy was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-AC02-76SF00515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' REFERENCES (1) Hoffmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Kabanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Trapp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Schluter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Gram-scale synthesis of two-dimensional polymer crystals and their structure analysis by X-ray diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2014, 6 (9), 779-784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1038/nchem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2007 (50) Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Lv, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Jena, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Electronic structures and bonding of graphyne sheet and its BN analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2011, 134 (17), 174701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3583476 1 Supplementary Materials for Scalable Synthesis and Characterization of Multilayer γ-Graphyne, New Carbon Crystals with a Small Direct Band Gap Victor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Desyatkin,1‡ William B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Martin,1‡ Ali E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Aliev,2 Nathaniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chapman,1 Alexandre F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Fonseca,3 Douglas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Galvão,3 Ericka Roy Miller,4 Kevin H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Stone,5 Zhong Wang,2 Dante Zakhidov,6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Ted Limpoco,7 Sarah R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Almahdali,1 Shane M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Parker,4 Ray H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Baughman,2 and Valentin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rodionov1* ‡These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1Department of Macromolecular Science and Engineering, Case Western Reserve University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2100 Adelbert Road, Cleveland, OH 44106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2Alan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' MacDiarmid NanoTech Institute, University of Texas at Dallas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 800 West Campbell Road, Richardson, TX 75080, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3Applied Physics Department, Institute of Physics “Gleb Wataghin”, University of Campinas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Campinas, SP, 13083-970, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4Department of Chemistry, Case Western Reserve University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 10900 Euclid Ave, Cleveland, OH 44106, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2575 Sand Hill Road, Menlo Park, CA 94025, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6Department of Materials Science and Engineering, Stanford University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 496 Lomita Mall, Stanford, CA 94305, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 7Oxford Instruments Asylum Research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 6310 Hollister Ave, Santa Barbara, CA 93117, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Correspondence to: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Valentin O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rodionov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Email: vor2@case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='edu 2 Materials and Methods Materials All reagents and solvents were acquired from commercial suppliers (Acros Organics, Sigma- Adrich, TCI Chemicals, Fisher Scientific, Oakwood Chemical and VWR International) and used without further purification, unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Tetrahydrofuran (THF) was distilled over Na/benzophenone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Triethylamine (TEA) was distilled over CaH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Anhydrous pyridine (Py) was purchased from Acros in AcroSeal packaging and used without further purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Copper foil, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mm thick Puratronic 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='999% (metals basis) with 25x50 mm lateral dimensions, was purchased from Alfa Aesar and cut into 5x10 mm pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These pieces of were then sequentially sonicated for 20 minutes in 3M HCl, water, ethanol, and acetone, dried under high vacuum at ambient temperature, and immediately used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Synthetic Methods Reactions were monitored by thin-layer chromatography (TLC) carried out on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 mm MilliporeSigma aluminum-backed silica gel plates (60F-254).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Plates were visualized using 254 nm UV light and basic potassium permanganate stain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 g KMnO4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 g NaOH, and 10 g K2CO3 in 150 ml water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' terminal alkynes stain yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Flash chromatography was performed on Luknova SuperSepTM (230-400 mesh) silica gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Reactions requiring anhydrous or air-free conditions were performed under positive pressure of N2 or Ar using standard Schlenk line techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Nuclear Magnetic Resonance (NMR) Spectrometry NMR spectra were recorded on a Bruker Avance III HD 500 spectrometer operating at 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='24 (1H), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='79 (13C), or 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='37 (29Si) MHz and equipped with Bruker Ascend 500 MHz US Narrow Bore Magnet and Broadband Prodigy TCI CryoProbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' NMR spectra were referenced to TMS (1H, 13C, 29Si) or residual solvent peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Chemical shifts (δ) are reported in parts per million (ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Gas Chromatography – Mass Spectrometry GC-MS analyses were performed on an Agilent 5977B GC/MSD instrument equipped with an Agilent 7890B automatic liquid sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Before injection of the sample, the 10 μL syringe was 3 cleaned with acetone and ethyl acetate (3x10 μL each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1 μL of the sample was then automatically injected into the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The method used a 3-minute solvent delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The oven was initially set at 60 °C and held at this temperature for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 minutes before increasing the temperature to 225 °C at 35 °C/min rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Data analysis was performed using Agilent MassHunter Qualitative Analysis Navigator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Infrared Spectroscopy Routine small molecule FTIR spectra were collected on an Agilent Cary 630 FTIR instrument equipped with single-reflection germanium or diamond attenuated total reflectance (ATR) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The instrument was calibrated before sampling against a newly cleaned (acetone) and dried crystal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Solid samples were placed directly on the crystal and secured with a needle press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 32 scans from 4000 to 550 cm-1 were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A background was collected for each sample (512 scans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' We attempted to obtain FTIR spectra of γ-graphyne using the routine spectrometer described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' When using the diamond ATR module, we observed primarily artifacts related to the diamond substrate itself (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S29a), suggesting that the refractive index (RI) of γ-graphyne in mid-IR is higher than that of diamond (n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Some IR bands in the fingerprint region (600- 1800 cm-1) and at ~2100 cm-1 could be observed using a germanium ATR crystal (n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S29b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, this spectrum featured low intensity and unfavorable signal-to-noise ratio because of the low penetration depth for ATR FTIR on germanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Likewise, we could not obtain satisfactory data using the KBr pellet technique due to the low absorbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Therefore, for all subsequent experiments we utilized a PerkinElmer Spotlight 200i FTIR Microscopy System equipped with a Spectrum Two spectrometer capable of both transmittance and reflectance measurements in the mid-IR to near-IR range (600-7800 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The micro-FTIR spectra were collected at ambient conditions in reflection mode using an 8 cm-1 resolution, 50x50 μm2 aperture, and 100 scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The typical polycrystalline particles chosen were on average ~300×300 μm and 20 μm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Spectra was deconvolved using CasaXPS software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1, 2 The peaks were modeled using a Gaussian/Lorentzian (SGL(10)) sum formula SGL(p) where the mixing between the two peak types are determined by m = p/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' SGL(100) is pure Lorentzian while SGL(0) is entirely Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A linear background was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4 Raman Spectroscopy Raman spectra were obtained using a Horiba Labram HR Evolution confocal Raman microscope with a 405 and 532 nm excitation laser source in reflection geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' An Edwards T- Station 85 turbo pump connected to a Janis ST-500 stage was used for vacuum control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Samples were measured in vacuum at a pressure reading of 1-2×10-5 mbar to control for possible oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The vacuum monitoring was done with a pressure gauge situated next to the turbo pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' It is most likely that the pressure in the Janis ST-500 stage was closer to 1×10-3 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A 50x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='55NA Mitutoyo long working-distance objective (13 mm) was used to collect Raman spectra from diffraction-limited spot sizes of ~620 nm (532 nm laser) and ~470 nm (405 nm laser).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A low laser power of less than ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW for the 532 nm laser was used for ideal measuring conditions to reduce beam damage and inhibit decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Powers up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW were used to explore laser light induced transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' For the 405 nm laser, laser powers between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='43 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Multiple spots were measured on each sample and no differences were found in the Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Melting Points Melting points were determined with a Mettler Toledo MP50 Melting Point System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Preparation of Exfoliated Samples Analyte dispersions in water (1 mg/mL) were prepared by ultrasonication using a Branson SFX550 Sonifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A 1/8" double-step microtip (Branson p/n 101-063-212) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Samples were processed for 15 minutes at 50% amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Scanning Electron Microscopy (SEM) The SEM images of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S20a-b and S21 were acquired on an FEI Apreo 2 SEM operating at 5kV and an FEI Inspect F-50 operating at 30kV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' For these images, dried material was added to carbon tape and mounted on SEM sample stands and sputtered with a thin layer of gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The SEM images of γ-graphyne in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2 (Main Text) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S20c-d were acquired by a high resolution Zeiss Supra-40 system, using an in-lens detector and a 10 kV accelerating voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5 The samples for these images were not coated with gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The small γ-graphyne particles were grounded to the gold-covered silicon substrate using silver paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Transmission Electron Microscopy (TEM) and Selected Area Electron Diffraction (SAED) Exfoliated samples were analyzed by TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Prior to sample preparation, 200-Cu C-B grids were plasma-treated for 30 seconds using an Emitech K100x glow discharger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3 µL of sample dispersion was added to the grid and allowed to absorb for 5 minutes before the excess solvent was wicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The grid was then transferred to a single-tilt sample holder and imaged on an FEI Tecnai 20 TEM operating at 200 kV in low-dose mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Images were recorded on a Tvips F416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Data was collected using SerialEM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Tilting was performed with the equipped alpha- rotation goniometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' SAED patterns were recorded on an FEI Tecnai 20 TEM using a 40 μm selected area aperture and in the absence of the selected area aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The obtained patterns were calibrated against the (111) planes of evaporated aluminum (plane spacing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2338 nm) on a 3 mm grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The calibration sample was purchased from Electron Microscopy Sciences (EMS p/n 80044).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' X-ray Photoelectron Spectroscopy (XPS) Samples were spread onto double-sided copper tape for XPS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Surveys and high- resolution spectra were acquired on a PHI VersaProbe II Scanning XPS Microprobe using a monochromatic Al X-ray at pressures of 10-10 to 10-7 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The data was smoothed by using the Savitzky-Golay method, with a smoothing width of five, and analyzed using CasaXPS software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1, 2 A Tougaard background3 was applied to each peak before deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' All peak fits used generalized Voigt-like peak shapes, as this function is most appropriate for fitting asymmetric XPS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 CasaXPS provides a generalized Voigt function described as Lorentzian Finite: LF(α, β, w, n, m), where the first three parameters (α, β, w) affect the Lorentzian line shape and its asymmetry and the final two (n, m) change the width of the Gaussian function and the number of times convolution with the Lorentzian component occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Symmetrical peak parameters for the LF line shape were used: LF(1, 1, 255, 360, 6), values derived from default symmetric peak shape settings for CasaXPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' All sub-peak widths were constrained to full width at half maximum 6 (FWHM) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6 eV or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The subpeaks are located at 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7 eV (terminal alkyne sp1),6 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6 eV (aromatic sp2), 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 eV (internal alkyne sp1), 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9 eV (aromatic C-Br), and 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 eV (carbonyl C=O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6, 7 All peaks were allowed a ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 eV padding to the peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Synchrotron Powder X-ray Diffraction Powder diffraction data were collected at the Stanford Synchrotron Radiation Lightsource (SSRL, SLAC National Accelerator Laboratory) beamline 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Samples were loaded into thin- walled glass capillaries of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5mm nominal diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The capillaries were spun during data collection to improve powder averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Measurements were made using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7281 Å wavelength X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Diffracted X-rays were collected using a Pilatus 100K area detector mounted approximately 700 mm from the sample on the 2-theta arm of the beamline diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Each image covers an angular range of approximately 5 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Images were collected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0625 degrees apart, providing substantial overlap between images, and subsequently merged and integrated into the final powder diffraction profile using custom software available at the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Scanning Probe Microscopy Scanning probe microscopy was performed on an Oxford Instruments Asylum Research Cypher VRS AFM instrument equipped with an Olympus RC800PSA probe with a nominal spring constant of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='39 N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Images were taken at 5 nm scan size, 16 Hz line rates, and 256 x 256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The analyte powder was suspended in HPLC grade water and cast on freshly cleaved highly oriented pyrolytic graphite (HOPG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The sample flakes were located optically using the AFM camera and the tip precisely positioned on the material using the scanner as inertial motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Lateral force microscopy (LFM) was the imaging mode used to obtained lattice resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In this mode, the tip is rastered orthogonal to the long axis of the cantilever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The torsional moment of the cantilever due to stick-slip friction was registered as the lateral signal in the photodetector, which is sensitive to the lattice corrugation of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 7 Computation and Modeling Density Functional Theory (DFT) Calculations All DFT calculations were performed using 2D periodic boundary conditions via the RIPER module of TURBOMOLE/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8-12 In all cases the PBE density functional13 with D3 dispersion corrections14 and Becke-Johnson damping15 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' All the calculated geometries were in broad agreement with prior computational studies of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='16-18 Potential energy surfaces of both a γ-graphyne bilayer and individual molecules/graphyne supercells were computed as a function of the horizontal offset of the upper γ-graphyne layer or molecule relative to the lower layer with a fixed interlayer distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='35 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Binding energies for each structure were calculated as adsorption energies (������������������������������������������������������������������������������������������������ = ������������������������������������������������������������������������������������������������ − ������������������������������������������������������������������������������������������������������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A 9x9 k-point grid was used for the γ-graphyne bilayer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The monomer/graphyne supercells were built using a 3x3 γ-graphyne monolayer to ensure 1 nm spacing between adjacent periodic images of monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Due to the resulting repetitiveness of these supercells, a coarser 3x3 k-point grid was used throughout the potential energy surface scan calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The def2-SVP basis set 19 was used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Local bilayer minima identified from the potential energy surfaces were refined by geometry optimization with the def2-TZVP basis set20 and a 9x9 k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Similarly, selected binding site supercell structures were optimized with def2-SVP and a 9x9 k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Final binding energies for all structures were calculated with def2-TZVP as well, along with a finer 17x17 k- point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Molecular Dynamics (MD) Simulations The energy and dynamics of a few different stackings of γ-graphynes were investigated using classical MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' LAMMPS package21 was used with the ReaxFF reactive force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='22, 23 ReaxFF is a state-of-the-art potential previously applied to simulations of structural, mechanical, and thermal properties of carbon nanostructures,24-28 including graphynes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29-32 The following γ-graphyne stackings were used as initial structures: AA, AB1 and AB2, and ABC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' AB1 and AB2 were previously proposed by Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33 AB1 and ABC structures were investigated by Ducéré and Chauvin34 as the most probable stackings of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, as we will see below, neither corresponds to the lowest energy stacking mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 8 All the structures were prepared with 3 and/or 6 layers and were first energy-minimized and then either equilibrated at 300 K or quenched from 1000 to 1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The energy minimizations were performed with periodic boundary conditions (PBC) imposed along all directions in space, including the possibility to relax the size along the PBC directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Combination of energy minimization with free evolution algorithms as suggested by Sihn35 was used to ensure the lowest- energy structure is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This protocol was recently used by Kanegae and Fonseca36 to study elastic properties of graphynes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Thermal equilibration of the structures was performed with PBC by applying a Langevin thermostat to all atoms with damping factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 fs and timestep of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='025 fs, for a total period of 500 ps (or 20 × 106 timesteps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Quenching of the structures was simulated at the same conditions as thermal equilibration except that a Boltzmann distribution of velocities corresponding to the initial temperature of 1000 K was initially attributed to the system, then the temperature was allowed to decrease from 1000 to 1 K over a period of 500 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' SAED and PXRD Simulations The lattice parameters and bond lengths were obtained from DFT calculations (vide supra) and a previously published computational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='16 SAED and PXRD simulations were performed using the CrystalMaker software suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='37 A model of a single γ-graphyne sheet was built in CrystalMaker using a hexagonal P6 lattice with parameters a and c set to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='86 Å and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The asymmetric unit comprised four atoms placed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='208, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='412, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='795 along the hexagonal P6 x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The basic models corresponding to various sheet stacking modes were constructed using Vesta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='38 Models shown in Fig S22c and S22e-h were generated from DFT simulations and were passed through Avogadro/spglib39 in an attempt to identify the space group associated with each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 9 Small Molecule Synthesis 1,3,5 tribromo 2,4,6 triiodobenzene Br Br Br Br Br Br I I I KI, H5IO6 H2SO4, rt, 72 h The synthesis procedure for 1,3,5-tribromo-2,4,6-triiodobenzene was adapted from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='40 To concentrated H2SO4 (500 mL) at room temperature was added periodic acid (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='03 g, 180 mmol) in small portions over 15 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After dissolution of the periodic acid, crushed KI (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='64 g, 540 mmol) was added in small portions at 0 °C over 1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' To the resulting deep purple solution at 0 °C was added 1,3,5-tribromobenzene (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='89 g, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mmol) in small portions over 25 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After the solution was stirred at room temperature for 72 h, the resulting thick mixture was poured onto ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The resulting precipitate was filtered and washed with H2O (5 × 200 mL) and then MeOH (5 × 200 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The product was recrystallized twice from pyridine/EtOH 1:4 (1000 mL) to yield a solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The solid was dried at under high vacuum for 1 day to give 1,3,5-tribromo- 2,4,6-triiodobenzene 2 (28 g, 67%) as a pale-yellow solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Mp > 300 °C (decomposition);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR (neat) νmax = 1488, 1354, 1262, 1227, 1147, 1002, 858, 771, 739, 554, 508 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR (126 MHz, DMSO-d6) δ 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='61 (CBr), 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='23 (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' EI-MS fragmentation: m/z 695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5, 693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5, 691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5, 689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5, 567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6, 566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6, 565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6, 564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6, 439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7, 437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' UV/vis (CHCl3, C =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='874 × 10-5 M): λmax (ɛ) = 227 (5000), 248 (27700), 283 (5600 M–1cm–1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ((2,4,6 tribromobenzene 1,3,5 triyl)tris(ethyne 2,1 diyl))tris(trimethylsilane) Br Br Br I I I Br Br Br Si Si Si Si PdCl2(PPh3)2, CuI, PPh3 Et3N, THF, 80°C, 48 h 1,3,5-tribromo-2,4,6-triiodobenzene 2 (346 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 mmol), [PdCl2(PPh3)2] (105 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='15 mmol, 30 mol%), CuI (19 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 mmol, 20 mol%), Et3N (50 mL) and THF (40 mL) were added to a dry three-necked flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Ethynyltrimethylsilane (736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7 mg, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='07 mL, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 mmol) and Ph3P (52 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 mmol, 40 mol%) were added to the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The mixture was stirred at 80 °C for 48 h 10 under argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After the removal of solvent on a rotary evaporator, DCM (100 mL) was added to the residue and filtered through Celite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The mixture was washed with water (20 mL) and NaCl(aq) (20 mL), dried over anhydrous Na2SO4, and the solvent was removed under reduced pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The residue was further purified by flash chromatography using n-hexane as the eluent to yield ((2,4,6- tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) 3 as a white solid (175 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29 mmol, yield: 58%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rf (hexane) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Mp = 110-111 °C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR (neat) νmax = 2958, 2160, 1376, 1340, 1245, 1019, 834, 758, 708, 658, 633, 539 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1H NMR (500 MHz, CDCl3): δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29 ppm [s, 27H, Si(CH3)3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR (126 MHz, CDCl3): d 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='09 (CBr), 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='49 (C6C≡C), 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='79 (C≡CSi), 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='83 (C6C≡C), –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='23 [Si(CH3)3] ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 29Si NMR (99 MHz, CDCl3) δ -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='87 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' EI- MS fragmentation: m/z 603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 602, 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' UV/vis (CHCl3, C = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='436 × 10-5 M): λmax (ɛ) = 260 (45700), 271 (44600), 289 (39600 M–1cm–1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1,3,5 tribromo 2,4,6 triethynylbenzene, TBTEB Br Br Br Si Si Si Br Br Br 0°C, 15 min TBAF, THF To a solution of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) (151 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 mmol) in THF (15 mL) was added 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='55 mL TBAF (75% solution in water, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 mmol) and stirred at 0 °C for 15 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The solution was then diluted with ethyl acetate and washed with distilled water and dried with anhydrous Na2SO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The solvent was removed on a rotary evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The residue was further purified by flash chromatography using n-hexane as the eluent to give TBTEB as a white solid (84 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='216 mmol, yield: 87%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rf (hexane) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR (neat) νmax = 3275, 2922, 2112, 1519, 1368, 1336, 965, 736, 681, 634 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1H NMR (500 MHz, CDCl3): δ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='16 (s, 3H, (C≡CH) ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR (126 MHz, CDCl3): δ 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='90 (CBr), 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33 (C6C≡CH), 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='87 (C≡CH), 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='97 (C6C≡CH) ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' EI-MS fragmentation: m/z 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8, 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9, 388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8, 387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8, 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8, 385,8, 384,8, 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' UV/vis (CHCl3, C = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='136 × 10-5 M): λmax (ɛ) = 212 (7100), 218 (8300), 260 (58100), 278 nm (33500 M–1cm–1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 11 Synthesis of Carbon Materials Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Representative Polymerization Reactions of TBTEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='a Entry [Pd] / mol% [Cu] / mol% Additive / mol% Solvent / Base Temp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' °C Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Pd(PPh3)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='CuI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Pyridine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='crystallinity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Pd(PPh3)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Cu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='foil1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Pyridine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='crystallinity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='Pd(PPh3)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='- ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='HCl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' water,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ethanol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' and acetone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' sequentially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' for 20 minutes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' dried under vacuum at rt and used immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a All the reactions were performed under positive pressure of Ar (Schlenk line with a Hg bubbler).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 12 General Synthetic Procedure for Carbon Materials In a typical procedure, TBTEB, Pd(PPh3)4, and Cu were charged to a Schlenk tube under argon atmosphere and solvent was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The tube was sealed, and the contents degassed by three freeze- pump-thaw cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The reaction mixture was stirred under argon atmosphere and heated for 72 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The reaction mixture was concentrated on a rotary evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The solid product was washed with methanol, ethanol, isopropanol, toluene, hexanes, ethyl acetate and acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The washing procedure involved dispersing the material in the corresponding solvent by gentle sonication, followed by centrifugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Conditions for selected experiments from Table S1 are detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Table S1, Entry 1 TBTEB (116 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol), Pd(PPh3)4 (347 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol) and CuI (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='024 mmol) reacted in anhydrous pyridine (50 mL) using the general procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Typical mass of the crude product after centrifugation and drying on low vacuum (1-2 Torr) over 10 hours is ~90% of the monomer mass (104 mg for the scale above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After extensive drying at high vacuum (10 mTorr) and/or heating to 100°C for 72 hours the mass decreased to ~60% of the original monomer mass (68 mg for the scale above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' TLC indicated monomer conversion is quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Table S1, Entry 2 TBTEB (116 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol), Pd(PPh3)4 (347 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol) and several pieces of copper foil reacted in a mixture of anhydrous pyridine (50 mL) using the general procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Table S1, Entry 3 TBTEB (116 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol) and Pd(PPh3)4 (347 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol) reacted in pyridine (50 mL) using the general procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' No Cu was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Table S1, Entry 4 TBTEB (116 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mmol) refluxed in pyridine (50 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Neither Pd(PPh3)4 nor copper was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13 Spectroscopic and Imaging Data and Supplementary Discussion -10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 f1 (ppm) -2E+07 0 2E+07 4E+07 6E+07 8E+07 1E+08 1E+08 1E+08 2E+08 2E+08 2E+08 2E+08 2E+08 3E+08 3E+08 3E+08 3E+08 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='02 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='19 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='35 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='52 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='69 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='85 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='02 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='23 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='61 Br Br Br I I I a b a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR spectrum of 1,3,5-tribromo-2,4,6-triiodobenzene in DMSO-d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 14 Br Br Br Si Si Si 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 f1 (ppm) 0 5E+07 1E+08 2E+08 2E+08 2E+08 3E+08 4E+08 4E+08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='26 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1H NMR spectrum of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris- (trimethylsilane) in CDCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR spectrum of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris- (trimethylsilane) in CDCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='23 4E+08 4E+08 Br Br 3E+08 a Br d 2E+08 2E+08 2E+08 1E+08 a C b 5E+07 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 f1 (ppm) 16 Br Br Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='50 DMSO d6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 1H NMR spectrum of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB in DMSO-d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 17 Br Br Br 10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 f1 (ppm) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5×108 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0×108 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='19 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='35 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='52 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='52 DMSO d6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='69 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='85 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='97 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='85 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='89 a b c d a b c d Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 13C NMR spectrum of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB in DMSO-d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR of 1,3,5-tribromo-2,4,6-triiodobenzene (ATR-FTIR on diamond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 70 60 w 50 (%) 40 30 20 Br 10 Br Br 0 4000 3500 3000 2500 2000 1500 1000 500 Wavenumber (cm 1) 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR of ((2,4,6-tribromobenzene-1,3,5-triyl)tris(ethyne-2,1-diyl))tris(trimethylsilane) (ATR-FTIR on germanium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 100 90 80 2901 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='(CH3) 70 2160 v(C三C) 2958 Vas(CH3) 09 50 T 1019 Si Ar 40 1340 (CH3) in TMS Br 30 633 v(C Br) 1245 Si Ar 20 Br Br 10 Si 834 v(Si CH3) 0 4000 3500 3000 2500 2000 1500 1000 500 Wavenumber (cm 1) 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' FTIR of 1,3,5-tribromo-2,4,6-triethynylbenzene, TBTEB (ATR-FTIR on germanium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 90 80 70 2113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 09 1C"C1: (%) 50 T 40 3281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9 C=CIH 30 Br 1338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 20 10 Br Br C 631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7 Br 0 4000 3500 3000 2500 2000 1500 1000 500 Wavenumber (cm 1) 21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS survey corresponding to Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c C 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='51% 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='62% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1 Br 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='87% Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0 Br 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 Br Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1000 750 500 250 0 Binding Energy (eV) 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS survey corresponding to Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 C 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='49% 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='73% Br 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='02% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='05% Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') Pd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0 Pd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Br Br p Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1000 750 500 250 0 Binding Energy (eV) 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS survey corresponding to Table S1, Entry 3 (TBTEB and Pd(PPh3)4 in pyridine, no Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c C 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='80% 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='31% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 JBr 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='79% P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='59% Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 IPd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5% 0 Pd 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Br Br Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1000 750 500 250 0 Binding Energy (eV) 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' XPS survey corresponding to Table S1, Entry 4 (thermal decomposition of TBTEB in refluxing pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' C C 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='41% N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='38% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='41% Br 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='80% Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 N 0 Br Br M Br 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 1000 750 500 250 0 Binding Energy (eV) 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' High resolution XPS data for C1s peak regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Table S1, Entry 3 (control experiment with TBTEB and Pd(PPh3)4 in pyridine, no Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, Table S1, Entry 4 (control experiment with no catalysts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a 4 d Residuals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Fit Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') C H(sp) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 3 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 C C (sp) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' C C (sp2) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 C Br (sp2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 C=O Tt nt 1 1 1 0 0 0 0 280 285 290 280 284 288 292 280 284 288 292 280 284 288 292 Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) 26 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' High resolution Br 3d region XPS spectra of selected carbon material samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Table S1, Entry 3 (control experiment with TBTEB and Pd(PPh3)4 in pyridine, no Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, Table S1, Entry 4 (control experiment with no catalysts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 5 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Residuals Fit Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 2 C-Br 3 3 3 Br 2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 0 0 0 0 65 70 75 65 70 75 65 70 75 65 70 75 Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) 27 High Resolution Br 3d XPS High-resolution Br 3d XPS spectra of the carbon materials from Table S1, Entries 1-4 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The Br signal manifests as a pair of peaks due to spin-orbit coupling, yielding a Br3d3/2 peak at a higher binding energy and Br3d5/2 peak at a lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' These coupled peaks are well-defined in XPS, with a separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='04 eV and intensity ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Prior to the reaction, bromine species constitute ~20% atomic percent (at%) of the monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' A decrease in this fraction is observed in all cases after the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The residual Br 3d peak can be deconvolved to two distinctive species: Br covalently bonded to aromatic carbon (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 eV),7, 41, 42 corresponding to partially unreacted sites, and weakly coordinated/anionic Br (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5- 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 eV)42-46 trapped within the carbon matrix or on the edges or surfaces of γ-graphyne sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' After the reaction catalyzed by Pd(PPh3)4/CuI in pyridine (Table S1, Entry 1) XPS survey indicates an apparent 12 at% residual Br (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S9), consisting almost entirely of covalent C(Ar)- Br (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The structure of the fit includes contributions from at least two C(Ar)-Br species, presumably mono- and di-substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' As this material consists of micron-scale crystallites (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S16a-d), and the surface penetration depth for XPS in carbon materials is 10 nm or less, we attribute this C(Ar)-Br signal to the unreacted edge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Because of the composition difference between edge and bulk of the material, a survey XPS measurement likely results in a significant overestimation of the Br content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' It is important to note that the number of terminal alkyne C-H groups estimated from fitting of the C1s XPS signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2d, Main Text and S13a) correlates well with the apparent C(Ar)-Br content, as would be expected for complementary edge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' For the rest of the reactions surveyed (Table S1, Entries 2-4) the carbon products are significantly contaminated with the weakly bound Br species (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S14b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' As these products are significantly less crystalline than the product of the optimized Pd(PPh3)4/CuI protocol, small- molecule impurities get trapped in disordered carbon matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This is also evidenced through observation of Pd, P, and N contaminants derived from solvents and/or catalysts in the corresponding survey XPS spectra (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S9-S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' While the small molecule contamination precludes quantitative interpretation of the results, important chemical insights can be obtained by examining the speciation of Br in the products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' It can be clearly seen that thermal decomposition of TBTEB is accompanied by Br loss, likely through spontaneous hydrodebromination (Fig S14d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Pd(PPh3)4 alone is capable of activating the C(Ar)-Br sites, accelerating the rate of hydrodebromination (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S14c and S14d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 28 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Photo of the carbon material corresponding to Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Photo of the carbon material corresponding to Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in refluxing pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b 29 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-d, Representative bright field TEM images of the carbon product corresponding to Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b 100nm 100nm C d 100nm 100nm 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-f, Representative bright field TEM images of the carbon product corresponding to Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' (e-f) Representative SAED ring patterns for the same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b 2 nm 1 2 nm 31 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-c, Representative bright field TEM images of the carbon product corresponding to Table S1, Entry 3 (control experiment with TBTEB and Pd(PPh3)4 in pyridine, no Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b 100nm 100nm 100nm 32 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-c, Representative bright field TEM images of the carbon product corresponding to Table S1, Entry 4 (thermal decomposition of TBTEB in refluxing pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b 100nm 100nm 33 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-d, Representative SEM images of the carbon product corresponding to Table S1, Entry 1 (TBTEB and Pd(PPh3)4/CuI in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b 10μm 10 μm C d 10 μm um 34 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a-b, Representative SEM images of the carbon product corresponding to Table S1, Entry 2 (TBTEB and Pd(PPh3)4/Cu foil in pyridine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b 10 μm 10 μm 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED patterns in the c orientation (sheets perpendicular to the incident beam, a-h, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Interatomic distances are obtained from DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a AA b AB c AB d ABC P6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='mc P6 Cmcm R3m 3 2 nm 2 nml 2 nmr 2 nm ABC ABC ABC ABC h Q P1 pi P3,12 (P1 2 nm 2 nmrl 2 nml 2 nml 36 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED patterns for 45° rotation from the (0001) pole to a viewing direction (matching the experimental diffraction pattern rotated 45°, a-h, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Interatomic distances are obtained from DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a AA b AB c AB d ABC 2 nml 2 nml 2 nm 2 nm ABC f ABC e g ABC h ABC 2 nml 2 nml 2 nm 2 nm 37 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Possible stacking modes of γ-graphyne sheets (a-h, top) and their simulated SAED patterns in the b orientation (sheets parallel to the incident beam, a-h, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Interatomic distances are obtained from DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a AA b AB c AB d ABC 2 nm 2 nm 2 nmr 2 nmrl e ABC f ABC g ABC h ABC 2 nml 2 nml 2 nmt 2 nml 38 Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for the structure in Fig 22-24a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' AA Stacking File: AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Hexagonal P 6 a = b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 Å α = β = 90°, γ = 120° Atom X Y Z C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='205 0 0 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='412 0 0 39 Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for the structure in Fig 22-24b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' AB Stacking File: AB_1_P_63_mmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Hexagonal P 63 m c a = b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8 Å α = β = 90°, γ = 120° Atom X Y Z C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='12833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='66667 0 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='9213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='66667 0 40 Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for structure in Fig 22-24c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' AB Stacking File: AB_2_C_mcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Orthorhombic C m c m a = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='86455 b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8 Å α = β = γ = 90° Atom X Y Z C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='60351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='27153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='7056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='16932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='08195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='28643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='39653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='47845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='79423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='08036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='66825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 C8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='46396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 41 Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for the structure in Fig 22-24d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ABC Stacking File: ABC_1_R_3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Trigonal R 3 m a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 Å α = β = 90°, γ = 120° Atom X Y Z C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='205 0 0 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='412 0 0 42 Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for the structure in Fig 22-24e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ABC Stacking File: ABC_2_P_31_1_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Trigonal P31 1 2 a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 Å α = β = 90°, γ = 120° Atom X Y Z C1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33416 C2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='15811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33675 C3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='74697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='45999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33221 C4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='74719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='66451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33033 C5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='04594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33418 C6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='84148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='33416 43 Table S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Crystallographic information for the structure in Fig 22-24f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ABC Stacking File: ABC_3_P_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='cif Triclinic P 1 a = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8826 Å c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 Å α = β = 90°, γ = 120° Atom X Y Z C1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2082 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='69353 47 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' LFM lattice images of the carbon product corresponding to Table S1, Entry 1 (γ-graphyne) and reference HOPG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Raw LFM image of a 10x10 nm flat area of a γ-graphyne flake (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4a, Main Text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Low-pass filtered version of the LFM image in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Raw LFM image of a 10x10 nm area of the reference HOPG substrate obtained several nm from the edge of the sample flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, Low-pass filtered version of the LFM image in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Y graphyne Lateral Response (mV) 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' HOPG substrate measurements used for the calibration of the LFM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Lateral deflection map of the HOPG surface, raw image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Plot of lateral response along the linear trace from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, FFT of the lateral response map from a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' d, Low-pass filtered lateral deflection map from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b uV 500 μV 750 Lateral Signal 0 500 0 2 nm c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='00 750 49 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Analysis of periodicity through FFT of scanning microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, SAED of the product corresponding to Table S1, Entry 1 (γ-graphyne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Reflections corresponding to the (2110) crystallographic planes are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, FFT of the AFM height image of the same material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 4a, Main Text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The bright-spot pattern is a perfect match for the reflections highlighted in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, FFT of the LFM image of HOPG substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S25b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The bright-spot pattern is consistent with the expected periodicity for a graphene sheet (distance of ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1 Å between (1010) crystallographic planes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The periodicity of HOPG standard is clearly different and from that of γ- graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b C 2110 2 nm 1 2 nm 1 2 nm 1 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Multi-power and multi-wavelength Raman spectroscopy of γ-graphyne and structural transformations under laser light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, Spectra obtained over 10s (#1, black) and 500s (#2, red) intervals with 532 nm excitation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The spectra were collected sequentially (10s, then 500s) from the same sample region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Spectra of the sample region in a obtained over 10s at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW (magenta, #3) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW (green, #4) incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The spectra were obtained sequentially (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW, then 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The experiment was performed after the low-power experiments in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Sequentially obtained spectra of γ-graphyne with 532 nm excitation (10s, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power, yellow, #1), 405 nm excitation (10s, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power, black, #2), and 532 nm excitation (10s, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power, red, #3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The spectra were acquired from the same sample region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a #1:532nm@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3mW10s b #3:532nm@3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0mW10s c #1: 532 nm @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW #2:532nm@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3mW500s #4:532nm@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3mW10s #2:405nm@4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3mW #3:532nm@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3mW (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') Normalized Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') Normalized Intensity ( Normalized ] ] 1 10001200 1400 1600 1800 2000 2200 2400 1000 1200 1400 1600 1800 2000 2200 2400 1000120014001600 180020002200 2400 Raman Shift (cm-1) Raman Shift (cm-1) Raman Shift (cm-1) 51 Discussion of the Multi-Power Raman Experiments Figure S28 shows a series of measurements performed to probe whether the transformation of γ-graphyne observed under laser light (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 5b, Main Text) is photochemical or photothermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' First, a spectrum was obtained over 10s with 532 nm excitation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28a, black trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Following this, a spectrum was collected for 500s from the same region at the same power setting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28a, red trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This experiment delivered 50× the photon exposure dose compared to the baseline 10s experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' No change in the Raman signature was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Then, the incident power was increased to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW, and a spectrum of the same sample region was collected over 10s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28b, magenta trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This spectrum is dominated by the high fluorescent background characteristic of disordered/polymeric carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='47 Finally, a spectrum of the same sample region is collected over 10s at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW incident power (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28, green trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' This spectrum is similar to the typical spectra of disordered graphitic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' No γ-graphyne- specific features are discernible, indicating a near-instant structural transformation during the high- power exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The photon exposure dose delivered in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW high power experiment is 10× compared to the baseline experiment, but only 20% that of the 500s low-power experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28a, black trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Thus, the degree of transformation does not linearly correlate with the exposure dose, indicating thresholding photochemical behavior or a photothermal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The experiment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S28c provides evidence against a purely photothermal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In this experiment, the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW high power 405 nm light exposure does not induce a structural transformation, with stable power 532 nm spectra taken before and after the 405 nm exposure showing no changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' If the process was purely photothermal, then the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3 mW 405 nm exposure should have induced transformation similar to the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 mW 532 nm exposure (Fig S29b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Our current hypothesis is that γ-graphyne is photochemically sensitive to 532 nm light above a certain power threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, we cannot conclusively eliminate a photothermal mechanism and speculate that a combination of both photochemical and photothermal transformations is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 52 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ATR-FTIR spectroscopy of bulk γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, diamond crystal, and b, Ge crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The prominent feature at ~2100 cm-1 in spectrum a is a diamond crystal artifact and not the acetylenic absorbance band of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 b Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content="5 4000 3800 3600 3400 3200 3000 2800 2600 2400 2200 2000 1800 1600 1400 1200 1000 800 Wavenumber (cm'1) 53 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Evaluation of thermal stability of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a, FTIR spectra of a polycrystalline γ- graphyne particle that was heated to increasing set-point temperatures under vacuum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3×10-3 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The sample was kept at every temperature set-point for one hour, and then cooled down to room temperature for spectroscopic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The indicated temperature is the set-point temperature before the spectral measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The sample was suspended in a round hole (D ~200 μm) fabricated in mica film (10x10x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='01mm), and fixed at the edges using bundles from a thin forest-drawn carbon nanotube sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' b, Intensity of the internal alkyne peak at 2188 cm-1 as a function of the maximum setpoint temperature for successive annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' c, Picture series for the polycrystalline particle that was heated in vacuum from ambient temperature up to an upper set- point temperature of 450°C, showing no noticeable changes in particle dimensions or texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' a b 25℃ 320°℃ 2188 cm1 150°℃ 340℃ 200°C 360°℃ 240°C 400°℃ Absorbance (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 280°C 450°℃ Absorbance (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 300°C 600°℃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='25 0 0 4000 3000 2000 1000 0 100 200 300 400 500 600 Wavenumber (cm1) Temperature (°C) C 25°℃ 150℃ 200°℃ 240 °C 280°℃ 300 μm 200 100 0 100 200 300 400 μm 300°℃ 320°C 340°C 360°C 450°C 54 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Micro-FTIR spectra of γ-graphyne and TBTEB monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Top panel: absorbance spectrum of γ-graphyne for a ~300x300 µm polycrystalline pellet 20 µm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Bottom panel: absorbance spectrum of TBTEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The spectra were collected in reflection mode using a 50x50 µm aperture, 8 cm-1 resolution, and 100 scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 Graphyne a Absorbance, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Precursor (TBTEB) =( 3 Absorbance, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 H II H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 7000 6000 5000 4000 3000 2000 1000 Wavenumber, cm 1 55 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Raman spectra of TBTEB and biphenyl acetylene demonstrating the frequency shift of terminal alkyne compared to internal alkyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' TBTEB Biphenyl Acetylene 2222 cm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='75 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 2114 cm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content="25 0 2060 2080 2100 2120 2140 2160 2180 2200 2220 2240 2260 2280 Raman Shift (cm'1) 56 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' DFT-generated potential energy surfaces for binding of TBTEB monomer to a γ- graphyne monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 5 Binding Energy (kcal/mol) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 Y Shift (A) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0 Br 5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='5 10 5 0 10 X Shift (A) 57 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Upper views of the γ-graphyne stacking modes used for molecular dynamics (MD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' From top-left to bottom-right, the packing modes are: AA, AB1, AB2 and ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Colors are used to highlight different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 58 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Final frames of MD quench simulations of AA (left), AB1 (middle) and ABC (right) stacking structures of γ-graphynes with 6 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The top row shows all 6 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' The middle and bottom rows show pairs (or triples in the bottommost right structure) of layers to help visualize the local mode of stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 59 Discussion of MD Simulation Results Table S10 shows the cohesion energies of the initial stacking structures (referred to as AA, AB1, AB2, and ABC, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S34) computed using the ReaxFF potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ReaxFF predicts that the lowest energy stacking mode is AB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' However, the differences in the formation energies are so small that the structures might be considered energetically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Furthermore, it is likely that these stacking modes are easily interconvertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Table S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' ReaxFF energies for possible stacking modes of γ-graphyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Structure Energy [eV/atom] AA -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0010 AB1 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0018 ABC -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0026 AB2 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0052 The dynamics of these stacking structures is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Thermal equilibration or quenching simulations starting from the AA, AB1 or ABC stacking structures end with structures where pairs or triples of layers adopt one of the two smallest energy stackings, AB2 or ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Figure S35 shows that under thermal fluctuations, the layers are relatively free to move and find out the best local stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' S35 confirm that the AB2 is the preferable stacking for γ-graphyne bilayer, which agrees with DFT calculation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' In this stacking mode, the upper layer is bound at an A-type site (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 3e, Main Text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' 60 References (1) CasaXPS v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='24PR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Guillot-Deudon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Walton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Smith, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Flahaut, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Greiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Biesinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Tougaard, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Resonant Raman spectroscopy of disordered, amorphous, and diamondlike carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' B 2001, 64 (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} +page_content='075414' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQf1g16/content/2301.05291v1.pdf'} diff --git a/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/2301.03071v1.pdf.txt b/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/2301.03071v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..622dabb66ee6737bf6854789101d859f91cea6f0 --- /dev/null +++ b/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/2301.03071v1.pdf.txt @@ -0,0 +1,1866 @@ +arXiv:2301.03071v1 [math.DG] 8 Jan 2023 +Curves of Constant Breadth According to Darboux +Frame in a Strict Walker 3-Manifold +Ameth Ndiaye* +D´epartement de Math´ematiques, FASTEF, UCAD, Dakar, Senegal. +Abstract +In this paper, we investigate the differential geometry properties of curves of constant +breadth according to Darboux frame in a given strict Walker 3-manifold. The considered curves +are lying on a timelike surface in the Walker 3-manifold. +MSC: 53B25 ; 53C40. +Keywords: Darboux frame, curvature, torsion, constant breadth curve, Walker 3-manifolds. +1 +Introduction +The study of curves of constant breadth were defined first in 1778 by Euler. Then, Solow [11] +investigated the curves of constant breadth. Kose, Magden and Yilmaz in [9, 10] studied plane +curves of constant breadth in Euclidean spaces E3 and E4. Fujiwara [7] defined constant breadth +for space curves and obtained a problem to determine whether there exists space curve of con- +stant breadth or not. Furthermore, Blaschke [3] defined the curves of constant breadth on a sphere. +In [2], Altunkaya et al. defined null curves of constant breadth in Minkowski 4-space and obtain +a characterization of these curves. Also Altunkaya et al. in [1] investigate constant breadth curves +on a surface according to Darboux frame and give some characterizations of these curves. +Motivated by the above papers, we investigate the geometries of curves of constant breadth accord- +ing to Darboux frame in a Strict Walker 3-manifold which is a Lorentzian three-manifold admitting +a parallel null vector field. It is known that Walker metrics have served as a powerful tool of con- +structing interesting indefinite metrics which exhibit various aspects of geometric properties not +given by any positive definite metrics. For more details about Walker 3-manifold see [5,6,8]. +2 +Preliminaries +A Walker n-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r- +planes, with r ≤ n +2. The canonical forms of the metrics were investigated by A. G. Walker ( [4]). +* E–mail: ameth1.ndiaye@ucad.edu.sn (A. Ndiaye) +1 + +Walker has derived adapted coordinates to a parallel plan field. Hence, the metric of a three- +dimensional Walker manifold (M, gǫ +f) with coordinates (x, y, z) is expressed as +gǫ +f = dx ◦ dz + ǫdy2 + f(x, y, z)dz2 +(1) +and its matrix form as +gǫ +f = + + +0 +0 +1 +0 +ǫ +0 +1 +0 +f + + +with inverse (gǫ +f)−1 = + + +−f +0 +1 +0 +ǫ +0 +1 +0 +0 + + +for some function f(x, y, z), where ǫ = ±1 and thus D = Span∂x as the parallel degenerate line +field. Notice that when ǫ = 1 and ǫ = −1 the Walker manifold has signature (2, 1) and (1, 2) +respectively, and therefore is Lorentzian in both cases. In this study we take ǫ = 1. +It follows after a straightforward calculation that the Levi-Civita connection of any metric (1) +is given by: +∇∂x∂z += +1 +2fx∂x, +∇∂y∂z = 1 +2fy∂x, +∇∂z∂z += +1 +2(ffx + fz)∂x + 1 +2fy∂y − 1 +2fx∂z +(2) +where ∂x, ∂y and ∂z are the coordinate vector fields +∂ +∂x, +∂ +∂y and +∂ +∂z , respectively. Hence, if (M, gǫ +f) +is a strict Walker manifolds i.e., f(x, y, z) = f(y, z), then the associated Levi-Civita connection +satisfies +∇∂y∂z = 1 +2fy∂x, +∇∂z∂z = 1 +2fz∂x − 1 +2fy∂y. +(3) +Note that the existence of a null parallel vector field (i.e f = f(y, z)) simplifies the non-zero +components of the Christoffel symbols and the curvature tensor of the metric gǫ +f as follows: +Γ1 +23 = Γ1 +32 = 1 +2fy, Γ1 +33 = 1 +2fz, Γ2 +33 = −1 +2fy +(4) +Let now u and v be two vectors in M. Denoted by (⃗i,⃗j,⃗k) the canonical frame in R3. +The vector product of u and v in (M, gǫ +f) with respect to the metric gǫ +f is the vector denoted by u×v +in M defined by +gǫ +f(u × v, w) = det(u, v, w) +(5) +for all vector w in M, where det(u, v, w) is the determinant function associated to the canonical +basis of R3. If u = (u1, u2, u3) and v = (v1, v2, v3) then by using (5), we have: +u × v = +����� +u1 +v1 +u2 +v2 +���� − f +���� +u2 +v2 +u3 +v3 +���� +� +⃗i − ǫ +���� +u1 +v1 +u3 +v3 +����⃗j + +���� +u2 +v2 +u3 +v3 +����⃗k +(6) +2 + +3 +Darboux equations in Walker 3-manifold +Let α : I ⊂ R −→ (M, gǫ +f) be a curve parametrized by its arc-length s. The Frenet frame of α is +the vectors T, N and B along α where T is the tangent, N the principal normal and B the binormal +vector. They satisfied the Frenet formulas + + + +∇TT(s) += +ǫ2κ(s)N(s) +∇TN(s) += +−ǫ1κT(s) − ǫ3τB(s) +∇TB(s) += +ǫ2τ(s)N(s) +(7) +where κ and τ are respectively the curvature and the torsion of the curve α, with ǫ1 = gf(T; T); ǫ2 = +gf(N; N) and ǫ3 = gf(B, B). +Starting from local coordinates (x, y, z) for which (1) holds, it is easy to check that +e1 = ∂y, e2 = 2 − f +2 +√ +2 ∂x + 1 +√ +2∂z, e3 = 2 + f +2 +√ +2 ∂x − 1 +√ +2∂z +are local pseudo-orthonormal frame fields on (M, gǫ +f), with gǫ +f(e1, e1) = ǫ, gǫ +f(e2, e2) = 1 and +gǫ +f(e3, e3) = −1. Thus the signature of the metric gǫ +f is (1, ǫ, −1). If we choose ǫ = 1 then, +pseudo-orthonormal frame is formed by two spacelike vectors and one timelike vector and If we +choose ǫ = −1 then, pseudo-orthonormal frame is formed by one spacelike vector and two timelike +vectors. For both cases we obtain Lorentzian manifold. In this work we assume that ǫ = 1 +Now we suppose that the curve α lies on a timelike surface S in M. Let U be the unit normal vector +of S, then the Darboux frame is given by {T, Y, U}, where T is the tangent vector of the curve α(s) +and Y = U × T. +Case 1: Let α be timelike curve. Then the tangent vector T is timelike (ǫ1 = −1), the normal +vector N and the binormal vector B are spacelike, that is (ǫ2 = ǫ3 = 1). +Since S is timelike, the unit normal vector U is spacelike and so Y becomes spacelike. The usual +transformations between the Walker Frenet frame and the Darboux takes the form +Y = cos θN + sin θB +(8) +U = − sin θN + cos θB, +(9) +where θ is an angle between the vector Y and the vector N. +Derivating Y along the curve alpha we get +∇TY = cos θ∇TN − θ′ sin θN + sin θ∇TB + θ′ cos θB. +Using the Frenet equation in (2.7) we have +∇T Y = cos θ(κT − ǫ3τB) − θ′ sin θN + sin θ(ǫ2τN) + θ′ cos θB. +Now we suppose that the principal normal and the binormal have the same sign. then we get +∇TY = κ cos θT + (θ′ − τ)U +(10) +The same calculus gives +∇TU = −κ sin θT − (θ′ − τ)Y. +(11) +3 + +Then the Walker Darboux equation is expressed as + + + +∇TT = κgY + κnU +∇TY = κgT + τgU +∇TU = κnT − τgY, +(12) +where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on +S, respectively. Also, (12) gives +gǫ +f (∇T Y, U) = τg = θ′ − τ, +(13) +gǫ +f (∇TT, Y ) = κg = κ cos θ, +(14) +gǫ +f (∇TT, U) = κn = −κ sin θ. +(15) +Case 2: Let α be spacelike curve. Then the tangent vector T is spacelike (ǫ1 = 1), the normal +vector N is spacelike (ǫ2 = 1) and the binormal vector B is timelike (ǫ3 = −1) or normal vector N +is timelike (ǫ2 = −1) and the binormal vector B is spacelike (ǫ3 = 1). So we have two following +subcases: +i): ǫ2 = 1 and ǫ3 = −1. +Then the usual transformations between the Walker Frenet frame and the Darboux takes the form +Y = cosh θN + sinh θB +(16) +U = sinh θN + cosh θB, +(17) +where θ is an angle between the vector Y and the vector N. +Since ∇TT = κN, we have +∇TT = −κ sinh θY + κ cosh θU. +(18) +Derivating Y along the curve alpha we get +∇T Y = −κ sinh θT + (θ′ + τ)U +(19) +The same calculus gives +∇TU = −κ cosh θT + (θ′ + τ)Y. +(20) +Then the Walker Darboux equation is expressed as + + + +∇TT = −κgY + κnU +∇TY = −κgT + τgU +∇TU = −κnT + τgY, +(21) +where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on +S, respectively. Also, (21) gives +gǫ +f (∇TY, U) = τg = θ′ + τ, +(22) +gǫ +f (∇TT, Y ) = κg = κ sinh θ, +(23) +gǫ +f (∇TT, U) = κn = κ cosh θ. +(24) +4 + +ii): ǫ2 = −1 and ǫ3 = 1. +Then the usual transformations between the Walker Frenet frame and the Darboux takes the form +Y = sinh θN + cosh θB +(25) +U = cosh θN + sinh θB, +(26) +where θ is an angle between the vector Y and the vector N. +Since ∇TT = −κN, we have +∇TT = −κ cosh θY + κ sinh θU. +(27) +Derivating Y with respect to s we get +∇TY = −κ cosh θT + (θ′ − τ)U +(28) +Derivating Y with respect to s alpha we get +∇TU = −κ sinh θT + (θ′ − τ)Y. +(29) +Then the Walker Darboux equation is expressed as + + + +∇TT = −κgY + κnU +∇TY = −κgT + τgU +∇TU = −κnT + τgY, +(30) +where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on +S, respectively. Also, (30) gives +gǫ +f (∇T Y, U) = τg = θ′ − τ, +(31) +gǫ +f (∇TT, Y ) = κg = κ cosh θ, +(32) +gǫ +f (∇TT, U) = κn = κ sinh θ. +(33) +4 +Space curves of constant breadth According to Darboux Frame +in Walker manifold +In this section, we define space curves of constant breadth in the three dimensional Walker mani- +fold. +Definition 4.1. A curve α : I → (M, gǫ +f) in the three-dimensional Walker manifold (M, gǫ +f) is +called a curve of constant breadth if there exists a curve β : I → Mf such that, at the corresponding +points of curves, the parallel tangent vectors of α and β at α(s) and β(s⋆) at s; s⋆ ∈ I are opposite +directions and the distance gǫ +f(β − α, β − α) is constant. In this case, (α; β) is called a pair curve +of constant breadth. +Let now (α; β) be a pair of unit speed curves of constant breadth and s, s⋆ be arc-length of α +and β, respectively. +We suppose that the curve α lies on a timelike surface in Mf, then it has Darboux frame in addition +to Frenet frame. Then we may write the following equation: +β(s⋆) = α(s) + m1(s)T(s) + m2(s)Y (s) + m3(s)U(s); +(34) +where mi(i = 1, 2, 3) are smooth functions of s. +5 + +4.1 +Case where α is timelike. +Differentiating (34) with respect to s and using (12) we obtain +dβ +ds += +dβ +ds⋆ +ds⋆ +ds += +T ⋆(s⋆)ds⋆ +ds = (1 + m′ +1 + m2κg + m3κn)T(s) ++(m′ +2 + m1κg − m3τg)Y (s) ++(m′ +3 + m2τg + m1κn)U(s), +(35) +where T ⋆ denotes the unit tangent vector of β. +Since T = −T ∗, from the equations in (35) we have + + + +m′ +1 += +−m2κg − m3κn − h(s) +m′ +2 += +−m1κg + m3τg +m′ +3 += +−m2τg − m1κn, +(36) +where h(s) = +ds⋆ +ds + 1. We assume that (α, β) is a curve pair of constant breadth. Since α is a +timelike curve and the vectors Y and U are spacelike vectors, we have +∥β − α∥ = −m2 +1 + m2 +2 + m2 +3 = constant, +(37) +which imlplies that +−m1 +dm1 +ds + m2 +dm2 +ds + m3 +dm3 +ds = 0. +(38) +If we combine (36) and (38), we get +m1h(s) = 0. +(39) +If α and β are curves of constant breadth then m1 = 0 or h(s) = 0. If m1 ̸= 0 (that is h(s) = 0) +then d = m1T(s) + m2Y (s) + m3U(s) becomes a constant vector. So β(s∗) is a translation of α +along the constant vector d. Also h(s) = 0 gives s∗ = −s + c, where c is constant. +Now, we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special case. +4.1.1 +Case (For geodesic curves) +Let α be non-straight line geodesic curve on a timelike surface. Then κg = κ cos θ = 0. As κ ̸= 0, +we get cos θ = 0. So it implies that κn = −κ, τg = −τ. From (36), we have following differential +equation system + + + +m′ +1 += +m3κ − h(s) +m′ +2 += +−m3τ +m′ +3 += +m1κ + m2τ. +(40) +By using (40), we obtain the following differential equation. +1 +κ +�1 +κ(m′ +1 + h) +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κ(m′ +1 + h) +�′ ++ +�τ +κ +�2 +(m′ +1+h)+ +�τ +κ +�′ κ +τ m1−m′ +1 = 0. +(41) +6 + +Subcase 1: m1 ̸= 0 (h(s) = 0). +If we write h(s) = 0 in equation (41), we have. +1 +κ +�1 +κm′ +1 +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κm′ +1 +�′ ++ +��τ +κ +�2 +− 1 +� +m′ +1 + +�τ +κ +�′ κ +τ m1 = 0. +(42) +Theorem 4.2. Let α be a timelike geodesic curve lying a timelike surface in M and let (α, β) be a +pair of unit speed curves of constant breadth. If m1 is a non-zero constant then α is a general helix +in the three dimensional Walker manifold (M, gǫ +f). Also the curve β is given as: +β(s⋆) = α(s) + m1T(s) + m2Y (s) +(43) +where m2 is a real constant and s∗ = −s + c. +Proof. If m1 is non zero constant, then from (42) we obtain that +� τ +κ +�′ = 0. So α is a general +helix. Also from the first and second equations of (40) we get m3 = 0 and m2 is a real constant, +respectively. +Theorem 4.3. Let α be a timelike geodesic curve and a general helix lying a timelike surface in +M. Let (α, β) be a pair of unit speed curves of constant breadth. If m1 is not zero, then the curve +β can be expressed as one of the following cases: +β(s∗) = α(s) + m1T(s) + 1 +c0 +( ¨m1 − m1)Y (s) + ˙m1U(s) +(44) +where +i) m1 = +1 +√ +c2 +0−1 +� +a1 sin( +� +c2 +0 − 1z) − a2 cos( +� +c2 +0 − 1z) +� ++ a3, +c2 +0 − 1 > 0 +ii) m1 = a1 +2 z2 + a2z + a3, +c2 +0 − 1 = 0 +iii) m1 = +1 +√ +1−c2 +0 +� +a1 sinh( +� +1 − c2 +0z) + a2 cosh( +� +1 − c2 +0z) +� ++ a3, +c2 +0 − 1 < 0 +where z = +� +κds and a1, a2, a3 are real constants. +Proof. Let us consider that α is timelike geodesic curve and a general helix in Wlaker 3-manifold. +Then we have τ +κ = c0 = constant. From (42), we have +�1 +κ +�1 +κm′ +1 +�′�′ ++ +� +c2 +0 − 1 +� +m′ +1 = 0. +(45) +By means of changing of the independant variable s with z = +� +κds, from (45) we obtain +m′ +1 = dm1 +ds = dm1 +dz +dz +ds = ˙m1κ. +... +m1 + (c2 +0 − 1) ˙m1 = 0. +(46) +7 + +If we solve this equation we get +m1 = + + + + + + + +1 +√ +c2 +0−1 +� +a1 sin( +� +c2 +0 − 1z) − a2 cos( +� +c2 +0 − 1z) +� ++ a3, if c2 +0 − 1 > 0 +a1 +2 z2 + a2z + a2, if c2 +0 − 1 = 0 +1 +√ +1−c2 +0 +� +a1 sinh( +� +1 − c2 +0z) + a2 cosh( +� +1 − c2 +0z) +� ++ a3, if c2 +0 − 1 < 0. +From (40) we obtain m3 = ˙m1 and m2 = +1 +c0( ¨m1 − m1). +Subcase 2: m1 = 0. +If we take m1 = 0 in the equation (40), we get + + + +h(s) += +m3κ +m′ +2 += +−m3τ +m′ +3 += +m2τ. +(47) +Since m3 = h +κ, m2 = 1 +τ m′ +3 = 1 +τ +�h +κ +�′, we get +�1 +τ +�h +κ +�′�′ ++ +�h +κ +� +τ = 0. +(48) +If we put y = h +κ, the equation (48) becomes +y′′ − τ ′ +τ y′ + τ 2y = 0. +(49) +For solving the equation (49), we put the new variable dw +ds = τ. Then +� +y′ = dy +dw +dw +ds = ˙yτ +y′′ = d2y +dw2τ 2 + dy +dwτ ′ +(50) +If we put the equation (50) in the equation (49) we obtain +d2y +dw2 + y = 0. +(51) +and the solution of (51) is y = b1 cos w + b2 sin w. Then we have +h(s) = κ +� +b1 cos +�� +τds +� ++ b2 sin +�� +τds +�� +(52) +m2 = h +κ = b1 cos +�� +τds +� ++ b2 sin +�� +τds +� +(53) +m3 = 1 +τ +�h +κ +�′ += −b1 sin +�� +τds +� ++ b2 cos +�� +τds +� +. +(54) +So we give the following theorem +Theorem 4.4. Let (α, β) be a pair of constant breadth curve in (M, gf) where α is a timelike +geodesic curve lying in a timelike surface in M. If m1 = 0, then the curve β is given by +β(s∗) = α(s)+ +� +b1 cos +�� +τds +� ++ b2 sin +�� +τds +�� +Y (s)+ +� +−b1 sin +�� +τds +� ++ b2 cos +�� +τds +�� +U(s). +8 + +4.1.2 +Case (For asymptotic lines) +Let α be non-straight line asymptotic line on a timelike surface. Then κn = −κ sin θ = 0. As +κ ̸= 0, we get sin θ = 0. So it implies that κg = κ, τg = −τ. From (36), we have following +differential equation system + + + +m′ +1 += +−m2κ − h(s) +m′ +2 += +−m1κ − m3τ +m′ +3 += +m2τ. +(55) +By using (55), we get +1 +κ +�1 +κ(m′ +1 + h) +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κ(m′ +1 + h) +�′ ++ +�τ +κ +�2 +(m′ +1+h)+ +�τ +κ +�′ κ +τ m1−m′ +1 = 0. +(56) +Subcase 1: m1 ̸= 0 (h(s) = 0). +If we take as h(s) = 0 in equation (56), we get following differential equation +1 +κ +�1 +κm′ +1 +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κm′ +1 +�′ ++ +��τ +κ +�2 +− 1 +� +m′ +1 + +�τ +κ +�′ κ +τ m1 = 0. +(57) +Theorem 4.5. Let α be a timelike asymptotic line lying a timelike surface in M. Let (α, β) be a +pair of unit speed curves of constant breadth. If m1 is non-zero constant then α is a general helix +in the three dimensional Walker manifold (M, gǫ +f). Also the curve β is given as: +β(s⋆) = α(s) + m1T(s) + m3U(s) +(58) +where m3 is a real constant and s∗ = −s + c. +Proof. If m1 is non zero constant, then from (57) we obtain that +� τ +κ +�′ = 0. So α is a general +helix. Also from the first and third equation of (55) we get m2 = 0 and m3 is a real constant, +respectively. +Theorem 4.6. Let α be a timelike asymptotic line lying in a timelike surface in M. Let (α, β) be a +pair of unit speed curves of constant breadth. If m1 is not zero, then the curve β can be expressed +as one of the following cases: +β(s∗) = α(s) + m1T(s) − ˙m1Y (s) + 1 +c0 +( ¨m1 − m1)U(s), +(59) +where +i) m1 = +1 +√ +c2 +0−1 +� +a1 sin( +� +c2 +0 − 1z) − a2 cos( +� +c2 +0 − 1z) +� ++ a3, c2 +0 − 1 > 0 +ii) m1 = a1 +2 z2 + a2z + a3, c2 +0 − 1 = 0 +iii) m1 = +1 +√ +1−c2 +0 +� +a1 sinh( +� +1 − c2 +0z) + a2 cosh( +� +1 − c2 +0z) +� ++ a3, c2 +0 − 1 < 0 +where z = +� +κds and a1, a2, a3 are constants. +9 + +Proof. The proof of Theorem (4.6) is done similarly to the proof of Theorem (4.3) +Subcase 2: m1 = 0. +If we take as m1 = 0 in (55) we get following differential equation system + + + +h(s) += +−m2κ +m′ +2 += +−m3τ +m′ +3 += +m2τ. +(60) +Then we give the following theorem. +Theorem 4.7. Let (α; β) be a curve pair of constant breadth in (M, gf) where α is a timelike +asymptotic curve lying in a timelike surface in M. If m1 = 0, then the curve β is given by +β(s∗) = α(s)+ +� +−b1 cos +�� +τds +� +− b2 sin +�� +τds +�� +Y (s)+ +� +−b1 sin +�� +τds +� ++ b2 cos +�� +τds +�� +U(s). +Proof. The proof of Theorem (4.7) is done similarly to the proof of Theorem (4.4). +4.1.3 +Case (For Principal line) +We suppose that α is a non-planar timelike principal line. Then we have τg = 0. Then it follows +that τ = θ′. By using (36), we have the following differential equation system + + + +m′ +1 += +m3κ sin θ − m2κ cos θ − h(s) +m′ +2 += +−m1κ cos θ +m′ +3 += +m1κ sin θ. +(61) +By mean of changing of the independant variable s with θ = +� +τds, we get + + + +˙m1 += +φ(m3 sin θ − m2 cos θ) − g(θ) +˙m2 += +−m1φ cos θ +˙m3 += +m1φ sin θ. +(62) +where g(θ) = (− ds +dθ − ds∗ +dθ ) and φ = κ +τ . In here we denote the derivative with respect to θ with ”.”. +From the equations in (62) we have +... +m1 + ¨g − d +dθ +� ˙φ +φ( ˙m1 + g) +� +− d +dθ(φ2m1) + ( ˙m1 + g) +− ˙φ +� +− sin θ +� +m1φ cos θdθ + cos θ +� +m1φ sin θdθ +� += 0. +(63) +Subcase 1: m1 ̸= 0 (h(s) = 0). +In this case, we give the following theorem: +Theorem 4.8. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a non-planar +timelike principal line and a general helix then β is given by one of the following cases: +β(s∗) = α(s) + m1T(s) − c +� +m1 cos θdθY (s) + c +� +m1 sin θdθU(s), +(64) +where +10 + +i) m1 = +1 +√ +1−c2 +� +a1 sin( +√ +1 − c2θ) − a2 cos( +√ +1 − c2θ) +� ++ a3, +1 − c2 > 0 +ii) m1 = a1 +2 θ2 + a2θ + a3, +c2 − 1 = 0 +iii) m1 = +1 +√ +c2−1 +� +a1 sinh( +√ +c2 − 1θ) + a2 cosh( +√ +c2 − 1θ) +� ++ a3, +1 − c2 < 0 +Proof. If h(s) = 0 then g(θ) = 0 and from (63) we have +... +m1 − d +dθ +� ˙φ +φ ˙m1 +� +− d +dθ(φ2m1) + ˙m1 − ˙φ +� +− sin θ +� +m1φ cos θdθ + cos θ +� +m1φ sin θdθ +� += 0.(65) +If α is helix curve then φ = κ +τ = c = constant. From (65) we have +... +m1 + (1 − c2) ˙m1 = 0. +(66) +Then the solution is +m1 = + + + + + +1 +√ +1−c2 +� +a1 sin( +√ +1 − c2θ) − a2 cos( +√ +1 − c2θ) +� ++ a3, if 1 − c2 > 0 +a1 +2 θ2 + a2θ + a3, +if +1 − c2 = 0 +1 +√ +c2−1 +� +a1 sinh( +√ +c2 − 1θ) + a2 cosh( +√ +c2 − 1θ) +� ++ a3, if 1 − c2 < 0, +where θ = +� +τdθ. +Subcase 2: m1 = 0. +The case where m1 = 0, we have the following the following theorem: +Theorem 4.9. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a non-planar +timelike principal line. If m1 = 0 then α is general helix. The curve β is expressed as +β(s∗) = α(s) + c2Y (s) + c3U(s), +(67) +where c2 and c3 are constants. +Proof. From (63) we have +¨g − d +dθ +� ˙φ +φg +� ++ g = 0. +(68) +On the other hand, from (61) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0 and +from (62) +g = φ(−c2 cos θ + c3 sin θ). +(69) +By considering (68) and (69) with together, we get +˙φ(c2 sin θ + c3 cos θ) = 0. +(70) +Then we have ˙φ = 0 or c2 sin θ + c3 cos θ = 0. If c2 sin θ + c3 cos θ = 0 then we have that θ is a +constant. So α becomes a planar curve. It is a contridiction. So ˙φ = 0. Then we obtain that φ = κ +τ +is a constant. Thus α is a general helix. +11 + +4.2 +Case where α is spacelike and ǫ2 = 1 and ǫ3 = −1. +Here we suppose that the curve α is spacelike and lying on a timelike surface in Mf. +Differentiating (34) with respect to s and using (21) we obtain +dβ +ds += +dβ +ds⋆ +ds⋆ +ds += +T ⋆ds⋆ +ds = (1 + m′ +1 − m2κg − m3κn)T ++(m′ +2 − m1κg + m3τg)Y ++(m′ +3 + m2τg + m1κn)U, +(71) +where T ⋆ denotes the tangent vector of β. +Since T = −T ∗, from the equation in (35) we have + + + +m′ +1 += +m2κg + m3κn − h(s) +m′ +2 += +m1κg − m3τg +m′ +3 += +−m2τg − m1κn, +(72) +where h(s) = ds∗ +ds + 1. +Since α is spacelike and ǫ2 = 1 andǫ3 = −1, then, if we assume that (α, β) is a curve pair of +constant breadth, we have +∥β − α∥ = m2 +1 + m2 +2 − m2 +3 = constant, +(73) +which imlplies that +m1 +dm1 +ds + m2 +dm2 +ds − m3 +dm3 +ds = 0. +(74) +If we combine (72) and (74) we get +m1(2m′ +1 + h(s)) = 0. +(75) +If α and β are curves of constant breadth then m1 = 0 or 2m′ +1 − h(s) = 0. +Now we investigate the case where α is geodesic curve or principal line curve because κn ̸= 0. +4.2.1 +Case (For geodesic curves) +Let α be non-straight line geodesic curve on a timelike surface. Then κg = κ sinh θ = 0. As κ ̸= 0, +we get sinh θ = 0. So it implies that κn = κ, τg = τ. From (72), we have the following differential +equation system + + + +m′ +1 += +m3κ − h(s) +m′ +2 += +−m3τ +m′ +3 += +−m1κ − m2τ. +(76) +From (76) we have + + + +m3 += +1 +κ(m′ +1 + h) +m′ +2 += +− τ +κ(m′ +1 + h) +m2 += +− 1 +τ +� +( 1 +κ(m′ +1 + h))′ + m1κ +� +. +(77) +12 + +Differentiating the third equation of (76) with respect to s and using the first, the second and the +third equations of (77), we obtain the following equation: +1 +κ +�1 +κ(m′ +1 + h) +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κ(m′ +1 + h) +�′ +− +�τ +κ +�2 +(m′ +1+h)− +�τ +κ +�′ κ +τ m1+m′ +1 = 0. +(78) +Subcase 1: m1 ̸= 0 (h(s) = −2m′ +1). +The equation (78) becomes +1 +κ +�1 +κm′ +1 +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κm′ +1 +�′ +− +��τ +κ +�2 ++ 1 +� +m′ +1 + +�τ +κ +�′ κ +τ m1 = 0. +(79) +Theorem 4.10. Let α be a geodesic curve. Let (α; β) be a pair of unit speed curves of constant +breadth where α is spacelike (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface in Mf. If m1 is +non-zero constant then m3 = 0 and α is a general helix in the three dimensional Walker manifold +(M, gǫ +f). Also the curve β is given as: +β(s⋆) = α(s) + m1T + cY +(80) +where c is a real constant and s∗ = −s + c. +Proof. If m1 is non zero constant, then from (79) we obtain that +� τ +κ +�′ = 0. So α is a general helix. +Also from the second and third equation of (76) we get m3 = 0 because h = 0 and m2 is a real +constant. +Theorem 4.11. Let α be a geodesic curve. Let (α, β) be a pair of unit speed curves of constant +breadth where α is spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface Mf. If m1 is +not zero, then the curve β can be expressed as one of the following cases: +β(s∗) = α(s) + m1T + 1 +c0 +( ¨m1 − m1)Y + ˙m1U, +(81) +where m1 = +1 +√ +1+c2 +0 +� +a1e +√ +1+c2 +0θ − a2e−√ +1+c2 +0θ� +, m3 = − ˙m1 and m2 = 1 +c0( ¨m1 − m1). +Proof. Let us consider that α is a general helix in Wlaker 3-manifold. Then we have τ +κ = c0 = +constant. From (79), we have +�1 +κ +�1 +κm′ +1 +�′�′ +− +� +c2 +0 + 1 +� +m′ +1 = 0. +(82) +By means of changing of the independant variable s with z = +� +κds, we obtain +m′ +1 = dm1 +ds = dm1 +dz +dz +ds = ˙m1κ. +From (82), we get +... +m1 − (c2 +0 + 1) ˙m1 = 0. +(83) +If we solve this equation we get +m1 = +1 +� +1 + c2 +0 +� +a1e +√ +1+c2 +0θ − a2e−√ +1+c2 +0θ� +. +(84) +From (77) we have m3 = − ˙m1 and m2 = +1 +c0( ¨m1 − m1). +13 + +Subcase 2: m1 = 0. +Theorem 4.12. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a geodesic +spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface on Mf. If m1 = 0 then the curve +β is expressed as +β(s∗) = α(s) + cY, +(85) +where c is a constant real. +Proof. If m′ +1 = 0 then h = 0 and from (76) we have m3 = 0 and m2 = constant. +4.2.2 +Case (For Principal line) +If α is principal line, then τg = 0 and τ = −θ′. From (72) + + + +m′ +1 += +m2κ sinh θ + m3κ cosh θ − h(s) +m′ +2 += +m1κ sinh θ +m′ +3 += +−m1κ cosh θ, +(86) +By mean of changing of the independant variable s with θ = +� +τds, we get + + + +˙m1 += +m3 +κ +τ cosh θ + m2 +κ +τ sinh θ − h(s) +τ(s) +˙m2 += +m1 +κ +τ sinh θ +˙m3 += +−m1 +κ +τ cosh θ. +(87) +Denoted by h(s) +τ(s) = g(θ) and κ +τ = φ, we have + + + +˙m1 += +φ(m3 cosh θ + m2 sinh θ) − g(θ) +˙m2 += +m1φ sinh θ +˙m3 += +−m1φ cosh θ. +(88) +From the equations in (88) we have + + + +1 +φ( ˙m1 + g) += +m3 cosh θ + m2 sinh θ +˙m2 sinh θ + ˙m3 cosh θ += +−m1φ +˙m2 cosh θ += +−m3 sinh θ. +(89) +Differentiating the first equation in (88), we get +... +m1 + ¨g − d +dθ +� ˙φ +φ( ˙m1 + g) +� ++ d +dθ(φ2m1) − ( ˙m1 + g) +− ˙φ +� +cosh θ +� +m1φ sinh θdθ − sinh θ +� +m1φ cosh θdθ +� += 0. +(90) +Subcase 1: m1 ̸= 0 (m′ +1 = −h +2). +If m′ +1 = −h +2 then ˙m1 = −g +2. From (90) we obtain +−... +m1 + d +dθ +� ˙φ +φ ˙m1 +� ++ d +dθ(φ2m1) + ˙m1 − ˙φ +� +cosh θ +� +m1φ sinh θdθ − sinh θ +� +m1φ cosh θdθ +� += 0.(91) +14 + +Theorem 4.13. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line +and a general helix then β is given by +β(s∗) = α(s) + m1T + m2Y + m3U, +(92) +where +m1 = +1 +√ +1 + c2 +� +a1e +√ +1+c2θ − a2e− +√ +1+c2θ� +, +m2 = c +� +m1 sinh θdθ and m3 = −c +� +m1 cosh θdθ. +Proof. If α is helix curve then φ = κ +τ = c = constant. From (91) we have +... +m1 − (1 + c2) ˙m1 = 0. +(93) +m1 = +1 +√ +1 + c2 +� +a1e +√ +1+c2θ − a2e− +√ +1+c2θ� +. +(94) +Subcase 2: m1 = 0. +From the equations in (72) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0. The first +equation in (72) gives +tanh θ = −c2 +c3 +. +(95) +Then θ is a constant and we have τ = 0. +Theorem 4.14. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line. +If m1 = 0 then α is planar curve. The curve β is expressed as +β(s∗) = α(s) + c2Y + c3U, +(96) +where c2 and c3 are constants. +4.3 +Case where α is spacelike and ǫ2 = −1 and ǫ3 = 1. +Let α be a spacelike with ǫ2 = −1 and ǫ3 = 1 lying on a timelike surface in Mf. +Differentiating (34) with respect to s and using (30) we obtain + + + +m′ +1 += +m2κg + m3κn − h(s) +m′ +2 += +m1κg − m3τg +m′ +3 += +−m2τg − m1κn, +(97) +where h(s) = ds∗ +ds + 1. +Since α is spacelike and ǫ2 = −1 andǫ3 = 1, then, if we assume that (α, β) is a curve pair of +constant breadth, we have +∥β − α∥ = m2 +1 − m2 +2 + m2 +3 = constant, +(98) +15 + +which imlplies that +m1 +dm1 +ds + m2 +dm2 +ds − m3 +dm3 +ds = 0. +(99) +If we combine (97) and (99) we get +m1h(s) = 0. +(100) +If α and β are curves of constant breadth then m1 = 0 or h(s) = 0. If m1 ̸= 0 (that is h(s) = 0) +then d = m1T + m2Y + m3U becomes a constant vector because d′ = 0. So β(s∗) is a translation +of α along the constant vector d. Also h(s) = 0 gives s∗ = −s + c, where c is constant. +Since κg ̸= 0, here we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special +case (asymptotic line or principal line). +4.3.1 +Case (For Asymptotic line) +Let α be non-straight line asymptotic line on a timelike surface. Then κn = κ sinh θ = 0. As +κ ̸= 0, we get cosh θ = 0. So it implies that κg = κ, τg = −τ. From (97), we have following +differential equation system + + + +m′ +1 += +m2κ − h(s) +m′ +2 += +m1κ + m3τ +m′ +3 += +−m2τ. +(101) +By differentiating the second equation in (101) with respect to s and using the first and third equa- +tions in (101), we get +1 +κ +�1 +κ(m′ +1 + h) +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κ(m′ +1 + h) +�′ +− +�τ +κ +�2 +(m′ +1+h)+ +�τ +κ +�′ κ +τ m1 −m′ +1 = 0. +(102) +Subcase 1: m1 ̸= 0 (h(s) = 0). +The equation (102) is given by +1 +κ +�1 +κm′ +1 +�′′ ++ +��1 +κ +�′ +− 1 +τ +�τ +κ +�′� �1 +κm′ +1 +�′ +− +��τ +κ +�2 ++ 1 +� +m′ +1 + +�τ +κ +�′ κ +τ m1 = 0. +(103) +Theorem 4.15. Let α be a asymptotic curve. Let (α; β) be a pair of unit speed curves of constant +breadth where α is spacelike (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 is +non-zero constant then m2 = 0 and α is a general helix in the three dimensional Walker manifold +(M, gǫ +f). Also the curve β is given as: +β(s⋆) = α(s) + m1T + m3U +(104) +where m3 is a real constant and s∗ = −s + c. +Proof. If m1 is non zero constant, then from (103) we obtain that +� τ +κ +�′ = 0. So α is a general helix. +Also from the first and third equation of (101) we get m2 = 0 and m3 is a real constant. +16 + +Theorem 4.16. Let α be a asymptotic line. Let (α, β) be a pair of unit speed curves of constant +breadth where α is timelike curve and lying in a timelike surface Mf. If m1 is not zero, then the +curve β can be expressed as one of the following cases: +β(s∗) = α(s) + m1T + ˙m1Y + 1 +c0 +( ¨m1 + m1)U, +(105) +where +m1 = +1 +� +c2 +0 + 1 +� +a1e +√ +c2 +0+1z − a2e +√ +c2 +0+1z� +. +Proof. Let us consider that α is a general helix in Walker 3-manifold. Then we have τ +κ = c0 = +constant. From (103), we have +�1 +κ +�1 +κm′ +1 +�′�′ +− +� +c2 +0 + 1 +� +m′ +1 = 0. +(106) +By means of changing of the independant variable s with z = +� +κds, we obtain +... +m1 − (c2 +0 + 1) ˙m1 = 0. +(107) +If we solve this equation we get +m1 = +1 +� +c2 +0 + 1 +� +a1e +√ +c2 +0+1z − a2e +√ +c2 +0+1z� +(108) +From (101) we obtain m2 = ˙m1 and m3 = 1 +c0( ¨m1 + m1). +Subcase 2: m1 = 0 +With the same computation as above, we have the following theorem: +Theorem 4.17. Let (α; β) be a curve pair of constant breadth in (M, gf). If α is a spacelike +asymptotic curve (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 = 0, then the +curve β is given by +β(s∗) = α(s)+ +� +b1 cos +�� +τds +� ++ b2 sin +�� +τds +�� +Y (s)+ +� +−b1 sin +�� +τds +� ++ b2 cos +�� +τds +�� +U(s). +4.3.2 +Case (For Principal line) +In this case we have the two following theorems: +Theorem 4.18. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be spacelike +principal line (with ǫ2 = −1 and ǫ3 = 1) and a general helix then β is given by +β(s∗) = α(s) + m1T + m2Y + m3U, +(109) +where +m1 = +1 +√ +1 + c2 +� +a1e +√ +1+c2θ − a2e− +√ +1+c2θ� +, +m2 = c +� +m1 cosh θdθ and m3 = −c +� +m1 sinh θdθ. +17 + +Theorem 4.19. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line +(with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 = 0 then α is general helix or +α is planar curve and the curve β is expressed as +β(s∗) = α(s) + c2Y + c3U, +(110) +where c2 and c3 are constants. +Acknowledgments +The author would like to thank the anonymous Referees for their comments and suggestions. All +many thanks to professor Ferdag Kahraman from Ahi Evran University (Turkish) for their remarks +and suggestions. +References +[1] B. Altunkaya, F. Kahraman, Curves of constant breadth according to Darboux frame. Com- +mun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66, (2), 44–52 (2017). +[2] B. 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J. 5, 179–184 (1914). +[8] M. Gningue, A. Ndiaye, R. Nkunzimana, Biharmonic Curves in a Strict Walker 3-Manifold. +Int. J. Math. Math. Sci., 1–6 (2022). +[9] O. Kose, Some properties of ovals and curves of constant width in a plane, Doga Sci. J. Serial +B (8), 2, 119-126 (1984). +[10] A. Magden and O. Kose, On the curves of constant breadth in E4 space, Turkish J. Math., 21, +277-284 (1997). +[11] R. M. Solow, Quarterly Journal of Economics, 70, 6594 (1956). +18 + diff --git a/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/load_file.txt b/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f9f9a2e729af3651dd2cbc5beb27a89ba22fdf1 --- /dev/null +++ b/5NE1T4oBgHgl3EQfSwPe/content/tmp_files/load_file.txt @@ -0,0 +1,462 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf,len=461 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='03071v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='DG] 8 Jan 2023 Curves of Constant Breadth According to Darboux Frame in a Strict Walker 3-Manifold Ameth Ndiaye* D´epartement de Math´ematiques, FASTEF, UCAD, Dakar, Senegal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Abstract In this paper, we investigate the differential geometry properties of curves of constant breadth according to Darboux frame in a given strict Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The considered curves are lying on a timelike surface in the Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' MSC: 53B25 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 53C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Keywords: Darboux frame, curvature, torsion, constant breadth curve, Walker 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 1 Introduction The study of curves of constant breadth were defined first in 1778 by Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then, Solow [11] investigated the curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Kose, Magden and Yilmaz in [9, 10] studied plane curves of constant breadth in Euclidean spaces E3 and E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Fujiwara [7] defined constant breadth for space curves and obtained a problem to determine whether there exists space curve of con- stant breadth or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Furthermore, Blaschke [3] defined the curves of constant breadth on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In [2], Altunkaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' defined null curves of constant breadth in Minkowski 4-space and obtain a characterization of these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also Altunkaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' in [1] investigate constant breadth curves on a surface according to Darboux frame and give some characterizations of these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Motivated by the above papers, we investigate the geometries of curves of constant breadth accord- ing to Darboux frame in a Strict Walker 3-manifold which is a Lorentzian three-manifold admitting a parallel null vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' It is known that Walker metrics have served as a powerful tool of con- structing interesting indefinite metrics which exhibit various aspects of geometric properties not given by any positive definite metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' For more details about Walker 3-manifold see [5,6,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 2 Preliminaries A Walker n-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r- planes, with r ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The canonical forms of the metrics were investigated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Walker ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' E–mail: ameth1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='ndiaye@ucad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='sn (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Ndiaye) 1 Walker has derived adapted coordinates to a parallel plan field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Hence, the metric of a three- dimensional Walker manifold (M, gǫ f) with coordinates (x, y, z) is expressed as gǫ f = dx ◦ dz + ǫdy2 + f(x, y, z)dz2 (1) and its matrix form as gǫ f = \uf8eb \uf8ed 0 0 1 0 ǫ 0 1 0 f \uf8f6 \uf8f8 with inverse (gǫ f)−1 = \uf8eb \uf8ed −f 0 1 0 ǫ 0 1 0 0 \uf8f6 \uf8f8 for some function f(x, y, z), where ǫ = ±1 and thus D = Span∂x as the parallel degenerate line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Notice that when ǫ = 1 and ǫ = −1 the Walker manifold has signature (2, 1) and (1, 2) respectively, and therefore is Lorentzian in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In this study we take ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' It follows after a straightforward calculation that the Levi-Civita connection of any metric (1) is given by: ∇∂x∂z = 1 2fx∂x, ∇∂y∂z = 1 2fy∂x, ∇∂z∂z = 1 2(ffx + fz)∂x + 1 2fy∂y − 1 2fx∂z (2) where ∂x, ∂y and ∂z are the coordinate vector fields ∂ ∂x, ∂ ∂y and ∂ ∂z , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Hence, if (M, gǫ f) is a strict Walker manifolds i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=', f(x, y, z) = f(y, z), then the associated Levi-Civita connection satisfies ∇∂y∂z = 1 2fy∂x, ∇∂z∂z = 1 2fz∂x − 1 2fy∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (3) Note that the existence of a null parallel vector field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='e f = f(y, z)) simplifies the non-zero components of the Christoffel symbols and the curvature tensor of the metric gǫ f as follows: Γ1 23 = Γ1 32 = 1 2fy, Γ1 33 = 1 2fz, Γ2 33 = −1 2fy (4) Let now u and v be two vectors in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Denoted by (⃗i,⃗j,⃗k) the canonical frame in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The vector product of u and v in (M, gǫ f) with respect to the metric gǫ f is the vector denoted by u×v in M defined by gǫ f(u × v, w) = det(u, v, w) (5) for all vector w in M, where det(u, v, w) is the determinant function associated to the canonical basis of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If u = (u1, u2, u3) and v = (v1, v2, v3) then by using (5), we have: u × v = ����� u1 v1 u2 v2 ���� − f ���� u2 v2 u3 v3 ���� � ⃗i − ǫ ���� u1 v1 u3 v3 ����⃗j + ���� u2 v2 u3 v3 ����⃗k (6) 2 3 Darboux equations in Walker 3-manifold Let α : I ⊂ R −→ (M, gǫ f) be a curve parametrized by its arc-length s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The Frenet frame of α is the vectors T, N and B along α where T is the tangent, N the principal normal and B the binormal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' They satisfied the Frenet formulas \uf8f1 \uf8f2 \uf8f3 ∇TT(s) = ǫ2κ(s)N(s) ∇TN(s) = −ǫ1κT(s) − ǫ3τB(s) ∇TB(s) = ǫ2τ(s)N(s) (7) where κ and τ are respectively the curvature and the torsion of the curve α, with ǫ1 = gf(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' ǫ2 = gf(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' N) and ǫ3 = gf(B, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Starting from local coordinates (x, y, z) for which (1) holds, it is easy to check that e1 = ∂y, e2 = 2 − f 2 √ 2 ∂x + 1 √ 2∂z, e3 = 2 + f 2 √ 2 ∂x − 1 √ 2∂z are local pseudo-orthonormal frame fields on (M, gǫ f), with gǫ f(e1, e1) = ǫ, gǫ f(e2, e2) = 1 and gǫ f(e3, e3) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Thus the signature of the metric gǫ f is (1, ǫ, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If we choose ǫ = 1 then, pseudo-orthonormal frame is formed by two spacelike vectors and one timelike vector and If we choose ǫ = −1 then, pseudo-orthonormal frame is formed by one spacelike vector and two timelike vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' For both cases we obtain Lorentzian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In this work we assume that ǫ = 1 Now we suppose that the curve α lies on a timelike surface S in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let U be the unit normal vector of S, then the Darboux frame is given by {T, Y, U}, where T is the tangent vector of the curve α(s) and Y = U × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Case 1: Let α be timelike curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then the tangent vector T is timelike (ǫ1 = −1), the normal vector N and the binormal vector B are spacelike, that is (ǫ2 = ǫ3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since S is timelike, the unit normal vector U is spacelike and so Y becomes spacelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The usual transformations between the Walker Frenet frame and the Darboux takes the form Y = cos θN + sin θB (8) U = − sin θN + cos θB, (9) where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Derivating Y along the curve alpha we get ∇TY = cos θ∇TN − θ′ sin θN + sin θ∇TB + θ′ cos θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Using the Frenet equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='7) we have ∇T Y = cos θ(κT − ǫ3τB) − θ′ sin θN + sin θ(ǫ2τN) + θ′ cos θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Now we suppose that the principal normal and the binormal have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' then we get ∇TY = κ cos θT + (θ′ − τ)U (10) The same calculus gives ∇TU = −κ sin θT − (θ′ − τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (11) 3 Then the Walker Darboux equation is expressed as \uf8f1 \uf8f2 \uf8f3 ∇TT = κgY + κnU ∇TY = κgT + τgU ∇TU = κnT − τgY, (12) where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also, (12) gives gǫ f (∇T Y, U) = τg = θ′ − τ, (13) gǫ f (∇TT, Y ) = κg = κ cos θ, (14) gǫ f (∇TT, U) = κn = −κ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (15) Case 2: Let α be spacelike curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then the tangent vector T is spacelike (ǫ1 = 1), the normal vector N is spacelike (ǫ2 = 1) and the binormal vector B is timelike (ǫ3 = −1) or normal vector N is timelike (ǫ2 = −1) and the binormal vector B is spacelike (ǫ3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So we have two following subcases: i): ǫ2 = 1 and ǫ3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then the usual transformations between the Walker Frenet frame and the Darboux takes the form Y = cosh θN + sinh θB (16) U = sinh θN + cosh θB, (17) where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since ∇TT = κN, we have ∇TT = −κ sinh θY + κ cosh θU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (18) Derivating Y along the curve alpha we get ∇T Y = −κ sinh θT + (θ′ + τ)U (19) The same calculus gives ∇TU = −κ cosh θT + (θ′ + τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (20) Then the Walker Darboux equation is expressed as \uf8f1 \uf8f2 \uf8f3 ∇TT = −κgY + κnU ∇TY = −κgT + τgU ∇TU = −κnT + τgY, (21) where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also, (21) gives gǫ f (∇TY, U) = τg = θ′ + τ, (22) gǫ f (∇TT, Y ) = κg = κ sinh θ, (23) gǫ f (∇TT, U) = κn = κ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (24) 4 ii): ǫ2 = −1 and ǫ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then the usual transformations between the Walker Frenet frame and the Darboux takes the form Y = sinh θN + cosh θB (25) U = cosh θN + sinh θB, (26) where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since ∇TT = −κN, we have ∇TT = −κ cosh θY + κ sinh θU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (27) Derivating Y with respect to s we get ∇TY = −κ cosh θT + (θ′ − τ)U (28) Derivating Y with respect to s alpha we get ∇TU = −κ sinh θT + (θ′ − τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (29) Then the Walker Darboux equation is expressed as \uf8f1 \uf8f2 \uf8f3 ∇TT = −κgY + κnU ∇TY = −κgT + τgU ∇TU = −κnT + τgY, (30) where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also, (30) gives gǫ f (∇T Y, U) = τg = θ′ − τ, (31) gǫ f (∇TT, Y ) = κg = κ cosh θ, (32) gǫ f (∇TT, U) = κn = κ sinh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (33) 4 Space curves of constant breadth According to Darboux Frame in Walker manifold In this section, we define space curves of constant breadth in the three dimensional Walker mani- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' A curve α : I → (M, gǫ f) in the three-dimensional Walker manifold (M, gǫ f) is called a curve of constant breadth if there exists a curve β : I → Mf such that, at the corresponding points of curves, the parallel tangent vectors of α and β at α(s) and β(s⋆) at s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' s⋆ ∈ I are opposite directions and the distance gǫ f(β − α, β − α) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In this case, (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) is called a pair curve of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let now (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) be a pair of unit speed curves of constant breadth and s, s⋆ be arc-length of α and β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' We suppose that the curve α lies on a timelike surface in Mf, then it has Darboux frame in addition to Frenet frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we may write the following equation: β(s⋆) = α(s) + m1(s)T(s) + m2(s)Y (s) + m3(s)U(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (34) where mi(i = 1, 2, 3) are smooth functions of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1 Case where α is timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Differentiating (34) with respect to s and using (12) we obtain dβ ds = dβ ds⋆ ds⋆ ds = T ⋆(s⋆)ds⋆ ds = (1 + m′ 1 + m2κg + m3κn)T(s) +(m′ 2 + m1κg − m3τg)Y (s) +(m′ 3 + m2τg + m1κn)U(s), (35) where T ⋆ denotes the unit tangent vector of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since T = −T ∗, from the equations in (35) we have \uf8f1 \uf8f2 \uf8f3 m′ 1 = −m2κg − m3κn − h(s) m′ 2 = −m1κg + m3τg m′ 3 = −m2τg − m1κn, (36) where h(s) = ds⋆ ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' We assume that (α, β) is a curve pair of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since α is a timelike curve and the vectors Y and U are spacelike vectors, we have ∥β − α∥ = −m2 1 + m2 2 + m2 3 = constant, (37) which imlplies that −m1 dm1 ds + m2 dm2 ds + m3 dm3 ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (38) If we combine (36) and (38), we get m1h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (39) If α and β are curves of constant breadth then m1 = 0 or h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 ̸= 0 (that is h(s) = 0) then d = m1T(s) + m2Y (s) + m3U(s) becomes a constant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So β(s∗) is a translation of α along the constant vector d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also h(s) = 0 gives s∗ = −s + c, where c is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Now, we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1 Case (For geodesic curves) Let α be non-straight line geodesic curve on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then κg = κ cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' As κ ̸= 0, we get cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So it implies that κn = −κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (36), we have following differential equation system \uf8f1 \uf8f2 \uf8f3 m′ 1 = m3κ − h(s) m′ 2 = −m3τ m′ 3 = m1κ + m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (40) By using (40), we obtain the following differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 1 κ �1 κ(m′ 1 + h) �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κ(m′ 1 + h) �′ + �τ κ �2 (m′ 1+h)+ �τ κ �′ κ τ m1−m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (41) 6 Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If we write h(s) = 0 in equation (41), we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 1 κ �1 κm′ 1 �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κm′ 1 �′ + ��τ κ �2 − 1 � m′ 1 + �τ κ �′ κ τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (42) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a timelike geodesic curve lying a timelike surface in M and let (α, β) be a pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is a non-zero constant then α is a general helix in the three dimensional Walker manifold (M, gǫ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also the curve β is given as: β(s⋆) = α(s) + m1T(s) + m2Y (s) (43) where m2 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non zero constant, then from (42) we obtain that � τ κ �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also from the first and second equations of (40) we get m3 = 0 and m2 is a real constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a timelike geodesic curve and a general helix lying a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is not zero, then the curve β can be expressed as one of the following cases: β(s∗) = α(s) + m1T(s) + 1 c0 ( ¨m1 − m1)Y (s) + ˙m1U(s) (44) where i) m1 = 1 √ c2 0−1 � a1 sin( � c2 0 − 1z) − a2 cos( � c2 0 − 1z) � + a3, c2 0 − 1 > 0 ii) m1 = a1 2 z2 + a2z + a3, c2 0 − 1 = 0 iii) m1 = 1 √ 1−c2 0 � a1 sinh( � 1 − c2 0z) + a2 cosh( � 1 − c2 0z) � + a3, c2 0 − 1 < 0 where z = � κds and a1, a2, a3 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let us consider that α is timelike geodesic curve and a general helix in Wlaker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we have τ κ = c0 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (42), we have �1 κ �1 κm′ 1 �′�′ + � c2 0 − 1 � m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (45) By means of changing of the independant variable s with z = � κds, from (45) we obtain m′ 1 = dm1 ds = dm1 dz dz ds = ˙m1κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 + (c2 0 − 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (46) 7 If we solve this equation we get m1 = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 √ c2 0−1 � a1 sin( � c2 0 − 1z) − a2 cos( � c2 0 − 1z) � + a3, if c2 0 − 1 > 0 a1 2 z2 + a2z + a2, if c2 0 − 1 = 0 1 √ 1−c2 0 � a1 sinh( � 1 − c2 0z) + a2 cosh( � 1 − c2 0z) � + a3, if c2 0 − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (40) we obtain m3 = ˙m1 and m2 = 1 c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If we take m1 = 0 in the equation (40), we get \uf8f1 \uf8f2 \uf8f3 h(s) = m3κ m′ 2 = −m3τ m′ 3 = m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (47) Since m3 = h κ, m2 = 1 τ m′ 3 = 1 τ �h κ �′, we get �1 τ �h κ �′�′ + �h κ � τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (48) If we put y = h κ, the equation (48) becomes y′′ − τ ′ τ y′ + τ 2y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (49) For solving the equation (49), we put the new variable dw ds = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then � y′ = dy dw dw ds = ˙yτ y′′ = d2y dw2τ 2 + dy dwτ ′ (50) If we put the equation (50) in the equation (49) we obtain d2y dw2 + y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (51) and the solution of (51) is y = b1 cos w + b2 sin w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we have h(s) = κ � b1 cos �� τds � + b2 sin �� τds �� (52) m2 = h κ = b1 cos �� τds � + b2 sin �� τds � (53) m3 = 1 τ �h κ �′ = −b1 sin �� τds � + b2 cos �� τds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (54) So we give the following theorem Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of constant breadth curve in (M, gf) where α is a timelike geodesic curve lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0, then the curve β is given by β(s∗) = α(s)+ � b1 cos �� τds � + b2 sin �� τds �� Y (s)+ � −b1 sin �� τds � + b2 cos �� τds �� U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2 Case (For asymptotic lines) Let α be non-straight line asymptotic line on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then κn = −κ sin θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' As κ ̸= 0, we get sin θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So it implies that κg = κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (36), we have following differential equation system \uf8f1 \uf8f2 \uf8f3 m′ 1 = −m2κ − h(s) m′ 2 = −m1κ − m3τ m′ 3 = m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (55) By using (55), we get 1 κ �1 κ(m′ 1 + h) �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κ(m′ 1 + h) �′ + �τ κ �2 (m′ 1+h)+ �τ κ �′ κ τ m1−m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (56) Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If we take as h(s) = 0 in equation (56), we get following differential equation 1 κ �1 κm′ 1 �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κm′ 1 �′ + ��τ κ �2 − 1 � m′ 1 + �τ κ �′ κ τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (57) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a timelike asymptotic line lying a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non-zero constant then α is a general helix in the three dimensional Walker manifold (M, gǫ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also the curve β is given as: β(s⋆) = α(s) + m1T(s) + m3U(s) (58) where m3 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non zero constant, then from (57) we obtain that � τ κ �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also from the first and third equation of (55) we get m2 = 0 and m3 is a real constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a timelike asymptotic line lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is not zero, then the curve β can be expressed as one of the following cases: β(s∗) = α(s) + m1T(s) − ˙m1Y (s) + 1 c0 ( ¨m1 − m1)U(s), (59) where i) m1 = 1 √ c2 0−1 � a1 sin( � c2 0 − 1z) − a2 cos( � c2 0 − 1z) � + a3, c2 0 − 1 > 0 ii) m1 = a1 2 z2 + a2z + a3, c2 0 − 1 = 0 iii) m1 = 1 √ 1−c2 0 � a1 sinh( � 1 − c2 0z) + a2 cosh( � 1 − c2 0z) � + a3, c2 0 − 1 < 0 where z = � κds and a1, a2, a3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='6) is done similarly to the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3) Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If we take as m1 = 0 in (55) we get following differential equation system \uf8f1 \uf8f2 \uf8f3 h(s) = −m2κ m′ 2 = −m3τ m′ 3 = m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (60) Then we give the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) be a curve pair of constant breadth in (M, gf) where α is a timelike asymptotic curve lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0, then the curve β is given by β(s∗) = α(s)+ � −b1 cos �� τds � − b2 sin �� τds �� Y (s)+ � −b1 sin �� τds � + b2 cos �� τds �� U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='7) is done similarly to the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3 Case (For Principal line) We suppose that α is a non-planar timelike principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we have τg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then it follows that τ = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' By using (36), we have the following differential equation system \uf8f1 \uf8f2 \uf8f3 m′ 1 = m3κ sin θ − m2κ cos θ − h(s) m′ 2 = −m1κ cos θ m′ 3 = m1κ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (61) By mean of changing of the independant variable s with θ = � τds, we get \uf8f1 \uf8f2 \uf8f3 ˙m1 = φ(m3 sin θ − m2 cos θ) − g(θ) ˙m2 = −m1φ cos θ ˙m3 = m1φ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (62) where g(θ) = (− ds dθ − ds∗ dθ ) and φ = κ τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In here we denote the derivative with respect to θ with ”.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From the equations in (62) we have .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 + ¨g − d dθ � ˙φ φ( ˙m1 + g) � − d dθ(φ2m1) + ( ˙m1 + g) − ˙φ � − sin θ � m1φ cos θdθ + cos θ � m1φ sin θdθ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (63) Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' In this case, we give the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a non-planar timelike principal line and a general helix then β is given by one of the following cases: β(s∗) = α(s) + m1T(s) − c � m1 cos θdθY (s) + c � m1 sin θdθU(s), (64) where 10 i) m1 = 1 √ 1−c2 � a1 sin( √ 1 − c2θ) − a2 cos( √ 1 − c2θ) � + a3, 1 − c2 > 0 ii) m1 = a1 2 θ2 + a2θ + a3, c2 − 1 = 0 iii) m1 = 1 √ c2−1 � a1 sinh( √ c2 − 1θ) + a2 cosh( √ c2 − 1θ) � + a3, 1 − c2 < 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If h(s) = 0 then g(θ) = 0 and from (63) we have .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 − d dθ � ˙φ φ ˙m1 � − d dθ(φ2m1) + ˙m1 − ˙φ � − sin θ � m1φ cos θdθ + cos θ � m1φ sin θdθ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (65) If α is helix curve then φ = κ τ = c = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (65) we have .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 + (1 − c2) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (66) Then the solution is m1 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 √ 1−c2 � a1 sin( √ 1 − c2θ) − a2 cos( √ 1 − c2θ) � + a3, if 1 − c2 > 0 a1 2 θ2 + a2θ + a3, if 1 − c2 = 0 1 √ c2−1 � a1 sinh( √ c2 − 1θ) + a2 cosh( √ c2 − 1θ) � + a3, if 1 − c2 < 0, where θ = � τdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The case where m1 = 0, we have the following the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a non-planar timelike principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0 then α is general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The curve β is expressed as β(s∗) = α(s) + c2Y (s) + c3U(s), (67) where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (63) we have ¨g − d dθ � ˙φ φg � + g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (68) On the other hand, from (61) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0 and from (62) g = φ(−c2 cos θ + c3 sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (69) By considering (68) and (69) with together, we get ˙φ(c2 sin θ + c3 cos θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (70) Then we have ˙φ = 0 or c2 sin θ + c3 cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If c2 sin θ + c3 cos θ = 0 then we have that θ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So α becomes a planar curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' It is a contridiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So ˙φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we obtain that φ = κ τ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Thus α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2 Case where α is spacelike and ǫ2 = 1 and ǫ3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Here we suppose that the curve α is spacelike and lying on a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Differentiating (34) with respect to s and using (21) we obtain dβ ds = dβ ds⋆ ds⋆ ds = T ⋆ds⋆ ds = (1 + m′ 1 − m2κg − m3κn)T +(m′ 2 − m1κg + m3τg)Y +(m′ 3 + m2τg + m1κn)U, (71) where T ⋆ denotes the tangent vector of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since T = −T ∗, from the equation in (35) we have \uf8f1 \uf8f2 \uf8f3 m′ 1 = m2κg + m3κn − h(s) m′ 2 = m1κg − m3τg m′ 3 = −m2τg − m1κn, (72) where h(s) = ds∗ ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since α is spacelike and ǫ2 = 1 andǫ3 = −1, then, if we assume that (α, β) is a curve pair of constant breadth, we have ∥β − α∥ = m2 1 + m2 2 − m2 3 = constant, (73) which imlplies that m1 dm1 ds + m2 dm2 ds − m3 dm3 ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (74) If we combine (72) and (74) we get m1(2m′ 1 + h(s)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (75) If α and β are curves of constant breadth then m1 = 0 or 2m′ 1 − h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Now we investigate the case where α is geodesic curve or principal line curve because κn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1 Case (For geodesic curves) Let α be non-straight line geodesic curve on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then κg = κ sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' As κ ̸= 0, we get sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So it implies that κn = κ, τg = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (72), we have the following differential equation system \uf8f1 \uf8f2 \uf8f3 m′ 1 = m3κ − h(s) m′ 2 = −m3τ m′ 3 = −m1κ − m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (76) From (76) we have \uf8f1 \uf8f2 \uf8f3 m3 = 1 κ(m′ 1 + h) m′ 2 = − τ κ(m′ 1 + h) m2 = − 1 τ � ( 1 κ(m′ 1 + h))′ + m1κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (77) 12 Differentiating the third equation of (76) with respect to s and using the first, the second and the third equations of (77), we obtain the following equation: 1 κ �1 κ(m′ 1 + h) �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κ(m′ 1 + h) �′ − �τ κ �2 (m′ 1+h)− �τ κ �′ κ τ m1+m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (78) Subcase 1: m1 ̸= 0 (h(s) = −2m′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The equation (78) becomes 1 κ �1 κm′ 1 �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κm′ 1 �′ − ��τ κ �2 + 1 � m′ 1 + �τ κ �′ κ τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (79) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a geodesic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) be a pair of unit speed curves of constant breadth where α is spacelike (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non-zero constant then m3 = 0 and α is a general helix in the three dimensional Walker manifold (M, gǫ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also the curve β is given as: β(s⋆) = α(s) + m1T + cY (80) where c is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non zero constant, then from (79) we obtain that � τ κ �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also from the second and third equation of (76) we get m3 = 0 because h = 0 and m2 is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a geodesic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of unit speed curves of constant breadth where α is spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is not zero, then the curve β can be expressed as one of the following cases: β(s∗) = α(s) + m1T + 1 c0 ( ¨m1 − m1)Y + ˙m1U, (81) where m1 = 1 √ 1+c2 0 � a1e √ 1+c2 0θ − a2e−√ 1+c2 0θ� , m3 = − ˙m1 and m2 = 1 c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let us consider that α is a general helix in Wlaker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we have τ κ = c0 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (79), we have �1 κ �1 κm′ 1 �′�′ − � c2 0 + 1 � m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (82) By means of changing of the independant variable s with z = � κds, we obtain m′ 1 = dm1 ds = dm1 dz dz ds = ˙m1κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (82), we get .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 − (c2 0 + 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (83) If we solve this equation we get m1 = 1 � 1 + c2 0 � a1e √ 1+c2 0θ − a2e−√ 1+c2 0θ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (84) From (77) we have m3 = − ˙m1 and m2 = 1 c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 13 Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a geodesic spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface on Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0 then the curve β is expressed as β(s∗) = α(s) + cY, (85) where c is a constant real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m′ 1 = 0 then h = 0 and from (76) we have m3 = 0 and m2 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2 Case (For Principal line) If α is principal line, then τg = 0 and τ = −θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (72) \uf8f1 \uf8f2 \uf8f3 m′ 1 = m2κ sinh θ + m3κ cosh θ − h(s) m′ 2 = m1κ sinh θ m′ 3 = −m1κ cosh θ, (86) By mean of changing of the independant variable s with θ = � τds, we get \uf8f1 \uf8f2 \uf8f3 ˙m1 = m3 κ τ cosh θ + m2 κ τ sinh θ − h(s) τ(s) ˙m2 = m1 κ τ sinh θ ˙m3 = −m1 κ τ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (87) Denoted by h(s) τ(s) = g(θ) and κ τ = φ, we have \uf8f1 \uf8f2 \uf8f3 ˙m1 = φ(m3 cosh θ + m2 sinh θ) − g(θ) ˙m2 = m1φ sinh θ ˙m3 = −m1φ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (88) From the equations in (88) we have \uf8f1 \uf8f2 \uf8f3 1 φ( ˙m1 + g) = m3 cosh θ + m2 sinh θ ˙m2 sinh θ + ˙m3 cosh θ = −m1φ ˙m2 cosh θ = −m3 sinh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (89) Differentiating the first equation in (88), we get .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 + ¨g − d dθ � ˙φ φ( ˙m1 + g) � + d dθ(φ2m1) − ( ˙m1 + g) − ˙φ � cosh θ � m1φ sinh θdθ − sinh θ � m1φ cosh θdθ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (90) Subcase 1: m1 ̸= 0 (m′ 1 = −h 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m′ 1 = −h 2 then ˙m1 = −g 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (90) we obtain −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 + d dθ � ˙φ φ ˙m1 � + d dθ(φ2m1) + ˙m1 − ˙φ � cosh θ � m1φ sinh θdθ − sinh θ � m1φ cosh θdθ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (91) 14 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be principal line and a general helix then β is given by β(s∗) = α(s) + m1T + m2Y + m3U, (92) where m1 = 1 √ 1 + c2 � a1e √ 1+c2θ − a2e− √ 1+c2θ� , m2 = c � m1 sinh θdθ and m3 = −c � m1 cosh θdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If α is helix curve then φ = κ τ = c = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (91) we have .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 − (1 + c2) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (93) m1 = 1 √ 1 + c2 � a1e √ 1+c2θ − a2e− √ 1+c2θ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (94) Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From the equations in (72) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The first equation in (72) gives tanh θ = −c2 c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (95) Then θ is a constant and we have τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0 then α is planar curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The curve β is expressed as β(s∗) = α(s) + c2Y + c3U, (96) where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3 Case where α is spacelike and ǫ2 = −1 and ǫ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a spacelike with ǫ2 = −1 and ǫ3 = 1 lying on a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Differentiating (34) with respect to s and using (30) we obtain \uf8f1 \uf8f2 \uf8f3 m′ 1 = m2κg + m3κn − h(s) m′ 2 = m1κg − m3τg m′ 3 = −m2τg − m1κn, (97) where h(s) = ds∗ ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since α is spacelike and ǫ2 = −1 andǫ3 = 1, then, if we assume that (α, β) is a curve pair of constant breadth, we have ∥β − α∥ = m2 1 − m2 2 + m2 3 = constant, (98) 15 which imlplies that m1 dm1 ds + m2 dm2 ds − m3 dm3 ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (99) If we combine (97) and (99) we get m1h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (100) If α and β are curves of constant breadth then m1 = 0 or h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 ̸= 0 (that is h(s) = 0) then d = m1T + m2Y + m3U becomes a constant vector because d′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So β(s∗) is a translation of α along the constant vector d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also h(s) = 0 gives s∗ = −s + c, where c is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Since κg ̸= 0, here we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special case (asymptotic line or principal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='1 Case (For Asymptotic line) Let α be non-straight line asymptotic line on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then κn = κ sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' As κ ̸= 0, we get cosh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So it implies that κg = κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (97), we have following differential equation system \uf8f1 \uf8f2 \uf8f3 m′ 1 = m2κ − h(s) m′ 2 = m1κ + m3τ m′ 3 = −m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (101) By differentiating the second equation in (101) with respect to s and using the first and third equa- tions in (101), we get 1 κ �1 κ(m′ 1 + h) �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κ(m′ 1 + h) �′ − �τ κ �2 (m′ 1+h)+ �τ κ �′ κ τ m1 −m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (102) Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' The equation (102) is given by 1 κ �1 κm′ 1 �′′ + ��1 κ �′ − 1 τ �τ κ �′� �1 κm′ 1 �′ − ��τ κ �2 + 1 � m′ 1 + �τ κ �′ κ τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (103) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a asymptotic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) be a pair of unit speed curves of constant breadth where α is spacelike (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non-zero constant then m2 = 0 and α is a general helix in the three dimensional Walker manifold (M, gǫ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also the curve β is given as: β(s⋆) = α(s) + m1T + m3U (104) where m3 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is non zero constant, then from (103) we obtain that � τ κ �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Also from the first and third equation of (101) we get m2 = 0 and m3 is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 16 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be a asymptotic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair of unit speed curves of constant breadth where α is timelike curve and lying in a timelike surface Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 is not zero, then the curve β can be expressed as one of the following cases: β(s∗) = α(s) + m1T + ˙m1Y + 1 c0 ( ¨m1 + m1)U, (105) where m1 = 1 � c2 0 + 1 � a1e √ c2 0+1z − a2e √ c2 0+1z� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let us consider that α is a general helix in Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Then we have τ κ = c0 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' From (103), we have �1 κ �1 κm′ 1 �′�′ − � c2 0 + 1 � m′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (106) By means of changing of the independant variable s with z = � κds, we obtain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' m1 − (c2 0 + 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' (107) If we solve this equation we get m1 = 1 � c2 0 + 1 � a1e √ c2 0+1z − a2e √ c2 0+1z� (108) From (101) we obtain m2 = ˙m1 and m3 = 1 c0( ¨m1 + m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Subcase 2: m1 = 0 With the same computation as above, we have the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' β) be a curve pair of constant breadth in (M, gf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If α is a spacelike asymptotic curve (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0, then the curve β is given by β(s∗) = α(s)+ � b1 cos �� τds � + b2 sin �� τds �� Y (s)+ � −b1 sin �� τds � + b2 cos �� τds �� U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='2 Case (For Principal line) In this case we have the two following theorems: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be spacelike principal line (with ǫ2 = −1 and ǫ3 = 1) and a general helix then β is given by β(s∗) = α(s) + m1T + m2Y + m3U, (109) where m1 = 1 √ 1 + c2 � a1e √ 1+c2θ − a2e− √ 1+c2θ� , m2 = c � m1 cosh θdθ and m3 = −c � m1 sinh θdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 17 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Let α be principal line (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' If m1 = 0 then α is general helix or α is planar curve and the curve β is expressed as β(s∗) = α(s) + c2Y + c3U, (110) where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Acknowledgments The author would like to thank the anonymous Referees for their comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' All many thanks to professor Ferdag Kahraman from Ahi Evran University (Turkish) for their remarks and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Altunkaya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Kahraman, Curves of constant breadth according to Darboux frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Ank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' A1 Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 66, (2), 44–52 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Altunkaya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Kahraman, Null curves of constant breadth in Minkowski 4-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Ank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' 68, (1), 451–456 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' [3] Blaschke W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=', Einige Bemerkungen uber Kurven und Flachen konstanter Breite, Ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' Verh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'} +page_content=' sachs.' metadata={'source': 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sha256:c32958deacc25d24d014cbe60bb156fecc9d4efb66f775470fa7ea7d059d4427 +size 167348 diff --git a/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/2301.13448v1.pdf.txt b/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/2301.13448v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..af865e5eea2ef6acfded79e72bac8c73084d5118 --- /dev/null +++ b/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/2301.13448v1.pdf.txt @@ -0,0 +1,747 @@ +arXiv:2301.13448v1 [nlin.AO] 31 Jan 2023 +Delay, resonance and the Lambert W function +Kenta Ohira1 and Toru Ohira2 +1Future Value Creation Research Center, +Graduate School of Informatics, Nagoya University, Japan +2Graduate School of Mathematics, Nagoya University, Japan +February 1, 2023 +Abstract +We present here a connection between the solutions of the transcenden- +tal trigonometric equation and the Lambert W function. This connection +emerged through an analysis of resonant conditions with a recently pro- +posed simple delay differential equation that shows transient oscillatory +behaviors. We investigate and present the connection both analytically +and numerically. +1 +Introduction +In the various fields including mathematics, biology, physics, engineering, eco- +nomics, and so on, there have been interests in investigating the effect of delays +in the system.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). Typically, delays intro- +duce oscillations and complex behaviors to otherwise simple and well-behaved +systems. Longer delays are known to induce an increase in the complexity of +dynamics. The representative example is the Mackey–Glass equation[8] which +shows the sequence of the monotonic convergence, transient oscillations, persis- +tent oscillations, and chaotic dynamics as the delay parameter in the feedback +function becomes longer. +The main mathematical approaches and modeling tools are “Delay Differen- +tial Equations”. Mathematical analysis of such delay systems has been posing +challenges. Though understanding of the delay systems have been gradually +gained (e.g.[15]), there is more to be investigated and explored. One recent an- +alytical approach to simple delay differential equations is an application of the +Lambert W function. It has been shown that the solution of the simple delay +differential equation can be expressed using the W function. +In this paper, we follow this path of analyzing a simple delay differential +equation using the W function. Specifically, we analyze the transient oscillatory +behaviors of a delay differential equation that shows a resonant behavior[16]. In +this resonance, the optimal height of the power spectrum of the dynamical +trajectory is observed with the suitably tuned value of the delay parameter. +1 + +We focus on the appearance of the power spectrum peaks and found that it +relates to the transcendental trigonometric equation. That condition, on the +other hand, can be also expressed in terms of the W function. We connect these +two approaches both analytically and numerically to provide a new way that +the W function can be useful. +2 +Delay Differential Equation +Recently, we have proposed and studied the following delay differential equation: +dX(t) +dt ++ atX(t) = bX(t − τ) +(1) +where a ≥ 0, b ≥ 0, τ ≥ 0 are real parameters with τ interpreted as the delay. +This equation is a slight extension of much-studied Hayes’s equation[4], +dX(t) +dt ++ αX(t) = βX(t − τ) +(2) +where α, β are real constants. +Even though the apparent change from (2) to (1) is small with only the +second term changed to a linear function of time, their behaviors are quite +different. Particularly for (1), we have shown oscillatory transient dynamics +appear and disappear as the value of delay increases without losing asymptotic +stability of X = 0. +2.1 +Analysis +Let us first review some properties of (1). +When b = 0. +With the initial +condition X(t = 0) = X0, the solution to the equation is given as +X(t) = X0e− 1 +2 at2 +(3) +Thus, this solution is a gaussian shape. We also note in this case that (1) is the +equation for the ground state of the quantum simple harmonic oscillator with +the interpretation of t as a position rather than time (e.g.[18]). +The case that a = 0 is a special case of (2). In this case, the origin X = 0 is +asymptotically stable only in the range of +− π/2τ < b < 0. +(4) +For the general case with a > 0, b > 0 with the delay τ = 0, the solution +X(t = 0) = X0 is obtained as +X(t) = X0e− 1 +2 at2+bt +(5) +This is again a Gaussian with its peak at b/a. +2 + +For the case with a > 0, b > 0 with the delay τ → ∞, the dynamics is +influenced by the initial function for all 0 ≤ t. Thus for the initial function +X(t) = X0, (−τ ≤ t ≤ 0), we can replace the right hand side of equation (1) as +bX(t − τ) → X0. The solution can be obtained as +X(t) = X0e− 1 +2 at2(1 + b +� t +0 +e +1 +2 as2ds) = X0e− 1 +2 at2(1 + b +� π +2aerfi( +� a +2 t)) +(6) +where erfi(x) is the imaginary error function defined as +erfi(x) = +2 +√π +� x +0 +es2 ds +(7) +The shape of this function is also a single peaked function approaching to the +origin X = 0. +We now see one of the major differences between the equations (1) and (2). +In the latter, the asymptotic stability of X = 0 is lost for the larger delay with +0 < α < β, while in the former, it is kept even with the large delay for all +a > 0, b > 0. Also, even though both exhibit transient oscillations, (1) shows +coherent oscillations with the tuned value of the delay τ. We now turn our +attention to these resonating phenomena. +2.2 +Power Spectrum and Resonance +The transient dynamics of equation (1) are investigated through numerical sim- +ulations. Some examples are shown in Fig. 1. With zero delays, the shape of +the dynamics is the gaussian as derived in the previous subsection. The oscil- +latory behaviors arise on top of the gaussian trajectory with increasing delay. +Further increase of delay changes the oscillatory shape into trains of pulses with +decreasing height at the delay interval, and asymptotically the pulses disappear +leading to the gaussian shape (5). As mentioned, the asymptotic stability of +X = 0 does not change by the increasing delay in this parameter set. +This property is in contrast to that of equation (2) where the onset of the +oscillation by the increasing delay leads to the loss of stability. Thus, it is differ- +ent from stability switching phenomena (e.g.[17]) with the delay as the bifurca- +tion parameter. It is also different from the delay induced transient oscillation +(DITO)[19, 20]. The phenomena arise in coupled delay differential equations +exhibiting the prolonged duration of oscillatory behaviors with increasing delay. +We investigate these oscillatory behaviors for the case a > 0, b > 0, and +finite τ ̸= 0 by taking the Fourier transform of the equation (1). +iω ˆX(ω) + iad ˆX(ω) +dω += −b ˆX(ω)eiωτ +(8) +where +ˆX(ω) = +� ∞ +−∞ +eiωtX(t)dt +(9) +3 + +(B) +X(t) +t +(A) +X(t) +t +(E) +X(t) +t +(F) +X(t) +t +(C) +X(t) +t +(D) +X(t) +t +50 +100 +150 +200 +0 +20000 +40000 +60000 +80000 +100000 +120000 +50 +100 +150 +200 +0 +50 +100 +150 +200 +250 +300 +350 +50 +100 +150 +200 +0 +5 +10 +15 +20 +50 +100 +150 +200 +0 +2 +4 +6 +50 +100 +150 +200 +0.0 +0.5 +1.0 +1.5 +2.0 +50 +100 +150 +200 +0.0 +0.5 +1.0 +1.5 +Figure 1: Representative dynamics of the main equation (1) with different values +of the delays, τ. The parameters are set at a = 0.15, b = 6.0 with the initial +interval condition as X(t) = 0.1(−τ ≤ t ≤ 0). The values of the delays τ are +(A)2, (B)4, (C)7, (D)10, (E)20, (F)25. +4 + +The solution is given as +ˆX(ω) = CExp[− 1 +2aω2 + b +τaeiωτ] +(10) +with C as the integration constant. We can calculate the power spectrum from +equation(9). +S(ω) = | ˆX(ω)|2 = ˆX(ω) ˆ +X∗(ω) = C2Exp[−1 +aω2 + 2b +τa cos ωτ] +(11) +We have plotted this equation for the power spectrum for the various delays. +Results with the same parameter setting as in Fig. 1 are shown in Fig. 2. +In the previous work, we noted that the peak of the power spectrum shows +a maximum height with the tuned value of the delay. The higher peak indicates +a more coherent oscillation. It is in this sense that the resonance exists with the +delay as a tuning parameter. +2.3 +Peaks of the Power Spectrum +We now focus on the analysis of these power spectrum peaks. The appearance +and disappearance of the peaks in the power spectrum correspond to those of +oscillatory behavior. By taking the derivative of (11), we see the maximum and +minimum points of the power spectrum function occur at ω satisfying, +ω = −b sin ωτ +(12) +They are given by the intersection points of the two functions from both +sides of this condition. The position of the first peak corresponds to the second +non-zero smallest intersection point (the first one corresponds to the minimum +before the peak). +We can also infer the condition for the appearance of the series of power +spectrum peaks. Each peak appears when the intersection point is the tangent +point (Fig.3). This gives the following conditions for the n-th peak. +ω = −b sin ωτ, +1 = −bτ cos ωτ, +(2n − 1)π +τ +< ω < (2n − 1 +2)π +τ +, (n = 1, 2, . . .). +(13) +If we set λ = bτ,θ = ωτ, the above condition leads to +θ = −λ sin θ, +1 = −λ cosθ, +(2n − 1)π < θ < (2n − 1 +2)π, (n = 1, 2, . . . ). +(14) +From this set of equations, we can numerically estimate the solutions (θn, λn) +for each n. First, we can derive that θn and λn are related by +λ2 +n = θ2 +n + 1 +(15) +Then, the followings are obtained: +θ = − +� +θ2 + 1 sin θ, +(2n − 1)π < θ < (2n − 1 +2)π, (n = 1, 2, . . .), +(16) +5 + +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +5 +10 +15 +20 +25 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +5.0×1016 +1.0×1017 +1.5×1017 +2.0×1017 +(A) +ω +S(ω) +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1×108 +2×108 +3×108 +4×108 +5×108 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +200 +400 +600 +800 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10 +20 +30 +40 +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +200 +400 +600 +800 +(B) +ω +S(ω) +(C) +ω +S(ω) +(F) +ω +S(ω) +(E) +ω +S(ω) +(D) +ω +S(ω) +Figure 2: Representative power spectrums given by the equation (1) with dif- +ferent values of the delays τ. +The parameters are set as the same as Fig.1; +a = 0.15, b = 6.0, C = 1 with the initial interval condition as X(t) = 0.1(−τ ≤ +t ≤ 0). The values of the delays τ are (A)2, (B)4, (C)7, (D)10, (E)20, (F)25. +6 + +5 +10 +15 +20 +- 20 +- 10 +10 +20 +5 +10 +15 +20 +- 20 +- 10 +10 +20 +30 +(A) +(B) +θ1 +θ2 +θ3 +θ1 +θ2 +θ3 +Figure 3: The plots of equations (A) (16) and (B) (17). +or +θ = tan θ, +(2n − 1)π < θ < (2n − 1 +2)π, (n = 1, 2, . . . ), +(17) +or +− 1 +λ = cos +� +λ2 − 1, +� +((2n − 1)π)2 + 1 < λ < +� +((2n − 1 +2)π)2 + 1. +(18) +The values of the solutions (θn, λn) are listed in the Table 1. +n +θn +λn +tan θn +1 +4.49341 +4.60334 +4.49341 +2 +10.9041 +10.9499 +10.9041 +3 +17.2208 +17.2498 +17.2208 +4 +23.5195 +23.5407 +23.5195 +5 +29.8116 +29.8284 +29.8116 +6 +36.1006 +36.1145 +36.1006 +7 +42.3879 +42.3997 +42.3879 +8 +48.6741 +48.6844 +48.6741 +9 +54.9597 +54.9688 +54.9597 +10 +61.2447 +61.2529 +61.2447 +Table 1: Numerically estimated values of θn, λn and tan θn +2.4 +Lambert W function +We present here that we can alternatively obtain the solutions (θn, λn) discussed +in the previous subsection using the Lambert W function. +W function is defined as a multivalued complex function with a complex +variable z satisfying +z = Wk(z)eWk(z), +(k = 0, 1, 2, . . .), +(19) +7 + +where k is the branch number. It has been pointed out that the W function +can be used to express the solution of simple delay differential equations[21, 22]. +What we present here is another way the W function can be utilized. +We start with (14) and use e−iθ = cos θ − i sin θ to obtain +1 − iθ = −λe−iθ. +(20) +By defining Q ≡ −1 + iθ we can rewrite (20) as +QeQ = λ +e . +(21) +We can now use the W function on (21), and Q can be expressed as +Q = Wk(λ +e ). +(22) +By the constraints that θ is real, the real part of Q must be equal to −1, or +Re[Q] = Re[Wk(λ +e )] = −1. +(23) +Also by the definition of Q, we have θ from the imaginary part, +θ = Im[Q] = Im[Wk(λ +e )]. +(24) +Further, we can prove the following. +Lemma +Re[Wk(λ +e )] = −1 ⇐⇒ |Wk(λ +e )| = λ +(25) +Proof +Necessary Part: +By the definition of the W function, +Wk(λ +e )Exp[Wk(λ +e )] = λ +e +(26) +Also, by the assumption of +Re[Wk(λ +e )] = −1, +(27) +we can write +Wk(λ +e ) = −1 + iµ, +(µ ∈ R). +(28) +8 + +Then, (26) leads to +λ +e = Wk(λ +e )Exp[Wk(λ +e )] = Wk(λ +e )Exp[−1 + iµ]. +(29) +Thus, +Wk(λ +e )Exp[iµ] = λ, +(30) +which is equivalent to +|Wk(λ +e )| = λ. +(31) +Sufficient Part: +By (26) and λ > 0 we have +λ +e = |Wk(λ +e )Exp[Wk(λ +e )]| = |Wk(λ +e )||Exp[Wk(λ +e )]|. +(32) +Also, by the assumption +|Wk(λ +e )| = λ, +(33) +this leads to +λ +e = λ|Exp[Wk(λ +e )]| +(34) +If we set +Wk(λ +e ) = η + iµ, +(η, µ ∈ R) +(35) +(34) can be re-writen as +1 +e = |Exp[Wk(λ +e )]| = eη, +(36) +leading to +η = Re[Wk(λ +e )] = −1 +(37) +We are now in a position to put together pieces obtained through the analysis +of resonant peaks. They can be summarized as follows. +9 + +Theorem +Let (θ, λ) satisfy the following, +θ = −λ sin θ, +1 = −λ cosθ, +(2n − 1)π < θ < (2n − 1 +2)π, (n = 1, 2, . . . ), +(38) +then they also satisfy the following for some k +Re[Wk(λ +e )] = −1, +(39) +and +θ = Im[Wk(λ +e )], +λ = |Wk(λ +e )| +(40) +Based on the above, we further want to investigate between the n-th root and +the n-th branch of the W function. With numerical estimations, we conjecture +the following. +Conjecture +The n-th root θn of the following, +θn = tan θn, +(2n − 1)π < θn < (2n − 1 +2)π, (n = 1, 2, . . .), +(41) +is given by the n-th branch of the W function +θn = Im[Wn(λn +e )], +(42) +where λn satisfies +Re[Wn(λn +e )] = −1. +(43) +In Table 2, we show the results of estimated related numerical values. Com- +paring Tables 1 and 2 supports the above theorems and conjecture. +Thus, +through the analysis of resonant peaks, we have connected the solutions of the +trigonometric transcendental function with a specific value of the n-th branch of +the W function. To the author’s knowledge, this relation has not been explicitly +pointed out. +3 +Discussion +In this paper, we presented some properties of the Lambert W function through +the analysis of resonant behaviors of a simple delay differential equation. The +connection between the solutions of trigonometric transcendental equation and +that of the W function is revealed. It remains to be explored if these properties +of the W function can be utilized in more broader context. +10 + +n +λn +Wn( λn +e ) +|Wn( λn +e )| +1 +4.60334 +-1.0 + i 4.49341 +4.60334 +2 +10.9499 +-1.0 + i 10.9041 +10.9499 +3 +17.2498 +-1.0 + i 17.2208 +17.2498 +4 +23.5407 +-1.0 + i 23.5195 +23.5407 +5 +29.8284 +-1.0 + i 29.8116 +29.8284 +6 +36.1145 +-1.0 + i 36.1006 +36.1145 +7 +42.3997 +-1.0 + i 42.3879 +42.3997 +8 +48.6844 +-1.0 + i 48.6741 +48.6844 +9 +54.9688 +-1.0 + i 54.9597 +54.9688 +10 +61.2529 +-1.0 + i 61.2447 +61.2529 +Table 2: Numerically estimated values of λn, Wn( λn +e ) and |Wn( λn +e )| +Acknowledgments +The authors would like to thank useful discussions with Prof. Hideki Ohira +and members of his research group at Nagoya University. This work was sup- +ported by ”Yocho-gaku” Project sponsored by Toyota Motor Corporation, JSPS +Topic-Setting Program to Advance Cutting-Edge Humanities and Social Sci- +ences Research Grant Number JPJS00122674991, JSPS KAKENHI Grant Num- +ber 19H01201, and the Research Institute for Mathematical Sciences, an Inter- +national Joint Usage/Research Center located in Kyoto University. +References +[1] U. an der Heiden. Delays in physiological systems. J. Math. Biol., 8:345–364, +1979. +[2] R. Bellman and K. Cooke. Differential-Difference Equations. Academic +Press, New York, 1963. +[3] J. L. Cabrera and J. G. Milton. On–off intermittency in a human balancing +task. Phys. Rev. Lett., 89:158702, 2002. +[4] N. D. Hayes. Roots of the transcendental equation associated with a certain +difference–differential equation. J. Lond. Math. Soc., 25:226–232, 1950. +[5] T. Insperger. Act-and-wait concept for continuous-time control systems with +feedback delay. IEEE Trans. Control Sys. 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Retarded dynamical systems: Stability and characteristic func- +tions. Wiley & Sons, New York, 1989. +[13] G. St´ep´an and T. Insperger. Stability of time-periodic and delayed systems: +a route to act-and-wait control. Ann. Rev. Control, 30:159–168, 2006. +[14] M. Szydlowski and A. Krawiec. The Kaldor–Kalecki model of business cycle +as a two-dimensional dynamical system. J. Nonlinear Math. Phys., 8: 266– +271, 2010. +[15] S. R.Taylor and S. A. Campbell. Approximating chaotic saddles for delay +differential equations. Phys. Rev. E, 75: 046215, 2007. +[16] K. Ohira. Resonating Delay Equation. EPL, 137: 23001, 2022. +[17] X. Yan, F. Liu and C. Zhang. Multiple stability switches and Hopf bifur- +cation in a damped harmonic oscillator with delayed feedback. Nonlinear +Dynamics, 99:2011, 2020. +[18] J. J. Sakurai. Modern Quantum Mechanics. Benjamin/Cummings, Menlo +Park, California, 1985. +[19] J. Milton, P. Naik, C. Chan, and S. A. Campbell. Indecision in neural +decision making models. Math. Model. Nat. Phenom., 5:125–145, 2010. +[20] K. Pakdaman, C. Grotta-Ragazzo, and C. P. Malta. Transient regime dura- +tion in continuous-time neural networks with delay. Phys. Rev. E, 58:3623– +3627, 1998. +[21] R. Pusenjak. Application of Lambert function in the control of production +systems with delay. Int. J. Eng. Sci, 6:28?–38, 2017. +[22] H. Shinozaki and T. Mori. Robust stability analysis of linear time delay +system by Lambert W function. Automatica, 42: 1791–1799, 2006. +12 + diff --git a/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/load_file.txt b/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c923235c341e3a9fd9b3417c6cfdd1bb40a37dc --- /dev/null +++ b/AdFQT4oBgHgl3EQf8zdx/content/tmp_files/load_file.txt @@ -0,0 +1,437 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf,len=436 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='13448v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='AO] 31 Jan 2023 Delay, resonance and the Lambert W function Kenta Ohira1 and Toru Ohira2 1Future Value Creation Research Center, Graduate School of Informatics, Nagoya University, Japan 2Graduate School of Mathematics, Nagoya University, Japan February 1, 2023 Abstract We present here a connection between the solutions of the transcenden- tal trigonometric equation and the Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' This connection emerged through an analysis of resonant conditions with a recently pro- posed simple delay differential equation that shows transient oscillatory behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We investigate and present the connection both analytically and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 1 Introduction In the various fields including mathematics, biology, physics, engineering, eco- nomics, and so on, there have been interests in investigating the effect of delays in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Typically, delays intro- duce oscillations and complex behaviors to otherwise simple and well-behaved systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Longer delays are known to induce an increase in the complexity of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The representative example is the Mackey–Glass equation[8] which shows the sequence of the monotonic convergence, transient oscillations, persis- tent oscillations, and chaotic dynamics as the delay parameter in the feedback function becomes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The main mathematical approaches and modeling tools are “Delay Differen- tial Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Mathematical analysis of such delay systems has been posing challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Though understanding of the delay systems have been gradually gained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' [15]), there is more to be investigated and explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' One recent an- alytical approach to simple delay differential equations is an application of the Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' It has been shown that the solution of the simple delay differential equation can be expressed using the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' In this paper, we follow this path of analyzing a simple delay differential equation using the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Specifically, we analyze the transient oscillatory behaviors of a delay differential equation that shows a resonant behavior[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' In this resonance, the optimal height of the power spectrum of the dynamical trajectory is observed with the suitably tuned value of the delay parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 1 We focus on the appearance of the power spectrum peaks and found that it relates to the transcendental trigonometric equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' That condition, on the other hand, can be also expressed in terms of the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We connect these two approaches both analytically and numerically to provide a new way that the W function can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2 Delay Differential Equation Recently, we have proposed and studied the following delay differential equation: dX(t) dt + atX(t) = bX(t − τ) (1) where a ≥ 0, b ≥ 0, τ ≥ 0 are real parameters with τ interpreted as the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' This equation is a slight extension of much-studied Hayes’s equation[4], dX(t) dt + αX(t) = βX(t − τ) (2) where α, β are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Even though the apparent change from (2) to (1) is small with only the second term changed to a linear function of time, their behaviors are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Particularly for (1), we have shown oscillatory transient dynamics appear and disappear as the value of delay increases without losing asymptotic stability of X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1 Analysis Let us first review some properties of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' When b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' With the initial condition X(t = 0) = X0, the solution to the equation is given as X(t) = X0e− 1 2 at2 (3) Thus, this solution is a gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We also note in this case that (1) is the equation for the ground state of the quantum simple harmonic oscillator with the interpretation of t as a position rather than time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The case that a = 0 is a special case of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' In this case, the origin X = 0 is asymptotically stable only in the range of − π/2τ < b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (4) For the general case with a > 0, b > 0 with the delay τ = 0, the solution X(t = 0) = X0 is obtained as X(t) = X0e− 1 2 at2+bt (5) This is again a Gaussian with its peak at b/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2 For the case with a > 0, b > 0 with the delay τ → ∞, the dynamics is influenced by the initial function for all 0 ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Thus for the initial function X(t) = X0, (−τ ≤ t ≤ 0), we can replace the right hand side of equation (1) as bX(t − τ) → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The solution can be obtained as X(t) = X0e− 1 2 at2(1 + b � t 0 e 1 2 as2ds) = X0e− 1 2 at2(1 + b � π 2aerfi( � a 2 t)) (6) where erfi(x) is the imaginary error function defined as erfi(x) = 2 √π � x 0 es2 ds (7) The shape of this function is also a single peaked function approaching to the origin X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We now see one of the major differences between the equations (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' In the latter, the asymptotic stability of X = 0 is lost for the larger delay with 0 < α < β, while in the former, it is kept even with the large delay for all a > 0, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Also, even though both exhibit transient oscillations, (1) shows coherent oscillations with the tuned value of the delay τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We now turn our attention to these resonating phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 Power Spectrum and Resonance The transient dynamics of equation (1) are investigated through numerical sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Some examples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' With zero delays, the shape of the dynamics is the gaussian as derived in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The oscil- latory behaviors arise on top of the gaussian trajectory with increasing delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Further increase of delay changes the oscillatory shape into trains of pulses with decreasing height at the delay interval, and asymptotically the pulses disappear leading to the gaussian shape (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' As mentioned, the asymptotic stability of X = 0 does not change by the increasing delay in this parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' This property is in contrast to that of equation (2) where the onset of the oscillation by the increasing delay leads to the loss of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Thus, it is differ- ent from stability switching phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' [17]) with the delay as the bifurca- tion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' It is also different from the delay induced transient oscillation (DITO)[19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The phenomena arise in coupled delay differential equations exhibiting the prolonged duration of oscillatory behaviors with increasing delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We investigate these oscillatory behaviors for the case a > 0, b > 0, and finite τ ̸= 0 by taking the Fourier transform of the equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' iω ˆX(ω) + iad ˆX(ω) dω = −b ˆX(ω)eiωτ (8) where ˆX(ω) = � ∞ −∞ eiωtX(t)dt (9) 3 (B) X(t) t (A) X(t) t (E) X(t) t (F) X(t) t (C) X(t) t (D) X(t) t 50 100 150 200 0 20000 40000 60000 80000 100000 120000 50 100 150 200 0 50 100 150 200 250 300 350 50 100 150 200 0 5 10 15 20 50 100 150 200 0 2 4 6 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5 Figure 1: Representative dynamics of the main equation (1) with different values of the delays, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The parameters are set at a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='15, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 with the initial interval condition as X(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1(−τ ≤ t ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The values of the delays τ are (A)2, (B)4, (C)7, (D)10, (E)20, (F)25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 4 The solution is given as ˆX(ω) = CExp[− 1 2aω2 + b τaeiωτ] (10) with C as the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We can calculate the power spectrum from equation(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' S(ω) = | ˆX(ω)|2 = ˆX(ω) ˆ X∗(ω) = C2Exp[−1 aω2 + 2b τa cos ωτ] (11) We have plotted this equation for the power spectrum for the various delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Results with the same parameter setting as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' In the previous work, we noted that the peak of the power spectrum shows a maximum height with the tuned value of the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The higher peak indicates a more coherent oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' It is in this sense that the resonance exists with the delay as a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3 Peaks of the Power Spectrum We now focus on the analysis of these power spectrum peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The appearance and disappearance of the peaks in the power spectrum correspond to those of oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' By taking the derivative of (11), we see the maximum and minimum points of the power spectrum function occur at ω satisfying, ω = −b sin ωτ (12) They are given by the intersection points of the two functions from both sides of this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The position of the first peak corresponds to the second non-zero smallest intersection point (the first one corresponds to the minimum before the peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We can also infer the condition for the appearance of the series of power spectrum peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Each peak appears when the intersection point is the tangent point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' This gives the following conditions for the n-th peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ω = −b sin ωτ, 1 = −bτ cos ωτ, (2n − 1)π τ < ω < (2n − 1 2)π τ , (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (13) If we set λ = bτ,θ = ωτ, the above condition leads to θ = −λ sin θ, 1 = −λ cosθ, (2n − 1)π < θ < (2n − 1 2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (14) From this set of equations, we can numerically estimate the solutions (θn, λn) for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' First, we can derive that θn and λn are related by λ2 n = θ2 n + 1 (15) Then, the followings are obtained: θ = − � θ2 + 1 sin θ, (2n − 1)π < θ < (2n − 1 2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ), (16) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0×1016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0×1017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5×1017 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0×1017 (A) ω S(ω) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 1×108 2×108 3×108 4×108 5×108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 200 400 600 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 200 400 600 800 (B) ω S(ω) (C) ω S(ω) (F) ω S(ω) (E) ω S(ω) (D) ω S(ω) Figure 2: Representative power spectrums given by the equation (1) with dif- ferent values of the delays τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The parameters are set as the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='15, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0, C = 1 with the initial interval condition as X(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1(−τ ≤ t ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The values of the delays τ are (A)2, (B)4, (C)7, (D)10, (E)20, (F)25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 6 5 10 15 20 20 10 10 20 5 10 15 20 20 10 10 20 30 (A) (B) θ1 θ2 θ3 θ1 θ2 θ3 Figure 3: The plots of equations (A) (16) and (B) (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' or θ = tan θ, (2n − 1)π < θ < (2n − 1 2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ), (17) or − 1 λ = cos � λ2 − 1, � ((2n − 1)π)2 + 1 < λ < � ((2n − 1 2)π)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (18) The values of the solutions (θn, λn) are listed in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' n θn λn tan θn 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='49341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='60334 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='49341 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9041 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9499 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9041 3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2208 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2498 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2208 4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5195 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5407 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5195 5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8116 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8284 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8116 6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1006 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1145 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1006 7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3879 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3997 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3879 8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6741 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6844 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6741 9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9597 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9688 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9597 10 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2447 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2529 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2447 Table 1: Numerically estimated values of θn, λn and tan θn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='4 Lambert W function We present here that we can alternatively obtain the solutions (θn, λn) discussed in the previous subsection using the Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' W function is defined as a multivalued complex function with a complex variable z satisfying z = Wk(z)eWk(z), (k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ), (19) 7 where k is the branch number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' It has been pointed out that the W function can be used to express the solution of simple delay differential equations[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' What we present here is another way the W function can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' We start with (14) and use e−iθ = cos θ − i sin θ to obtain 1 − iθ = −λe−iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (20) By defining Q ≡ −1 + iθ we can rewrite (20) as QeQ = λ e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (21) We can now use the W function on (21), and Q can be expressed as Q = Wk(λ e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (22) By the constraints that θ is real, the real part of Q must be equal to −1, or Re[Q] = Re[Wk(λ e )] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (23) Also by the definition of Q, we have θ from the imaginary part, θ = Im[Q] = Im[Wk(λ e )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (24) Further, we can prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Lemma Re[Wk(λ e )] = −1 ⇐⇒ |Wk(λ e )| = λ (25) Proof Necessary Part: By the definition of the W function, Wk(λ e )Exp[Wk(λ e )] = λ e (26) Also, by the assumption of Re[Wk(λ e )] = −1, (27) we can write Wk(λ e ) = −1 + iµ, (µ ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (28) 8 Then, (26) leads to λ e = Wk(λ e )Exp[Wk(λ e )] = Wk(λ e )Exp[−1 + iµ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (29) Thus, Wk(λ e )Exp[iµ] = λ, (30) which is equivalent to |Wk(λ e )| = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (31) Sufficient Part: By (26) and λ > 0 we have λ e = |Wk(λ e )Exp[Wk(λ e )]| = |Wk(λ e )||Exp[Wk(λ e )]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (32) Also, by the assumption |Wk(λ e )| = λ, (33) this leads to λ e = λ|Exp[Wk(λ e )]| (34) If we set Wk(λ e ) = η + iµ, (η, µ ∈ R) (35) (34) can be re-writen as 1 e = |Exp[Wk(λ e )]| = eη, (36) leading to η = Re[Wk(λ e )] = −1 (37) We are now in a position to put together pieces obtained through the analysis of resonant peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' They can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 9 Theorem Let (θ, λ) satisfy the following, θ = −λ sin θ, 1 = −λ cosθ, (2n − 1)π < θ < (2n − 1 2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ), (38) then they also satisfy the following for some k Re[Wk(λ e )] = −1, (39) and θ = Im[Wk(λ e )], λ = |Wk(λ e )| (40) Based on the above, we further want to investigate between the n-th root and the n-th branch of the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' With numerical estimations, we conjecture the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Conjecture The n-th root θn of the following, θn = tan θn, (2n − 1)π < θn < (2n − 1 2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' ), (41) is given by the n-th branch of the W function θn = Im[Wn(λn e )], (42) where λn satisfies Re[Wn(λn e )] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' (43) In Table 2, we show the results of estimated related numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Com- paring Tables 1 and 2 supports the above theorems and conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Thus, through the analysis of resonant peaks, we have connected the solutions of the trigonometric transcendental function with a specific value of the n-th branch of the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' To the author’s knowledge, this relation has not been explicitly pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 3 Discussion In this paper, we presented some properties of the Lambert W function through the analysis of resonant behaviors of a simple delay differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' The connection between the solutions of trigonometric transcendental equation and that of the W function is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' It remains to be explored if these properties of the W function can be utilized in more broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' 10 n λn Wn( λn e ) |Wn( λn e )| 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='60334 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='49341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='60334 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9499 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9041 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9499 3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2498 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2208 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2498 4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5407 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5195 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='5407 5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8284 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8116 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='8284 6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1145 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1006 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='1145 7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3997 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3879 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='3997 8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6844 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6741 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='6844 9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9688 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9597 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='9688 10 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2529 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='0 + i 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2447 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content='2529 Table 2: Numerically estimated values of λn, Wn( λn e ) and |Wn( λn e )| Acknowledgments The authors would like to thank useful discussions with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Hideki Ohira and members of his research group at Nagoya University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' This work was sup- ported by ”Yocho-gaku” Project sponsored by Toyota Motor Corporation, JSPS Topic-Setting Program to Advance Cutting-Edge Humanities and Social Sci- ences Research Grant Number JPJS00122674991, JSPS KAKENHI Grant Num- ber 19H01201, and the Research Institute for Mathematical Sciences, an Inter- national Joint Usage/Research Center located in Kyoto University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' References [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' an der Heiden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' Delays in physiological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'} +page_content=' J.' metadata={'source': 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sha256:4c1f05f79425b5f210ccf09b91e76aba2fe9907f9645f84d5dc5ea4eca363d43 +size 1835053 diff --git a/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/2301.02907v1.pdf.txt b/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/2301.02907v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c3f9642062470f33421796a715d95235c98683c --- /dev/null +++ b/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/2301.02907v1.pdf.txt @@ -0,0 +1,690 @@ +David Maurice Brink +20 July 1930 - 8 March 2021 +Elected FRS 1981 +C. V. Sukumar1∗, A. Bonaccorso2 † +1Wadham College, Oxford OX1 3PN, U.K, 2INFN, Sezione di Pisa, 56127 Pisa, Italy. +January 10, 2023 +Abstract +David Brink was one of the leading theoretical nuclear physicists of his generation. He +made major contributions to the study of all aspects of nuclear physics embracing nuclear +structure, nuclear scattering, and nuclear instability. His wide ranging interests and inter- +actions with theorists and experimentalists alike helped him in providing both theoretical +analysis and interpretations and suggesting experiments. +He had the gift of visualising +complex problems in simple terms and provided clear analysis of the underlying processes. +He was an expert on the use of semi-classical methods which provided an intuitively clear +picture of complex phenomena. His research work and books are characterised by scientific +clarity, transparency, and depth. David possessed outstanding skills in mathematical com- +putation, and he was an expert on special functions, group theory, and the Feynman path +integral method. David had many research students and collaborated with a large number +of scientists from across the world, for whom he was a source of scientific and human in- +spiration and admiration. His most fundamental belief was that research was a means of +trying to discover and understand the beauties of Nature and explain them in simple terms +to others. His absolute belief in the value of truth and his unselfish and generous attitude +in sharing knowledge makes him an outstanding figure in contemporary Nuclear Physics. +1 +Early years and family memories +David Maurice Brink was born on 20 July 1930 in Hobart, Tasmania. His father, Maurice Brink +had been born in the village of Bjuv in Sweden in 1900. David’s grandparents emigrated to +Australia in July 1900. At the age of 14 David’s father moved to Sydney where he trained to +become an accountant. After this he went to Tasmania and joined an accountancy firm Wise, +Lord and Ferguson, where he eventually became a partner. In 1929 he married Victoria Finlayson, +(born in 1900). Her father David had emigrated with his parents from Scotland in 1884. They had +an engineering firm in Devonport, Tasmania whose main activity was maintaining and repairing +machinery for mining, shipping, and timber companies. David’s grandfather and his colleagues +built the first steam car in Tasmania and between 1900 and 1904 built nine vehicles including +three passenger cars and one 12-passenger bus. +David visited his grandparents often during +vacations. He saw the casting floor and other parts of the factory and enjoyed playing amongst +the remains of old steam traction engines. +∗candadi.sukumar@wadham.ox.ac.uk +†bonac@df.unipi.it +1 +arXiv:2301.02907v1 [physics.hist-ph] 7 Jan 2023 + +Figure 1: David M. Brink +2 + +David was the eldest of three brothers. The Brink brothers went to a Quaker school in Hobart, +Australia 1936 to 1948. David attended the University of Tasmania during 1948-51 studying +Physics, Mathematics, and Chemistry, graduating with a BSc in December 1950 and was elected +as a Rhodes Scholar at Magdalen College, Oxford, from October 1951. From February 1951 to +September 1951 he studied for BSc Honours in Hobart but did not complete the course because +he moved to Oxford in September 1951. +As a student at the University of Tasmania David joined the Hobart Walking Club. With this +club he went on many trips to the interior of the island. When he arrived in Oxford he became +a member of the Oxford University Alpine Club. Its activities took him to the Alps where he +climbed in the Valais and the Engadine in Switzerland. It was in Switzerland that he met his +future wife Verena. Verena and David married in 1958 and had three children together. His love +for walking was transmitted to his three children who continue to enjoy walking in urban, rural, +and mountainous settings. While always very committed and absorbed with his Physics he was +also a devoted husband and father, transmitting his joy for walking and travel to his family. He +often helped his children with their homework and was very patient with them, even when they +were not! Together David and his family travelled to, and lived in many countries across the +world, where their horizons were broadened and they were introduced to the idea that there are +many different ways of living and being. When his children had left home and travelled to other +countries he would often be found in front of an atlas studying their exact whereabouts. +David was very open minded and curious, always accepting other people’s opinions and points of +view. David and Verena were very close, shared everything and had full respect for each other. +Verena was a wonderful host and the Brinks often organised tea and dinner parties for students, +visitors, and their families. Verena also helped visitors find accommodation, and with other +issues related to living in Oxford. They were also very generous in offering accommodation at +their place whenever possible. +In Oxford David developed an interest in birds, initially just birds he saw in Oxford, but when +he travelled he always liked to look for birds and made lists of species he saw. This curiosity +in nature extended to other species as well, including trees. When in 1993 he moved to Trento, +Italy, he became a member of the SOSAT, a branch of the alpine club, and went regularly with +them on Sunday trekking trips. +2 +Graduate studies and Oxford beginnings +David started his studies at Oxford in October 1951. When he arrived at Magdalen College there +was no tutor in Theoretical Physics at the college. His maths tutor was David Kendal who sent +him for tutorials to Jack De Wet at Balliol College. Jack asked David to read Von Neumann’s +book on the foundations of Quantum Mechanics in German. He also encouraged David to change +his studies from a BA in Mathematics to a D. Phil in Theoretical Physics. Maurice H. L. Pryce +(FRS 1951) was the Wykeham Professor and head of the Theoretical Physics Department in +Oxford from 1946 to 1954. He was David’s supervisor. Pryce was also the part-time leader of the +Theoretical Physics Division of the Atomic Energy Research Establishment (AERE) at Harwell, +not far from Oxford, where nuclear theory was very much in the forefront and Rudolf E. Peierls +(FRS 1945) was a consultant. At Harwell there was a very productive theory group including +Tony Skyrme and J. P. (Phil) Elliott (FRS 1980). Skyrme organized regular informal meetings +known as ’Skyrmishes’. Important papers in the latest journals were presented and discussed. +Members of the group attended Oxford seminars and while the local group including Roger Blin- +Stoyle (FRS 1976), David Brink, and Pryce attended the Harwell meetings. Elliott gave some +3 + +Figure 2: David (right) in Tasmania 1950. +4 + +EVENT +RESlectures at Oxford on Racah algebra. Later on his best-known work brought together the shell +and collective models to explain rotational bands in deformed nuclei using the unitary group +SU(3). During this time he wrote a long article in Handbuch der Physik with A. M.(Tony)Lane +(FRS 1975) [1] on the shell model. +The foundations of David Brink’s lifelong research, can all be found in his thesis ”Some Aspects +of the Interactions of Fields with Matter” [2] which was submitted in May 1955. It is a remark- +able document for its breadth and early contributions to the field of nuclear physics. M. Pryce, +his thesis adviser was interested largely in atomic spectroscopy but also studied the spectroscopy +of nuclear energy levels. The advent of the shell model around 1950 opened the door to new +theoretical approaches for understanding the properties of nuclei and applying quantum mechan- +ical tools to calculate them. There was also a great interest in reactions involving heavy nuclei +and which could only be treated by statistical methods that had been developed much earlier. +Brink’s two-part thesis contained contributions to both areas, reflecting the interactions between +the Harwell and Oxford groups. The first part was inspired by the shell model and the second +contains important contributions to the statistical theory of nuclear reactions. +In the first part of his thesis, dealing with Nuclear Structure, David analyzed the spectroscopic +consequences of the nucleon-nucleon interaction acting on the valence nucleons in nuclei close to +the doubly-magic 208Pb. David was able to estimate the order of magnitude of the interaction +matrix elements from the properties of the deuteron. He also proposed treating the interac- +tion through a density matrix expansion. This would figure prominently in later work in the +field. +The second part of his thesis dealt with reactions involving heavy nuclei. It was probably inspired +by the work of experimental group at Harwell. There, a Van de Graaff accelerator was used to +measure energy levels, moments and transition rates in nuclei. +David was also fortunate to +have contact with the strong experimental group working on neutron resonances. While David +was working on gamma widths of neutron resonances he benefited from contacts with Prof. +Hughes [3] and Prof. Weisskopf who were visiting Oxford. Weisskopf was very much interested +in applying the detailed balance theory to nuclear reaction and interactions with him must +have influenced David because at the end of the thesis he acknowledges discussions with Victor +Weisskopf. The first subject in this part was the theory of inelastic scattering on deformed nuclei. +David constructed a theory for the excitation of rotational bands in deformed nuclei based on +two new ideas, namely Bohr’s model of deformed nuclei and the optical model of Weisskopf et al. +[4] published the previous year. David was able to carry out the calculations to a point where +the relative importance of this mechanism in the total cross section could be estimated. This +was an impressive achievement at a time before computers were available to carry out the full +calculations. +The final section of his thesis deals with the decays of the compound-nucleus resonances produced +in reactions on heavy nuclei. The formulas he presented here are still in use for modeling the +spectra and reactions in heavy nuclei [5]. The best known is the formula for gamma decay rates +in compound-nucleus resonances. This formula is based on a treatment widely known as the +”Brink-Axel” hypothesis. At a fundamental level, the theory was derived from the principle of +detailed balance which Weisskopf had used very successfully in other contexts. The principle +gives a formula to relate decay rates to absorption cross sections in the inverse reaction. The +Brink-Axel hypothesis simply states that the absorption cross sections for gamma radiation on +excited states of heavy nuclei can be estimated by the corresponding cross sections on the ground +states. Axel and Brink worked independently. Peter Axel’s paper appeared in 1962 [6]. The +important statement is made on page 101 of David’s thesis and is expressed in equation (11) of +5 + +Figure 3: David and his children (left to right), Barbara, Thomas, and Anne-Katherine 1969. +Axel’s paper. The prediction of the statistics of the widths of nuclear resonances, based on the +generalization of the central limit theorem which David had learned about in his statistics course +in Tasmania. David published the results in [7] where he showed the close connection between +the shell-model description of the giant dipole resonance and the collective model of Goldhaber +and Teller [8] and Steinwedel and Jensen [9]. After his paper, theory of the giant resonances +used the shell model as a starting point. Confirmation of the Brink-Axel hypothesis first came +from the Berkeley experiments in 1981 [10]. +The last part of thesis has formulas related to another important topic in compound-nucleus +theory, the fluctuations in decay widths of individual resonances. Here, David speculated that +the fluctuations would follow a chi-squared distribution with one or two degrees of freedom. +This is borne out experimentally and is now considered one of the hallmark properties of the +compound nucleus. It also became a part of random matrix theory in mathematical physics. +Unlike the early parts of the thesis, David never published the parts on compound-nucleus decay +widths. However, physicists at the Harwell Laboratory knew about David’s results and J.E. +Lynn explained them in his book [11]. Unfortunately, David’s treatment of fluctuations was not +recognized until very recently [12] and the distributions are known today under other author’s +names [13]. +6 + +3 +Research areas +David’s interactions with the physicists mentioned earlier were reflected not only in David’s thesis +but also in his early publications. One paper [14], which dealt with angular momentum couplings +and angular distributions of γ-rays and other particles, is still the ”Bible” most experimentalist +use when they analyse their data, as we have been told by Peter Butler (FRS 2019) (Liverpool) +and Yorick Blumenfeld (Orsay), and others. Early in his research career David wrote the textbook +Angular Momentum [15] with Ray Satchler. This textbook was prominent among several texts +published in this time period. +It was widely used by graduate students and post-graduates +working in nuclear theory. David also published a book on Nuclear Forces [16]. +3.1 +Effective interactions and calculations tools +In his thesis David had laid the basis for the use of effective interactions in the calculations +of matrix elements for nuclear structure studies. The idea was greatly advanced in three later +papers. The first proposes a gaussian form for the effective nucleon-nucleon interaction known +as the ”Brink-Boeker” interaction [17] that all nuclear physicists have used at least once in +their lives. This paper was very influential at the time and was later developed by Gogny and +collaborators in the interaction that is widely used even today [18, 19]. +In 1959 Tony Skyrme proposed modelling the effective interaction between nucleons in nuclei +by a short-range potential, an idea which is useful in nuclear structure and the equation of +state of neutron stars [20]. The Skyrme force is an effective interaction depending on a small +number of parameters whose strength could be fitted to reproduce various bulk properties of +nuclei as well as selected properties of some nuclei, especially the doubly magic nuclei. At the +beginning of the 1970s David was a frequent visitor to the Theoretical Division at the Institut +de Physique Nucl´eaire, Orsay where his sixty-fifth birthday was celebrated (figure 4). The work +done there produced two papers with Dominique Vautherin [21, 22] which were the basis for the +intense use of the so-called Skyrme interactions, in all their many present variants. The papers +revived a general interest in using Skyrme’s parametrization of the nucleon-nucleon interaction +to calculate nuclear binding energies, and later to other aspects of nuclear structure. In effect, +the interaction is treated as an energy-density functional theory in the spirit of the Kohn-Sham +theory in condensed matter physics. +The Hartree-Fock calculations in [21] for spherical nuclei used Skyrme’s density dependent effec- +tive interaction. This seminal paper showed how the Skyrme force could be used to make accurate +calculations of certain nuclear properties and Vautherin and Brink developed these ideas further +in a series of papers which had a strong impact on nuclear structure calculations. T. Otsuka +comments: “The paper [21] has had a huge impact, as verified by the number of citations >2000. +In nuclear theory, papers having the citation index >1000 are rather few, which implies how +important the Vautherin-Brink paper is. This year is the 50 year anniversary of this paper, and +it is amazing that the basic formulation within the mean-field approach has not changed too +much, implying that the scheme presented in this paper is so solid”. +The calculations of Vautherin and Brink were extended by many other physicists during the +subsequent period. +In particular at Oxford, Micky Engel, Klaus Goeke and Steve Krieger, +together with Dominique Vautherin derived the energy density using a Slater determinant where +the single particle states were no longer invariant under time reversal, as it is in the Hartree-Fock +method. With the Skyrme interaction the TDHF approach leads to an equation of continuity +for the single particle density [22]. This paper showed how Dirac’s time-dependent Hartree-Fock +theory could be applied to nuclear dynamics in a light nucleus. In the year immediately following +7 + +the publication, the theory was applied to collisions involving a large number of nucleons [23], +showing that the method would be a powerful one for heavy nuclei as well. +The method is +justified as a time-dependent density-functional theory, and it remains in widespread use. +In 1973 Ica Stancu came to Oxford as a post doctoral fellow and worked with David on heavy +ion reactions in deriving the interaction potential of two 16O nuclei starting from the Skyrme +energy density formalism [24]. They included the previously ignored tensor part of the Skyrme +interaction. +Along with an additional effort from Hubert Flocard at Orsay, the Skyrme HF +calculations yielded single particle levels of spherical closed nuclei [25]. The role of the tensor +force is to contribute to the spin-orbit splitting of the single-particle levels. For spherical closed +shell nuclei the effect turned out to be small. Later it was found that in spherical spin unsaturated +nuclei it makes a dramatic difference, giving the correct order of single particle levels, as, for +example, in the Sn isotopes [26]. Many experiments on neutron-rich nuclei since 2006 have shown +that the Skyrme formalism including the tensor force was the simplest way to describe the shell +evolution of neutron-rich or proton-rich nuclei and indicated new magic numbers. +3.2 +Heavy-ions and Semi-classical methods in Nuclear Physics +As tandem accelerators and cyclotrons were built to study heavy-ion Physics, David started an +intense collaboration with the experimentalists at the Department of Nuclear Physics in Oxford. +The accelerators were used to study heavy-ion elastic scattering and direct reactions such as +transfer and measure masses and perform spectroscopy of neutron-rich matter. In those years +semiclassical methods were widely used in the Nuclear Physics community to analyse data. They +were particularly appropriate for heavy ions because of the high incident energies and the large +impact parameters involved. Thus David started the Oxford school on the subject, more or +less parallel in time to the Copenhagen school of Broglia and Winter and collaborators. +At +that time, these heavy-ion reactions were analyzed through the partial wave expansions of the +colliding partners, a methodology that was computationally demanding and giving little insight +to the underlying dynamics. David’s semi-classical treatment of the collision was much simpler. +Some of the early papers on the theory of peripheral reactions were based on his student’s thesis, +including Hashima Hasan and Luigi Lo Monaco [27, 28]. +David’s investigation of the kinematical effects in such reactions, for which there was concrete +experimental evidence from the work of Peter Twin (FRS 1993) and his collaborators at Liver- +pool, became a key element for experimentalists. In the paper by the title ”Kinematical effects in +heavy-ion reactions” [29] David introduced a ”semi-classical amplitude” [30] that could be used +in DWBA-like calculations of transfer [31] and proposed a matching condition to predict a large +reaction cross-sections, a condition that was beautifully adapted to understand spin-polarization +experiments. He showed that energy and angular momentum couplings in heavy-ion reactions led +to very selective matching rules by which high angular momentum single-particle states could +be populated. +High angular momentum single-particle states sometimes appear as low-lying +continuum resonances. They have been studied by the method of transfer-to-the-continuum [32] +which has helped disentangle single-particle from collective degrees of freedom and has also been +applied in the so called ”surrogate reactions” as a substitute for free neutron beams. +Semi-classical ideas have been helpful in studying breakup and dissociation of weakly bound +radioactive ions including halo nuclei and other such unstable nuclei whose dynamics is rather +involved and difficult to study experimentally due to the very low intensity of beams. David, +Angela Bonaccorso and her students got heavily involved in this new physics from the ’90s on, +with a long series of papers (see [33] and references therein), conference contributions, meeting +organization, some of them at the ECT* in Trento, spanning the last forty years of David’s +8 + +career. Finally it has recently been shown [34] that the semi-classical treatment of breakup by +David and his collaborators is fully consistent with a quantum mechanical treatment. +David studied microscopic models for the real and imaginary parts of the ion-ion optical potential +to be used in elastic scattering calculations with Ica Stancu. He also studied fusion with Neil +Rowley and N. Takigawa. David and Takigawa developed a semi-classical reaction theory with +three classical turning points which explained the anomalous large angle scattering of α particles +as a quantum-mechanical interference between the barrier wave and the internal wave, thereby +providing an intuitively clear picture of a complex phenomenon underlying nuclear reactions in +terms of classical and quantum ideas. David, Vautherin, and M.C. Nemes studied the effect of +intrinsic degrees of freedom on the quantum tunnelling of a collective variable. This work was +further developed by other theorists including Kouichi Hagino who studied the deviation from +adiabaticity in quantum tunnelling with many degrees of freedom. +David met Uzi Smilanski in Munich when they were both there on sabbatical. Both had worked +on semi-classical approximations and gave a joint series of lectures on this topic. David was con- +cerned that the standard WKB method was insufficient to explain tunnelling through a barrier +and was particularly bad near the barrier top. David and Uzi applied the uniform semi-classical +method evolved by Michael Berry (FRS 1982) to successfully address the problem [35]. +Uzi +remembers David as a physicist with excellent intuition and an ability to grasp the essence +of a problem before cracking the problem with rigorous mathematics and complex computa- +tion. +David, Massimo di Toro, and Alberto Dellafiore developed a semi-classical description of col- +lective responses with a mean field approach paving the way for a study of the dynamics of a +nuclear Hartree-Fock fluid. When the national heavy-ion laboratory started in Catania (LNS- +INFN) around an advanced superconducting cyclotron, David was a reference point for simple +physics suggestions. +3.3 +Path integral methods in Nuclear Physics +David’s expertise with semi-classical methods for tackling quantum problems naturally led him +towards the Feynman path integral approach to quantum mechanics which was based on a +Lagrangian approach. Hans Weidenm¨uller had met David at various conferences in the 1950s +and 1960s and spent 1977-78 on a sabbatical in Oxford. During this period David and Hans +worked on the application of the Feynman path integral method to the study of the heavy-ion +reactions and developed the Influence Functional approach to this problem which David and his +collaborators later used to establish master equations. Hans remembers that at a summer school +a few years later David delivered a series of lectures on nuclear reactions. In the first lecture +he developed the topic using a dozen transparencies and in subsequent lectures used the same +transparencies in a different order to display and illuminate aspects of the topic that had gone +unnoticed before. Hans remembers it as a display of the combination of simplicity and depth +that were hallmarks of David’s approach to Physics. +The path integral method was particularly well suited for studying problems with many degrees +of freedom in which classical description in terms of trajectories was good for some degrees of +freedom but not for all. Coulomb excitation in heavy-ion collisions is an example where the +relative motion of the ions could be described in terms of coulomb trajectories but the excitation +of the quantum states of the ions had to be treated using quantum mechanics. +David and +Sukumar [36] used the Feynman path integral method to evolve a systematic way of arranging +the correction terms for the quantum amplitudes for processes involving coupled degrees of +9 + +Figure 4: David and his wife Verena, May 2018. +freedom where the description in terms of classical trajectories was good for some degrees of +freedom. David, Sukumar, and Fernando Dos Aidos used this method to provide corrections +to the primitive semi-classical amplitude for Coulomb excitation of heavy-ions. Sukumar and +David used the path integral method to describe spin-orbit coupling effects and together with Ron +Johnson at Surrey and his group successfully explained the experimental data on polarization +effects. +4 +Other topics +David was very quick at grasping the core of a Physics problem and putting it in simple, calculable +terms. +Often the problem required somewhat involved analytical calculations, but he was a +master of that. Thus anytime a visitor went to Oxford with a new problem, David would start +a very successful line of research which he often followed up with his graduate students. +10 + +4.1 +Cluster models +It happened for example with the cluster model physics, starting with the seminal paper [37]. +This paper developed the generator coordinate method of Hill and Wheeler [38] to produce +a practical tool to reduce the many-particle Hamiltonian to an ordinary Schr¨odinger equation +for a collective variable. Thus the nuclear cluster model was related to the shell model. To +treat nuclear states in such different circumstances, a formulation which includes clustering at +one extreme and shell structure at the other extreme was needed. David proposed microscopic +multi-α-clusters treating four nucleons with different spin-isospin states as a single particle orbit. +Under anti-symmetrisation of nucleons the cluster model wave-functions approximate shell model +functions and enabled the description of both cluster and shell model structures in a unified way. +Their approach was adopted and is in widespread use even in present-day nuclear theory. The +main applications up to now are on spectroscopy and large-amplitude collective motion. +Y. Suzuki’s work on the cluster model was largely inspired by David’s paper on ”Do alpha +clusters exist in nuclei?” [39] presented at a meeting in Tokyo in 1975. This paper contained +all the essential components needed in the alpha particle model, the microscopic theory beyond +the shell model description based on many-particle many-hole excitations, the relation between +the resonating group method GCM, the equilibrium arrangement of clusters, extension of the +Hill-Wheeler method, the angular momentum projection, and the Slater determinant technique +for evaluating matrix elements. Suzuki remembers that David never forgot to mention that the +original model was proposed by H. Margenau and C.Bloch [40, 41, 42]. +At the Varenna School in 1955 David met S. Yoshida from Japan and they discussed inelastic +scattering of protons and neutrons by deformed nuclei. By chance David had a chapter in his +thesis on this topic and Yoshida had been studying the same subject. This interaction with +Yoshida helped David to develop strong connections with nuclear theory groups in Japan over +many years. +4.2 +Bose-Einstein condensation of atoms +During his period as Deputy Director of ECT* in Trento, 1993-1998, David interacted with many +members of the Physics Department in Trento. One such interaction with Sandro Stringari led to +David’s interest in Bose-Einstein condensation of alkali atoms in magnetic traps [43]. Sukumar +and David [44] developed an approximate method for calculating the rate of escape from the +magnetic trap thereby enabling an estimation of the duration for which the condensate atoms +can be held in the trap as a function of the ultra-cold temperature and the strength of the +magnetic field. +4.3 +Miscellaneous +David was interested in the role of pairing interaction in finite nuclei and this led to the study +of nuclear superfluids. His book with R. Broglia [45] is considered to be a wonderful exposition +of this subject. David’s knack for explaining detailed Physics in a simple and clear manner is +abundantly evident in this book. In the 1990s Ica Stancu raised David’s interest in the quark +structure of exotic hadrons named tetraquarks, a system of two quarks and two antiquarks, and +studied the stability of such systems containing heavy quarks/antiquarks in a QCD inspired +quark model. Even though David had not worked on the Interacting Boson Model (IBM) he +nevertheless provided supervision for doctoral students such as Martin Zirnbauer who chose +topics in this field. He also supervised Hans Peter Pavel’s thesis on Schwinger pair production +in a flux tube model containing a chromomagnetic field. +11 + +5 +Teaching and administrative roles +David’s doctoral students remember him for the gentle way he corrected them when they had +made errors. Many of the students learned from him how to take a critical approach to their +results and how it is possible to look at a complex problem from several different viewpoints and +find the one that gives the best physical insight. They also remember the immense support he +gave to their research and pastoral care. Many graduate students also remember how much they +had learned from the courses he taught at Oxford and at Summer schools. His book with Satchler +[15] and paper with Rose [14] on angular momentum algebra were found to be of immense value +in formulating and tackling problems in Nuclear Physics. Many researchers and students who +met David were astonished that someone with such towering achievements could be so humble, +nice and honest. David was very open-minded and we report a number of episodes to illustrate +this aspect of his character. +Future Nobel laureate Prof. Tony Leggett remembers: ” My undergraduate major at Balliol was +in Greats (classical languages, ancient history and philosophy) and I was set to graduate (and +eventually did so) in the summer of 1959. Towards the end of the academic year 1957-1958, +partly encouraged by the post-Sputnik cultural swing towards science in the UK, I conceived +the ambition of taking a second undergraduate degree in physics and perhaps eventually making +my career in academia in that subject. Given that I had essentially no meaningful exposure to +physics at the high-school level and only a brief and informal exposure to any kind of mathematics +beyond simple differential calculus (I’m not sure that I had even had that), such a drastic change +of academic direction was extremely unusual, indeed at the time almost unheard-of. My first +concern was to find a higher education institution which would accept me for it and I rapidly +concluded that my only hope was to apply to my existing Oxford college, Balliol. David had +just recently become the college’s first tutor in theoretical physics (most Oxford colleges did +not have such a thing in 1958), so it fell on him to take the decision on my application. To +this end he asked me to read over the summer vacation a few chapters from the book ”What +is Mathematics?” by Courant and Robbins [46], perhaps the most beautiful presentation I have +ever seen of mathematical topics for the layperson. When I returned to Oxford in the Fall of +1958 he gave me an informal mini-exam on that material, and on the basis of my performance +decided to recommend to Balliol to accept me. In the event I did my physics degree at Merton, +who offered me a scholarship, but since they did not at the time have a tutor in theoretical +physics David played that role for me for much of the two years which it took me to complete +the degree. I think it is virtually certain that had he made the opposite decision, I would never +have had a career in physics, and I am profoundly grateful to him for the imagination he showed +in going beyond my formal academic qualifications.” +Another story comes from Paul Stevenson: ”I was called up for interview at Balliol in December +1991. The office I was in for that interview was David Brink’s office, above the Senior Common +Room. In the interview were me, David Brink, David Wark, Jonathan Hodby (those three there +for physics) and Bill Newton-Smith (for philosophy). I don’t remember all the questions. I do +remember that David Brink showed me a postcard and asked me what, physically, was wrong +with the picture. It was a Japanese style print with a mountain in the background and a lake in +the foreground. There was a reflection of the mountain in the lake, but it was off to one side. I +saw what was wrong, and struggled to articulate it in the language of a physicist, and in the end +David prompted me by asking what is particular about an incident light ray, a reflected light +ray, and the normal to the surface at which it is reflected and I said the right thing - that they +are all in the same plane. I was duly accepted to Balliol and spent three years there studying +physics”. Danny Chapman remembers: ”I don’t think I’ll ever forget the ”sense” of David Brink’s +12 + +tutorials, and of being in the presence of such a sharp and insightful mind. I remember being +quite inspired once when my fellow student had tried to answer a question in what I thought was +an odd and probably wrong way, ending up with a sum, which he then attempted to turn into +an integral, which didn’t work out. Rather than saying ”don’t do it like that, do it like this”, +David was able to continue from there and make it work, which was a really positive experience +and encouragement to follow every path to its end. I feel lucky to have been at Balliol when he +was there.” +Angela Bonaccorso remembers daily life as one of David’s students: At the Department of The- +oretical Physics there was a coffee room where coffee was served between 11:00 and 11:30. We +would try to be there on time to sit around David who would be chatting with other senior +members of the department or some visitor. There would always be someone bringing up some +interesting and challenging new problem. Everyone gave an opinion, the atmosphere was com- +petitive. Most of the time David would win the argument and his students felt very proud.Not +all supervisors were so nice, helpful, and respectful of us as David was. But it was not at all +easy to be David’s student. First of all we needed to have detective skills. David was very busy +and very elusive. In those days there was no email or SMS. The only way to be sure that he was +inside was to look for his bicycle. If the bicycle was outside we would knock at the door of his +office and if we were lucky he would answer and let us in. In spite of all his many commitments +we always managed to have at least one chat per week with him. Another reason why it was +not easy to be his student was that David had a very original way of understanding things and +finding the way out of problems. During our conversations often he would stop talking and be +silent for five to ten minutes, rubbing his hand on his forehead. Then he came up with some +equation, or a drawing or something like that and he would tell us: I think it is like this...I think +we should get something like that...etc. I (we) would stare at him speechless and in wonder. +Where did the ’oracle’ come from? Most of the time this was the end of the meeting. I (we) +left his office rather puzzled, worked desperately hard for one week and if we had managed to +understand his line of thought, after pages and pages of calculations, we would find exactly what +he had predicted. We all knew it was like that, we all passed this information on to each other, +generation after generation: listen to David, he is always right, just try to reproduce the miracle +of his craftsmanship in physics. +A further proof of how much busy David was and how precious was for everyone the time spent in +conversation with him can be found in the comment Gerry Brown made in his review for Science +[47] of the Proceedings of the Varenna summer school [41] : ’Let me draw special attention also +to the article of David Brink, ”The alpha-particle model of light nuclei,” which is one of the most +beautiful developments in this subject. Brink likes to sit on his work for years and, on the whole, +doesn’t even answer letters inquiring about it, so that one must either adopt the expedient of +traveling to Oxford to talk with him, or invite him to lecture at summer schools. Both are worth +while.’ +David was a pillar of Balliol college and Department of Theoretical Physics for decades, an +immensely popular tutor and supervisor, a cheerful and always helpful colleague, and a wonderful +guide to younger colleagues and administrative staff who happened to be working with him. +David had another long and distinguished career in Italy after he left Oxford. Following an +invitation from Renzo Leonardi he moved to Trento as full professor of History of Physics and +helped in establishing the ECT*, European Center for Theoretical Studies in Nuclear Physics +and Related Areas. The Nobel laureate Ben Mottelson was the founding director and David +the vice-director, while Renzo Leonardi was the Scientific Secretary. In the five years David +spent at Trento he took care of organising various technical aspects of the secretarial offices, +13 + +Figure 5: David’s sixty-fifth birthday celebration. Orsay, 1995. +library, computer center and visitor hospitality. +At the same time he gave very productive +contributions to workshops with his constant presence, his huge knowledge of nuclear physics +and stimulating discussions. The superb reputation and international standing of this extremely +important European initiative is undoubtedly due in large part to David’s wisdom in its crucial, +formative years. +6 +Career, Honours and Awards +1954-55 Royal Society Rutherford Scholarship. +1957-1958 Instructor at the Massachusetts Institute of Technology (MIT). +1958 Fellow of Balliol College and Lecturer in Theoretical Physics, Oxford. +1976-1978 Vice-Master of Balliol College. +1981 Fellow of the Royal Society. +1982 Rutherford Medal of the Institute of Physics. +1988 H. J. G. Mosley Reader at Oxford. +1990-1993 Senior Tutor, Balliol College, academic planning and administration, Oxford. +1992 Foreign member of the Royal Society of Sciences, Uppsala. +1993-1998 ECT*, Trento, Vice-Director . +1993-1998 Full professor of History of Physics, University of Trento. +14 + +2006 Varenna Conference on Nuclear Reactions dedicated to him. +2006 Lise Meitner prize of the European Physical Society shared with H. J. Kluge. +Visiting scientist at : +• Niels Bohr Institute 1964, +• University of British Columbia 1975, +• Institut de Physique Nucl´eaire d’Orsay 1969 and 1981-1982, +• The Technical University of Munich 1982, +• University of Trento 1988, +• University of Catania 1988, +• Michigan State University 1988-1989. +7 +Acknowledgements +The authors are greatly indebted to the Brink family for sharing with them private memories +and photographs and for a critical reading of the manuscript. A large number of friends and +colleagues, too many to be individually mentioned, contributed with their appreciation of David’s +life and scientific career. Ica Stancu and Sharon McGrayne Bertsch read and commented the +manuscript. One of us (AB) gratefully acknowledges George F. Bertsch for his help in digging +out from David’s thesis and early work the roots of several founding pillars of modern Nuclear +Physics. +References +[1] Elliott J.P., Lane A.M. 1957 The Nuclear Shell-Model. Structure of Atomic Nuclei / Bau +der Atomkerne. Encyclopedia of Physics / Handbuch der Physik vol 8 / 39. Springer, Berlin, +Heidelberg. https://doi.org/10.1007/978-3-642-45872-9 4 +[2] Brink, D. M., 1955 +https://ora.ox.ac.uk/objects/uuid:334ec4a3-8a89-42aa-93f4-2e54d070ee09. +[3] Hughes, D. J., and Harvey, J. A. 1954 Radiation-widths of nuclear energy-levels Nature 173, +942 - 943. DOI: https://doi.org/10.1038/173942a0 +[4] Feshbach, H. Porter, C.E. and Weisskopf, V.F. 1954 Model for Nuclear Reactions with +Neutrons Phys. Rev. 96 448-464. DOI:https://doi.org/10.1103/PhysRev.96.448 +[5] Capote, R. et al., 2009 RIPL, Reference Input Parameter Library for Calculation of Nuclear +Reactions and Nuclear Data Evaluations +Nuclear Data Sheets 110 3107-3214. DOI: https://doi.org/10.1016/j.nds.2009.10.004 +[6] Axel, P. 1962 Electric Dipole ground-state transition width strength function and 7 Mev +photon interaction Phys. Rev. 126, 671-683. DOI:https://doi.org/10.1103/PhysRev.126.671 +15 + +[7] Brink, D. M., 1957 Individual Particle and Collective Aspects of the Nuclear Photoeffect +Nucl. Phys. 4 215 - 220. DOI: 10.1016/0029-5582(87)90021-6 +[8] Goldhaber, M. and Teller, E. 1948 On Nuclear dipole vibrations Phys. Rev. 74 1046 - 49. +DOI:https://doi.org/10.1103/PhysRev.74.1046 +[9] Steinwedel, H., Jensen, J. H. D. and Jensen, P. 1950 Nuclear dipole vibrations Phys. Rev. +79 1019. DOI:https://doi.org/10.1103/PhysRev.79.1019 +[10] Newton, +J. +O. +et +al. +1981 +Observation +of +Giant +dipole +resonances +built +on +states +of +high +energy +and +spin +Phys. +Rev. +Lett. +46 +, +1383-1386. +DOI:https://doi.org/10.1103/PhysRevLett.46.1383 +[11] Lynn, J.E., 1968 Theory of neutron resonance reactions, (OUP) 321. +[12] Hagino, K. and Bertsch, G. F., 2021, Porter-Thomas fluctuations in complex quantum sys- +tems Phys. Rev. E104 L052104. DOI:https://doi.org/10.1103/PhysRevE.104.L052104 and +references therein. +[13] Porter, C.E. and Thomas, R.G. 1956 Fluctuations of Nuclear Reaction widths Phys. Rev. +104, 483-491. DOI:https://doi.org/10.1103/PhysRev.104.483 +[14] Rose, H.J. and Brink, D.M. 1967 Angular Distributions of Gamma Rays in Terms of Phase- +Defined Reduced Matrix Elements Rev. Mod. Phys. 39 , 306-347. DOI: 10.1103/RevMod- +Phys.39.306 +[15] Brink, D. M. and Satchler, G. R., 1962 Angular Momentum, (OUP). +[16] Brink, D. M. 1965 Nuclear Forces, (Pergamon). +[17] Brink, D.M. and Boeker, E. 1967 Effective interactions for Hartree-Fock calculations Nucl. +Phys. A 91, 1-26. DOI: 10.1016/0375-9474(67)90446-0 +[18] Gogny, D., Pires, D. P. and De Tourreil, R. 1970 A smooth realistic nucleon-nucleon +force suitable for nuclear Hartree-Fock calculations Phys. Lett. B32 591-595. DOI: +https://doi.org/10.1016/0370-2693(70)90552-6 +[19] Decharge, +J. +and +Gogny, +D.1980 +Hartree-Fock-Bogolyubov +calculations +with +the +D1effective +interaction +on +spherical +nuclei +Phys. +Rev. +C21 +1568-1593. +DOI:https://doi.org/10.1103/PhysRevC.21.1568 +[20] Skyrme, +A. +1959 +The +effective +nuclear +potential +Nucl. +Phys. +9 +615-634. +DOI:https://doi.org/10.1016/0029-5582(58)90345-6 +[21] Vautherin , D. and Brink, D. M.1972 Hartree-Fock calculations with Skyrme’s interaction. +1. Spherical nuclei Phys. Rev. C5 626-647. DOI: 10.1103/PhysRevC.5.626 +[22] Engel, Y.M., Brink, D. M., Goeke, K., Kriege, S.J. and Vautherin, D. 1975 Time de- +pendent Hartree-Fock theory with Skyrme’s interaction Nucl. Phys. A249, 215-238. DOI: +10.1016/0375-9474(75)90184-0 +[23] Bonche, P., Koonin S., and Negele, J. W., 1976 One-dimensional nuclear dynam- +ics in the time-dependent Hartree-Fock approximation Phys. Rev. C13 1226-1258. +DOI:https://doi.org/10.1103/PhysRevC.13.1226 +[24] Brink, D.M.and Stancu, Fl.1975 Interaction potential between two O-16 nuclei derived from +the Skyrme interaction Nucl. Phys. A 243 175-188. +16 + +[25] Stancu, Fl., Brink, D.M. and Flocard, H. 1977 The tensor part of Skyrme’s interaction Phys. +Lett. B68 108-112. +[26] Brink, D.M. and Stancu, Fl., 2007 Evolution of nuclear shells with the Skyrme density +dependent interaction Phys. Rev. C 75 064311. +[27] Hasan, H. and Brink, D. M. 1979 The transfer amplitude and angular distributions in +heavy-ion reactions J. Phys. G: Nucl. Phys. 5 771. +[28] Lo Monaco, L. and Brink, D. M. 1985 Perturbation approach to nucleon transfer in heavy-ion +reactions J. Phys. G: Nucl. Phys. 11 935-952. +[29] Brink, D.M. 1972 Kinematical effects in heavy-ion reactions Phys .Lett. B40 37-40. DOI: +10.1016/0370-2693(72)90274-2 +[30] Brink, D. M., 1985. Semi-Classical Methods in Nucleus-Nucleus Scattering, (Cambridge +University Press). +[31] Anyas-Weiss, N., Cornell, J.C., Fisher, P.S. ,Hudson, P.N., Menchaca-Rocha, A., Millener, +D.J., Panagiotou, A.D., Scott, D.K., Strottman, D., Brink, D.M., Buck, B., Ellis, T.P. +and Engeland, J., 1974 Nuclear structure of light nuclei using the selectivity of high en- +ergy transfer reactions with heavy ions Physics Reports 12 201-272. DOI: 10.1016/0370- +1573(74)90045-3 +[32] Bonaccorso, A. and Brink, D. M., 1988 Nuclear transfer to continuum state Phys. Rev. C38 +1776-1786; 1991 Stripping to the continuum of 208Pb Phys. Rev. C44 1559-1568. +[33] Bonaccorso, A. and Brink, D. M., 2021 Models of breakup: a final state interaction problem +Eur. Phys. J.A57 171. +[34] Jin Lei, Bonaccorso, A. 2021 Comparison of semiclassical transfer to continuum model with +Ichimura-Austern-Vincent model in medium energy knockout reactions Phys. Lett. B813 +136032. DOI:https://doi.org/10.1016/j.physletb.2020.136032 +[35] Berry, M. V. 1966 Uniform approximation for potential scattering. Proc. Phys. Soc. 89, +479-490. +[36] Sukumar, C. V. and Brink, D.M. 1983 Path integral methods for inelastic scattering Nucl. +Phys. A404 121-141. +[37] Brink, D.M. and Weiguny, A. 1968 The generator coordinate theory of collective motion +Nucl. Phys. A120 59-93. DOI: 10.1016/0375-9474(68)90059-6 +[38] Hill, D. L. and Wheeler, J. A. 1953 Phys. Rev. 89, 1102. +[39] Brink, D. M. 1975 Do alpha-clusters exist in nuclei. Proceedings of the INS-IPCR Symposium +on Cluster Structure of Nuclei and Transfer Reactions Induced by Heavy-Ions, Tokyo, March +17-22, 1975. +[40] Margenau, H. 1940 Phys. Rev. 5, 37. +[41] Brink, D. M. 1966 The Alpha-Particle Model of Light Nuclei, Scuola Internazionale di Fisica +’Enrico FERMI’, XXXVI Corso, Ed. C. Bloch, 247. +[42] Brink, D. M. 2008 J. Phys.: Conf. Ser. 111, 012001. +[43] Brink, D.M., Stringari, S., 1990 Density of states and evaporation rate of helium clusters. +Z Phys D-Atoms, Molecules and Clusters 15 257-263 . +17 + +[44] Sukumar, C. V. and Brink, D.M. 1997 Spin flip transitions in a magnetic trap Phys. Rev. +A56 2451-2454. +[45] Brink, D. M. and Broglia, R. A., 2005 Nuclear Superfluidity Pairing in Finite Systems, +(CUP). +[46] Courant, R. and Robbins, H. 1941 What is mathematics? +Oxford, UK: Oxford University +Press. +[47] Brown, G. E., 1967 Nuclear Physics: Many-Body Description of Nuclear Structure and Re- +actions. Course 36, International School of Physics ’Enrico Fermi.’ C. Bloch, Ed. Academic +Press, New York, 1966. Science, 158 (3807), DOI: 10.1126/science.158.3807.1440 +18 + diff --git a/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/load_file.txt b/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f06d34624d7d89debca426f1f7ef9a0b5be489a --- /dev/null +++ b/ItE1T4oBgHgl3EQfFwPf/content/tmp_files/load_file.txt @@ -0,0 +1,709 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf,len=708 +page_content='David Maurice Brink 20 July 1930 - 8 March 2021 Elected FRS 1981 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Sukumar1∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Bonaccorso2 † 1Wadham College, Oxford OX1 3PN, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='K, 2INFN, Sezione di Pisa, 56127 Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' January 10, 2023 Abstract David Brink was one of the leading theoretical nuclear physicists of his generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He made major contributions to the study of all aspects of nuclear physics embracing nuclear structure, nuclear scattering, and nuclear instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His wide ranging interests and inter- actions with theorists and experimentalists alike helped him in providing both theoretical analysis and interpretations and suggesting experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He had the gift of visualising complex problems in simple terms and provided clear analysis of the underlying processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He was an expert on the use of semi-classical methods which provided an intuitively clear picture of complex phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His research work and books are characterised by scientific clarity, transparency, and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David possessed outstanding skills in mathematical com- putation, and he was an expert on special functions, group theory, and the Feynman path integral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David had many research students and collaborated with a large number of scientists from across the world, for whom he was a source of scientific and human in- spiration and admiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His most fundamental belief was that research was a means of trying to discover and understand the beauties of Nature and explain them in simple terms to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His absolute belief in the value of truth and his unselfish and generous attitude in sharing knowledge makes him an outstanding figure in contemporary Nuclear Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1 Early years and family memories David Maurice Brink was born on 20 July 1930 in Hobart, Tasmania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His father, Maurice Brink had been born in the village of Bjuv in Sweden in 1900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David’s grandparents emigrated to Australia in July 1900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At the age of 14 David’s father moved to Sydney where he trained to become an accountant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' After this he went to Tasmania and joined an accountancy firm Wise, Lord and Ferguson, where he eventually became a partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In 1929 he married Victoria Finlayson, (born in 1900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Her father David had emigrated with his parents from Scotland in 1884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They had an engineering firm in Devonport, Tasmania whose main activity was maintaining and repairing machinery for mining, shipping, and timber companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David’s grandfather and his colleagues built the first steam car in Tasmania and between 1900 and 1904 built nine vehicles including three passenger cars and one 12-passenger bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David visited his grandparents often during vacations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He saw the casting floor and other parts of the factory and enjoyed playing amongst the remains of old steam traction engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' ∗candadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='sukumar@wadham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='uk †bonac@df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='it 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='02907v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='hist-ph] 7 Jan 2023 Figure 1: David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Brink 2 David was the eldest of three brothers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The Brink brothers went to a Quaker school in Hobart, Australia 1936 to 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David attended the University of Tasmania during 1948-51 studying Physics, Mathematics, and Chemistry, graduating with a BSc in December 1950 and was elected as a Rhodes Scholar at Magdalen College, Oxford, from October 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' From February 1951 to September 1951 he studied for BSc Honours in Hobart but did not complete the course because he moved to Oxford in September 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' As a student at the University of Tasmania David joined the Hobart Walking Club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' With this club he went on many trips to the interior of the island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When he arrived in Oxford he became a member of the Oxford University Alpine Club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Its activities took him to the Alps where he climbed in the Valais and the Engadine in Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It was in Switzerland that he met his future wife Verena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Verena and David married in 1958 and had three children together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His love for walking was transmitted to his three children who continue to enjoy walking in urban, rural, and mountainous settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' While always very committed and absorbed with his Physics he was also a devoted husband and father, transmitting his joy for walking and travel to his family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He often helped his children with their homework and was very patient with them, even when they were not!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Together David and his family travelled to, and lived in many countries across the world, where their horizons were broadened and they were introduced to the idea that there are many different ways of living and being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When his children had left home and travelled to other countries he would often be found in front of an atlas studying their exact whereabouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was very open minded and curious, always accepting other people’s opinions and points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David and Verena were very close, shared everything and had full respect for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Verena was a wonderful host and the Brinks often organised tea and dinner parties for students, visitors, and their families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Verena also helped visitors find accommodation, and with other issues related to living in Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They were also very generous in offering accommodation at their place whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In Oxford David developed an interest in birds, initially just birds he saw in Oxford, but when he travelled he always liked to look for birds and made lists of species he saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This curiosity in nature extended to other species as well, including trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When in 1993 he moved to Trento, Italy, he became a member of the SOSAT, a branch of the alpine club, and went regularly with them on Sunday trekking trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 2 Graduate studies and Oxford beginnings David started his studies at Oxford in October 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When he arrived at Magdalen College there was no tutor in Theoretical Physics at the college.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His maths tutor was David Kendal who sent him for tutorials to Jack De Wet at Balliol College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Jack asked David to read Von Neumann’s book on the foundations of Quantum Mechanics in German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He also encouraged David to change his studies from a BA in Mathematics to a D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Phil in Theoretical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Maurice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Pryce (FRS 1951) was the Wykeham Professor and head of the Theoretical Physics Department in Oxford from 1946 to 1954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He was David’s supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Pryce was also the part-time leader of the Theoretical Physics Division of the Atomic Energy Research Establishment (AERE) at Harwell, not far from Oxford, where nuclear theory was very much in the forefront and Rudolf E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Peierls (FRS 1945) was a consultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At Harwell there was a very productive theory group including Tony Skyrme and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' (Phil) Elliott (FRS 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Skyrme organized regular informal meetings known as ’Skyrmishes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Important papers in the latest journals were presented and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Members of the group attended Oxford seminars and while the local group including Roger Blin- Stoyle (FRS 1976), David Brink, and Pryce attended the Harwell meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Elliott gave some 3 Figure 2: David (right) in Tasmania 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 4 EVENT RESlectures at Oxford on Racah algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Later on his best-known work brought together the shell and collective models to explain rotational bands in deformed nuclei using the unitary group SU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' During this time he wrote a long article in Handbuch der Physik with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='(Tony)Lane (FRS 1975) [1] on the shell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The foundations of David Brink’s lifelong research, can all be found in his thesis ”Some Aspects of the Interactions of Fields with Matter” [2] which was submitted in May 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It is a remark- able document for its breadth and early contributions to the field of nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Pryce, his thesis adviser was interested largely in atomic spectroscopy but also studied the spectroscopy of nuclear energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The advent of the shell model around 1950 opened the door to new theoretical approaches for understanding the properties of nuclei and applying quantum mechan- ical tools to calculate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' There was also a great interest in reactions involving heavy nuclei and which could only be treated by statistical methods that had been developed much earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Brink’s two-part thesis contained contributions to both areas, reflecting the interactions between the Harwell and Oxford groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The first part was inspired by the shell model and the second contains important contributions to the statistical theory of nuclear reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the first part of his thesis, dealing with Nuclear Structure, David analyzed the spectroscopic consequences of the nucleon-nucleon interaction acting on the valence nucleons in nuclei close to the doubly-magic 208Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was able to estimate the order of magnitude of the interaction matrix elements from the properties of the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He also proposed treating the interac- tion through a density matrix expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This would figure prominently in later work in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The second part of his thesis dealt with reactions involving heavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It was probably inspired by the work of experimental group at Harwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' There, a Van de Graaff accelerator was used to measure energy levels, moments and transition rates in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was also fortunate to have contact with the strong experimental group working on neutron resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' While David was working on gamma widths of neutron resonances he benefited from contacts with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Hughes [3] and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Weisskopf who were visiting Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Weisskopf was very much interested in applying the detailed balance theory to nuclear reaction and interactions with him must have influenced David because at the end of the thesis he acknowledges discussions with Victor Weisskopf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The first subject in this part was the theory of inelastic scattering on deformed nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David constructed a theory for the excitation of rotational bands in deformed nuclei based on two new ideas, namely Bohr’s model of deformed nuclei and the optical model of Weisskopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' [4] published the previous year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was able to carry out the calculations to a point where the relative importance of this mechanism in the total cross section could be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This was an impressive achievement at a time before computers were available to carry out the full calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The final section of his thesis deals with the decays of the compound-nucleus resonances produced in reactions on heavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The formulas he presented here are still in use for modeling the spectra and reactions in heavy nuclei [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The best known is the formula for gamma decay rates in compound-nucleus resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This formula is based on a treatment widely known as the ”Brink-Axel” hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At a fundamental level, the theory was derived from the principle of detailed balance which Weisskopf had used very successfully in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The principle gives a formula to relate decay rates to absorption cross sections in the inverse reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The Brink-Axel hypothesis simply states that the absorption cross sections for gamma radiation on excited states of heavy nuclei can be estimated by the corresponding cross sections on the ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Axel and Brink worked independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Peter Axel’s paper appeared in 1962 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The important statement is made on page 101 of David’s thesis and is expressed in equation (11) of 5 Figure 3: David and his children (left to right), Barbara, Thomas, and Anne-Katherine 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Axel’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The prediction of the statistics of the widths of nuclear resonances, based on the generalization of the central limit theorem which David had learned about in his statistics course in Tasmania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David published the results in [7] where he showed the close connection between the shell-model description of the giant dipole resonance and the collective model of Goldhaber and Teller [8] and Steinwedel and Jensen [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' After his paper, theory of the giant resonances used the shell model as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Confirmation of the Brink-Axel hypothesis first came from the Berkeley experiments in 1981 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The last part of thesis has formulas related to another important topic in compound-nucleus theory, the fluctuations in decay widths of individual resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Here, David speculated that the fluctuations would follow a chi-squared distribution with one or two degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This is borne out experimentally and is now considered one of the hallmark properties of the compound nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It also became a part of random matrix theory in mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Unlike the early parts of the thesis, David never published the parts on compound-nucleus decay widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' However, physicists at the Harwell Laboratory knew about David’s results and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Lynn explained them in his book [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Unfortunately, David’s treatment of fluctuations was not recognized until very recently [12] and the distributions are known today under other author’s names [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 6 3 Research areas David’s interactions with the physicists mentioned earlier were reflected not only in David’s thesis but also in his early publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' One paper [14], which dealt with angular momentum couplings and angular distributions of γ-rays and other particles, is still the ”Bible” most experimentalist use when they analyse their data, as we have been told by Peter Butler (FRS 2019) (Liverpool) and Yorick Blumenfeld (Orsay), and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Early in his research career David wrote the textbook Angular Momentum [15] with Ray Satchler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This textbook was prominent among several texts published in this time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It was widely used by graduate students and post-graduates working in nuclear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David also published a book on Nuclear Forces [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='1 Effective interactions and calculations tools In his thesis David had laid the basis for the use of effective interactions in the calculations of matrix elements for nuclear structure studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The idea was greatly advanced in three later papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The first proposes a gaussian form for the effective nucleon-nucleon interaction known as the ”Brink-Boeker” interaction [17] that all nuclear physicists have used at least once in their lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This paper was very influential at the time and was later developed by Gogny and collaborators in the interaction that is widely used even today [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In 1959 Tony Skyrme proposed modelling the effective interaction between nucleons in nuclei by a short-range potential, an idea which is useful in nuclear structure and the equation of state of neutron stars [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The Skyrme force is an effective interaction depending on a small number of parameters whose strength could be fitted to reproduce various bulk properties of nuclei as well as selected properties of some nuclei, especially the doubly magic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At the beginning of the 1970s David was a frequent visitor to the Theoretical Division at the Institut de Physique Nucl´eaire, Orsay where his sixty-fifth birthday was celebrated (figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The work done there produced two papers with Dominique Vautherin [21, 22] which were the basis for the intense use of the so-called Skyrme interactions, in all their many present variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The papers revived a general interest in using Skyrme’s parametrization of the nucleon-nucleon interaction to calculate nuclear binding energies, and later to other aspects of nuclear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In effect, the interaction is treated as an energy-density functional theory in the spirit of the Kohn-Sham theory in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The Hartree-Fock calculations in [21] for spherical nuclei used Skyrme’s density dependent effec- tive interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This seminal paper showed how the Skyrme force could be used to make accurate calculations of certain nuclear properties and Vautherin and Brink developed these ideas further in a series of papers which had a strong impact on nuclear structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Otsuka comments: “The paper [21] has had a huge impact, as verified by the number of citations >2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In nuclear theory, papers having the citation index >1000 are rather few, which implies how important the Vautherin-Brink paper is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This year is the 50 year anniversary of this paper, and it is amazing that the basic formulation within the mean-field approach has not changed too much, implying that the scheme presented in this paper is so solid”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The calculations of Vautherin and Brink were extended by many other physicists during the subsequent period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In particular at Oxford, Micky Engel, Klaus Goeke and Steve Krieger, together with Dominique Vautherin derived the energy density using a Slater determinant where the single particle states were no longer invariant under time reversal, as it is in the Hartree-Fock method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' With the Skyrme interaction the TDHF approach leads to an equation of continuity for the single particle density [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This paper showed how Dirac’s time-dependent Hartree-Fock theory could be applied to nuclear dynamics in a light nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the year immediately following 7 the publication, the theory was applied to collisions involving a large number of nucleons [23], showing that the method would be a powerful one for heavy nuclei as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The method is justified as a time-dependent density-functional theory, and it remains in widespread use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In 1973 Ica Stancu came to Oxford as a post doctoral fellow and worked with David on heavy ion reactions in deriving the interaction potential of two 16O nuclei starting from the Skyrme energy density formalism [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They included the previously ignored tensor part of the Skyrme interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Along with an additional effort from Hubert Flocard at Orsay, the Skyrme HF calculations yielded single particle levels of spherical closed nuclei [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The role of the tensor force is to contribute to the spin-orbit splitting of the single-particle levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' For spherical closed shell nuclei the effect turned out to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Later it was found that in spherical spin unsaturated nuclei it makes a dramatic difference, giving the correct order of single particle levels, as, for example, in the Sn isotopes [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Many experiments on neutron-rich nuclei since 2006 have shown that the Skyrme formalism including the tensor force was the simplest way to describe the shell evolution of neutron-rich or proton-rich nuclei and indicated new magic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='2 Heavy-ions and Semi-classical methods in Nuclear Physics As tandem accelerators and cyclotrons were built to study heavy-ion Physics, David started an intense collaboration with the experimentalists at the Department of Nuclear Physics in Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The accelerators were used to study heavy-ion elastic scattering and direct reactions such as transfer and measure masses and perform spectroscopy of neutron-rich matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In those years semiclassical methods were widely used in the Nuclear Physics community to analyse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They were particularly appropriate for heavy ions because of the high incident energies and the large impact parameters involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Thus David started the Oxford school on the subject, more or less parallel in time to the Copenhagen school of Broglia and Winter and collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At that time, these heavy-ion reactions were analyzed through the partial wave expansions of the colliding partners, a methodology that was computationally demanding and giving little insight to the underlying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David’s semi-classical treatment of the collision was much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Some of the early papers on the theory of peripheral reactions were based on his student’s thesis, including Hashima Hasan and Luigi Lo Monaco [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David’s investigation of the kinematical effects in such reactions, for which there was concrete experimental evidence from the work of Peter Twin (FRS 1993) and his collaborators at Liver- pool, became a key element for experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the paper by the title ”Kinematical effects in heavy-ion reactions” [29] David introduced a ”semi-classical amplitude” [30] that could be used in DWBA-like calculations of transfer [31] and proposed a matching condition to predict a large reaction cross-sections, a condition that was beautifully adapted to understand spin-polarization experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He showed that energy and angular momentum couplings in heavy-ion reactions led to very selective matching rules by which high angular momentum single-particle states could be populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' High angular momentum single-particle states sometimes appear as low-lying continuum resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They have been studied by the method of transfer-to-the-continuum [32] which has helped disentangle single-particle from collective degrees of freedom and has also been applied in the so called ”surrogate reactions” as a substitute for free neutron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Semi-classical ideas have been helpful in studying breakup and dissociation of weakly bound radioactive ions including halo nuclei and other such unstable nuclei whose dynamics is rather involved and difficult to study experimentally due to the very low intensity of beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David, Angela Bonaccorso and her students got heavily involved in this new physics from the ’90s on, with a long series of papers (see [33] and references therein), conference contributions, meeting organization, some of them at the ECT* in Trento, spanning the last forty years of David’s 8 career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Finally it has recently been shown [34] that the semi-classical treatment of breakup by David and his collaborators is fully consistent with a quantum mechanical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David studied microscopic models for the real and imaginary parts of the ion-ion optical potential to be used in elastic scattering calculations with Ica Stancu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He also studied fusion with Neil Rowley and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Takigawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David and Takigawa developed a semi-classical reaction theory with three classical turning points which explained the anomalous large angle scattering of α particles as a quantum-mechanical interference between the barrier wave and the internal wave, thereby providing an intuitively clear picture of a complex phenomenon underlying nuclear reactions in terms of classical and quantum ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David, Vautherin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Nemes studied the effect of intrinsic degrees of freedom on the quantum tunnelling of a collective variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This work was further developed by other theorists including Kouichi Hagino who studied the deviation from adiabaticity in quantum tunnelling with many degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David met Uzi Smilanski in Munich when they were both there on sabbatical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Both had worked on semi-classical approximations and gave a joint series of lectures on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was con- cerned that the standard WKB method was insufficient to explain tunnelling through a barrier and was particularly bad near the barrier top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David and Uzi applied the uniform semi-classical method evolved by Michael Berry (FRS 1982) to successfully address the problem [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Uzi remembers David as a physicist with excellent intuition and an ability to grasp the essence of a problem before cracking the problem with rigorous mathematics and complex computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David, Massimo di Toro, and Alberto Dellafiore developed a semi-classical description of col- lective responses with a mean field approach paving the way for a study of the dynamics of a nuclear Hartree-Fock fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When the national heavy-ion laboratory started in Catania (LNS- INFN) around an advanced superconducting cyclotron, David was a reference point for simple physics suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='3 Path integral methods in Nuclear Physics David’s expertise with semi-classical methods for tackling quantum problems naturally led him towards the Feynman path integral approach to quantum mechanics which was based on a Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Hans Weidenm¨uller had met David at various conferences in the 1950s and 1960s and spent 1977-78 on a sabbatical in Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' During this period David and Hans worked on the application of the Feynman path integral method to the study of the heavy-ion reactions and developed the Influence Functional approach to this problem which David and his collaborators later used to establish master equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Hans remembers that at a summer school a few years later David delivered a series of lectures on nuclear reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the first lecture he developed the topic using a dozen transparencies and in subsequent lectures used the same transparencies in a different order to display and illuminate aspects of the topic that had gone unnoticed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Hans remembers it as a display of the combination of simplicity and depth that were hallmarks of David’s approach to Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The path integral method was particularly well suited for studying problems with many degrees of freedom in which classical description in terms of trajectories was good for some degrees of freedom but not for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Coulomb excitation in heavy-ion collisions is an example where the relative motion of the ions could be described in terms of coulomb trajectories but the excitation of the quantum states of the ions had to be treated using quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David and Sukumar [36] used the Feynman path integral method to evolve a systematic way of arranging the correction terms for the quantum amplitudes for processes involving coupled degrees of 9 Figure 4: David and his wife Verena, May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' freedom where the description in terms of classical trajectories was good for some degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David, Sukumar, and Fernando Dos Aidos used this method to provide corrections to the primitive semi-classical amplitude for Coulomb excitation of heavy-ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Sukumar and David used the path integral method to describe spin-orbit coupling effects and together with Ron Johnson at Surrey and his group successfully explained the experimental data on polarization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 4 Other topics David was very quick at grasping the core of a Physics problem and putting it in simple, calculable terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Often the problem required somewhat involved analytical calculations, but he was a master of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Thus anytime a visitor went to Oxford with a new problem, David would start a very successful line of research which he often followed up with his graduate students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='1 Cluster models It happened for example with the cluster model physics, starting with the seminal paper [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This paper developed the generator coordinate method of Hill and Wheeler [38] to produce a practical tool to reduce the many-particle Hamiltonian to an ordinary Schr¨odinger equation for a collective variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Thus the nuclear cluster model was related to the shell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' To treat nuclear states in such different circumstances, a formulation which includes clustering at one extreme and shell structure at the other extreme was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David proposed microscopic multi-α-clusters treating four nucleons with different spin-isospin states as a single particle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Under anti-symmetrisation of nucleons the cluster model wave-functions approximate shell model functions and enabled the description of both cluster and shell model structures in a unified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Their approach was adopted and is in widespread use even in present-day nuclear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The main applications up to now are on spectroscopy and large-amplitude collective motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Suzuki’s work on the cluster model was largely inspired by David’s paper on ”Do alpha clusters exist in nuclei?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' [39] presented at a meeting in Tokyo in 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This paper contained all the essential components needed in the alpha particle model, the microscopic theory beyond the shell model description based on many-particle many-hole excitations, the relation between the resonating group method GCM, the equilibrium arrangement of clusters, extension of the Hill-Wheeler method, the angular momentum projection, and the Slater determinant technique for evaluating matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Suzuki remembers that David never forgot to mention that the original model was proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Margenau and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='Bloch [40, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At the Varenna School in 1955 David met S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Yoshida from Japan and they discussed inelastic scattering of protons and neutrons by deformed nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' By chance David had a chapter in his thesis on this topic and Yoshida had been studying the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' This interaction with Yoshida helped David to develop strong connections with nuclear theory groups in Japan over many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='2 Bose-Einstein condensation of atoms During his period as Deputy Director of ECT* in Trento, 1993-1998, David interacted with many members of the Physics Department in Trento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' One such interaction with Sandro Stringari led to David’s interest in Bose-Einstein condensation of alkali atoms in magnetic traps [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Sukumar and David [44] developed an approximate method for calculating the rate of escape from the magnetic trap thereby enabling an estimation of the duration for which the condensate atoms can be held in the trap as a function of the ultra-cold temperature and the strength of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='3 Miscellaneous David was interested in the role of pairing interaction in finite nuclei and this led to the study of nuclear superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His book with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Broglia [45] is considered to be a wonderful exposition of this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David’s knack for explaining detailed Physics in a simple and clear manner is abundantly evident in this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the 1990s Ica Stancu raised David’s interest in the quark structure of exotic hadrons named tetraquarks, a system of two quarks and two antiquarks, and studied the stability of such systems containing heavy quarks/antiquarks in a QCD inspired quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Even though David had not worked on the Interacting Boson Model (IBM) he nevertheless provided supervision for doctoral students such as Martin Zirnbauer who chose topics in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' He also supervised Hans Peter Pavel’s thesis on Schwinger pair production in a flux tube model containing a chromomagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 11 5 Teaching and administrative roles David’s doctoral students remember him for the gentle way he corrected them when they had made errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Many of the students learned from him how to take a critical approach to their results and how it is possible to look at a complex problem from several different viewpoints and find the one that gives the best physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' They also remember the immense support he gave to their research and pastoral care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Many graduate students also remember how much they had learned from the courses he taught at Oxford and at Summer schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' His book with Satchler [15] and paper with Rose [14] on angular momentum algebra were found to be of immense value in formulating and tackling problems in Nuclear Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Many researchers and students who met David were astonished that someone with such towering achievements could be so humble, nice and honest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was very open-minded and we report a number of episodes to illustrate this aspect of his character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Future Nobel laureate Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Tony Leggett remembers: ” My undergraduate major at Balliol was in Greats (classical languages, ancient history and philosophy) and I was set to graduate (and eventually did so) in the summer of 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Towards the end of the academic year 1957-1958, partly encouraged by the post-Sputnik cultural swing towards science in the UK, I conceived the ambition of taking a second undergraduate degree in physics and perhaps eventually making my career in academia in that subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Given that I had essentially no meaningful exposure to physics at the high-school level and only a brief and informal exposure to any kind of mathematics beyond simple differential calculus (I’m not sure that I had even had that), such a drastic change of academic direction was extremely unusual, indeed at the time almost unheard-of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' My first concern was to find a higher education institution which would accept me for it and I rapidly concluded that my only hope was to apply to my existing Oxford college, Balliol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David had just recently become the college’s first tutor in theoretical physics (most Oxford colleges did not have such a thing in 1958), so it fell on him to take the decision on my application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' To this end he asked me to read over the summer vacation a few chapters from the book ”What is Mathematics?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' by Courant and Robbins [46], perhaps the most beautiful presentation I have ever seen of mathematical topics for the layperson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' When I returned to Oxford in the Fall of 1958 he gave me an informal mini-exam on that material, and on the basis of my performance decided to recommend to Balliol to accept me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the event I did my physics degree at Merton, who offered me a scholarship, but since they did not at the time have a tutor in theoretical physics David played that role for me for much of the two years which it took me to complete the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I think it is virtually certain that had he made the opposite decision, I would never have had a career in physics, and I am profoundly grateful to him for the imagination he showed in going beyond my formal academic qualifications.” Another story comes from Paul Stevenson: ”I was called up for interview at Balliol in December 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The office I was in for that interview was David Brink’s office, above the Senior Common Room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the interview were me, David Brink, David Wark, Jonathan Hodby (those three there for physics) and Bill Newton-Smith (for philosophy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I don’t remember all the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I do remember that David Brink showed me a postcard and asked me what, physically, was wrong with the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' It was a Japanese style print with a mountain in the background and a lake in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' There was a reflection of the mountain in the lake, but it was off to one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I saw what was wrong, and struggled to articulate it in the language of a physicist, and in the end David prompted me by asking what is particular about an incident light ray, a reflected light ray, and the normal to the surface at which it is reflected and I said the right thing - that they are all in the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I was duly accepted to Balliol and spent three years there studying physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Danny Chapman remembers: ”I don’t think I’ll ever forget the ”sense” of David Brink’s 12 tutorials, and of being in the presence of such a sharp and insightful mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I remember being quite inspired once when my fellow student had tried to answer a question in what I thought was an odd and probably wrong way, ending up with a sum, which he then attempted to turn into an integral, which didn’t work out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Rather than saying ”don’t do it like that, do it like this”, David was able to continue from there and make it work, which was a really positive experience and encouragement to follow every path to its end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I feel lucky to have been at Balliol when he was there.” Angela Bonaccorso remembers daily life as one of David’s students: At the Department of The- oretical Physics there was a coffee room where coffee was served between 11:00 and 11:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' We would try to be there on time to sit around David who would be chatting with other senior members of the department or some visitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' There would always be someone bringing up some interesting and challenging new problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Everyone gave an opinion, the atmosphere was com- petitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Most of the time David would win the argument and his students felt very proud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='Not all supervisors were so nice, helpful, and respectful of us as David was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' But it was not at all easy to be David’s student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' First of all we needed to have detective skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David was very busy and very elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In those days there was no email or SMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The only way to be sure that he was inside was to look for his bicycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' If the bicycle was outside we would knock at the door of his office and if we were lucky he would answer and let us in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In spite of all his many commitments we always managed to have at least one chat per week with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Another reason why it was not easy to be his student was that David had a very original way of understanding things and finding the way out of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' During our conversations often he would stop talking and be silent for five to ten minutes, rubbing his hand on his forehead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Then he came up with some equation, or a drawing or something like that and he would tell us: I think it is like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='I think we should get something like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content='etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I (we) would stare at him speechless and in wonder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Where did the ’oracle’ come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Most of the time this was the end of the meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' I (we) left his office rather puzzled, worked desperately hard for one week and if we had managed to understand his line of thought, after pages and pages of calculations, we would find exactly what he had predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' We all knew it was like that, we all passed this information on to each other, generation after generation: listen to David, he is always right, just try to reproduce the miracle of his craftsmanship in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' A further proof of how much busy David was and how precious was for everyone the time spent in conversation with him can be found in the comment Gerry Brown made in his review for Science [47] of the Proceedings of the Varenna summer school [41] : ’Let me draw special attention also to the article of David Brink, ”The alpha-particle model of light nuclei,” which is one of the most beautiful developments in this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Brink likes to sit on his work for years and, on the whole, doesn’t even answer letters inquiring about it, so that one must either adopt the expedient of traveling to Oxford to talk with him, or invite him to lecture at summer schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Both are worth while.’ David was a pillar of Balliol college and Department of Theoretical Physics for decades, an immensely popular tutor and supervisor, a cheerful and always helpful colleague, and a wonderful guide to younger colleagues and administrative staff who happened to be working with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' David had another long and distinguished career in Italy after he left Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Following an invitation from Renzo Leonardi he moved to Trento as full professor of History of Physics and helped in establishing the ECT*, European Center for Theoretical Studies in Nuclear Physics and Related Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The Nobel laureate Ben Mottelson was the founding director and David the vice-director, while Renzo Leonardi was the Scientific Secretary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' In the five years David spent at Trento he took care of organising various technical aspects of the secretarial offices, 13 Figure 5: David’s sixty-fifth birthday celebration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Orsay, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' library, computer center and visitor hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' At the same time he gave very productive contributions to workshops with his constant presence, his huge knowledge of nuclear physics and stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' The superb reputation and international standing of this extremely important European initiative is undoubtedly due in large part to David’s wisdom in its crucial, formative years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 6 Career, Honours and Awards 1954-55 Royal Society Rutherford Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1957-1958 Instructor at the Massachusetts Institute of Technology (MIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1958 Fellow of Balliol College and Lecturer in Theoretical Physics, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1976-1978 Vice-Master of Balliol College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1981 Fellow of the Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1982 Rutherford Medal of the Institute of Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1988 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Mosley Reader at Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1990-1993 Senior Tutor, Balliol College, academic planning and administration, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1992 Foreign member of the Royal Society of Sciences, Uppsala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1993-1998 ECT*, Trento, Vice-Director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 1993-1998 Full professor of History of Physics, University of Trento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 14 2006 Varenna Conference on Nuclear Reactions dedicated to him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 2006 Lise Meitner prize of the European Physical Society shared with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Kluge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Visiting scientist at : Niels Bohr Institute 1964, University of British Columbia 1975, Institut de Physique Nucl´eaire d’Orsay 1969 and 1981-1982, The Technical University of Munich 1982, University of Trento 1988, University of Catania 1988, Michigan State University 1988-1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' 7 Acknowledgements The authors are greatly indebted to the Brink family for sharing with them private memories and photographs and for a critical reading of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' A large number of friends and colleagues, too many to be individually mentioned, contributed with their appreciation of David’s life and scientific career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' Ica Stancu and Sharon McGrayne Bertsch read and commented the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} +page_content=' One of us (AB) gratefully acknowledges George F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfFwPf/content/2301.02907v1.pdf'} 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file mode 100644 index 0000000000000000000000000000000000000000..0f6a7d89cfc3ee8e2f5a4e7dd1ec5367fdffb752 --- /dev/null +++ b/ItE5T4oBgHgl3EQfXA9_/content/tmp_files/2301.05563v1.pdf.txt @@ -0,0 +1,747 @@ + + +High-fidelity ptychographic imaging of highly periodic structures +enabled by vortex high harmonic beams +Bin Wang1*†, Nathan J. Brooks1*, Peter C. Johnsen1, Nicholas W. Jenkins1, Yuka Esashi1, Iona +Binnie1, Michael Tanksalvala1, Henry C. Kapteyn1,2, Margaret M. Murnane1 +1STROBE Science and Technology Center, JILA, University of Colorado, Boulder, CO 80309, USA +2KMLabs Inc., 4775 Walnut St. #102, Boulder, CO 80301, USA +*These authors contributed equally +†bin.wang-2@colorado.edu + +Abstract +Ptychographic Coherent Diffractive Imaging enables diffraction-limited imaging of nanoscale structures at extreme +ultraviolet and x-ray wavelengths, where high-quality image-forming optics are not available. However, its reliance +on a set of diverse diffraction patterns makes it challenging to use ptychography to image highly periodic samples, +limiting its application to defect inspection for electronic and photonic devices. Here, we use a vortex high harmonic +light beam driven by a laser carrying orbital angular momentum to implement extreme ultraviolet ptychographic +imaging of highly periodic samples with high fidelity and reliability. We also demonstrate, for the first time to our +knowledge, ptychographic imaging of an isolated, near-diffraction-limited defect in an otherwise periodic sample +using vortex high harmonic beams. This enhanced metrology technique can enable high-fidelity imaging and +inspection of highly periodic structures for next-generation nano, energy, photonic and quantum devices. +Introduction +In recent decades, a powerful coherent diffractive imaging (CDI) technique known as ptychography has enabled robust, +diffraction-limited, phase-contrast imaging of nanoscale structures [1-5]. Although ptychography has been +implemented using a range of illumination wavelengths, its use in the extreme-ultraviolet (EUV) and x-ray regions is +particularly attractive for achieving high spatial resolution with inherent elemental and chemical contrast [6-10]. In +ptychography, a coherent illumination (the probe) is focused and scanned across an extended sample. A series of far- +field diffraction patterns are recorded, while maintaining a large overlap between adjacent scan positions. Iterative +phase-retrieval algorithms [11-15] can then be used to robustly and uniquely reconstruct the complex-valued probe +field and sample transmission or reflection functions. However, successful reconstruction relies heavily on diversity +in the data provided by the lateral scanning of the probe relative to the sample, i.e., interferences at the detector plane +mix amplitude and phase, allowing the reconstruction algorithms to unravel both. this means that ptychographic +imaging of highly periodic samples with a sufficiently small period is extremely challenging due to the lack of +diversity in a series of diffraction patterns, leading to poor convergence of the reconstructed sample images. This +significantly limits ptychography’s application to a wide variety of nanoscale periodic structures such as photonic +crystals [16-17], semiconductor devices [18], and EUV photomasks [19-25]. Consequently, it is critical to fill this +characterization gap to aid the advancement of a host of next-generation nano-devices. +High harmonic upconversion of femtosecond lasers can produce bright coherent beams from the EUV to the soft x- +ray regions of the spectrum, in a tabletop-scale setup [26-28]. When combined with ptychography, high harmonic +generation (HHG) enables phase-sensitive lensless imaging with diffraction-limited nanoscale spatial resolution and +excellent elemental specificity [9,15,29-31]. Moreover, because of the quantum-coherent nature of the HHG +upconversion process, polarization and phase structure present in the driving laser beam can be transferred to the +generated harmonics, provided energy, spin and orbital angular momentum are conserved [32,33]. This makes it +possible to create designer short-wavelength structured light for a variety of applications in advanced spectro- +microscopies [34,35]. +Light beams carrying orbital angular momentum (OAM) have attracted considerable interest for super-resolution +imaging [36] and enhanced optical sensing, communication and inspection [37-39]. Recently, by considering + + + +conservation of OAM in HHG upconversion, additional routes for controlling the OAM, polarization, as well as the +spectral and temporal properties of HHG have been revealed [40-47]. A property particularly interesting for +ptychography is the relationship between OAM and the HHG beam divergence/propagation: the spiral phase structure +characteristic of OAM-HHG beams forces them to diverge more quickly from the focal point [46]. This means that +by using one or more OAM beams to drive the HHG process (referred to as OAM-HHG), one can control the +divergence of the emitted HHG probe without changing the focusing optics of the HHG driving laser. +In this article, we demonstrate a solution to a decade-long challenge by showing that high harmonic beams carrying +orbital angular momentum can be used to advantage in high-resolution, high-fidelity and fast-convergence +ptychographic imaging of highly periodic two-dimensional (2D) grating structures, using the standard extended +ptychographic iterative engine (ePIE) algorithm [13]. The key to this technique is that the increased divergence of the +OAM-HHG source, combined with the ring-shaped intensity distribution, introduces strong interference fringes +between adjacent diffraction orders in the far-field. These encode the non-measurable phase information into the +measurable intensity modulation in diffraction fields, significantly enhancing data diversity so that the phase of the +diffracted field can be reliably retrieved. We further show that using OAM-HHG beams for illumination provides +three significant advantages compared to standard Gaussian-HHG beams, all of which lead to enhanced signal-to- +noise-ratio (SNR) for imaging periodic structures: First, due to the conservation of OAM in the HHG process and the +resulting strong spiral phase structure in the generated EUV beams, OAM-HHG beams naturally have a significantly +increased divergence compared to that of Gaussian-HHG beams. This enhanced illumination NA makes it possible to +achieve overlap between different diffraction orders for small pitch periodic samples, beyond what is possible with a +Gaussian-HHG probe, and without making any changes to the focusing optics of the ptychographic EUV microscope. +Second, the unique ring-shaped OAM beam intensity distribution, which is determined by the strong spiral phase +structure in the EUV beams, leads to overlap between different diffraction orders in the high-intensity regions of the +beam. And third, OAM-HHG also allows a higher total number of photons to be collected by the detector given a +fixed detector dynamic range. Therefore, by leveraging OAM-HHG beams for ptychography, we successfully imaged +highly periodic samples with substantially reduced gridding artifacts, and reliably detected defects near the diffraction +limit. This new structured-EUV HHG metrology technique can support the advancement of next-generation EUV +lithography, nanoelectronics, photonic and quantum devices. +Methodology + +To date, imaging highly periodic structures has been extremely challenging for ptychography. In a conventional +implementation of ptychography using a Gaussian EUV beam to illuminate highly periodic 2D structures (see Fig. +1(a)), the far-field diffraction patterns consist of many isolated diffraction orders, each of which is a copy of the far- +field beam profile, and is modulated by an envelope in both amplitude and phase. The zoomed-in green circle in Fig. +1(a) shows this characteristic behavior, with the white circles indicating the edge of each diffraction order. In the +resulting ptychographic dataset, diffraction patterns taken at different positions on the highly periodic sample are +almost identical to each other. This is because, in contrast to diffraction from non-periodic structures, changes in the +far-field diffraction field happen almost entirely in the relative phase between the diffraction orders, but not in the +intensity (i.e., diffraction efficiency) of the diffracted orders. The phase information is thus totally lost in this case. +Ptychography, as a phase retrieval algorithm, tries to retrieve the phase distributions of diffraction patterns from their +intensity measurements. The fact that the phase information is totally lost for highly periodic samples with a +sufficiently short period, as opposed to being encoded in the intensity measurements as would be the case for non- +periodic structures, makes it extremely challenging to achieve successful ptychographic imaging of such highly +periodic structures. As expected, the reconstruction fails for ptychography using a Gaussian-HHG beam, as shown in +Fig. 1(b), in which the amplitude and phase of the reconstruction are plotted in brightness and hue, respectively. This +phase problem can also be understood through the convolution theorem, as discussed in detail in Supplementary +Section 1. +In 1969, Hoppe proposed to achieve electron diffraction imaging of periodic atomic lattices by encoding the non- +measurable phase information into the measurable intensity modulation in diffraction patterns, through interference +between neighboring diffraction orders [48]. As schematically shown in Fig. 1(c), OAM-HHG beams are able to +achieve overlap and interference between neighboring diffraction orders due to their intrinsically larger beam + + + +divergence and ring-shaped intensity distribution. The zoomed-in blue circle in Fig. 1(c) shows the interference fringes, +with the yellow circles indicating the edge of each diffraction order. As one scans the probe relative to the periodic +structures, the relative phase of each diffraction order changes accordingly, which then causes the interference fringes +to shift. In other words, the phase information in the diffraction patterns is now encoded in the intensity measurements +through interference. These interference fringes contain the missing phase information, and increase the diversity in +diffraction patterns, thereby enabling robust and reliable ptychographic reconstructions (see Supplementary Section +2). Figure 1(d) shows a high-fidelity ptychographic reconstruction of a 2D periodic structure under an OAM-HHG +illumination. + +Figure 1. Robust and reliable ptychographic imaging of highly periodic structures. (a) Schematic showing HHG +ptychographic imaging of a periodic structure using conventional Gaussian-HHG illumination. The resulting diffraction orders are +isolated (see zoomed-in green circle), where the white circles indicate the edges of each diffraction order. This leads to a complete +loss of the relative phase information between the orders in the far-field diffraction, which subsequently leads to the failure of the +ptychographic reconstruction in (b). (c) OAM-HHG illumination intrinsically has a larger source divergence and a ring-shaped +intensity profile, to support overlap and interference between diffraction orders (see zoomed-in blue circle), in which the yellow +circles indicate the edges of each diffraction order. This interference converts the relative phase between the diffraction orders into +measurable intensity modulation, enabling fast and robust ptychographic reconstruction of the 2D periodic structure in (d). In (b, +d), the complex-valued amplitude and phase are plotted as brightness and hue, respectively. + +5um + +The required NA for high fidelity imaging of periodic samples can be understood as follows. When a periodic structure +is illuminated by a focused coherent beam, the angular separation between two neighboring diffraction orders is given +by 𝛥𝜃 = 𝜆/𝛬, where 𝜆 is the illumination wavelength, and 𝛬 is the period of the structure. The illumination NA for +the microscope is defined as the half-cone angle of the focusing beam. Geometrically, for fixed 𝜆 and 𝛬, there exists +a critical value for illumination NA: +𝑁𝐴𝑐 = +1 +2 𝛥𝜃 = +𝜆 +2𝛬. (1) +Only for illumination NAs larger than 𝑁𝐴𝑐 will neighboring diffraction orders have sufficiently large footprints on +the detector to overlap with each other and generate interference fringes, thus enabling successful ptychographic +reconstructions. +Experimental configuration +We designed and built an EUV ptychographic microscope in a transmission geometry, as shown in Fig. 2. The driving +laser for the HHG process is a frequency-doubled Ti:sapphire laser amplifier system (𝜆~395 nm), with an intrinsic +near-Gaussian mode (vortex charge of ℓ = 0), that can be converted to an OAM beam of vortex charge ℓ = 1 using a +spiral phase plate. The 7th harmonic of the driving laser (𝜆~56 nm) exhibits either a Gaussian mode or an OAM of +vortex charge ℓ = 7 depending on whether a spiral phase plate is used. The EUV beam is then focused by a double- +toroid focusing system onto the periodic samples, with a spot size of ~13 × 18 μm (1/𝑒2 intensity) for Gaussian-mode +HHG, or ~27 × 32 μm (donut intensity peak-to-peak) for OAM-HHG. The reconstructed Gaussian- and OAM-HHG +beam profiles are shown in a complex representation in Fig. S4, with the beam amplitude and phase indicated by +brightness and hue. The test samples are three Quantifoil holey carbon films (~20 nm thick) which have various hole +sizes and shapes arranged in a periodic rectangular grid. The three Quantifoil holey carbon films have a pitch of 9 μm, +4.5 μm and 3 μm, respectively. These Quantifoil holey carbon films are mounted on standard Ted Pella Ø3mm Cu +200 mesh TEM grids. (See the Methods section for more information.) + +Figure 2. EUV ptychographic microscope using OAM-HHG EUV beams for imaging highly periodic structures. A spiral +phase plate (ℓ = 1 at 395 nm wavelength) converts the driving laser at 395 nm wavelength to an OAM beam, which is focused into +a semi-infinite gas cell to produce a nearly monochromatic 7th harmonic beam with a wavelength of 56 nm and an OAM charge +of ℓ = 7. A double-toroidal mirror focusing system focuses this OAM-HHG beam onto a 2D periodic sample, and an EUV-CCD +camera is used to record the far-field diffraction patterns. + +e=7 +入=56nm +Detector +Periodic +Sample +0=3.5mrad +To +l= 1 +入=395nm +do= 100 μm + +During the ptychography scans, the test samples are translated in the plane perpendicular to the beam path in 7 × 7 +rectangular grids (49 scan positions) with nominally 3.3 μm distance between adjacent scan positions. A random offset +of ±20% of the scan step size was added to each scan position to avoid gridding artifacts in the reconstructions [49]. +The far-field diffraction patterns are recorded by an EUV-CCD detector (Andor iKon-L, 2048 × 2048, 13.5 μm pixel +size) positioned 50 mm after the sample. To obtain the best ptychographic reconstructions possible for each +illumination case, we carefully pre-characterized each probe function in the sample plane by taking ptychographic +scans on a non-periodic sample and reconstructing both the sample and the probe functions through blind +deconvolution. These pre-characterized probe functions were used as initial guesses in the ptychographic +reconstructions of highly periodic structures. Other than using the pre-characterized probe as the initial guess, we used +the standard ePIE algorithm [13] for all reconstructions in this study, without the need for additional constraints such +as modulus enforced probe [15] or total variation regularization [25,50]. + +Results +OAM-HHG enables robust and reliable ptychographic imaging of periodic structures +We experimentally demonstrate that OAM-HHG beams enable robust and reliable ptychographic imaging of highly +periodic structures because of three intrinsic advantages compared to Gaussian-HHG illumination. First, due to the +conservation of OAM in the HHG process and the resulting strong spiral phase structure in the generated EUV beams, +OAM-HHG beams naturally exhibit a significantly increased divergence (i.e., increased illumination NA for the +microscope given the same focusing optics) compared with Gaussian-HHG beams. This enhanced illumination NA +allows us to achieve overlap between diffraction orders for smaller pitch periodic samples beyond what is possible +with a Gaussian-mode probe, without making any changes to the EUV microscope end-station. Second, the +characteristic ring-shaped intensity distribution of OAM-HHG beams ensures that the majority of photons fall in the +overlap area (in contrast to the Gaussian-HHG beams, for which the overlap between diffraction orders occurs at the +tails of the intensity distributions), which increases the SNR for the interference fringes. Third, the ring-shaped +intensity distribution of OAM-HHG beams allows one to collect a higher total number of photons by the detector +given a fixed dynamic range, which also leads to higher SNR without the need for high dynamic range (HDR). +We performed ptychographic imaging on three highly periodic structures with 9 μm, 4.5 μm and 3 μm pitches using +Gaussian- and OAM-HHG beams at a wavelength of 56 nm. Example diffraction patterns and reconstructed images +from each ptychography scan can be found in Fig. 3. Furthermore, each ptychography scan collected 49 far-field +diffraction intensity patterns, as shown in a log scale in Supplementary Video 1. Note that while there is a clear change +in the diffraction patterns from frame to frame for the OAM-HHG case, particularly in the interference fringes between +the adjacent diffraction orders, the diffraction patterns in the Gaussian-HHG case do not change very much for small +period samples. All ptychography datasets were taken without making any changes to the EUV microscope — the +difference in divergence between Gaussian- and OAM-HHG beams is intrinsic to the HHG upconversion process, +which conserves energy and OAM. +For the 9 μm pitch sample, the small diffraction angle means that successive orders largely overlap even for the +Gaussian-HHG beam, as shown in Fig. 1(a). This results in a reasonably good image, apart from some gridding +artifacts as shown in Fig. 1(d). In comparison, the ptychography scan using OAM-HHG illumination sees more overlap +resulting in improved SNR in the interference fringes and a much higher-fidelity image with greatly reduced gridding +artifacts, as shown in Fig. 1(g,j). +For the smaller 4.5 μm pitch sample, the diffraction orders are further apart, causing the Gaussian-HHG beams to lose +most of the interference in the diffraction patterns, as shown in Fig. 3(b). The low SNR in the interference fringes +results in reduced quality image reconstruction, as shown in Fig. 3(e). However, due to their higher intrinsic divergence +and the unique ring-shaped intensity distribution, OAM-HHG maintains a large area of overlap between neighboring +diffraction orders with more photons, as shown in Fig. 3(h). This results in higher-quality images of the periodic +structure with a 4.5 μm period, as shown in Fig. 3(k). Thus, simply by inserting a spiral phase plate to convert the + + + +driving laser to an OAM beam, while keeping everything else in the microscope the same, a greatly improved +reconstruction quality is obtained. +Lastly, for the smallest 3 μm period sample, Gaussian-HHG illumination totally fails due to the lack of interference +between diffraction orders, as shown in Fig. 3(c,f). In this case, OAM beams can reconstruct a reasonable image, +although the quality of the unit cell is degraded, as shown in Fig. 3(i,l). +We also evaluated the quality of these ptychographic reconstructions using complex histogram analysis and the results +can be found in Supplementary Section 3. We note that all reconstructions in Fig. 3 have the correct sample periodicity +because this information is directly available from the measured intensity of the diffracted fields — the success or +failure of the reconstructions of the unit cells depends on whether the relative phase between the diffraction orders +can be successfully retrieved or not. The ptychography reconstructed images in Fig. 3(d–f, j–l) are complex-valued +and are shown in a complex representation where the amplitude and phase information are represented by the +brightness and hue, respectively. The color wheel is shown in the bottom left corner of each panel. + + + + +Figure 3. High divergence OAM-HHG beams are able to produce higher-quality ptychography images of periodic +structures than low divergence Gaussian-HHG beams. Three test samples with different periods and shapes, i.e., 9 μm period +with square holes, 4.5 μm period with circular holes and 3 μm period with circular holes, are investigated. For Gaussian-HHG +beams, example diffraction patterns from the three test samples are shown in (a–c) and the corresponding ptychography +reconstructed images are shown in (d–f). The example diffraction patterns and ptychography images from OAM-HHG beams are +shown in (g–i) and (j–l). The complex-valued image in (d–f, j–l) are plotted in a complex representation where amplitude and +phase are shown in brightness and hue, respectively. + +9 μm pitch +4.5 μm pitch +3 μm pitch +(a) +1.0 +(b) +C +Intensity (a.u.) +Gaussian HHG +0.0 +1.0 +(g) +h +1 +Intensity (a.u.) +OAM HHG +0.0 +(k) +um + +OAM-HHG beams reveal nanoscale defects in otherwise periodic samples +A major motivation for imaging periodic structures is to reliably detect and pinpoint small areas where the periodicity +is broken, i.e., to locate defects. However, when the diffraction orders are insufficiently overlapped, the artifacts in +the ptychographic reconstructions make it difficult or even impossible to locate defects. In contrast, the increase in +reconstruction quality enabled by OAM-HHG beams, especially the suppression of periodic artifacts in the +reconstructions (inherent to ptychographic imaging of periodic structures), enables reliable location of nanoscale +defects in otherwise highly periodic structures. This can potentially find its application in metrology for micro- and +nano-fabrication and manufacturing, including in advanced metrologies in support of EUV lithography. +In the 9 μm pitch sample, a damaged carbon bar (~300 nm wide) can be seen in the scanning electron microscopy +(SEM) image in Fig. 4(e), indicated by the red arrow. We first performed ptychographic imaging of the corresponding +area of the sample using an OAM-HHG beam. During data acquisition at each scan position, we acquired two +diffraction patterns with exposure times of 0.1 and 1 second, and combined them to form a composite high dynamic +range (HDR) image to increase SNR. The reconstructed image of the transmitted amplitude is shown in Fig. 4(a), in +which the defect is clearly resolved and is indicated by the red arrow. Given that the pixel size in the ptychography +reconstruction images is 200 nm, the fact that our EUV microscope using OAM-HHG illuminations can clearly image +a defect with size of about 300 nm (i.e., 1.5× the pixel size in the reconstruction images) makes it very promising to +detect or image smaller defects down to 10’s of nanometers using shorter EUV wavelengths and increased imaging +NA. +Next, a similar experiment is performed using a Gaussian-HHG beam under the same conditions, resulting in the same +approximate maximum detector count in the diffraction patterns as for the OAM case. The reconstructed image of the +transmitted amplitude is shown in Fig. 4(b), where reconstruction artifacts heavily corrupt the image details and render +the defect unidentifiable. Furthermore, due to the different intensity distributions of the Gaussian- and OAM-HHG +beams, even though the two datasets have the same maximum detector count, the OAM dataset has 3 times more total +detector counts than the Gaussian one. +To confirm that the difference in reconstructed image quality is not simply due to this different in the total number of +photons collected, but is due to how those photons are distributed in the diffraction plane (i.e., in the area of overlap +between diffraction orders), we performed a third experiment using the Gaussian-HHG beam and triple HDR exposure +(0.1-, 1- and 3-second exposure time), which leads to longer data acquisition time by a factor of 1.67, to have +approximately equal total counts in the combined diffraction data compared to the OAM-HHG case. The resulting +amplitude image is shown in Fig. 4(c). There is significant improvement over the reconstructed image from Gaussian- +HHG beams with double HDR exposure, but the reconstruction artifacts still make it difficult to identify the nanoscale +defect. We further quantitatively analyzed the SNR of the defect in these three reconstruction amplitude images using +the transmission profiles of the thin carbon bar in the boxes in Fig. 4(a-c). These transmission profiles are obtained by +averaging the transmission images in the vertical direction, and are plotted along the horizontal direction, as shown in +Fig. 4(d). The SNRs of the defect in Figs. 4(a–c) are calculated (see Methods) and summarized in Table 1. The SNR +for the defect image from OAM-HHG illumination is improved by a factor of >135 compared to that from Gaussian- +HHG illumination with equal exposure time. Furthermore, we evaluated the quality of these ptychographic +reconstructions using complex histogram analysis and verified that OAM-HHG illuminations result in higher fidelity +images, as discussed in detail in Supplementary Section 3. It is worth emphasizing that when taking the SEM image +in Fig. 4(e), the high-energy electron beam at 300 keV severely damaged the thin carbon bar, causing shrinkage and +the appearance of the bright areas on the top and bottom edges. In contrast, the EUV HHG beam can non-destructively +image both the periodic sample and the defect. + + + + +Figure 4. Enhanced sensitivity to nanoscale defects in periodic structures using OAM-HHG beams. (a–c) Amplitude images +of ptychographic reconstruction of a 2D square periodic structure (9 µm period, with a nanoscale defect of ~300 nm in size) under +various conditions: (a) OAM-HHG beams with double HDR (0.1- and 1- second exposure times), (b) Gaussian-HHG beams with +equal exposure time as the OAM-HHG case using double HDR (0.1- and 1- second exposure times), and (c) Gaussian-HHG beams +with roughly equal number of photons as the OAM-HHG case using triple HDR (0.1-, 1- and 3-second exposure times). The red +arrows indicate the nano-defect in the thin carbon bar. (d) Ptychography reconstructed transmission profile of the thin carbon bar +containing a nano-defect (indicated by the red arrow) in the boxes in (a–c). The transmission profiles are averaged in the vertical +direction. The red arrow indicates the nano-defect. (e) An SEM image of the same sample area shows a 300-nm-wide defect. Bright +areas on the top and bottom edges are due to sample damage from the high energy electron (300 keV) beams. + +Table 1. Signal-to-noise ratio analysis for the three ptychographic reconstructions shown in Fig. 4(a–c). +Ptychographic reconstructions +signal +background +noise +SNR +Improvement +factor +OAM-HHG in Fig. 4(a) +1.46e-1 +1.52e-2 +5.62e-3 +23.25 +135.8 +Gaussian-HHG, equal +exposure time in Fig. 4(b) +7.34e-2 +6.94e-2 +2.35e-2 +0.17 +Benchmark +Gaussian-HHG, equal number +of photons in Fig. 4(c) +7.40e-2 +1.72e-2 +1.31e-2 +4.34 +25.53 + +(a) +(b) +C +5um +(p) +0.2 +(e) +OAM-HHG +Gaussian-HHG, equal exposure time +Gaussian-HHG, equal number of photons +300 nm +0.05 +6 +8 +10 +12 +14 +16 +Sample position (μm) + +Conclusion +In conclusion, we demonstrated that by incorporating illumination engineering via OAM-HHG beams into an EUV +ptychography microscope, we can address the long-standing challenge of high-fidelity coherent diffractive imaging +of periodic structures. The intrinsic large divergence and ring-shaped intensity distribution of OAM-HHG beams leads +to the formation of higher SNR interference fringes in the diffraction patterns — thus enabling faster and higher +fidelity image reconstructions using the basic ePIE algorithm, without extra algorithmic effort. Furthermore, the +improvement in image fidelity allowed sensitive detection of a 300 nm wide defect, which is 1.5× the pixel size of +the reconstructed images, in an otherwise periodic thin carbon mesh with 9 μm period. +Ptychographic imaging of highly periodic structures has been widely recognized to be challenging, which has +precluded its application in critical science and technology fields such as semiconductor metrology and EUV +photomask inspection. Future studies can employ coherent EUV and X-ray vortex beams to enable nanometer- or +even sub-nanometer-scale spatial resolution in a broad range of next-generation nanoelectronics, photonics and +quantum devices. A particularly interesting direction would be to use coherent EUV light at a wavelength of 13.5 nm +for actinic imaging and inspection of EUV photomasks [19-25]. Finally, this work can provide inspiration for the +electron ptychography community (e.g., cryo-EM and 4D-STEM), where recent work has explored the potential +benefits of engineered vortex electron beams for enhanced imaging fidelity and lower dose [51,52]. + +Methods +Experimental setup +A Ti:sapphire amplifier system (KMLabs Wyvern HE) with a 𝜆 = 790 nm central wavelength, 45 fs pulse duration, 8 +mJ pulse energy, and 1 kHz repetition rate was used for this demonstration. Part of the laser output is used for second +harmonic generation (SHG) in a 𝛽-barium borate crystal (𝛽-BBO), yielding a frequency doubled beam at 395 nm +central wavelength for driving the HHG process. This SHG beam is focused into a semi-infinite gas cell, which +consists of a Brewster-cut entrance window, a 20 cm length filled with 50 torr of argon gas, and a copper gasket placed +in the focal plane of the driving laser where a coherent HHG beam is generated [53,54]. The driving laser at 395 nm +central wavelength is separated from the high-harmonic beam by using a 200 nm aluminum filter. This filter also +blocks any harmonics with 𝜆 > 77 nm, while harmonics with 𝜆 < 39 nm exceed the HHG cutoff energy, and so are +not generated. Furthermore, due to the centrosymmetry of the medium, only odd-numbered harmonic orders are +generated. The resulting EUV beam after the aluminum filter thus consists of narrow peaks at the 7th (𝜆 = 56 nm) and +9th (𝜆 = 44 nm) harmonics. The intensity ratio of the two harmonics in our experimental setup is estimated to be +Iλ=56nm/Iλ=44nm ~30:1, which can be well-approximated as a monochromatic illumination suitable for ptychographic +imaging. For generating HHG beams with a Gaussian spatial profile, we used an SHG beam with pulse energy of +~500 µJ. For generating HHG beams carrying OAM, we increased the pulse energy of the SHG beam to ~1.5 mJ, and +inserted a spiral phase plate (Holo-Or, VL-214-395-Y-A, OAM charge number ℓ = 1 at 395 nm wavelength) right +after the focusing optics into the semi-infinite gas cell to generate a driving beam with OAM charge number ℓ = 1, +and 𝜆 = 395 nm. The increased pulse energy is necessary in order to make the peak intensity (located at a central point +for the Gaussian beam, but distributed in a ring for the OAM beam) equal for the two cases, thus matching HHG cutoff +energies and conversion efficiency. Due to the conservation of OAM in HHG, the resulting quasi-monochromatic 7th +harmonic beam (𝜆 = 56 nm) carries an OAM charge number of ℓ = 7. +The HHG beam at 56 nm wavelength is focused sequentially by two toroidal mirrors (1: B4C-coated, feff = 27 cm, θ = +15°; 2: Au-coated, feff = 50 cm, θ = 10°) in a Wolter configuration to create an imaging system with higher NA (feff = +17 cm) while managing coma aberration [55]. The resulting focusing beam is redirected towards the sample at normal +incidence using a glancing incidence mirror (B4C coating, fused silica substrate, 3° incidence angle from grazing, +nominal reflectivity 95%). The testing samples are three Quantifoil holey carbon films which have various hole sizes +and shapes arranged in a rectangular grid, and are mounted on standard Ted Pella Ø3mm Cu TEM grids with 200 +mesh (125 um pitch, 90 um hole width and 35 um bar width). More specifically, the three Quantifoil holey carbon + + + +films have a pitch of 9 μm (7 μm square hole and 2 μm bar, product number 656-200-CU), 4.5 μm (3.5 μm diameter +circular holes and 1 μm separation, product number 660-200-CU) and 3 μm (2 um diameter circular holes and 1 um +separation, product number 661-200-CU), respectively. The samples are positioned close to the beam focus, and are +mounted on a precision translation stage ensemble (SmarAct XYZ-SLC17:30). They are translated in the plane +perpendicular to the beam path to perform ptychographic scans in 7 × 7 rectangular grids (49 positions) with nominally +3.3 μm between adjacent positions. A random offset of ±20% of the scan step size was added to each scan position +to avoid artifacts originating from the scan grid itself. The far-field diffraction patterns are recorded by an EUV-CCD +detector (Andor iKon-L, 2048 × 2048, 13.5 μm pixel size) positioned 50 mm after the sample. In order to obtain the +best ptychographic reconstructions possible for each illumination case, we carefully characterized each probe function +in the sample planes by taking ptychographic scans on a non-periodic sample and reconstructing both the sample and +the probe function through blind deconvolution. The reconstructed probe functions were used in the ptychographic +reconstructions of highly periodic samples as initial guesses. + +Ptychographic data processing and image reconstructions +The diffraction patterns were recorded by an EUV-CCD detector with 2048 × 2048 pixels and 13.5 μm detector pixel +size. During data processing, we cropped them to 1024 × 1024 because very few photons were detected outside this +area. The resulting pixel size of the reconstructed images is +𝑑𝑟 = +𝜆∙𝑧 +𝑁∙𝑑𝑥 ≈ 200 nm, + + + + + +(2) +where λ = 56 nm is the wavelength, z = 50 cm is the distance from the sample to the CCD detector, N = 1024 is the +number of pixels in one direction and dx = 13.5 μm is the detector pixel size. +The ptychographic reconstructions were performed in two steps using only the ePIE algorithm [13]. In the first step, +the complex-valued probe functions (both Gaussian- and OAM-HHG beams) were characterized by performing +ptychography on a non-periodic sample and using the ePIE algorithm for reconstruction. In the second step, the pre- +characterized probe functions were used as initial guesses for reconstructing the periodic structures. For the first 100 +iterations, only the sample images were updated while the probe functions were kept fixed. Then, both the sample +images and probe functions were updated by the ePIE algorithm for another 900 iterations. The total number of +iterations for ptychographic reconstructions of periodic structures was 1000. This procedure was kept constant for all +Gaussian-HHG and OAM-HHG reconstructions in this paper. We also want to emphasize the fast convergence speed +of our technique compared to that in the work by Gardner et al. [15], which takes more than 10,000 iterations. + +SNR analysis of imaging of the nano-defect +The SNR of the defect detection in Table 1 is calculated as follows: We start from the three curves in Fig. 4(d). For +each curve, corresponding to an experimental condition shown in the legend, the signal level is the transmission value +in the defect, the background level and noise level are calculated as the average and the standard deviation, respectively, +of the transmission values excluding the defect. 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Exp. 12(19), 4430-4436 (2004). +[54] Steingrube, D. S. et al. Phase matching of high-order harmonics in a semi-infinite gas cell. Phys. Rev. A 80, +043819 (2009). +[55] Coudert-Alteirac, H. et al. Micro-focusing of broadband high-order radiation by a double toroidal mirror. Appl. +Sci. 7(11), 1159 (2017). + +Acknowledgements +This research was primarily supported by STROBE: a National Science Foundation (NSF) Science and Technology +Center (STC) under award DMR-1548924 for the setup and new illumination engineering and algorithms, and also +by a DARPA STTR grant 140D0419C0094 for imaging periodic samples. A Moore Foundation Grant No. 10784 +supported the low-dose imaging research. The authors thank Guan Gui, Drew Morrill, Yunzhe Shao, Chen-Ting Liao, +Emma Cating-Subramanian for comments on the text. + +Author contributions +B. W., N. J. B., M. M. M. and H. C. K. conceived the experiment. B. W., N. J. B. and P. J. built and maintained the +EUV source. B. W. and N. J. B. collected the data sets and performed the reconstructions and data analysis. N. J., Y. +E. and B. W. performed the SEM imaging of the test samples. M.T., Y.E. and N.J. advised on the phase retrieval + + + +algorithms and setup, while I.B. helped to develop the laser setup. All authors designed aspects of the experiment, +performed the research and wrote the paper. + +Competing financial interests +B. W., N. J. B., M. M. M. and H. C. K. have submitted a patent disclosure based on this work. M.M M. and H. C. K. +are partial owners of KMLabs Inc. who manufactured the ultrafast laser used in this study. + + + + + + +Supplementary information: +S1. Convolution theorem perspective on ptychographic imaging of highly periodic structures +In ptychography, the far-field diffraction fields are approximated as the Fourier transform of the product of the +complex probe and object functions, p(x,y) and o(x,y), i.e., +Ψ (u,v) = ℱ [p(x,y) × o(x,y)], (S1) +where ℱ is the Fourier transform operation, (x,y) are the real space coordinates and (u,v) are the reciprocal space +coordinates. According to the convolution theorem, this can also be represented as a convolution of the Fourier +transform of the probe function, P(u,v) = ℱ [p(x,y)], and that of the object function, O(u,v) = ℱ [o(x,y)], i.e., +Ψ(u,v) = P(u,v) ⁕ O(u,v), (S2) +where ⁕ is the convolution operation. Often, P(u,v), which is the complex beam in the detector plane when no sample +is in the way, has an edge resulting from apertures in the system, as indicated by the circle in the close-ups in Fig. +1(a,c). In the case of 2D highly periodic structures, O(u,v) consists of a 2D comb of 𝛿 functions (diffraction peaks) +arranged in a 2D periodic grid, the amplitudes and phases of which are modulated by the Fourier transform of the unit +cell of the periodic structure. This is a sparse function in the reciprocal space. The convolution operation in Eq. (S2) +puts a copy of P(u,v) at the location of each 𝛿 function with modulated amplitude and phase. +In cases where P(u,v) is small in size such that all diffraction orders are isolated, the modulated phase of each +diffraction order is totally lost when we collect intensity measurements, thus causing ptychography to fail. However, +in cases where P(u,v) is sufficiently large in size, the interference fringes in the overlapped regions between +neighboring diffraction orders encoded the relative phase of each diffraction orders into the intensity modulations that +are directly measurable with the EUV-CCD camera, thus enabling fast and robust ptychographic reconstructions of +the highly periodic structures. + +S2. A phase-change-like behavior in ptychography demonstrated by Gaussian-HHG +illuminations with controlled divergence +Since the key to successfully achieving ptychographic imaging of highly periodic structures is to obtain overlap and +interference between neighboring diffraction orders, an abrupt, phase-change-like behavior in reconstruction quality +is expected as one smoothly changes the illumination NA. We experimentally demonstrated this behavior in +ptychographic imaging of highly periodic structures, as shown in Fig. S1, using Gaussian-HHG illuminations with a +controlled illumination NA. This is achieved by installing an in-vacuum iris ~0.5 m after the semi-infinite gas cell, +which allows direct control of the divergence of the HHG beams, and thus of the illumination NA on the sample and +the overlap between neighboring diffraction orders given the same focusing optics. +We performed four ptychography scans on the same 2D square periodic structure with a 9 μm period under various +illumination NAs controlled by the in-vacuum iris. Fig. S1(a–d) shows example diffraction patterns from each scan +from small illumination NA in (a) to large illumination NA in (d). The close-ups in the blue circles show the effect of +illumination NA on the resulting diffraction patterns. We then reconstructed these four datasets using the standard +ePIE algorithm [13] and the corresponding results are shown in Fig. S1(e–h). It is clear that for ptychography scans +where diffraction orders are isolated, the periodic structure cannot be reliably reconstructed due to the lost phase +information, as shown in Fig. S1(e–f), while for ptychography scans where the illumination NA is large enough to +support overlap between diffraction orders, the ePIE algorithm can quickly and reliably reconstruct the periodic +structures. + + + + + +Figure S1. Experimental demonstrations of a phase-change-like behavior in ptychographic imaging of 2D square periodic +structures with 9 um period using Gaussian-HHG beams with controlled divergence. (a–d) Example diffractions from a 2D square +periodic structure using Gaussian-HHG beams with various divergences. The inserts show close-ups of the center of the diffraction +patterns. (e–h) The corresponding ptychographic reconstructions of the 2D square periodic structure under various illumination +conditions. The reconstructions are successful only when the diffraction orders have overlap, showing a phase-change-like behavior. + +S3. Image quality assessment using complex histogram analysis +We use a complex histogram analysis to evaluate the quality of the ptychographic reconstructions in Fig. 3(d–f), 3(j– +l) and 4(a–c). A 2D complex histogram is an extension of a normal histogram showing how many data points of a +complex field lie within a certain range of real and imaginary parts. For the approximately binary test samples used in +this study, ideally, the complex histograms consist of only two 𝛿-function peaks corresponding to the transmissive +and opaque areas. In reality, the two 𝛿-function peaks are broadened due to limited SNR and spatial resolution. The +quality of the ptychographic reconstructions can thus be assessed by examining the degree of broadening of these +peaks, where reconstructions with higher quality have narrower peaks. +We first evaluate the quality of the ptychographic reconstructions in Fig. 3. In the complex histograms shown in Fig. +S2, the two parts of the sample (free space and the carbon bars) are indicated by the red and yellow circles respectively. +The complex histograms for OAM-HHG images (d–f) have narrower peaks than those for Gaussian-HHG images (a– +c), which indicates that OAM-HHG images have better quality. The reconstruction in (c) (Gaussian-HHG +illuminations on a 3-um-pitch structure) failed, thus not showing the double-peak feature. +We then evaluate the quality of the ptychographic reconstructions in Fig. 4. As shown in Fig. S3(a–c), the +ptychographic reconstructions are shown in the complex representation with amplitude and phase indicated by +brightness and hue. Visually, the image from OAM-HHG illumination in a has the best quality in terms of a sharp +transition from free space area to thin carbon bar area and smoothness within free space or carbon bar areas. The +complex histograms in (d–f) confirmed this: the primary peaks (indicated by the red and yellow circles) in the complex +histogram from OAM-HHG illuminations (as shown in d) are the narrowest. + +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +10um + + + +Figure S2. Quality assessment of ptychographic reconstructions in Fig. 3 using complex histogram analysis. (a–c) Complex +histograms of ptychographic reconstructions of 9 μm, 4.5 μm and 3 μm pitch periodic structures using Gaussian-HHG illuminations. +The ptychographic reconstructions are shown in each bottom left corner and correspond to Fig. 3(d–f). (d–f) Complex histograms +of ptychographic reconstructions of 9 μm, 4.5 μm and 3 μm pitch periodic structures using OAM-HHG illuminations. The +ptychographic reconstructions are shown in each bottom left corner and correspond to Fig. 3(j–l). These complex histograms consist +of two primary peaks (except panel c because the reconstruction failed), which correspond to the open space area (indicated by the +red circles) and the thin carbon bar area (indicated by the yellow circles). The complex histograms from OAM-HHG illuminations +(the bottom row) have narrower primary peaks than those from Gaussian-HHG illuminations (the top row), which shows superior +image quality for OAM-HHG reconstructions. The ‘Re’ and ‘Im’ axes in (a) show the complex coordinate. + +Im + + +Figure S3. Quality assessment of ptychographic reconstructions in Fig. 4 using complex histogram analysis. (a–c) Complex +representations of ptychographic reconstructions of 2D square periodic structures with 9 μm period under three different +experimental conditions: (a) an OAM-HHG illumination, (b) a Gaussian-HHG illumination with equal exposure time, and (c) a +Gaussian-HHG illumination with equal number of photons. The amplitude and phase of these images are presented in brightness +and hue, respectively. The color wheel is shown in the bottom left corner of panel a. (d–f) Complex histograms of ptychographic +reconstructions are shown in (a–c). These complex histograms all consist of two primary peaks, which correspond to the open +space area, indicated by the red circles, and the thin carbon bar area, indicated by the yellow circles. The complex histogram in +panel d has the narrowest primary peaks, which indicates its superior image quality provided by the intrinsic advantages of OAM- +HHG illumination. + + +(a) +(b) +(c) +Im +Re + + +Figure S4. Complex representations of the ptychography reconstructed Gaussian-HHG and OAM-HHG beams in the +sample plane (a–b) and in the detector plane (c–d). The amplitude and phase of the beams are shown in brightness and hue, +respectively. The scale bars in (a–b) indicate beam size in the sample plane, and those in (c–d) indicate beam divergence angle in +the detector plane. The OAM-HHG beam in the detector plane in (d) shows a characteristic donut intensity profile, while the OAM- +HHG beam in the sample plane does not show a donut intensity profile due to aberrations introduced by the focusing optics. + +Gaussian-HHG +OAM-HHG +(a) +(b) +Sample plane +0 +元 +50 um +50 μm +(c) +(d) +Detectorplane +10mrad +10mrad \ No newline at end of file diff --git a/ItE5T4oBgHgl3EQfXA9_/content/tmp_files/load_file.txt b/ItE5T4oBgHgl3EQfXA9_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7932682aea843892a515813b38bb5f195e5eb4f5 --- /dev/null +++ b/ItE5T4oBgHgl3EQfXA9_/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf,len=773 +page_content='High-fidelity ptychographic imaging of highly periodic structures enabled by vortex high harmonic beams Bin Wang1*†, Nathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Brooks1*, Peter C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Johnsen1, Nicholas W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Jenkins1, Yuka Esashi1, Iona Binnie1, Michael Tanksalvala1, Henry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Kapteyn1,2, Margaret M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Murnane1 1STROBE Science and Technology Center, JILA, University of Colorado, Boulder, CO 80309, USA 2KMLabs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', 4775 Walnut St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' #102, Boulder, CO 80301, USA *These authors contributed equally †bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='wang-2@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='edu Abstract Ptychographic Coherent Diffractive Imaging enables diffraction-limited imaging of nanoscale structures at extreme ultraviolet and x-ray wavelengths, where high-quality image-forming optics are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' However, its reliance on a set of diverse diffraction patterns makes it challenging to use ptychography to image highly periodic samples, limiting its application to defect inspection for electronic and photonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Here, we use a vortex high harmonic light beam driven by a laser carrying orbital angular momentum to implement extreme ultraviolet ptychographic imaging of highly periodic samples with high fidelity and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We also demonstrate, for the first time to our knowledge, ptychographic imaging of an isolated, near-diffraction-limited defect in an otherwise periodic sample using vortex high harmonic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This enhanced metrology technique can enable high-fidelity imaging and inspection of highly periodic structures for next-generation nano, energy, photonic and quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Introduction In recent decades, a powerful coherent diffractive imaging (CDI) technique known as ptychography has enabled robust, diffraction-limited, phase-contrast imaging of nanoscale structures [1-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Although ptychography has been implemented using a range of illumination wavelengths, its use in the extreme-ultraviolet (EUV) and x-ray regions is particularly attractive for achieving high spatial resolution with inherent elemental and chemical contrast [6-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In ptychography, a coherent illumination (the probe) is focused and scanned across an extended sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A series of far- field diffraction patterns are recorded, while maintaining a large overlap between adjacent scan positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Iterative phase-retrieval algorithms [11-15] can then be used to robustly and uniquely reconstruct the complex-valued probe field and sample transmission or reflection functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' However, successful reconstruction relies heavily on diversity in the data provided by the lateral scanning of the probe relative to the sample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', interferences at the detector plane mix amplitude and phase, allowing the reconstruction algorithms to unravel both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' this means that ptychographic imaging of highly periodic samples with a sufficiently small period is extremely challenging due to the lack of diversity in a series of diffraction patterns, leading to poor convergence of the reconstructed sample images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This significantly limits ptychography’s application to a wide variety of nanoscale periodic structures such as photonic crystals [16-17], semiconductor devices [18], and EUV photomasks [19-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Consequently, it is critical to fill this characterization gap to aid the advancement of a host of next-generation nano-devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' High harmonic upconversion of femtosecond lasers can produce bright coherent beams from the EUV to the soft x- ray regions of the spectrum, in a tabletop-scale setup [26-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' When combined with ptychography, high harmonic generation (HHG) enables phase-sensitive lensless imaging with diffraction-limited nanoscale spatial resolution and excellent elemental specificity [9,15,29-31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Moreover, because of the quantum-coherent nature of the HHG upconversion process, polarization and phase structure present in the driving laser beam can be transferred to the generated harmonics, provided energy, spin and orbital angular momentum are conserved [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This makes it possible to create designer short-wavelength structured light for a variety of applications in advanced spectro- microscopies [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Light beams carrying orbital angular momentum (OAM) have attracted considerable interest for super-resolution imaging [36] and enhanced optical sensing, communication and inspection [37-39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Recently, by considering conservation of OAM in HHG upconversion, additional routes for controlling the OAM, polarization, as well as the spectral and temporal properties of HHG have been revealed [40-47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A property particularly interesting for ptychography is the relationship between OAM and the HHG beam divergence/propagation: the spiral phase structure characteristic of OAM-HHG beams forces them to diverge more quickly from the focal point [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This means that by using one or more OAM beams to drive the HHG process (referred to as OAM-HHG), one can control the divergence of the emitted HHG probe without changing the focusing optics of the HHG driving laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In this article, we demonstrate a solution to a decade-long challenge by showing that high harmonic beams carrying orbital angular momentum can be used to advantage in high-resolution, high-fidelity and fast-convergence ptychographic imaging of highly periodic two-dimensional (2D) grating structures, using the standard extended ptychographic iterative engine (ePIE) algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The key to this technique is that the increased divergence of the OAM-HHG source, combined with the ring-shaped intensity distribution, introduces strong interference fringes between adjacent diffraction orders in the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These encode the non-measurable phase information into the measurable intensity modulation in diffraction fields, significantly enhancing data diversity so that the phase of the diffracted field can be reliably retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We further show that using OAM-HHG beams for illumination provides three significant advantages compared to standard Gaussian-HHG beams, all of which lead to enhanced signal-to- noise-ratio (SNR) for imaging periodic structures: First, due to the conservation of OAM in the HHG process and the resulting strong spiral phase structure in the generated EUV beams, OAM-HHG beams naturally have a significantly increased divergence compared to that of Gaussian-HHG beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This enhanced illumination NA makes it possible to achieve overlap between different diffraction orders for small pitch periodic samples, beyond what is possible with a Gaussian-HHG probe, and without making any changes to the focusing optics of the ptychographic EUV microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Second, the unique ring-shaped OAM beam intensity distribution, which is determined by the strong spiral phase structure in the EUV beams, leads to overlap between different diffraction orders in the high-intensity regions of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' And third, OAM-HHG also allows a higher total number of photons to be collected by the detector given a fixed detector dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Therefore, by leveraging OAM-HHG beams for ptychography, we successfully imaged highly periodic samples with substantially reduced gridding artifacts, and reliably detected defects near the diffraction limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This new structured-EUV HHG metrology technique can support the advancement of next-generation EUV lithography, nanoelectronics, photonic and quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Methodology To date, imaging highly periodic structures has been extremely challenging for ptychography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In a conventional implementation of ptychography using a Gaussian EUV beam to illuminate highly periodic 2D structures (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(a)), the far-field diffraction patterns consist of many isolated diffraction orders, each of which is a copy of the far- field beam profile, and is modulated by an envelope in both amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The zoomed-in green circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(a) shows this characteristic behavior, with the white circles indicating the edge of each diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the resulting ptychographic dataset, diffraction patterns taken at different positions on the highly periodic sample are almost identical to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This is because, in contrast to diffraction from non-periodic structures, changes in the far-field diffraction field happen almost entirely in the relative phase between the diffraction orders, but not in the intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', diffraction efficiency) of the diffracted orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The phase information is thus totally lost in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Ptychography, as a phase retrieval algorithm, tries to retrieve the phase distributions of diffraction patterns from their intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The fact that the phase information is totally lost for highly periodic samples with a sufficiently short period, as opposed to being encoded in the intensity measurements as would be the case for non- periodic structures, makes it extremely challenging to achieve successful ptychographic imaging of such highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' As expected, the reconstruction fails for ptychography using a Gaussian-HHG beam, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(b), in which the amplitude and phase of the reconstruction are plotted in brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This phase problem can also be understood through the convolution theorem, as discussed in detail in Supplementary Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In 1969, Hoppe proposed to achieve electron diffraction imaging of periodic atomic lattices by encoding the non- measurable phase information into the measurable intensity modulation in diffraction patterns, through interference between neighboring diffraction orders [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' As schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(c), OAM-HHG beams are able to achieve overlap and interference between neighboring diffraction orders due to their intrinsically larger beam divergence and ring-shaped intensity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The zoomed-in blue circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(c) shows the interference fringes, with the yellow circles indicating the edge of each diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' As one scans the probe relative to the periodic structures, the relative phase of each diffraction order changes accordingly, which then causes the interference fringes to shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In other words, the phase information in the diffraction patterns is now encoded in the intensity measurements through interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These interference fringes contain the missing phase information, and increase the diversity in diffraction patterns, thereby enabling robust and reliable ptychographic reconstructions (see Supplementary Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Figure 1(d) shows a high-fidelity ptychographic reconstruction of a 2D periodic structure under an OAM-HHG illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Robust and reliable ptychographic imaging of highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a) Schematic showing HHG ptychographic imaging of a periodic structure using conventional Gaussian-HHG illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The resulting diffraction orders are isolated (see zoomed-in green circle), where the white circles indicate the edges of each diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This leads to a complete loss of the relative phase information between the orders in the far-field diffraction, which subsequently leads to the failure of the ptychographic reconstruction in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (c) OAM-HHG illumination intrinsically has a larger source divergence and a ring-shaped intensity profile, to support overlap and interference between diffraction orders (see zoomed-in blue circle), in which the yellow circles indicate the edges of each diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This interference converts the relative phase between the diffraction orders into measurable intensity modulation, enabling fast and robust ptychographic reconstruction of the 2D periodic structure in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In (b, d), the complex-valued amplitude and phase are plotted as brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 5um The required NA for high fidelity imaging of periodic samples can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' When a periodic structure is illuminated by a focused coherent beam, the angular separation between two neighboring diffraction orders is given by 𝛥𝜃 = 𝜆/𝛬, where 𝜆 is the illumination wavelength, and 𝛬 is the period of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The illumination NA for the microscope is defined as the half-cone angle of the focusing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Geometrically, for fixed 𝜆 and 𝛬, there exists a critical value for illumination NA: 𝑁𝐴𝑐 = 1 2 𝛥𝜃 = 𝜆 2𝛬.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (1) Only for illumination NAs larger than 𝑁𝐴𝑐 will neighboring diffraction orders have sufficiently large footprints on the detector to overlap with each other and generate interference fringes, thus enabling successful ptychographic reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Experimental configuration We designed and built an EUV ptychographic microscope in a transmission geometry, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The driving laser for the HHG process is a frequency-doubled Ti:sapphire laser amplifier system (𝜆~395 nm), with an intrinsic near-Gaussian mode (vortex charge of ℓ = 0), that can be converted to an OAM beam of vortex charge ℓ = 1 using a spiral phase plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The 7th harmonic of the driving laser (𝜆~56 nm) exhibits either a Gaussian mode or an OAM of vortex charge ℓ = 7 depending on whether a spiral phase plate is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The EUV beam is then focused by a double- toroid focusing system onto the periodic samples, with a spot size of ~13 × 18 μm (1/𝑒2 intensity) for Gaussian-mode HHG, or ~27 × 32 μm (donut intensity peak-to-peak) for OAM-HHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstructed Gaussian- and OAM-HHG beam profiles are shown in a complex representation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S4, with the beam amplitude and phase indicated by brightness and hue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The test samples are three Quantifoil holey carbon films (~20 nm thick) which have various hole sizes and shapes arranged in a periodic rectangular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The three Quantifoil holey carbon films have a pitch of 9 μm, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm and 3 μm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These Quantifoil holey carbon films are mounted on standard Ted Pella Ø3mm Cu 200 mesh TEM grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (See the Methods section for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=') Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' EUV ptychographic microscope using OAM-HHG EUV beams for imaging highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A spiral phase plate (ℓ = 1 at 395 nm wavelength) converts the driving laser at 395 nm wavelength to an OAM beam, which is focused into a semi-infinite gas cell to produce a nearly monochromatic 7th harmonic beam with a wavelength of 56 nm and an OAM charge of ℓ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A double-toroidal mirror focusing system focuses this OAM-HHG beam onto a 2D periodic sample, and an EUV-CCD camera is used to record the far-field diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' e=7 入=56nm Detector Periodic Sample 0=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5mrad To l= 1 入=395nm do= 100 μm During the ptychography scans, the test samples are translated in the plane perpendicular to the beam path in 7 × 7 rectangular grids (49 scan positions) with nominally 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='3 μm distance between adjacent scan positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A random offset of ±20% of the scan step size was added to each scan position to avoid gridding artifacts in the reconstructions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The far-field diffraction patterns are recorded by an EUV-CCD detector (Andor iKon-L, 2048 × 2048, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm pixel size) positioned 50 mm after the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' To obtain the best ptychographic reconstructions possible for each illumination case, we carefully pre-characterized each probe function in the sample plane by taking ptychographic scans on a non-periodic sample and reconstructing both the sample and the probe functions through blind deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These pre-characterized probe functions were used as initial guesses in the ptychographic reconstructions of highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Other than using the pre-characterized probe as the initial guess, we used the standard ePIE algorithm [13] for all reconstructions in this study, without the need for additional constraints such as modulus enforced probe [15] or total variation regularization [25,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Results OAM-HHG enables robust and reliable ptychographic imaging of periodic structures We experimentally demonstrate that OAM-HHG beams enable robust and reliable ptychographic imaging of highly periodic structures because of three intrinsic advantages compared to Gaussian-HHG illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' First, due to the conservation of OAM in the HHG process and the resulting strong spiral phase structure in the generated EUV beams, OAM-HHG beams naturally exhibit a significantly increased divergence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', increased illumination NA for the microscope given the same focusing optics) compared with Gaussian-HHG beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This enhanced illumination NA allows us to achieve overlap between diffraction orders for smaller pitch periodic samples beyond what is possible with a Gaussian-mode probe, without making any changes to the EUV microscope end-station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Second, the characteristic ring-shaped intensity distribution of OAM-HHG beams ensures that the majority of photons fall in the overlap area (in contrast to the Gaussian-HHG beams, for which the overlap between diffraction orders occurs at the tails of the intensity distributions), which increases the SNR for the interference fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Third, the ring-shaped intensity distribution of OAM-HHG beams allows one to collect a higher total number of photons by the detector given a fixed dynamic range, which also leads to higher SNR without the need for high dynamic range (HDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We performed ptychographic imaging on three highly periodic structures with 9 μm, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm and 3 μm pitches using Gaussian- and OAM-HHG beams at a wavelength of 56 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Example diffraction patterns and reconstructed images from each ptychography scan can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Furthermore, each ptychography scan collected 49 far-field diffraction intensity patterns, as shown in a log scale in Supplementary Video 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Note that while there is a clear change in the diffraction patterns from frame to frame for the OAM-HHG case, particularly in the interference fringes between the adjacent diffraction orders, the diffraction patterns in the Gaussian-HHG case do not change very much for small period samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' All ptychography datasets were taken without making any changes to the EUV microscope — the difference in divergence between Gaussian- and OAM-HHG beams is intrinsic to the HHG upconversion process, which conserves energy and OAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For the 9 μm pitch sample, the small diffraction angle means that successive orders largely overlap even for the Gaussian-HHG beam, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This results in a reasonably good image, apart from some gridding artifacts as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In comparison, the ptychography scan using OAM-HHG illumination sees more overlap resulting in improved SNR in the interference fringes and a much higher-fidelity image with greatly reduced gridding artifacts, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(g,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For the smaller 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm pitch sample, the diffraction orders are further apart, causing the Gaussian-HHG beams to lose most of the interference in the diffraction patterns, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The low SNR in the interference fringes results in reduced quality image reconstruction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' However, due to their higher intrinsic divergence and the unique ring-shaped intensity distribution, OAM-HHG maintains a large area of overlap between neighboring diffraction orders with more photons, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This results in higher-quality images of the periodic structure with a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm period, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Thus, simply by inserting a spiral phase plate to convert the driving laser to an OAM beam, while keeping everything else in the microscope the same, a greatly improved reconstruction quality is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Lastly, for the smallest 3 μm period sample, Gaussian-HHG illumination totally fails due to the lack of interference between diffraction orders, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(c,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In this case, OAM beams can reconstruct a reasonable image, although the quality of the unit cell is degraded, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(i,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We also evaluated the quality of these ptychographic reconstructions using complex histogram analysis and the results can be found in Supplementary Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We note that all reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3 have the correct sample periodicity because this information is directly available from the measured intensity of the diffracted fields — the success or failure of the reconstructions of the unit cells depends on whether the relative phase between the diffraction orders can be successfully retrieved or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The ptychography reconstructed images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(d–f, j–l) are complex-valued and are shown in a complex representation where the amplitude and phase information are represented by the brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The color wheel is shown in the bottom left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' High divergence OAM-HHG beams are able to produce higher-quality ptychography images of periodic structures than low divergence Gaussian-HHG beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Three test samples with different periods and shapes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', 9 μm period with square holes, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm period with circular holes and 3 μm period with circular holes, are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For Gaussian-HHG beams, example diffraction patterns from the three test samples are shown in (a–c) and the corresponding ptychography reconstructed images are shown in (d–f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The example diffraction patterns and ptychography images from OAM-HHG beams are shown in (g–i) and (j–l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The complex-valued image in (d–f, j–l) are plotted in a complex representation where amplitude and phase are shown in brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 9 μm pitch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm pitch 3 μm pitch (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='0 (b) C Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=') Gaussian HHG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='0 (g) h 1 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=') OAM HHG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='0 (k) um OAM-HHG beams reveal nanoscale defects in otherwise periodic samples A major motivation for imaging periodic structures is to reliably detect and pinpoint small areas where the periodicity is broken, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', to locate defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' However, when the diffraction orders are insufficiently overlapped, the artifacts in the ptychographic reconstructions make it difficult or even impossible to locate defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In contrast, the increase in reconstruction quality enabled by OAM-HHG beams, especially the suppression of periodic artifacts in the reconstructions (inherent to ptychographic imaging of periodic structures), enables reliable location of nanoscale defects in otherwise highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This can potentially find its application in metrology for micro- and nano-fabrication and manufacturing, including in advanced metrologies in support of EUV lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the 9 μm pitch sample, a damaged carbon bar (~300 nm wide) can be seen in the scanning electron microscopy (SEM) image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(e), indicated by the red arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We first performed ptychographic imaging of the corresponding area of the sample using an OAM-HHG beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' During data acquisition at each scan position, we acquired two diffraction patterns with exposure times of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='1 and 1 second, and combined them to form a composite high dynamic range (HDR) image to increase SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstructed image of the transmitted amplitude is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(a), in which the defect is clearly resolved and is indicated by the red arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Given that the pixel size in the ptychography reconstruction images is 200 nm, the fact that our EUV microscope using OAM-HHG illuminations can clearly image a defect with size of about 300 nm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5× the pixel size in the reconstruction images) makes it very promising to detect or image smaller defects down to 10’s of nanometers using shorter EUV wavelengths and increased imaging NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Next, a similar experiment is performed using a Gaussian-HHG beam under the same conditions, resulting in the same approximate maximum detector count in the diffraction patterns as for the OAM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstructed image of the transmitted amplitude is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(b), where reconstruction artifacts heavily corrupt the image details and render the defect unidentifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Furthermore, due to the different intensity distributions of the Gaussian- and OAM-HHG beams, even though the two datasets have the same maximum detector count, the OAM dataset has 3 times more total detector counts than the Gaussian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' To confirm that the difference in reconstructed image quality is not simply due to this different in the total number of photons collected, but is due to how those photons are distributed in the diffraction plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', in the area of overlap between diffraction orders), we performed a third experiment using the Gaussian-HHG beam and triple HDR exposure (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='1-, 1- and 3-second exposure time), which leads to longer data acquisition time by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='67, to have approximately equal total counts in the combined diffraction data compared to the OAM-HHG case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The resulting amplitude image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' There is significant improvement over the reconstructed image from Gaussian- HHG beams with double HDR exposure, but the reconstruction artifacts still make it difficult to identify the nanoscale defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We further quantitatively analyzed the SNR of the defect in these three reconstruction amplitude images using the transmission profiles of the thin carbon bar in the boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These transmission profiles are obtained by averaging the transmission images in the vertical direction, and are plotted along the horizontal direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The SNRs of the defect in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(a–c) are calculated (see Methods) and summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The SNR for the defect image from OAM-HHG illumination is improved by a factor of >135 compared to that from Gaussian- HHG illumination with equal exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Furthermore, we evaluated the quality of these ptychographic reconstructions using complex histogram analysis and verified that OAM-HHG illuminations result in higher fidelity images, as discussed in detail in Supplementary Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' It is worth emphasizing that when taking the SEM image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(e), the high-energy electron beam at 300 keV severely damaged the thin carbon bar, causing shrinkage and the appearance of the bright areas on the top and bottom edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In contrast, the EUV HHG beam can non-destructively image both the periodic sample and the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Enhanced sensitivity to nanoscale defects in periodic structures using OAM-HHG beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a–c) Amplitude images of ptychographic reconstruction of a 2D square periodic structure (9 µm period, with a nanoscale defect of ~300 nm in size) under various conditions: (a) OAM-HHG beams with double HDR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='1- and 1- second exposure times), (b) Gaussian-HHG beams with equal exposure time as the OAM-HHG case using double HDR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='1- and 1- second exposure times), and (c) Gaussian-HHG beams with roughly equal number of photons as the OAM-HHG case using triple HDR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='1-, 1- and 3-second exposure times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The red arrows indicate the nano-defect in the thin carbon bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (d) Ptychography reconstructed transmission profile of the thin carbon bar containing a nano-defect (indicated by the red arrow) in the boxes in (a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The transmission profiles are averaged in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The red arrow indicates the nano-defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (e) An SEM image of the same sample area shows a 300-nm-wide defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Bright areas on the top and bottom edges are due to sample damage from the high energy electron (300 keV) beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Signal-to-noise ratio analysis for the three ptychographic reconstructions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Ptychographic reconstructions signal background noise SNR Improvement factor OAM-HHG in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='46e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='52e-2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='62e-3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='25 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='8 Gaussian-HHG, equal exposure time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(b) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='34e-2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='94e-2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='35e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='17 Benchmark Gaussian-HHG, equal number of photons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(c) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='40e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='72e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='31e-2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='34 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='53 (a) (b) C 5um (p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='2 (e) OAM-HHG Gaussian-HHG, equal exposure time Gaussian-HHG, equal number of photons 300 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='05 6 8 10 12 14 16 Sample position (μm) Conclusion In conclusion, we demonstrated that by incorporating illumination engineering via OAM-HHG beams into an EUV ptychography microscope, we can address the long-standing challenge of high-fidelity coherent diffractive imaging of periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The intrinsic large divergence and ring-shaped intensity distribution of OAM-HHG beams leads to the formation of higher SNR interference fringes in the diffraction patterns — thus enabling faster and higher fidelity image reconstructions using the basic ePIE algorithm, without extra algorithmic effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Furthermore, the improvement in image fidelity allowed sensitive detection of a 300 nm wide defect, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5× the pixel size of the reconstructed images, in an otherwise periodic thin carbon mesh with 9 μm period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Ptychographic imaging of highly periodic structures has been widely recognized to be challenging, which has precluded its application in critical science and technology fields such as semiconductor metrology and EUV photomask inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Future studies can employ coherent EUV and X-ray vortex beams to enable nanometer- or even sub-nanometer-scale spatial resolution in a broad range of next-generation nanoelectronics, photonics and quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A particularly interesting direction would be to use coherent EUV light at a wavelength of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 nm for actinic imaging and inspection of EUV photomasks [19-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Finally, this work can provide inspiration for the electron ptychography community (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', cryo-EM and 4D-STEM), where recent work has explored the potential benefits of engineered vortex electron beams for enhanced imaging fidelity and lower dose [51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Methods Experimental setup A Ti:sapphire amplifier system (KMLabs Wyvern HE) with a 𝜆 = 790 nm central wavelength, 45 fs pulse duration, 8 mJ pulse energy, and 1 kHz repetition rate was used for this demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Part of the laser output is used for second harmonic generation (SHG) in a 𝛽-barium borate crystal (𝛽-BBO), yielding a frequency doubled beam at 395 nm central wavelength for driving the HHG process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This SHG beam is focused into a semi-infinite gas cell, which consists of a Brewster-cut entrance window, a 20 cm length filled with 50 torr of argon gas, and a copper gasket placed in the focal plane of the driving laser where a coherent HHG beam is generated [53,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The driving laser at 395 nm central wavelength is separated from the high-harmonic beam by using a 200 nm aluminum filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This filter also blocks any harmonics with 𝜆 > 77 nm, while harmonics with 𝜆 < 39 nm exceed the HHG cutoff energy, and so are not generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Furthermore, due to the centrosymmetry of the medium, only odd-numbered harmonic orders are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The resulting EUV beam after the aluminum filter thus consists of narrow peaks at the 7th (𝜆 = 56 nm) and 9th (𝜆 = 44 nm) harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The intensity ratio of the two harmonics in our experimental setup is estimated to be Iλ=56nm/Iλ=44nm ~30:1, which can be well-approximated as a monochromatic illumination suitable for ptychographic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For generating HHG beams with a Gaussian spatial profile, we used an SHG beam with pulse energy of ~500 µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For generating HHG beams carrying OAM, we increased the pulse energy of the SHG beam to ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 mJ, and inserted a spiral phase plate (Holo-Or, VL-214-395-Y-A, OAM charge number ℓ = 1 at 395 nm wavelength) right after the focusing optics into the semi-infinite gas cell to generate a driving beam with OAM charge number ℓ = 1, and 𝜆 = 395 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The increased pulse energy is necessary in order to make the peak intensity (located at a central point for the Gaussian beam, but distributed in a ring for the OAM beam) equal for the two cases, thus matching HHG cutoff energies and conversion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Due to the conservation of OAM in HHG, the resulting quasi-monochromatic 7th harmonic beam (𝜆 = 56 nm) carries an OAM charge number of ℓ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The HHG beam at 56 nm wavelength is focused sequentially by two toroidal mirrors (1: B4C-coated, feff = 27 cm, θ = 15°;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 2: Au-coated, feff = 50 cm, θ = 10°) in a Wolter configuration to create an imaging system with higher NA (feff = 17 cm) while managing coma aberration [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The resulting focusing beam is redirected towards the sample at normal incidence using a glancing incidence mirror (B4C coating, fused silica substrate, 3° incidence angle from grazing, nominal reflectivity 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The testing samples are three Quantifoil holey carbon films which have various hole sizes and shapes arranged in a rectangular grid, and are mounted on standard Ted Pella Ø3mm Cu TEM grids with 200 mesh (125 um pitch, 90 um hole width and 35 um bar width).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' More specifically, the three Quantifoil holey carbon films have a pitch of 9 μm (7 μm square hole and 2 μm bar, product number 656-200-CU), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm diameter circular holes and 1 μm separation, product number 660-200-CU) and 3 μm (2 um diameter circular holes and 1 um separation, product number 661-200-CU), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The samples are positioned close to the beam focus, and are mounted on a precision translation stage ensemble (SmarAct XYZ-SLC17:30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' They are translated in the plane perpendicular to the beam path to perform ptychographic scans in 7 × 7 rectangular grids (49 positions) with nominally 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='3 μm between adjacent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A random offset of ±20% of the scan step size was added to each scan position to avoid artifacts originating from the scan grid itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The far-field diffraction patterns are recorded by an EUV-CCD detector (Andor iKon-L, 2048 × 2048, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm pixel size) positioned 50 mm after the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In order to obtain the best ptychographic reconstructions possible for each illumination case, we carefully characterized each probe function in the sample planes by taking ptychographic scans on a non-periodic sample and reconstructing both the sample and the probe function through blind deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstructed probe functions were used in the ptychographic reconstructions of highly periodic samples as initial guesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Ptychographic data processing and image reconstructions The diffraction patterns were recorded by an EUV-CCD detector with 2048 × 2048 pixels and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm detector pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' During data processing, we cropped them to 1024 × 1024 because very few photons were detected outside this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The resulting pixel size of the reconstructed images is 𝑑𝑟 = 𝜆∙𝑧 𝑁∙𝑑𝑥 ≈ 200 nm, (2) where λ = 56 nm is the wavelength, z = 50 cm is the distance from the sample to the CCD detector, N = 1024 is the number of pixels in one direction and dx = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm is the detector pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The ptychographic reconstructions were performed in two steps using only the ePIE algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the first step, the complex-valued probe functions (both Gaussian- and OAM-HHG beams) were characterized by performing ptychography on a non-periodic sample and using the ePIE algorithm for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the second step, the pre- characterized probe functions were used as initial guesses for reconstructing the periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For the first 100 iterations, only the sample images were updated while the probe functions were kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Then, both the sample images and probe functions were updated by the ePIE algorithm for another 900 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The total number of iterations for ptychographic reconstructions of periodic structures was 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This procedure was kept constant for all Gaussian-HHG and OAM-HHG reconstructions in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We also want to emphasize the fast convergence speed of our technique compared to that in the work by Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' [15], which takes more than 10,000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' SNR analysis of imaging of the nano-defect The SNR of the defect detection in Table 1 is calculated as follows: We start from the three curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For each curve, corresponding to an experimental condition shown in the legend, the signal level is the transmission value in the defect, the background level and noise level are calculated as the average and the standard deviation, respectively, of the transmission values excluding the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The SNR is then calculated using the following formula: 𝑆𝑁𝑅 = 𝑠𝑖𝑔𝑛𝑎𝑙 − 𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑 𝑛𝑜𝑖𝑠𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (3) Data availability The data that supports the plots and other findings within this paper are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' References [1] Miao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non- crystalline specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Nature 400, 342–344 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' [2] Rodenburg, J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The authors thank Guan Gui, Drew Morrill, Yunzhe Shao, Chen-Ting Liao, Emma Cating-Subramanian for comments on the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Author contributions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' conceived the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' built and maintained the EUV source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' collected the data sets and performed the reconstructions and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' performed the SEM imaging of the test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' advised on the phase retrieval algorithms and setup, while I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' helped to develop the laser setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' All authors designed aspects of the experiment, performed the research and wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Competing financial interests B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' have submitted a patent disclosure based on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='M M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' are partial owners of KMLabs Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' who manufactured the ultrafast laser used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Supplementary information: S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Convolution theorem perspective on ptychographic imaging of highly periodic structures In ptychography, the far-field diffraction fields are approximated as the Fourier transform of the product of the complex probe and object functions, p(x,y) and o(x,y), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', Ψ (u,v) = ℱ [p(x,y) × o(x,y)], (S1) where ℱ is the Fourier transform operation, (x,y) are the real space coordinates and (u,v) are the reciprocal space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' According to the convolution theorem, this can also be represented as a convolution of the Fourier transform of the probe function, P(u,v) = ℱ [p(x,y)], and that of the object function, O(u,v) = ℱ [o(x,y)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=', Ψ(u,v) = P(u,v) ⁕ O(u,v), (S2) where ⁕ is the convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Often, P(u,v), which is the complex beam in the detector plane when no sample is in the way, has an edge resulting from apertures in the system, as indicated by the circle in the close-ups in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 1(a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the case of 2D highly periodic structures, O(u,v) consists of a 2D comb of 𝛿 functions (diffraction peaks) arranged in a 2D periodic grid, the amplitudes and phases of which are modulated by the Fourier transform of the unit cell of the periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This is a sparse function in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The convolution operation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (S2) puts a copy of P(u,v) at the location of each 𝛿 function with modulated amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In cases where P(u,v) is small in size such that all diffraction orders are isolated, the modulated phase of each diffraction order is totally lost when we collect intensity measurements, thus causing ptychography to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' However, in cases where P(u,v) is sufficiently large in size, the interference fringes in the overlapped regions between neighboring diffraction orders encoded the relative phase of each diffraction orders into the intensity modulations that are directly measurable with the EUV-CCD camera, thus enabling fast and robust ptychographic reconstructions of the highly periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A phase-change-like behavior in ptychography demonstrated by Gaussian-HHG illuminations with controlled divergence Since the key to successfully achieving ptychographic imaging of highly periodic structures is to obtain overlap and interference between neighboring diffraction orders, an abrupt, phase-change-like behavior in reconstruction quality is expected as one smoothly changes the illumination NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We experimentally demonstrated this behavior in ptychographic imaging of highly periodic structures, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S1, using Gaussian-HHG illuminations with a controlled illumination NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' This is achieved by installing an in-vacuum iris ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 m after the semi-infinite gas cell, which allows direct control of the divergence of the HHG beams, and thus of the illumination NA on the sample and the overlap between neighboring diffraction orders given the same focusing optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We performed four ptychography scans on the same 2D square periodic structure with a 9 μm period under various illumination NAs controlled by the in-vacuum iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S1(a–d) shows example diffraction patterns from each scan from small illumination NA in (a) to large illumination NA in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The close-ups in the blue circles show the effect of illumination NA on the resulting diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We then reconstructed these four datasets using the standard ePIE algorithm [13] and the corresponding results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S1(e–h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' It is clear that for ptychography scans where diffraction orders are isolated, the periodic structure cannot be reliably reconstructed due to the lost phase information, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S1(e–f), while for ptychography scans where the illumination NA is large enough to support overlap between diffraction orders, the ePIE algorithm can quickly and reliably reconstruct the periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Experimental demonstrations of a phase-change-like behavior in ptychographic imaging of 2D square periodic structures with 9 um period using Gaussian-HHG beams with controlled divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a–d) Example diffractions from a 2D square periodic structure using Gaussian-HHG beams with various divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The inserts show close-ups of the center of the diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (e–h) The corresponding ptychographic reconstructions of the 2D square periodic structure under various illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstructions are successful only when the diffraction orders have overlap, showing a phase-change-like behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Image quality assessment using complex histogram analysis We use a complex histogram analysis to evaluate the quality of the ptychographic reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(d–f), 3(j– l) and 4(a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' A 2D complex histogram is an extension of a normal histogram showing how many data points of a complex field lie within a certain range of real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' For the approximately binary test samples used in this study, ideally, the complex histograms consist of only two 𝛿-function peaks corresponding to the transmissive and opaque areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In reality, the two 𝛿-function peaks are broadened due to limited SNR and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The quality of the ptychographic reconstructions can thus be assessed by examining the degree of broadening of these peaks, where reconstructions with higher quality have narrower peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We first evaluate the quality of the ptychographic reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' In the complex histograms shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S2, the two parts of the sample (free space and the carbon bars) are indicated by the red and yellow circles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The complex histograms for OAM-HHG images (d–f) have narrower peaks than those for Gaussian-HHG images (a– c), which indicates that OAM-HHG images have better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The reconstruction in (c) (Gaussian-HHG illuminations on a 3-um-pitch structure) failed, thus not showing the double-peak feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' We then evaluate the quality of the ptychographic reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' S3(a–c), the ptychographic reconstructions are shown in the complex representation with amplitude and phase indicated by brightness and hue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Visually, the image from OAM-HHG illumination in a has the best quality in terms of a sharp transition from free space area to thin carbon bar area and smoothness within free space or carbon bar areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The complex histograms in (d–f) confirmed this: the primary peaks (indicated by the red and yellow circles) in the complex histogram from OAM-HHG illuminations (as shown in d) are the narrowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) (g) (h) 10um Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Quality assessment of ptychographic reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3 using complex histogram analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a–c) Complex histograms of ptychographic reconstructions of 9 μm, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm and 3 μm pitch periodic structures using Gaussian-HHG illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The ptychographic reconstructions are shown in each bottom left corner and correspond to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(d–f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (d–f) Complex histograms of ptychographic reconstructions of 9 μm, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content='5 μm and 3 μm pitch periodic structures using OAM-HHG illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The ptychographic reconstructions are shown in each bottom left corner and correspond to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 3(j–l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These complex histograms consist of two primary peaks (except panel c because the reconstruction failed), which correspond to the open space area (indicated by the red circles) and the thin carbon bar area (indicated by the yellow circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The complex histograms from OAM-HHG illuminations (the bottom row) have narrower primary peaks than those from Gaussian-HHG illuminations (the top row), which shows superior image quality for OAM-HHG reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The ‘Re’ and ‘Im’ axes in (a) show the complex coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Im Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Quality assessment of ptychographic reconstructions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' 4 using complex histogram analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a–c) Complex representations of ptychographic reconstructions of 2D square periodic structures with 9 μm period under three different experimental conditions: (a) an OAM-HHG illumination, (b) a Gaussian-HHG illumination with equal exposure time, and (c) a Gaussian-HHG illumination with equal number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The amplitude and phase of these images are presented in brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The color wheel is shown in the bottom left corner of panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (d–f) Complex histograms of ptychographic reconstructions are shown in (a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' These complex histograms all consist of two primary peaks, which correspond to the open space area, indicated by the red circles, and the thin carbon bar area, indicated by the yellow circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The complex histogram in panel d has the narrowest primary peaks, which indicates its superior image quality provided by the intrinsic advantages of OAM- HHG illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' (a) (b) (c) Im Re Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Complex representations of the ptychography reconstructed Gaussian-HHG and OAM-HHG beams in the sample plane (a–b) and in the detector plane (c–d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The amplitude and phase of the beams are shown in brightness and hue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The scale bars in (a–b) indicate beam size in the sample plane, and those in (c–d) indicate beam divergence angle in the detector plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' The OAM-HHG beam in the detector plane in (d) shows a characteristic donut intensity profile, while the OAM- HHG beam in the sample plane does not show a donut intensity profile due to aberrations introduced by the focusing optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} +page_content=' Gaussian HHG OAM HHG (a) (b) Sample plane 0 元 50 um 50 μm (c) (d) Detectorplane 10mrad 10mrad' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE5T4oBgHgl3EQfXA9_/content/2301.05563v1.pdf'} diff --git a/JdE1T4oBgHgl3EQfsAXy/vector_store/index.pkl b/JdE1T4oBgHgl3EQfsAXy/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..46136abbd8c3bf9a08784f1980c9d54300730d6d --- /dev/null +++ b/JdE1T4oBgHgl3EQfsAXy/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9b7164b0325f7fecb51f3c369022730022b82f4a7eed9608cf944a82851f605d +size 514042 diff --git a/KNE3T4oBgHgl3EQfAQkY/content/2301.04255v1.pdf 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Computacional, Universidad de Granada, +E-18071 Granada, Spain +January 3, 2023 +Abstract +We have studied a massive U(1) gauge holographic model with pure +gauge and mixed gauge-gravitational Chern-Simons terms. The full +backreaction of the gauge field on the metric tensor has been consid- +ered in order to explore the vortical and energy transport sector. The +background solution has been computed numerically. On this back- +ground, we have considered the fluctuation of the fields and evaluated +the different correlators. We have found that all the correlators depend +on the mass of the gauge field. Correlators such as the current-current +one, ⟨JxJx⟩, which were completely absent in the massless case, in the +presence of a finite gauge boson mass start picking up some finite value +even at zero chemical potential. Similarly, the energy-current corre- +lator, ⟨T0xJx⟩, which was also absent in the massless theory, has now +a non-vanishing value but for finite values of the chemical potential. +Using Kubo formulae we have evaluated the chiral magnetic and chiral +vortical conductivities and studied their behaviour with the variation +of the mass of the gauge field. Our findings for the chiral vortical con- +ductivity, σV , and the chiral magnetic/vortical conductivity of energy +current, σε +B = σε +V , are completely new results. In addition to this, we +have found that these anomalous transport coefficients depend linearly +both on the pure Chern-Simon coupling, κ, and on the mixed gauge- +gravity Chern-Simon coupling, λ. One of the results which we would +like to highlight is the contribution to σV induced by λ in the massive +theory, which was not present in the massless case. +0 +arXiv:2301.00361v1 [hep-th] 1 Jan 2023 + +1 +Introduction +The AdS/CFT correspondence [1, 2] has been one of the most prominent +theoretical handles for studying systems which were very hard to tackle pre- +viously. It states that, in the low energy limit, the large-Nc, N = 4 super +Yang-Mills field theory in four-dimensional space is equivalent to the type +IIB string theory in AdS5 × S5 space. It has been widely applied for the +study of strongly coupled systems such as condensed matter systems, QCD +and hydrodynamics. Our current objective is to study the hydrodynamical +approach using this correspondence. +Quantum chiral anomalies are very fascinating properties which arise in +the context of relativistic field theories of chiral fermions beyond perturba- +tion theory [3–5]. Chiral anomalies have played a very crucial role in the +formulation of relativistic hydrodynamics [6]. Anomaly-induced transport +mechanisms have appeared on many occasions since the 80’s [7]. The ax- +ial current was the main topic in [8], and AdS/CFT correspondence was +first used to anomalous hydrodynamics in [9]. Recently a lot of attention is +gained by the effect of quantum anomalies on the hydrodynamics of otherwise +conserved currents. The chiral magnetic effect [10] and the chiral vortical ef- +fect [11–14] are two of such effects. In the former, the axial anomaly induces +a current parallel to the external magnetic field, while in the latter a current +is generated due to the presence of a vortex in the charged relativistic fluid. +It has been argued that these and other anomaly-induced effects may be pro- +duced in non-central heavy ion collisions at RHIC and LHC [15], inducing in +particular an event-by-event parity violation. These effects can also lead to +anomalous transport properties in some condensed matter systems, such as +the Weyl semi-metals [16,17]. +In the past few years, these anomalous effects has been implemented +in holography giving a lot of insights. +One of such works is [18], where +they considered a holographic model with a pure Chern-Simon term, and +they computed the chiral magnetic conductivity which exactly matches with +the results of the weakly coupled system. This is due to the fact that the +anomalous conductivities have non-renormalization properties so that they +are independent of the coupling constant. Later on, this model was extended +to incorporate the effect of the energy-momentum tensor related to the energy +current as well, and the mixed gauge-gravitational Chern-Simon term was +added in the gravitational action [19–21]. In these references the gauge fields +were considered to be massless. +In a similar line of work, the authors of [22] have studied the depen- +dence of the anomalous transport properties with the mass of the gauge field +which is introduced via the Stückelberg mechanism. In their case, they have +1 + +considered the probe limit. As a consequence the sectors comprising of the +correlators related to the energy-momentum tensor were not accessible, in +particular: i) the chiral vortical conductivity, ii) the chiral vortical conduc- +tivity of energy current, and iii) the chiral magnetic conductivity of energy +current. In a sense, this model only comprises a pure gauge Chern-Simon +term. Our goal in the present work is to access those sectors and to study +the chiral vortical effects as well. To this end, we have considered the full +backreaction of the gauge field onto the metric tensor, and included in the +action of the model a mixed gauge-gravitational Chern-Simons term. +The paper has been organized as follows. In Section 2, we will discuss the +model under consideration and get the full backreacted numerical solution +for the background. Next, we will discuss in Section 3 the Kubo formulae +and their relation with the retarded Green’s functions, i.e. the correlators. +Using the AdS/CFT dictionary we will define these correlators in terms of +the boundary terms. In Section 4 we will start presenting our results; first, +we will compare the results with the known results for the massless case [19], +and after that, we will present our main results regarding the behaviour of the +two-point correlators including the mass term for the gauge boson. We will +discuss in the same section the effect of the mixed gauge-gravitational Chern- +Simons term in these correlators, and finally we will show how the gauge +boson mass affects the anomalous conductivities, namely the chiral vortical +conductivity, σV , the chiral vortical conductivity of energy current, σε +V , the +chiral magnetic conductivity, σB, and the chiral magnetic conductivity for +energy current, σε +B. Finally, we end with a discussion in Section 5. +2 +Holographic massive U(1) gauge theory +We consider a holographic model with a massive U(1) gauge boson that +includes both a pure gauge and a mixed gauge-gravitational Chern-Simon +term in the action [19,22]. The action of the model is +S += +1 +16πG +� +d5x√−g +� +R + 2Λ − 1 +4FMNF MN +−m2 +2 (AM − ∂Mθ)(AM − ∂Mθ) ++ϵMNPQR(AM − ∂Nθ) +�κ +3FNPFQR + λRA +BNPRB +AQR +� � ++SGH + SCSK , +(2.1) +2 + +where +SGH += +1 +8πG +� +∂ +d4x +√ +hK , +(2.2) +SCSK += +− 1 +2πG +� +∂ +d4x +√ +hKλnMϵMNPQRANKPLDQKL +R , +(2.3) +are the Gibbons-Hawking boundary term, and a boundary term induced by +the mixed gauge-gravitational anomaly, respectively, which have been well +discussed in [19]. θ is a field which ensures gauge invariance (up to gauge +anomalies), and thus the mass term enters in a consistent way. As it men- +tioned in [23–25], the Stückelberg term arises as the holographic realization +of dynamical anomalies. A comparison of the consistent form of the anomaly +for chiral fermions [3] with the variation of the action under axial gauge +transformations, allows to fix the anomaly coefficients to +κ = κp ≡ − G +2π , +λ = λp ≡ − G +48π . +(2.4) +See e.g. Ref. [19] for a discussion. In the following we will refer to the values +of Eq. (2.4) as the physical values of the anomaly coefficients. +The bulk equations of motion for the action of Eq. (2.1) turn out to be +GMN − ΛgMN += +1 +2FMLFN +L − 1 +8gMNF 2 + m2 +2 BMBN − m2 +4 gMNBPBP ++2λϵLPQR(M▽B +� +F PLRB +N) +QR� +, +(2.5) +▽NF NM += +−ϵMNPQR � +κFNPFQR + λRA +BNPRB +AQR +� ++ m2BM ,(2.6) +where we have defined a new field BM ≡ AM − ∂Mθ, so that in the following +θ will not appear explicitly anywhere. We have used the notation X(MN) ≡ +1 +2(XMN + XNM). +The ansatz for the background metric is a black hole solution in Fefferman- +Graham coordinates, which is given by [26,27] +ds2 = −ℓ2 +ρ gττ(ρ)dτ 2 + ℓ2 +ρ gxx(ρ)d⃗x2 + ℓ2 +4ρ2dρ2, +(2.7) +where the boundary lies at ρ = 0 and the horizon at ρ = ρh, while ℓ is the +radius of AdS. The horizon ρh is chosen in such a way that gττ(ρh) = 0, and +the temperature of the black hole turns out to be +T = 1 +2π +� +2ρhg′′ +ττ(ρh) . +(2.8) +3 + +The asymptotic expansion (ρ → 0) of the solution of Eq. (2.6) shows that +the gauge field near the boundary behaves as +BM(ρ) = a0ρ− ∆ +2 + a1ρ +∆ +2 +1 + · · · , +(2.9) +where m2ℓ2 = ∆(∆ + 2), with ∆ the anomalous dimension of the dual +current [22]. +The first(second) term in Eq. (2.9) corresponds to a non- +normalizable(normalizable) mode. The scaling dimension of the normaliz- +able mode is (3 + ∆), and this puts an upper bound on the value ∆ = 1. For +∆ > 1 the dual operators become irrelevant (in the IR), and so we will be +working in the range of values of ∆ below this bound. +2.1 +Numerical solution for the background +In order to account for the chiral vortical effects within the present model, we +will be considering the full backreaction of the gauge field onto the metric. +Plugging Eq. (2.7) into Eqs. (2.5) and (2.6), the equations of motion for the +background metric and gauge field turn out to be +g′′ +xx(ρ) − g′ +xx(ρ) +ρ ++ +1 +6ℓ2ρ +gxx(ρ) +gττ(ρ) +�m2ℓ2 +4 +Bt(ρ)2 + ρ2B′ +t(ρ)2 +� += 0 ,(2.10) +g′ +ττ(ρ) +� +1 − ρg′ +xx(ρ) +gxx(ρ) +� ++ gττ(ρ)g′ +xx(ρ) +gxx(ρ) +� +3 − ρg′ +xx(ρ) +gxx(ρ) +� +−1 +3 +ρ2 +ℓ2 B′ +t(ρ)2 + 1 +12m2Bt(ρ)2 = 0 , +(2.11) +B′′ +t (ρ) + 1 +2 +� +3g′ +xx(ρ) +gxx(ρ) − g′ +ττ(ρ) +gττ(ρ) +� +B′ +t(ρ) − ℓ2m2 +4ρ2 Bt(ρ) = 0 , +(2.12) +where the gauge field has been chosen in the following way +BMdxM = Bt(ρ)dt , +(2.13) +so that Br = 0. We will solve numerically the above coupled differential +equations with the following boundary conditions +Bt(ρh) = 0 , +lim +ρ→0 +� +ρ∆/2Bt(ρ) +� += µ5 , +(2.14) +with µ5 being the source. As it is discussed in [22], in the presence of a finite +gauge boson mass µ5 does not correspond to a thermodynamic parameter, +but it is instead a coupling in the Hamiltonian. As a result, different values of +chemical potential correspond to different theories. For completeness, we will +4 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +ρ +gxx +gττ +ρΔ/2·Bt +Figure 2.1: (color) Dependence of the background metric and gauge field +with ρ. +We display the results for gxx(ρ) (blue), gττ(ρ) (orange), and +ρ∆/2Bt(ρ) (green). We have chosen ∆ = 0.1 and µ5 = 0.5. +provide the analytical solution of the background equations of motion (2.10)- +(2.12) for vanishing µ5. These are +gττ(ρ) = 1 +ρ2 +h +(ρ2 +h − ρ2)2 +ρ2 +h + ρ2 +, +gxx(ρ) = 1 + ρ2 +ρ2 +h +, +Bt(ρ) = 0 , +(2.15) +while the temperature turns out to be T = 1 +π +� +2 +ρh. For the metric tensor, we +demand that the solution is regular at the horizon, while at the boundary +it reaches some constant value which we can always scale to set it to 1. +Hereafter we will set the values of ℓ = 1 and ρh = 1 for our numerical +calculations, which one can fix as such by using the scaling symmetry of the +metric tensor. This will set the units of all the quantities, i.e. µ5, Q, M, +etc. We have plotted in Fig. 2.1 the numerical solution of all the background +fields, i.e. gxx(ρ), gττ(ρ) and Bt(ρ). One may note from this figure that +limρ→0 +� +ρ∆/2Bt(ρ) +� += µ5. +3 +Kubo formulae and correlators +In this section, we will discuss the Kubo formulae needed to compute the +anomalous transport coefficients in our model, and set up the equations to +evaluate these transport properties. The Kubo formulae for the anomalous +conductivities have been well studied [28]. The authors of this reference have +shown that the chiral vortical conductivity for charge and energy transport +5 + +can be obtained from the following two-point functions +σV = lim +kc→0 +i +2kc +� +a,b +ϵabc⟨JaT 0b⟩|w=0 , +σε +V = lim +kc→0 +i +2kc +� +a,b +ϵabc⟨T 0aT 0b⟩|w=0 , +(3.1) +where σV is the chiral vortical conductivity and σε +V the chiral vortical con- +ductivity of energy current, respectively. The chiral magnetic conductivities +for charge, σB, and energy, σε +B, current are given by +σB = lim +kc→0 +i +2kc +� +a,b +ϵabc⟨JaJb⟩|w=0 , +σε +B = lim +kc→0 +i +2kc +� +a,b +ϵabc⟨T 0aJb⟩|w=0 . +(3.2) +To compute these correlators one can use the AdS/CFT dictionary [19, +29,30]. Keeping this in mind, we proceed with the perturbation of the fields, +where the background is set by the numerical solution as shown in Fig. 2.1. +We will study the linear response of the fluctuation, so that we split the +metric and gauge field into a background and a linear perturbation part, i.e. +gMN = g(0) +MN + ϵhMN, +BM = B(0) +M + ϵbM . +(3.3) +Then, we will follow the general procedure of Fourier mode decomposition [28] +hMN(ρ, xµ) += +� +ddk +(2π)dhMN(ρ)e−iωt+i⃗k.⃗x , +(3.4) +bM(ρ, xµ) += +� +ddk +(2π)dbM(ρ)e−iωt+i⃗k.⃗x . +(3.5) +Without the loss of generality, one can consider perturbations of frequency ω +and momentum k in the z-direction. In order to study the anomalous effect +we will switch on the fluctuations Bi, hi +t and hi +z, where i = x, y. Following +this, we will substitute (3.3) in the equations of motion (2.5) and (2.6), and +consider the resulting expressions at order O(ϵ). +Since we are interested in computing correlators at zero frequency, we +can set the frequency-dependent parts as zero in the equations, and solve +the system up to first order in k. In this limit, the fields hi +z decouple from +the system and take a constant value. Finally, we can write the system of +6 + +differential equations for the shear sector as +b′′ +i (ρ) + 1 +2 +�g′ +xx(ρ) +gxx(ρ) + g′ +ττ(ρ) +gττ(ρ) +� +b′ +i(ρ) − ∆(∆ + 2) +4ρ2 +bi(ρ) +(3.6) ++ +� +4iκkϵijbj(ρ) +� +gxx(ρ)gττ(ρ) ++ gxx(ρ)hi′ +t(ρ) +gττ(ρ) +� +B′ +t(ρ) + iλkϵijhj′ +t(ρ)Ω(ρ) = 0 , +hi′′ +t (ρ) − +� g′ +ττ(ρ) +2gττ(ρ) − 5g′ +xx(ρ) +2gxx(ρ) + 1 +ρ +� +hi′ +t(ρ) + ρB′ +t(ρ) +gxx(ρ) b′ +i(ρ) ++∆(∆ + 2)Bt(ρ) +4ρgxx(ρ) +bi(ρ) + iλkϵijΦj(ρ) = 0 , +(3.7) +where i, j = x, y. The explicit expressions of the functions Ω(ρ) and Φj(ρ) +are given in Appendix A. +Asymptotic analysis of the fluctuations near the boundary (ρ → 0) up to +the first subleading order shows +bi(ρ) += +b(0) +i ρ− ∆ +2 + b(1) +i ρ +∆ +2 +1 + · · · , +(3.8) +hi +t(ρ) += +hi +t +(0) + hi +t +(1)ρ2 + · · · , +(3.9) +where the leading order terms b(0) +i +and hi +t +(0) are the sources. From the holo- +graphic description of the correlation functions, one can evaluate the one- +point functions as +⟨Ja⟩ += +δSren +δb(0) +a += − +2 +16πG(∆ + 1)b(1) +a , +(a = x, y) , +(3.10) +⟨T0a⟩ += +δSren +δha +t (0) = +1 +16πG +� +2ha +t +(0) + ha +t +(1)� +, +(a = x, y) , +(3.11) +where Sren = S + Sct is the renormalized action, with S the action given +in Eq. (2.1) and Sct the counterterm. The procedure to evaluate this coun- +terterm is given in [19] and [22]. We find that the counterterm needed to +renormalize this theory is the same as the one given in [22], i.e. the mixed +gauge-gravitational Chern-Simons term does not introduce new divergences, +and so the renormalization is not modified by it (see e.g. Ref. [19] for a discus- +sion in the massless case). In this regards, we are not writing the counterterm +Sct explicitly. ⟨Ja⟩ and ⟨T0a⟩ correspond to current and energy-momentum +tensor one-point functions, respectively 1. Similarly, the two-point functions +can be obtained by taking the variation of one-point function with respect +1Ji and T0i are related with the fluctuations bi and hi +t, respectively, with i = x, y. +7 + +to the corresponding source term, i.e. +⟨JaJb⟩ += +δ⟨Ja⟩ +δb(0) +b +, +(a, b = x, y) , +(3.12) +⟨JaT0b⟩ += +δ⟨Ja⟩ +δhb +t(0) , +(a, b = x, y) , +(3.13) +⟨T0aJb⟩ += +δ⟨T0a⟩ +δb(0) +b +, +(a, b = x, y) , +(3.14) +⟨T0aT0b⟩ += +δ⟨T0a⟩ +δhb +t(0) , +(a, b = x, y) . +(3.15) +From the above expressions, it is clear that it is required the leading and +subleading parts of the asymptotic expansion of the fluctuations to evaluate +the two-point functions we are interested in. To do so we have solved numer- +ically the coupled differential equations of the fluctuations (3.6) and (3.7) +and imposed suitable boundary conditions, i.e. i) regularity at the horizon, +and ii) sourceless condition at the asymptotic boundary. +4 +Results +In this section, we will start presenting our results. Firstly, we will start +with the massless case (∆ = 0) and compare the results with the previous +work done in [19]. In the second part of this section, we will consider the +massive case ∆ ̸= 0, and study the dependence of the two-point functions +with ∆ for different values of µ5. In both cases, we will set G = 1/(16π) so +that the physical values of the anomalous couplings are κ = −1/(32π2) and +λ = −1/(768π2), cf. Eq. (2.4). Later on, we will study the dependence of the +two-point functions with the parameters κ and λ. This is done to show that +the parametric dependence of the correlators is linear in these parameters, +but values of κ and λ different from κ/λ = 24 are non-physical. In addition +to this, to make a direct comparison with the previous work in [22], all the +anomalous correlators have been displayed normalized by |κ|−1. +8 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +μ5 +-Im  < Jx Jy > +k κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +μ5 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +2.5 +3.0 +3.5 +4.0 +μ5 +-Im  < Jx T0 y > +k κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +1.00 +1.05 +1.10 +1.15 +μ5 + +Figure 4.1: Upper panel: Plots of the correlators ⟨JxJy⟩ (left) and ⟨JyT0y⟩ +(right) vs µ5. Lower panel: Plots of the correlators ⟨JxT0y⟩(left) and ⟨T0yT0y⟩ +(right) vs µ5. These plots are obtained in the massless case (∆ = 0). +4.1 +Massless case +In the absence of mass, the correlators have been evaluated in [19,22], leading +to +⟨JxT0x⟩ += +⟨JyT0y⟩ = +√ +3Q +4πGℓ3 , +⟨JxJy⟩ += +−⟨JyJx⟩ = κi +√ +3kQ +2πGr2 +h +− κ ikα +6πG = −ik(3µ5 − α) +12π2 +, +⟨JxT0y⟩ += +−⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ = κ 3ikQ2 +4πGr4 +h ++ λ2ikπT 2 +G +, += −ik +� µ2 +5 +8π2 + T 2 +24 +� +, +(4.1) +⟨T0xT0x⟩ += +⟨T0yT0y⟩ = +M +16πGℓ3 , +⟨T0xT0y⟩ += +−⟨T0yT0x⟩ = κi +√ +3kQ3 +2πGr6 +h ++ λ4πi +√ +3kQT 2 +Gr2 +h += −ik +� µ3 +5 +12π2 + µ5T 2 +12 +� +, +9 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +1 +2 +3 +4 +μ5 +-Im  < T0 x T0 y > +k κ +Figure 4.2: Plot of the correlator ⟨T0xT0y⟩ vs µ5 in the massless case (∆ = 0). +with M = r4 +h +ℓ2 + Q2 +r2 +h +and Q = µ5r2 +h +√ +3 the mass and charge of the black hole +solution computed in Poincaré coordinates, with blackening factor +f(r) = 1 − Mℓ2 +r4 ++ Q2ℓ2 +r6 +. +(4.2) +The Hawking temperature is given in terms of these black hole parameters +as +T = +r2 +h +4πℓ2f ′(rh) = (2r2 +hM − 3Q2) +2πr5 +h +. +(4.3) +The parameter α in Eq. (4.1) corresponds to the asymptotic value of the +gauge field At for ρ → 0. In our case, we are assuming α = µ5 for ∆ = 0, cf. +Eq. (2.14). The other correlators are vanishing in the massless case, i.e. +⟨JxJx⟩ = ⟨JyJy⟩ = 0 , +⟨T0xJx⟩ = ⟨T0yJy⟩ = 0 . +(4.4) +While the correlators with the same indices are not induced by quantum +anomalies (i.e. they are non-anomalous) and they become real, the correla- +tors with different indexes are anomalous and they become imaginary. We +will be comparing the numerical results with the analytical expressions given +in the above equations, Eq. (4.1). We plot in Figs. 4.1 and 4.2 five inde- +pendent non-vanishing correlators, while the other correlators are related to +them through the expressions given in Eq. (4.1). In these and subsequent +plots, it is understood that it has been taken the limit k → 0 with k ≡ kz. +In these figures the dots stand for the numerical results, and the solid lines +correspond to the analytic results of Eq. (4.1). One may observe that the +numerical results are in good agreement with the analytic expression. +10 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +1 +2 +3 +4 +5 +6 +7 +Δ +- +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +Δ + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +2 +4 +6 +8 +10 +Δ +- +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +1.000 +1.005 +1.010 +1.015 +1.020 +1.025 +1.030 +Δ + +Figure 4.3: (color) Plots for non-anomalous correlators vs ∆. Upper panel: +Plot of the correlators ⟨JxJx⟩ (left) and ⟨JxT0x⟩ (right) vs ∆. Lower panel: +Plot of the correlators ⟨T0xJx⟩ (left) and ⟨T0xT0x⟩ (right) vs ∆. We have +considered in all the panels µ5 = {0, 0.1, 0.2} (blue, orange and green). +4.2 +Massive case +We will split our discussion into anomalous and non-anomalous correlators. +We have found that the above mentioned relations between different corre- +lators still hold in the massive case, i.e. +⟨JxT0x⟩ = ⟨JyT0y⟩ , +⟨JxJy⟩ = −⟨JyJx⟩ , +⟨JxT0y⟩ = −⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ , +⟨T0xT0x⟩ = ⟨T0yT0y⟩ , +⟨T0xT0y⟩ = −⟨T0yT0x⟩ . +(4.5) +In addition to this, there are two more independent correlators, i.e ⟨T0xJx⟩ = +⟨T0yJy⟩ and ⟨JxJx⟩ = ⟨JyJy⟩. In this regard, we will be plotting only seven +independent correlators. +11 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +1 +2 +3 +4 +5 +Δ +Im  < Jy Jx > +k κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Δ +Im  < T0 y T0 x > +k κ +Figure 4.4: (color) Plot of the correlators ⟨JyJx⟩ (left) and ⟨T0yT0x⟩ (right) +vs ∆ with µ5 = {0, 0.1, 0.2} (blue, orange and green). +4.2.1 +Non-anomalous correlators +While the correlator ⟨JxJx⟩ is vanishing for ∆ = 0 (cf. Section 4.1), we can +see from the Fig. 4.3 (upper-left panel) that this correlator starts picking +up some finite value in the massive case (∆ ̸= 0). With the increase of ∆ +the absolute value of this correlator increases quite sharply, and gets even +shaper with the increase in µ5. This property, i.e. an increasing value of +the (absolute value of the) correlator for increasing ∆ and for finite µ5, is a +general feature for all the non-anomalous coefficients as we will discuss below. +We can see from Fig. 4.3 (upper-right panel) that for µ5 = 0 the correlator +⟨JxT0x⟩ is zero for all values of ∆. +As the value of µ5 increases, ⟨JxT0x⟩ +becomes finite and its value increases with ∆ in a somewhat linear fashion. +The slope of ⟨JxT0x⟩ vs ∆ also increases with the increase of µ5. +In Fig. 4.3 (lower panel-left) we can see that even though the correlator +⟨T0xJx⟩ is vanishing for ∆ = 0, for finite values of ∆ and µ5 this corre- +lator is non-vanishing. More in details, for a given finite value of µ5, the +absolute value |⟨T0xJx⟩| increases quite sharply with ∆. Notice that ⟨T0xJx⟩ +was completely absent in the previous work [19], but we find now that it is +non-vanishing at finite µ5 in the massive theory. +Finally, we can see in Fig. 4.3 (lower-right panel) that ⟨T0xT0x⟩ is inde- +pendent of ∆ for µ5 = 0, i.e. it has a constant value corresponding to the +pressure term, a feature that has been well discussed in [18–20, 22]. At fi- +nite chemical potential, this correlator increases with ∆, a behavior which is +sharper for larger values of µ5. +4.2.2 +Anomalous correlators +We display in Fig. 4.4 (left) the behaviour of ⟨JyJx⟩ vs ∆. One can see that +the absolute value of this correlator increases with ∆, and the change is quite +12 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +2.5 +3.0 +3.5 +4.0 +4.5 +Δ +-Im  < Jx T0 y > +k κ +Figure 4.5: (color) Plot of the correlator ⟨JxT0y⟩ vs ∆ with µ5 = {0, 0.15, 0.3} +(blue, orange and green). +-0.0002 +-0.0001 +0.0000 +0.0001 +0.0002 +-0.0010 +-0.0005 +0.0000 +0.0005 +0.0010 +-0.2 +-0.1 +0.0 +0.1 +0.2 +-2 +-1 +0 +1 +2 +λ +κp +Im  < T0 y T0 x > +k κp +-0.0010 +-0.0005 +0.0000 +0.0005 +0.0010 +-0.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +0.03 +-0.2 +-0.1 +0.0 +0.1 +0.2 +-15 +-10 +-5 +0 +5 +10 +15 +λ +κp +Im  < Jy T0 x > +k κp +Figure 4.6: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs λ/|κp|. +The inset figures correspond to zooms of the main figures in the small λ +regime. We have considered in both panels, µ5 = 0.1, ∆ = 0.1 and κp = +−1/(32π2). +subtle. It is plotted in Fig. 4.4 (right) the correlator ⟨T0yT0x⟩ vs ∆, and unlike +the other correlator, its absolute value decreases with the increase of ∆. +In Fig. 4.5 we have plotted ⟨JxT0y⟩ vs ∆. We find that the absolute value +of this correlator increases with the increase in ∆ and µ5. We have taken a +different value of µ5 as compared to the other correlators, because for those +values of µ5 the correlator did not have any substantial changes. The new +values of µ5 = {0, 0.15, 0.3} are chosen to make these changes distinct in +the figure. We can see from the figure that even in the absence of µ5 this +correlator is non-zero. This can be traced back to the temperature term, as +the temperature does not vanish for µ5 = 0. Finally, one may notice that +in all the cases the values of two point correlators tend toward the analytic +values as given in (4.1) when considering the limit ∆ → 0. This is also shown +in the figures for the massless case. +13 + +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.674 +0.676 +0.678 +0.680 +0.682 +λ +κp +Im  < Jy Jx > +k κp +-1.0 +-0.5 +0.0 +0.5 +1.0 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +κ +κp +Im  < Jy Jx > +k κp +Figure 4.7: Left: Plot of the correlator ⟨JyJx⟩ vs λ/|κp|. Right: Plot of the +correlator ⟨JyJx⟩ vs κ/|κp| for λ = −1/(768π2). We have considered in both +panels µ5 = 0.1 and ∆ = 0.1, while κp = −1/(32π2). +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.492 +0.493 +0.494 +0.495 +κ +κp +Im  < T0 y T0 x > +k κp +-1.0 +-0.5 +0.0 +0.5 +1.0 +2.82 +2.84 +2.86 +2.88 +2.90 +κ +κp +Im  < Jy T0 x > +k κp +Figure 4.8: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs κ/|κp|. +In both cases, µ5 = 0.1, ∆ = 0.1, |κp| = 1/(32π2) and λ = −1/(768π2). +λ and κ dependence: +To study the dependence of the two-point functions +with the parameter λ, we will consider the case where we fix the values as +µ5 = ∆ = 0.1 and κ = −1/(32π2), and vary λ. Here we will only present the +correlators that have a dependence on λ, while the λ independent correlators +are given in Fig. B.1 of Appendix B. We have plotted in Figs. 4.6 and 4.7 +(left) the dependence of the anomalous correlators with λ. One can see that +the behaviour is linear with λ in all the cases. The inset figures are given +to show that the corresponding correlators do not vanish at λ = 0. This is +in fact true, as the non-vanishing values arise due to the κ coupling, which +leads to ⟨T0xT0y⟩ ∼ µ3 +5κ and ⟨J0yT0x⟩ ∼ µ2 +5κ at λ = 0, with some contribution +from ∆. In the case of ⟨JyJx⟩ the λ dependence only arises in the massive +case. +Setting the values of ∆ = µ5 = 0.1, λ = −1/(768π2) and varying κ, we +see a similar kind of linear behaviour with κ. The effect of κ is only seen in +⟨T0xT0y⟩, ⟨J0yT0x⟩ and ⟨JyJx⟩ as shown in Fig. 4.7 (right) and Fig. 4.8. The +non-anomalous correlators are independent of κ, and they are displayed in +14 + +Fig. B.2 of Appendix B. These correlators are in fact independent of both the +parameters κ and λ, and hence they are non-anomalous in nature even in the +massive theory. This means that they do not contribute to anomalous trans- +port, unlike the correlators studied above which are associated to anoma- +lous conductivities. This can be seen in the Kubo formulae for anomalous +conductivities, Eqs. +(3.1) and (3.2), as these formulae involve Levi-Civita +(ϵijz) symbols which runs over i = j = {x, y}. Hence, the correlators with +i = j = x and i = j = y do not lead to anomalous transport effects. +Anomalous conductivities: +Finally, as a summary of the previous nu- +merical results, we now present the anomalous conductivities which are com- +puted with the Kubo formulas (3.1) and (3.2), i.e. +σV = − lim +k→0 +1 +kIm⟨JxT0y⟩ , +σε +V = − lim +k→0 +1 +kIm⟨T0xT0y⟩ , +(4.6) +σB = − lim +k→0 +1 +kIm⟨JxJy⟩ , +σε +B = − lim +k→0 +1 +kIm⟨T0xJy⟩ . +(4.7) +The results are displayed in Fig. 4.9. We can see from this figure that the +chiral vortical conductivity and the chiral magnetic conductivity for energy +current are the same either at zero or finite mass, i.e. σV = σε +B, and these +quantities increase with ∆. We also see in this figure that the chiral vortical +conductivity of energy current, σε +V , decreases with ∆ but the rate decreases +rapidly. In the case of the chiral magnetic conductivity, σB, it increases with +∆ as shown in Fig. 4.9. +Regarding the other dependences of the anomalous conductivities, for +instance the dependence in the parameters κ and λ, it would be sufficient to +study them from Fig. 4.6 and Fig. 4.7, as the two-point functions and the +anomalous conductivities are related through Kubo formulae. We conclude +that for a given value of µ5 and ∆, the anomalous transport coefficients: +σV , σε +B, σB and σε +B; change linearly with the pure (κ) and mixed (λ) gauge- +gravitational Chern-Simon couplings. At the limit of vanishing mass, our +results lead to +σB +µ5|κ| ≃ 16/3, which exactly coincides with the results in [18, +19,22] where α has been set to µ5 in both references 2. In order to reproduce +the results of [18] where they have set α = 0, our κ needs to be rescaled by +a factor 3/2. Finally, let us emphasize that all the correlators involving the +energy-momentum tensor are completely new results at finite mass (∆ ̸= 0), +i.e. σV , σε +V and σε +B. +2α corresponds to the asymptotic value of the gauge field At for ρ → 0. In our case, +we assume α = µ5 for ∆ = 0. +15 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +2.8 +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +Δ +σV +κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +Δ +σε +V +κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Δ +σB +κ +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +2.8 +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +Δ +σε +B +κ +Figure 4.9: Upper panel: Plot of σV (left) and σε +V (right) vs ∆. Lower panel: +Plot of σB (left) and σε +B (right) vs ∆. We have considered µ5 = 0.15 in all +the panels. +5 +Discussion +We have studied the anomalous and non-anomalous conductivities in the +holographic Stückelberg model including both pure gauge and mixed gauge- +gravitational anomaly terms. To access the sectors concerning the energy- +momentum tensor we have to consider the full backreaction of the massive +gauge field onto the metric tensor. We have evaluated the numerical back- +ground solution and on this background, we have considered the fluctuations +of the fields. From these fluctuations, we have calculated the different corre- +lators and studied their behaviors with the relevant parameters of the model +(µ5, ∆, κ and λ). +We have found that the correlators in the massless case match with previ- +ous results in the literature [28]. Later on, we have studied the dependence of +these correlators with the mass of the gauge field, m2 = ∆(∆+2), and found +that all the correlators explicitly depend on the mass for a given non-zero +value of µ5. One of the results that it is important to emphasize here is that +the non-anomalous correlators such as ⟨JxJx⟩ and ⟨T0xJx⟩ are non-vanishing +in the massive theory for finite values of µ5. Moreover ⟨JxJx⟩ is non-zero in +this theory even for µ5 = 0, while ⟨T0xJx⟩ is vanishing for µ5 = 0 indepen- +dently of the mass. These correlators are vanishing in the massless theory, +independently of µ5. The mass of the gauge field highly enhances the abso- +16 + +lute value of the correlators, and this gets translated into an enhancement +of the anomalous conductivities. The behaviours of the correlators on the +pure gauge and mixed gauge-gravitational Chern-Simon couplings, κ and λ, +were also studied. We found that the correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and +⟨T0xT0x⟩ are independent of κ and λ, and hence they are non-anomalous in +nature. They do not contribute to the anomalous conductivities, as it can +be seen from the Kubo formulae (3.1) and (3.2) as well. +Finally, we have computed the anomalous conductivities and studied their +dependence with the mass of the gauge field (m). We have found that the +chiral vortical conductivity, σV , and the chiral magnetic conductivity for +energy current, σε +B, are equal and increase with ∆. One interesting result is +that there are contributions to σB coming from λ in the massive theory, which +was completely absent in the massless case. The conductivities σB, σV and +σε +B increase with ∆, while the chiral vortical conductivity of energy current, +σε +V , decreases with ∆. We have explicitly checked that all our numerical +results for the conductivities at finite mass tend to the known results at zero +mass in the limit ∆ → 0. For instance, it is known that at zero mass, the +chiral magnetic conductivity is σB = − 16 +3 κµ5 when α = µ5, which implies +that the ratio − σB +κµ5 = 16/3, independently of κ and µ5. As one can see +from Fig. 4.9 (left), our numerics produces in this limit σB/|κ| = 0.8 for +µ5 = 0.15, in agreement with the expected result. We have also checked the +ratio − σB +κµ5 = 16/3 for other values of κ and µ5. +This work can be extended in several ways. One possible extension could +be to consider the U(1)V × U(1)A gauge group. There are some studies in +holography with this gauge group, see e.g. Refs. [22,31,32]. However, in these +works: i) either the probe limit has been considered so that the chiral vortical +effect and the transport conductivities in the energy-momentum tensor are +not accessible, or ii) they correspond to studies for massless gauge bosons. +In particular, it would be interesting to study the interplay between the +anomalous and non-anomalous currents in the set-up of the full backreacted +background of Ref. [32], both for massless and massive gauge bosons. We +will explore these and other issues in future works. +Acknowledgments +We would like to thank Karl Landsteiner for enlightening discussions. E.M. +is grateful to Manuel Valle for collaboration in the early stages of this work. +N.R. thanks the Instituto de Física Teórica UAM/CSIC, Spain, for its hos- +pitality and partial support during his research visits in the final stages +of this work. +The works of N.R. and E.M. are supported by the project +17 + +PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/501100011033, by +the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento +2014-2020 Operational Program under Grant A-FQM-178-UGR18, and by +the Ramón y Cajal Program of the Spanish MCIN under Grant RYC-2016- +20678. The work of E.M. is also supported by Junta de Andalucía under +Grant FQM-225. +Appendix A +Explicit expressions for the func- +tions Ω(ρ) and Φj(ρ) +These functions have been introduced in the equations of motion of the fluc- +tuations (3.6)-(3.7). Their explicit expressions are given by +Ω(ρ) = +4 +� +gττ(ρ) (g′ +xx(ρ) + 2ρg′′ +xx(ρ)) + ρg′ +xx(ρ)g′ +ττ(ρ) +� +� +gxx(ρ)gττ(ρ)3/2 +− 8ρ g′ +xx(ρ)2 +gxx(ρ)3/2 +� +gττ(ρ) ++ +� +gxx(ρ) +gττ(ρ)5/2 +� +4ρg′ +ττ(ρ)2 − 4gττ(ρ) (g′ +ττ(ρ) + 2ρg′′ +ττ(ρ)) +� +, +(A.1) +and +Φj(ρ) = b′ +j(ρ) +� +− 8ρ2� +gττ(ρ)g′ +xx(ρ)2 +gxx(ρ)7/2 ++ +4ρ +� +gττ(ρ) +� +g′ +xx(ρ) + 2ρg′′ +xx(ρ) +� ++ ρg′ +xx(ρ)g′ +ττ(ρ) +� +gxx(ρ)5/2� +gττ(ρ) +− +4ρ +� +gττ(ρ) +� +g′ +ττ(ρ) + 2ρg′′ +ττ(ρ) +� +− ρg′ +ττ(ρ)2� +gxx(ρ)3/2gττ(ρ)3/2 +� ++ bj(ρ) +� +8ρ2� +gττ(ρ)g′ +xx(ρ)3 +gxx(ρ)9/2 +− 8ρ +� +gττ(ρ)g′ +xx(ρ) +gxx(ρ)7/2 +� +g′ +xx(ρ) + 2ρg′′ +xx(ρ) +� ++ +4ρ +� +ρg′′ +xx(ρ)g′ +ττ(ρ) − ρg′ +xx(ρ)g′′ +ττ(ρ) + 3gττ(ρ)g′′ +xx(ρ) + 2ρgxx(3)(ρ)gττ(ρ) +� +gxx(ρ)5/2� +gττ(ρ) +− +4ρ +� +− 2gττ(ρ)g′ +ττ(ρ) +� +g′ +ττ(ρ) + 2ρg′′ +ττ(ρ) +� ++ 2ρg′ +ττ(ρ)3 + gττ(ρ)2� +3g′′ +ττ(ρ) + 2ρgττ (3)(ρ) +�� +gxx(ρ)3/2gττ(ρ)5/2 +� +(A.2) +18 + ++ hj′ +t(ρ) +� +B′ +t(ρ) +� +� +16ρ2g′ +xx(ρ) +gxx(ρ)3/2� +gττ(ρ) ++ +8ρ +� +gττ(ρ) − ρg′ +ττ(ρ) +� +� +gxx(ρ)gττ(ρ)3/2 +� +� + +8ρ2B′′ +t (ρ) +� +gxx(ρ) +� +gττ(ρ) +� ++ hj′′ +t (ρ) +8ρ2B′ +t(ρ) +� +gxx(ρ) +� +gττ(ρ) +. +Appendix B +Some additional results for the non- +anomalous correlators +We show in this Appendix the numerical results for the non-anomalous cor- +relators as a function of the anomalous parameters κ and λ, in the massive +case (∆ ̸= 0). The correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and ⟨T0xT0x⟩, are +displayed in Figs. B.2 and B.1. These correlators turn out to be constant in +both κ and λ. The lack of dependence in these parameters implies that they +lead to non-anomalous transport effects. +19 + +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.0 +0.1 +0.2 +0.3 +0.4 +λ +κp +- +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +λ +κp +- +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.00 +0.05 +0.10 +0.15 +λ +κp +- +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.0 +0.5 +1.0 +1.5 +2.0 +λ +κp + +Figure B.1: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩ +(right) vs λ/|κp|. +Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and +⟨T0xT0x⟩ (right) vs λ/|κp|. +We have considered µ5 = 0.1, ∆ = 0.1 and +κp = −1/(32π2) in all the panels. +20 + +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.1 +0.2 +0.3 +0.4 +κ +κp +- +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +κ +κp + +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.00 +0.05 +0.10 +0.15 +κ +κp +- +-2 +-1 +0 +1 +2 +0.0 +0.5 +1.0 +1.5 +2.0 +κ +κp + +Figure B.2: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩ +(right) vs κ/|κp|. +Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and +⟨T0xT0x⟩ (right) vs κ/|κp|. 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D 90, no.6, +065026 (2014) doi:10.1103/PhysRevD.90.065026 [arXiv:1312.1204 [hep- +ph]]. +[32] E. +Megías, +EPJ +Web +Conf. +164 +(2017), +08001 +doi:10.1051/epjconf/201716408001 [arXiv:1701.00087 [hep-th]]. +24 + diff --git a/M9AyT4oBgHgl3EQfgfjF/content/tmp_files/load_file.txt b/M9AyT4oBgHgl3EQfgfjF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d59d7caf47227ebf6cdbb5f97d772ce0c6eb61c --- /dev/null +++ b/M9AyT4oBgHgl3EQfgfjF/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf,len=1171 +page_content='Anomalous conductivities in the holographic Stückelberg model Nishal Rai1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 and Eugenio Megías1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' SRM University Sikkim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Upper Tadong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Sikkim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' India 3 Instituto Carlos I de Física Teórica y Computacional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Universidad de Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' E-18071 Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Spain January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2023 Abstract We have studied a massive U(1) gauge holographic model with pure gauge and mixed gauge-gravitational Chern-Simons terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The full backreaction of the gauge field on the metric tensor has been consid- ered in order to explore the vortical and energy transport sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The background solution has been computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' On this back- ground, we have considered the fluctuation of the fields and evaluated the different correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have found that all the correlators depend on the mass of the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Correlators such as the current-current one, ⟨JxJx⟩, which were completely absent in the massless case, in the presence of a finite gauge boson mass start picking up some finite value even at zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Similarly, the energy-current corre- lator, ⟨T0xJx⟩, which was also absent in the massless theory, has now a non-vanishing value but for finite values of the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Using Kubo formulae we have evaluated the chiral magnetic and chiral vortical conductivities and studied their behaviour with the variation of the mass of the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Our findings for the chiral vortical con- ductivity, σV , and the chiral magnetic/vortical conductivity of energy current, σε B = σε V , are completely new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In addition to this, we have found that these anomalous transport coefficients depend linearly both on the pure Chern-Simon coupling, κ, and on the mixed gauge- gravity Chern-Simon coupling, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One of the results which we would like to highlight is the contribution to σV induced by λ in the massive theory, which was not present in the massless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 0 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='00361v1 [hep-th] 1 Jan 2023 1 Introduction The AdS/CFT correspondence [1, 2] has been one of the most prominent theoretical handles for studying systems which were very hard to tackle pre- viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' It states that, in the low energy limit, the large-Nc, N = 4 super Yang-Mills field theory in four-dimensional space is equivalent to the type IIB string theory in AdS5 × S5 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' It has been widely applied for the study of strongly coupled systems such as condensed matter systems, QCD and hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Our current objective is to study the hydrodynamical approach using this correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Quantum chiral anomalies are very fascinating properties which arise in the context of relativistic field theories of chiral fermions beyond perturba- tion theory [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Chiral anomalies have played a very crucial role in the formulation of relativistic hydrodynamics [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Anomaly-induced transport mechanisms have appeared on many occasions since the 80’s [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The ax- ial current was the main topic in [8], and AdS/CFT correspondence was first used to anomalous hydrodynamics in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Recently a lot of attention is gained by the effect of quantum anomalies on the hydrodynamics of otherwise conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The chiral magnetic effect [10] and the chiral vortical ef- fect [11–14] are two of such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the former, the axial anomaly induces a current parallel to the external magnetic field, while in the latter a current is generated due to the presence of a vortex in the charged relativistic fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' It has been argued that these and other anomaly-induced effects may be pro- duced in non-central heavy ion collisions at RHIC and LHC [15], inducing in particular an event-by-event parity violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These effects can also lead to anomalous transport properties in some condensed matter systems, such as the Weyl semi-metals [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the past few years, these anomalous effects has been implemented in holography giving a lot of insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One of such works is [18], where they considered a holographic model with a pure Chern-Simon term, and they computed the chiral magnetic conductivity which exactly matches with the results of the weakly coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This is due to the fact that the anomalous conductivities have non-renormalization properties so that they are independent of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Later on, this model was extended to incorporate the effect of the energy-momentum tensor related to the energy current as well, and the mixed gauge-gravitational Chern-Simon term was added in the gravitational action [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In these references the gauge fields were considered to be massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In a similar line of work, the authors of [22] have studied the depen- dence of the anomalous transport properties with the mass of the gauge field which is introduced via the Stückelberg mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In their case, they have 1 considered the probe limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As a consequence the sectors comprising of the correlators related to the energy-momentum tensor were not accessible, in particular: i) the chiral vortical conductivity, ii) the chiral vortical conduc- tivity of energy current, and iii) the chiral magnetic conductivity of energy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In a sense, this model only comprises a pure gauge Chern-Simon term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Our goal in the present work is to access those sectors and to study the chiral vortical effects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' To this end, we have considered the full backreaction of the gauge field onto the metric tensor, and included in the action of the model a mixed gauge-gravitational Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The paper has been organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In Section 2, we will discuss the model under consideration and get the full backreacted numerical solution for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Next, we will discuss in Section 3 the Kubo formulae and their relation with the retarded Green’s functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' the correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Using the AdS/CFT dictionary we will define these correlators in terms of the boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In Section 4 we will start presenting our results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' first, we will compare the results with the known results for the massless case [19], and after that, we will present our main results regarding the behaviour of the two-point correlators including the mass term for the gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We will discuss in the same section the effect of the mixed gauge-gravitational Chern- Simons term in these correlators, and finally we will show how the gauge boson mass affects the anomalous conductivities, namely the chiral vortical conductivity, σV , the chiral vortical conductivity of energy current, σε V , the chiral magnetic conductivity, σB, and the chiral magnetic conductivity for energy current, σε B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, we end with a discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2 Holographic massive U(1) gauge theory We consider a holographic model with a massive U(1) gauge boson that includes both a pure gauge and a mixed gauge-gravitational Chern-Simon term in the action [19,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The action of the model is S = 1 16πG � d5x√−g � R + 2Λ − 1 4FMNF MN −m2 2 (AM − ∂Mθ)(AM − ∂Mθ) +ϵMNPQR(AM − ∂Nθ) �κ 3FNPFQR + λRA BNPRB AQR � � +SGH + SCSK , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) 2 where SGH = 1 8πG � ∂ d4x √ hK , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2) SCSK = − 1 2πG � ∂ d4x √ hKλnMϵMNPQRANKPLDQKL R , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3) are the Gibbons-Hawking boundary term, and a boundary term induced by the mixed gauge-gravitational anomaly, respectively, which have been well discussed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' θ is a field which ensures gauge invariance (up to gauge anomalies), and thus the mass term enters in a consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As it men- tioned in [23–25], the Stückelberg term arises as the holographic realization of dynamical anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' A comparison of the consistent form of the anomaly for chiral fermions [3] with the variation of the action under axial gauge transformations, allows to fix the anomaly coefficients to κ = κp ≡ − G 2π , λ = λp ≡ − G 48π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4) See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' [19] for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the following we will refer to the values of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4) as the physical values of the anomaly coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The bulk equations of motion for the action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) turn out to be GMN − ΛgMN = 1 2FMLFN L − 1 8gMNF 2 + m2 2 BMBN − m2 4 gMNBPBP +2λϵLPQR(M▽B � F PLRB N) QR� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5) ▽NF NM = −ϵMNPQR � κFNPFQR + λRA BNPRB AQR � + m2BM ,(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6) where we have defined a new field BM ≡ AM − ∂Mθ, so that in the following θ will not appear explicitly anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have used the notation X(MN) ≡ 1 2(XMN + XNM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The ansatz for the background metric is a black hole solution in Fefferman- Graham coordinates, which is given by [26,27] ds2 = −ℓ2 ρ gττ(ρ)dτ 2 + ℓ2 ρ gxx(ρ)d⃗x2 + ℓ2 4ρ2dρ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7) where the boundary lies at ρ = 0 and the horizon at ρ = ρh, while ℓ is the radius of AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The horizon ρh is chosen in such a way that gττ(ρh) = 0, and the temperature of the black hole turns out to be T = 1 2π � 2ρhg′′ ττ(ρh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8) 3 The asymptotic expansion (ρ → 0) of the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6) shows that the gauge field near the boundary behaves as BM(ρ) = a0ρ− ∆ 2 + a1ρ ∆ 2 +1 + · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9) where m2ℓ2 = ∆(∆ + 2), with ∆ the anomalous dimension of the dual current [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The first(second) term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9) corresponds to a non- normalizable(normalizable) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The scaling dimension of the normaliz- able mode is (3 + ∆), and this puts an upper bound on the value ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' For ∆ > 1 the dual operators become irrelevant (in the IR), and so we will be working in the range of values of ∆ below this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 Numerical solution for the background In order to account for the chiral vortical effects within the present model, we will be considering the full backreaction of the gauge field onto the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6), the equations of motion for the background metric and gauge field turn out to be g′′ xx(ρ) − g′ xx(ρ) ρ + 1 6ℓ2ρ gxx(ρ) gττ(ρ) �m2ℓ2 4 Bt(ρ)2 + ρ2B′ t(ρ)2 � = 0 ,(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10) g′ ττ(ρ) � 1 − ρg′ xx(ρ) gxx(ρ) � + gττ(ρ)g′ xx(ρ) gxx(ρ) � 3 − ρg′ xx(ρ) gxx(ρ) � −1 3 ρ2 ℓ2 B′ t(ρ)2 + 1 12m2Bt(ρ)2 = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='11) B′′ t (ρ) + 1 2 � 3g′ xx(ρ) gxx(ρ) − g′ ττ(ρ) gττ(ρ) � B′ t(ρ) − ℓ2m2 4ρ2 Bt(ρ) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='12) where the gauge field has been chosen in the following way BMdxM = Bt(ρ)dt , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='13) so that Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We will solve numerically the above coupled differential equations with the following boundary conditions Bt(ρh) = 0 , lim ρ→0 � ρ∆/2Bt(ρ) � = µ5 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='14) with µ5 being the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As it is discussed in [22], in the presence of a finite gauge boson mass µ5 does not correspond to a thermodynamic parameter, but it is instead a coupling in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As a result, different values of chemical potential correspond to different theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' For completeness, we will 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 ρ gxx gττ ρΔ/2·Bt Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1: (color) Dependence of the background metric and gauge field with ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We display the results for gxx(ρ) (blue), gττ(ρ) (orange), and ρ∆/2Bt(ρ) (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have chosen ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' provide the analytical solution of the background equations of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='12) for vanishing µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These are gττ(ρ) = 1 ρ2 h (ρ2 h − ρ2)2 ρ2 h + ρ2 , gxx(ρ) = 1 + ρ2 ρ2 h , Bt(ρ) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15) while the temperature turns out to be T = 1 π � 2 ρh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' For the metric tensor, we demand that the solution is regular at the horizon, while at the boundary it reaches some constant value which we can always scale to set it to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Hereafter we will set the values of ℓ = 1 and ρh = 1 for our numerical calculations, which one can fix as such by using the scaling symmetry of the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This will set the units of all the quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' µ5, Q, M, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 the numerical solution of all the background fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' gxx(ρ), gττ(ρ) and Bt(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One may note from this figure that limρ→0 � ρ∆/2Bt(ρ) � = µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 3 Kubo formulae and correlators In this section, we will discuss the Kubo formulae needed to compute the anomalous transport coefficients in our model, and set up the equations to evaluate these transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The Kubo formulae for the anomalous conductivities have been well studied [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The authors of this reference have shown that the chiral vortical conductivity for charge and energy transport 5 can be obtained from the following two-point functions σV = lim kc→0 i 2kc � a,b ϵabc⟨JaT 0b⟩|w=0 , σε V = lim kc→0 i 2kc � a,b ϵabc⟨T 0aT 0b⟩|w=0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) where σV is the chiral vortical conductivity and σε V the chiral vortical con- ductivity of energy current, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The chiral magnetic conductivities for charge, σB, and energy, σε B, current are given by σB = lim kc→0 i 2kc � a,b ϵabc⟨JaJb⟩|w=0 , σε B = lim kc→0 i 2kc � a,b ϵabc⟨T 0aJb⟩|w=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2) To compute these correlators one can use the AdS/CFT dictionary [19, 29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Keeping this in mind, we proceed with the perturbation of the fields, where the background is set by the numerical solution as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We will study the linear response of the fluctuation, so that we split the metric and gauge field into a background and a linear perturbation part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' gMN = g(0) MN + ϵhMN, BM = B(0) M + ϵbM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3) Then, we will follow the general procedure of Fourier mode decomposition [28] hMN(ρ, xµ) = � ddk (2π)dhMN(ρ)e−iωt+i⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='⃗x , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4) bM(ρ, xµ) = � ddk (2π)dbM(ρ)e−iωt+i⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='⃗x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5) Without the loss of generality, one can consider perturbations of frequency ω and momentum k in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In order to study the anomalous effect we will switch on the fluctuations Bi, hi t and hi z, where i = x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Following this, we will substitute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3) in the equations of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6), and consider the resulting expressions at order O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Since we are interested in computing correlators at zero frequency, we can set the frequency-dependent parts as zero in the equations, and solve the system up to first order in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In this limit, the fields hi z decouple from the system and take a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, we can write the system of 6 differential equations for the shear sector as b′′ i (ρ) + 1 2 �g′ xx(ρ) gxx(ρ) + g′ ττ(ρ) gττ(ρ) � b′ i(ρ) − ∆(∆ + 2) 4ρ2 bi(ρ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6) + � 4iκkϵijbj(ρ) � gxx(ρ)gττ(ρ) + gxx(ρ)hi′ t(ρ) gττ(ρ) � B′ t(ρ) + iλkϵijhj′ t(ρ)Ω(ρ) = 0 , hi′′ t (ρ) − � g′ ττ(ρ) 2gττ(ρ) − 5g′ xx(ρ) 2gxx(ρ) + 1 ρ � hi′ t(ρ) + ρB′ t(ρ) gxx(ρ) b′ i(ρ) +∆(∆ + 2)Bt(ρ) 4ρgxx(ρ) bi(ρ) + iλkϵijΦj(ρ) = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7) where i, j = x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The explicit expressions of the functions Ω(ρ) and Φj(ρ) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Asymptotic analysis of the fluctuations near the boundary (ρ → 0) up to the first subleading order shows bi(ρ) = b(0) i ρ− ∆ 2 + b(1) i ρ ∆ 2 +1 + · · · , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8) hi t(ρ) = hi t (0) + hi t (1)ρ2 + · · · , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9) where the leading order terms b(0) i and hi t (0) are the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' From the holo- graphic description of the correlation functions, one can evaluate the one- point functions as ⟨Ja⟩ = δSren δb(0) a = − 2 16πG(∆ + 1)b(1) a , (a = x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10) ⟨T0a⟩ = δSren δha t (0) = 1 16πG � 2ha t (0) + ha t (1)� , (a = x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='11) where Sren = S + Sct is the renormalized action, with S the action given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) and Sct the counterterm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The procedure to evaluate this coun- terterm is given in [19] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We find that the counterterm needed to renormalize this theory is the same as the one given in [22], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' the mixed gauge-gravitational Chern-Simons term does not introduce new divergences, and so the renormalization is not modified by it (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' [19] for a discus- sion in the massless case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In this regards, we are not writing the counterterm Sct explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' ⟨Ja⟩ and ⟨T0a⟩ correspond to current and energy-momentum tensor one-point functions, respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Similarly, the two-point functions can be obtained by taking the variation of one-point function with respect 1Ji and T0i are related with the fluctuations bi and hi t, respectively, with i = x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 7 to the corresponding source term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' ⟨JaJb⟩ = δ⟨Ja⟩ δb(0) b , (a, b = x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='12) ⟨JaT0b⟩ = δ⟨Ja⟩ δhb t(0) , (a, b = x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='13) ⟨T0aJb⟩ = δ⟨T0a⟩ δb(0) b , (a, b = x, y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='14) ⟨T0aT0b⟩ = δ⟨T0a⟩ δhb t(0) , (a, b = x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15) From the above expressions, it is clear that it is required the leading and subleading parts of the asymptotic expansion of the fluctuations to evaluate the two-point functions we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' To do so we have solved numer- ically the coupled differential equations of the fluctuations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7) and imposed suitable boundary conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' i) regularity at the horizon, and ii) sourceless condition at the asymptotic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4 Results In this section, we will start presenting our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Firstly, we will start with the massless case (∆ = 0) and compare the results with the previous work done in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the second part of this section, we will consider the massive case ∆ ̸= 0, and study the dependence of the two-point functions with ∆ for different values of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In both cases, we will set G = 1/(16π) so that the physical values of the anomalous couplings are κ = −1/(32π2) and λ = −1/(768π2), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Later on, we will study the dependence of the two-point functions with the parameters κ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This is done to show that the parametric dependence of the correlators is linear in these parameters, but values of κ and λ different from κ/λ = 24 are non-physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In addition to this, to make a direct comparison with the previous work in [22], all the anomalous correlators have been displayed normalized by |κ|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 μ5 Im \uf000 < Jx Jy >\uf006 k \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 μ5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 μ5 Im \uf000 < Jx T0 y >\uf006 k \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15 μ5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1: Upper panel: Plots of the correlators ⟨JxJy⟩ (left) and ⟨JyT0y⟩ (right) vs µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Lower panel: Plots of the correlators ⟨JxT0y⟩(left) and ⟨T0yT0y⟩ (right) vs µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These plots are obtained in the massless case (∆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 Massless case In the absence of mass, the correlators have been evaluated in [19,22], leading to ⟨JxT0x⟩ = ⟨JyT0y⟩ = √ 3Q 4πGℓ3 , ⟨JxJy⟩ = −⟨JyJx⟩ = κi √ 3kQ 2πGr2 h − κ ikα 6πG = −ik(3µ5 − α) 12π2 , ⟨JxT0y⟩ = −⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ = κ 3ikQ2 4πGr4 h + λ2ikπT 2 G , = −ik � µ2 5 8π2 + T 2 24 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) ⟨T0xT0x⟩ = ⟨T0yT0y⟩ = M 16πGℓ3 , ⟨T0xT0y⟩ = −⟨T0yT0x⟩ = κi √ 3kQ3 2πGr6 h + λ4πi √ 3kQT 2 Gr2 h = −ik � µ3 5 12π2 + µ5T 2 12 � , 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0 1 2 3 4 μ5 Im \uf000 < T0 x T0 y >\uf006 k \uf603κ\uf604 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2: Plot of the correlator ⟨T0xT0y⟩ vs µ5 in the massless case (∆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' with M = r4 h ℓ2 + Q2 r2 h and Q = µ5r2 h √ 3 the mass and charge of the black hole solution computed in Poincaré coordinates, with blackening factor f(r) = 1 − Mℓ2 r4 + Q2ℓ2 r6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2) The Hawking temperature is given in terms of these black hole parameters as T = r2 h 4πℓ2f ′(rh) = (2r2 hM − 3Q2) 2πr5 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3) The parameter α in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) corresponds to the asymptotic value of the gauge field At for ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In our case, we are assuming α = µ5 for ∆ = 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The other correlators are vanishing in the massless case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' ⟨JxJx⟩ = ⟨JyJy⟩ = 0 , ⟨T0xJx⟩ = ⟨T0yJy⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4) While the correlators with the same indices are not induced by quantum anomalies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' they are non-anomalous) and they become real, the correla- tors with different indexes are anomalous and they become imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We will be comparing the numerical results with the analytical expressions given in the above equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We plot in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 five inde- pendent non-vanishing correlators, while the other correlators are related to them through the expressions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In these and subsequent plots, it is understood that it has been taken the limit k → 0 with k ≡ kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In these figures the dots stand for the numerical results, and the solid lines correspond to the analytic results of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One may observe that the numerical results are in good agreement with the analytic expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0 1 2 3 4 5 6 7 Δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 Δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0 2 4 6 8 10 Δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='030 Δ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3: (color) Plots for non-anomalous correlators vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Upper panel: Plot of the correlators ⟨JxJx⟩ (left) and ⟨JxT0x⟩ (right) vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Lower panel: Plot of the correlators ⟨T0xJx⟩ (left) and ⟨T0xT0x⟩ (right) vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered in all the panels µ5 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2} (blue, orange and green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 Massive case We will split our discussion into anomalous and non-anomalous correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have found that the above mentioned relations between different corre- lators still hold in the massive case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' ⟨JxT0x⟩ = ⟨JyT0y⟩ , ⟨JxJy⟩ = −⟨JyJx⟩ , ⟨JxT0y⟩ = −⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ , ⟨T0xT0x⟩ = ⟨T0yT0y⟩ , ⟨T0xT0y⟩ = −⟨T0yT0x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5) In addition to this, there are two more independent correlators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e ⟨T0xJx⟩ = ⟨T0yJy⟩ and ⟨JxJx⟩ = ⟨JyJy⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In this regard, we will be plotting only seven independent correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0 1 2 3 4 5 Δ Im \uf000 < Jy Jx >\uf006 k \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 Δ Im \uf000 < T0 y T0 x >\uf006 k \uf603κ\uf604 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4: (color) Plot of the correlators ⟨JyJx⟩ (left) and ⟨T0yT0x⟩ (right) vs ∆ with µ5 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2} (blue, orange and green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 Non-anomalous correlators While the correlator ⟨JxJx⟩ is vanishing for ∆ = 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1), we can see from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 (upper-left panel) that this correlator starts picking up some finite value in the massive case (∆ ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' With the increase of ∆ the absolute value of this correlator increases quite sharply, and gets even shaper with the increase in µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' an increasing value of the (absolute value of the) correlator for increasing ∆ and for finite µ5, is a general feature for all the non-anomalous coefficients as we will discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 (upper-right panel) that for µ5 = 0 the correlator ⟨JxT0x⟩ is zero for all values of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As the value of µ5 increases, ⟨JxT0x⟩ becomes finite and its value increases with ∆ in a somewhat linear fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The slope of ⟨JxT0x⟩ vs ∆ also increases with the increase of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 (lower panel-left) we can see that even though the correlator ⟨T0xJx⟩ is vanishing for ∆ = 0, for finite values of ∆ and µ5 this corre- lator is non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' More in details, for a given finite value of µ5, the absolute value |⟨T0xJx⟩| increases quite sharply with ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Notice that ⟨T0xJx⟩ was completely absent in the previous work [19], but we find now that it is non-vanishing at finite µ5 in the massive theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 (lower-right panel) that ⟨T0xT0x⟩ is inde- pendent of ∆ for µ5 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' it has a constant value corresponding to the pressure term, a feature that has been well discussed in [18–20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' At fi- nite chemical potential, this correlator increases with ∆, a behavior which is sharper for larger values of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 Anomalous correlators We display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 (left) the behaviour of ⟨JyJx⟩ vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One can see that the absolute value of this correlator increases with ∆, and the change is quite 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 Δ Im \uf000 < Jx T0 y >\uf006 k \uf603κ\uf604 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5: (color) Plot of the correlator ⟨JxT0y⟩ vs ∆ with µ5 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3} (blue, orange and green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0001 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 2 1 0 1 2 λ \uf0c7κp\uf0ca Im \uf000 < T0 y T0 x >\uf006 k \uf0c7κp\uf0ca 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 15 10 5 0 5 10 15 λ \uf0c7κp\uf0ca Im \uf000 < Jy T0 x >\uf006 k \uf0c7κp\uf0ca Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs λ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The inset figures correspond to zooms of the main figures in the small λ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered in both panels, µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and κp = −1/(32π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' It is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 (right) the correlator ⟨T0yT0x⟩ vs ∆, and unlike the other correlator, its absolute value decreases with the increase of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 we have plotted ⟨JxT0y⟩ vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We find that the absolute value of this correlator increases with the increase in ∆ and µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have taken a different value of µ5 as compared to the other correlators, because for those values of µ5 the correlator did not have any substantial changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The new values of µ5 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3} are chosen to make these changes distinct in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We can see from the figure that even in the absence of µ5 this correlator is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This can be traced back to the temperature term, as the temperature does not vanish for µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, one may notice that in all the cases the values of two point correlators tend toward the analytic values as given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) when considering the limit ∆ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This is also shown in the figures for the massless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='682 λ \uf0c7κp\uf0ca Im \uf000 < Jy Jx >\uf006 k \uf0c7κp\uf0ca 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 κ \uf0c7κp\uf0ca Im \uf000 < Jy Jx >\uf006 k \uf0c7κp\uf0ca Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7: Left: Plot of the correlator ⟨JyJx⟩ vs λ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Right: Plot of the correlator ⟨JyJx⟩ vs κ/|κp| for λ = −1/(768π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered in both panels µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, while κp = −1/(32π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='495 κ \uf0c7κp\uf0ca Im \uf000 < T0 y T0 x >\uf006 k \uf0c7κp\uf0ca 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='90 κ \uf0c7κp\uf0ca Im \uf000 < Jy T0 x >\uf006 k \uf0c7κp\uf0ca Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs κ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In both cases, µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, |κp| = 1/(32π2) and λ = −1/(768π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' λ and κ dependence: To study the dependence of the two-point functions with the parameter λ, we will consider the case where we fix the values as µ5 = ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and κ = −1/(32π2), and vary λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Here we will only present the correlators that have a dependence on λ, while the λ independent correlators are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7 (left) the dependence of the anomalous correlators with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One can see that the behaviour is linear with λ in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The inset figures are given to show that the corresponding correlators do not vanish at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This is in fact true, as the non-vanishing values arise due to the κ coupling, which leads to ⟨T0xT0y⟩ ∼ µ3 5κ and ⟨J0yT0x⟩ ∼ µ2 5κ at λ = 0, with some contribution from ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the case of ⟨JyJx⟩ the λ dependence only arises in the massive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Setting the values of ∆ = µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, λ = −1/(768π2) and varying κ, we see a similar kind of linear behaviour with κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The effect of κ is only seen in ⟨T0xT0y⟩, ⟨J0yT0x⟩ and ⟨JyJx⟩ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7 (right) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The non-anomalous correlators are independent of κ, and they are displayed in 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These correlators are in fact independent of both the parameters κ and λ, and hence they are non-anomalous in nature even in the massive theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This means that they do not contribute to anomalous trans- port, unlike the correlators studied above which are associated to anoma- lous conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This can be seen in the Kubo formulae for anomalous conductivities, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2), as these formulae involve Levi-Civita (ϵijz) symbols which runs over i = j = {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Hence, the correlators with i = j = x and i = j = y do not lead to anomalous transport effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Anomalous conductivities: Finally, as a summary of the previous nu- merical results, we now present the anomalous conductivities which are com- puted with the Kubo formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' σV = − lim k→0 1 kIm⟨JxT0y⟩ , σε V = − lim k→0 1 kIm⟨T0xT0y⟩ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6) σB = − lim k→0 1 kIm⟨JxJy⟩ , σε B = − lim k→0 1 kIm⟨T0xJy⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7) The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We can see from this figure that the chiral vortical conductivity and the chiral magnetic conductivity for energy current are the same either at zero or finite mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' σV = σε B, and these quantities increase with ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We also see in this figure that the chiral vortical conductivity of energy current, σε V , decreases with ∆ but the rate decreases rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In the case of the chiral magnetic conductivity, σB, it increases with ∆ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Regarding the other dependences of the anomalous conductivities, for instance the dependence in the parameters κ and λ, it would be sufficient to study them from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7, as the two-point functions and the anomalous conductivities are related through Kubo formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We conclude that for a given value of µ5 and ∆, the anomalous transport coefficients: σV , σε B, σB and σε B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' change linearly with the pure (κ) and mixed (λ) gauge- gravitational Chern-Simon couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' At the limit of vanishing mass, our results lead to σB µ5|κ| ≃ 16/3, which exactly coincides with the results in [18, 19,22] where α has been set to µ5 in both references 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In order to reproduce the results of [18] where they have set α = 0, our κ needs to be rescaled by a factor 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, let us emphasize that all the correlators involving the energy-momentum tensor are completely new results at finite mass (∆ ̸= 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' σV , σε V and σε B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2α corresponds to the asymptotic value of the gauge field At for ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In our case, we assume α = µ5 for ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 Δ σV \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='80 Δ σε V \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 Δ σB \uf603κ\uf604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 Δ σε B \uf603κ\uf604 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9: Upper panel: Plot of σV (left) and σε V (right) vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Lower panel: Plot of σB (left) and σε B (right) vs ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15 in all the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 5 Discussion We have studied the anomalous and non-anomalous conductivities in the holographic Stückelberg model including both pure gauge and mixed gauge- gravitational anomaly terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' To access the sectors concerning the energy- momentum tensor we have to consider the full backreaction of the massive gauge field onto the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have evaluated the numerical back- ground solution and on this background, we have considered the fluctuations of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' From these fluctuations, we have calculated the different corre- lators and studied their behaviors with the relevant parameters of the model (µ5, ∆, κ and λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have found that the correlators in the massless case match with previ- ous results in the literature [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Later on, we have studied the dependence of these correlators with the mass of the gauge field, m2 = ∆(∆+2), and found that all the correlators explicitly depend on the mass for a given non-zero value of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One of the results that it is important to emphasize here is that the non-anomalous correlators such as ⟨JxJx⟩ and ⟨T0xJx⟩ are non-vanishing in the massive theory for finite values of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Moreover ⟨JxJx⟩ is non-zero in this theory even for µ5 = 0, while ⟨T0xJx⟩ is vanishing for µ5 = 0 indepen- dently of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These correlators are vanishing in the massless theory, independently of µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The mass of the gauge field highly enhances the abso- 16 lute value of the correlators, and this gets translated into an enhancement of the anomalous conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The behaviours of the correlators on the pure gauge and mixed gauge-gravitational Chern-Simon couplings, κ and λ, were also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We found that the correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and ⟨T0xT0x⟩ are independent of κ and λ, and hence they are non-anomalous in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' They do not contribute to the anomalous conductivities, as it can be seen from the Kubo formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Finally, we have computed the anomalous conductivities and studied their dependence with the mass of the gauge field (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have found that the chiral vortical conductivity, σV , and the chiral magnetic conductivity for energy current, σε B, are equal and increase with ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One interesting result is that there are contributions to σB coming from λ in the massive theory, which was completely absent in the massless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The conductivities σB, σV and σε B increase with ∆, while the chiral vortical conductivity of energy current, σε V , decreases with ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have explicitly checked that all our numerical results for the conductivities at finite mass tend to the known results at zero mass in the limit ∆ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' For instance, it is known that at zero mass, the chiral magnetic conductivity is σB = − 16 3 κµ5 when α = µ5, which implies that the ratio − σB κµ5 = 16/3, independently of κ and µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' As one can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='9 (left), our numerics produces in this limit σB/|κ| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 for µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15, in agreement with the expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have also checked the ratio − σB κµ5 = 16/3 for other values of κ and µ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' This work can be extended in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' One possible extension could be to consider the U(1)V × U(1)A gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' There are some studies in holography with this gauge group, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' [22,31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' However, in these works: i) either the probe limit has been considered so that the chiral vortical effect and the transport conductivities in the energy-momentum tensor are not accessible, or ii) they correspond to studies for massless gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' In particular, it would be interesting to study the interplay between the anomalous and non-anomalous currents in the set-up of the full backreacted background of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' [32], both for massless and massive gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We will explore these and other issues in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Acknowledgments We would like to thank Karl Landsteiner for enlightening discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' is grateful to Manuel Valle for collaboration in the early stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' thanks the Instituto de Física Teórica UAM/CSIC, Spain, for its hos- pitality and partial support during his research visits in the final stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The works of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' are supported by the project 17 PID2020-114767GB-I00 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='13039/501100011033, by the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020 Operational Program under Grant A-FQM-178-UGR18, and by the Ramón y Cajal Program of the Spanish MCIN under Grant RYC-2016- 20678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The work of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' is also supported by Junta de Andalucía under Grant FQM-225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Appendix A Explicit expressions for the func- tions Ω(ρ) and Φj(ρ) These functions have been introduced in the equations of motion of the fluc- tuations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Their explicit expressions are given by Ω(ρ) = 4 � gττ(ρ) (g′ xx(ρ) + 2ρg′′ xx(ρ)) + ρg′ xx(ρ)g′ ττ(ρ) � � gxx(ρ)gττ(ρ)3/2 − 8ρ g′ xx(ρ)2 gxx(ρ)3/2 � gττ(ρ) + � gxx(ρ) gττ(ρ)5/2 � 4ρg′ ττ(ρ)2 − 4gττ(ρ) (g′ ττ(ρ) + 2ρg′′ ττ(ρ)) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='Φj(ρ) = b′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='j(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− 8ρ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) + 2ρg′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='+ ρg′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)5/2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) + 2ρg′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− ρg′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)3/2gττ(ρ)3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='+ bj(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8ρ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)9/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− 8ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) + 2ρg′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ρg′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) − ρg′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ)g′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) + 3gττ(ρ)g′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='xx(ρ) + 2ρgxx(3)(ρ)gττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)5/2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='− 2gττ(ρ)g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) + 2ρg′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='+ 2ρg′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ)3 + gττ(ρ)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3g′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='ττ(ρ) + 2ρgττ (3)(ρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='gxx(ρ)3/2gττ(ρ)5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2) 18 + hj′ t(ρ) � B′ t(ρ) � � 16ρ2g′ xx(ρ) gxx(ρ)3/2� gττ(ρ) + 8ρ � gττ(ρ) − ρg′ ττ(ρ) � � gxx(ρ)gττ(ρ)3/2 � � + 8ρ2B′′ t (ρ) � gxx(ρ) � gττ(ρ) � + hj′′ t (ρ) 8ρ2B′ t(ρ) � gxx(ρ) � gττ(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Appendix B Some additional results for the non- anomalous correlators We show in this Appendix the numerical results for the non-anomalous cor- relators as a function of the anomalous parameters κ and λ, in the massive case (∆ ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and ⟨T0xT0x⟩, are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' These correlators turn out to be constant in both κ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' The lack of dependence in these parameters implies that they lead to non-anomalous transport effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 λ \uf0c7κp\uf0ca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 λ \uf0c7κp\uf0ca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15 λ \uf0c7κp\uf0ca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 λ \uf0c7κp\uf0ca Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩ (right) vs λ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and ⟨T0xT0x⟩ (right) vs λ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 and κp = −1/(32π2) in all the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 κ \uf0c7κp\uf0ca 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='8 κ \uf0c7κp\uf0ca 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='15 κ \uf0c7κp\uf0ca 2 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='0 κ \uf0c7κp\uf0ca Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='2: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩ (right) vs κ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and ⟨T0xT0x⟩ (right) vs κ/|κp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' We have considered µ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1, |κp| = 1/(32π2) and λ = −1/(768π2) in all the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 21 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Maldacena, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 38, 1113 (1999) [Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' 2, 231 (1998)] doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content='1023/A:1026654312961 [hep- th/9711200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQfgfjF/content/2301.00361v1.pdf'} +page_content=' [2] E.' metadata={'source': 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Most of these practices include the +images that medical assistance uses to identify different pathologies of the human body. One of them is X-ray images +which cover much of our work in this paper. Chest x-rays have played an important role in Covid 19 identification and +diagnosis. The Covid 19 virus has been declared a global pandemic since 2020 after the first case found in Wuhan +China in December 2019. Our goal in this project is to be able to classify different chest X-ray images containing Covid +19, viral pneumonia, lung opacity and normal images. We used CNN architecture and different pre-trained models. The +best result is obtained by the use of the ResNet 18 architecture with 94.1% accuracy. We also note that The GPU +execution time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge +much faster than the CNN. The time saving is very considerable. +With these results not only will solve the diagnosis time for patients, but will provide an interesting tool for practitioners, +thus helping them in times of strong pandemic in particular. + +Keywords: Deep learning, Image classification, CNN, Covid-19, Chest Xray, Pre-trained models. + + + + +1. Introduction + +Computerized Tomography (CT) and X-ray scans are frequently used for chest imaging. An X-ray is a scan of +the body that looks for pneumonia, tumors, fractures, and lung infections. An upgraded X-ray machine called a +CT scan can produce sharper images of bones, tissue, and organs. Compared to CT, the X-ray approach is +simpler, faster, and more affordable, but it is also more dangerous. Doctors can visually diagnose viral +bacterial infections, viruses like covid 19 [1], and other infections by examining chest X-ray images. The +technique of visual diagnosis is typically unappealing, time-consuming, and inaccurate, because it can result +in low accuracy and requires specialized human resources. +Coronavirus disease 2019 (COVID-19) is an infectious disease brought on by the coronavirus strain known as +severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) [2]. It is a lung infection that is respiratory in +nature. The root of the coronavirus word is Greek (κορώνη) which means "crown or halo." It relates to the +virus's appearance under an electron microscope, which resembles a royal crown. Because of this, +coronavirus is also known as the crowned virus. The purpose of this paper is therefore to provide a decision +making tool that will lighten the burden on medical staff, especially during pandemic peaks. + +2. Related work + +Since Corona was announced as a pandemic, different projects were carried out since 2020 to 2022 that it +became among an interesting subject to learn from, some of the works related to image classification for +different reasons are discussed in this paragraph. +The COVID-CT dataset of 2560 images was the database used in [3], 2214 of which were used for training +and the remaining 246 for testing. By employing WOA to optimize the network's hyperparameters, the model +used to train ResNet-50 became the WOANet model. This last experiment looked at the accuracy of the +classification using the suggested method on 246 CT scans, and found that 98.37% of them were categorized +as COVID-19, while 99.18% were identified as non-COVID-19. The radiologists will be greatly assisted by this +proposed WOANet in reducing the burden on the healthcare system and hospitals. +In this study [4], patients' X-ray images are used to classify patients using CNN deep learning. One of the +most powerful algorithms with generative and deterministic capabilities is the capsule network (CapsNet). +However, compared to the basic CNN structures, this network has been relatively more responsive to images. +The dataset utilized was the NIH complete Chest X-rays [5] collection. VDSNet has a validation accuracy +value of 73%, which is higher than the sample dataset's score of 70.8%. +Using a dataset of 6432 images, the DLH COVID [6] model is distinct, trustworthy, and independently created +without any input from the transfer learning approach. The experimental findings from the prospective +validation phase suggest that the DLH MODEL outperformed the majority of the pre-trained models since it +distinguished COVID-19, pneumonia, and healthy/unhealthy patients from the image dataset with a promising +accuracy of 96%. + +3. Dataset + +As seen in figure 1, a database of chest X-ray images for COVID-19 positive cases as well as images of +normal and viral pneumonia was created in collaboration with medical professionals by a group of researchers +from Qatar University, Doha, Qatar, and the University of Dhaka, Bangladesh, as well as their collaborators +from Pakistan and Malaysia. This dataset contains 3616 COVID-19 positive cases, 10,192 Normal, 6012 Lung +Opacity (Non-COVID lung infection), and 1345 Viral Pneumonia images. The COVID-19 x-ray image database +was created using different sources [7, 8, 9]. + +Fig. 1 CXR scans with four categories of pathology. + +COVID +Lung_Opacity +Normal +ViralPneumonia + + + +For training of datasets, we employed Matlab 2021 installed on computer with 64-bit operating system, +windows 10 Pro, 24 GB of Random Access Memory (RAM), with an Intel(R) Xeon(R) CPU E5-2620 v3 @ +2.40GHz and Graphical Processing Unit (GPU). Eighty percent of the datasets are used for training and 20% +for testing (evaluating the model performance). + +4. Proposed Model + +In this study, we analyzed the different techniques for image classification of COVID-19 using X-Ray +radiographic images of the chest, then examined CNN’s architecture that is based on research on the visual +cortex of the cat by Hubel and Wisiel [10], and different pre-trained models: AlexNet, ResNet18 and +GoogleNet in order to see the variation of answers in our work. + + +Fig. 2 The proposed model with different algorithms. + +The use of deep learning methods not only allows us to process a very large number of images but at the +same time allows us to skip the feature extraction step, such a cumbersome step because it is done by hand +crucially. +Due to their capacity to extract features (see figure 2) and learn to distinguish between various classes, +convolutional neural networks (CNNs) are the top DL tool that are widely employed in several fields of the +healthcare system (i.e., positive and negative, infected). Transfer learning (TL) has made it simpler to quickly +and accurately retrain neural networks on chosen datasets. + +5. Experiments and Results +5.1 Experiment 1: Application of the CNN model + +The structure of our CNN includes a number of layers, as shown in table I. CNN receives a CXR image with a +size of 299 by 299 pixels as its input, and the rest of the architecture is mentioned in the table I. + + + + + + + + + + + + + + + +Model: CNN/AlexNet/GoogleNet/ResNet 18 +Fully +Convolution +Connected +Pooling +Output +Input +O +O +O +299*299*1 +4 classes +FeatureExtraction +Classification + + +Table 1 The architecture of the CNN model. +Name +Type +Description of output size +Input layer +Input data +299*299 +Conv 1 +Convolution +ReLU +32*32*8 +S1 +Max pooling +3,2 +Conv 2 +Convolution +ReLU +64*64*3 +S2 +Max pooling +3,2 +Conv 3 +Convolution +ReLU +128*128*5 +S3 +Max pooling +3,2 +Conv 4 +Convolution +ReLU +256*256*5 +S4 +Max pooling +3,2 +Conv 5 +Convolution +ReLU +512*512*5 +S5 +Max pooling +3,2 +Conv 6 +Convolution +ReLU +1024*1024*5 +S6 +Max pooling +3,2 +Fc +Fully connected +1 Fc (4) + +The trained parameters used in this model are in the options side where all the hyperparameters used were +defined including the number of epochs used (1 or 5), the mini batch (64), the learning rate is 0.001 and +frequency validation is 20. The given CNN was trained using different parameters to test the accuracy for this +model. We utilized the accuracy parameter to evaluate how well the trained models performed. The +percentage of correctly classified images over all the images is what is referred to as accuracy. The following +formula is utilized: +𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = +𝑇𝑃 + 𝑇𝑁 +𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 + +Table 2 The results of the CNN model. +The CNN model + +1 epoch +5 epochs +Accuracy +75.61% +89.13% +Time GPU +execution +146 min 58s +703 min 16s + +The first top accuracy after training the model using one epoch provided us with 75.61% accuracy for our 4 +classes classification. As training the model using only one epoch did not provide the best result, we had to +increase the number of epochs and see the performance of our model and the results for our model gave us +89.13%, which is a lot better compared to our first experiment with one epoch. + +5.2 Experiment 2: Application of the pretrained models + +AlexNet: With 5 convolutional layers and convolutional filter sizes of 3*3 and 2*2 for max pooling operation, +AlexNet is an 8-layer convolutional neural network [11]. Fully connected layers are the final three layers. The +AlexNet model's standard input size is 227*227*3. +GoogleNet (Inception v3): A convolutional neural network with 50 layers in depth is called GoogleNet [12]. +The program, titled "Going deeper with convolutions," was developed and taught by Google. Up to 1000 +objects can be classified using the pre-trained Inceptionv3 model with the ImageNet dataset [13] weights. This +network's image input size was 299x299 pixels. +ResNet18: A convolutional neural network with 18 layers in depth is called ResNet18. Deep Residual +Learning for Image Recognition, as it is known, was developed and trained by Microsoft in 2015 [14]. To +address the issue of vanishing gradient that may affect the weightage change in neural networks, ResNet +architectures introduced the use of residual layers and skip connections. This made training easier and +allowed neural networks to get much deeper with greater performance. The network was trained on colored +images with a resolution of 224x224 pixels. + + + + + +In addition to the accuracy parameters, we estimated the time GPU execution for each model. The results +obtained are shown in Table 3. +Table 3 The results of the pretrained models. + +Pretrained models + +AlexNet +GoogleNet +ResNet18 +Accuracy +89.93% +91.87% +94.1% +Time +GPU +execution +14 min +58s +41 min +34s +33 min +13s + +Confusion matrix is the common approach used for evaluation of model performance based on true positive +(TP), true negative (TN), false positive (FP), and false negative (FN). + +The figure 3 represents the confusion matrix of the Resnet 18 model which gave the best result in terms +of accuracy. + + +Fig. 3 The confusion matrix for the RestNet 18 model with the best result. +When the same dataset was used using the same hyper parameters the accuracy found was 89.93%, 91.87% +and 94.1 % for AlexNet, GoogleNet and ResNet18 respectively. Note that the pretrained models used only +one epoch. We see that the results can be improved by using pretrained architectures, attaining an accuracy +of 94.1%. The increased classification rate attained by Resnet 18 can be attributed to the network's use of +novel techniques to lessen over-fitting in its model. +The first method involved artificially enlarging the dataset with the aid of a label-preserving transformation. +This involved extracting random patches (224x224 for ResNet 18) and training the network on them while +varying the intensities of the RGB channels in the training images. The result was the generation of image +translations and horizontal reflections. The second strategy was "dropout," which involves removing neurons +that do not participate in the forward pass or the backward propagation. As a result, the model is forced to +learn more robust characteristics and decreases the complex co-adaptations of neurons. The GPU execution +time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge +much faster than the CNN. The time saving is very considerable. + + +707 +COVID +6 +6 +0 +97.9% +16.7% +0.1% +0.2% +0.0% +2.1% +Lungopacity +S +1060 +LL +1 +92.7% +0.1% +25.1% +1.8% +0.0% +7.3% +Output Class +Normal +8 +136 +1949 +5 +92.9% +0.2% +3.2% +46.1% +0.1% +7.1% +1 +0 +3 +263 +ViralPneumonia +98.5% +0.0% +%00 +0.1% +6.2% +1.5% +981% +88.2% +95.6% +97.8% +94.1% +1.9% +118% +4.4% +2.2% +5.9% +alPpr +Target Class + + +6. Conclusion +This work aimed at developing a convolutional neural network (CNN) model that will help classify COVID-19 +and non-COVID 19 disease such as viral pneumonia cases using chest X-ray images in the period caused by +the pandemic. The model used in this work was CNN as well as pre-trained models including AlexNet, +GoogleNet, and ResNet18. The CNN gave a result with 89.13% accuracy for classifying the four classes after +training 80% of the dataset and testing on 20%. This motivated us not only to keep changing settings, but also +to work on pretrained model. In the latter, the pre-trained models were used on the same dataset but with just +one epoch for each model. And the results were 89.93%, 91.87% and 94.1 % for AlexNet, GoogleNet and +ResNet18 respectively. + +7. References +[1] Kutlu, Yakup, and Yunus Camgözlü. "Detection of coronavirus disease (COVID-19) from X-ray images +using deep convolutional neural networks." Natural and Engineering Sciences 6, no. 1 (2021): 60-74. +[2] Alakus, T.B. and Turkoglu, I., 2020. Comparison of deep learning approaches to predict COVID-19 +infection. Chaos, Solitons & Fractals, 140, p.110120 +[3] Murugan, R., Goel, T., Mirjalili, S., & Chakrabartty, D. K. (2021). WOANet: Whale optimized deep neural +network for the classification of COVID-19 from radiography images. Biocybernetics and Biomedical +Engineering, 41(4), 1702-1718. +[4] Apostolopoulos, I.D., Mpesiana, T.A. Covid-19: automatic detection from X-ray images utilizing transfer +learning with convolutional neural networks. Phys Eng Sci Med 43, 635–640 (2020). +[5] Patel +P. +Chest +X-ray +(COVID-19 +& +Pneumonia). +Kaggle. +(2020); +https://www.kaggle.com/prashant268/chest-xray-covid19- pneumonia +[6] CDey, S., Bacellar, G. C., Chandrappa, M. B., & Kulkarni, R. (2021). COVID-19 Chest X-Ray Image +Classification +Using +Deep +Learning. +medRxiv +2021.07.15.21260605; +doi: +https://doi.org/10.1101/2021.07.15.21260605 +[7] https://bimcv.cipf.es/bimcv-projects/bimcv covid19/#1590858128006-9e640421-6711 +[8] https://github.com/ml-workgroup/covid-19-image-repository/tree/master/png +[9] https://sirm.org/category/senza-categoria/covid-19/ +[10] D. H. Hubel, T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex, J. Physiol, +vol. 195, pp. 215-243, 1968. +[11] Li, Shaojuan, Lizhi Wang, Jia Li, and Yuan Yao. "Image classification algorithm based on improved +AlexNet." In Journal of Physics: Conference Series, vol. 1813, no. 1, p. 012051. IOP Publishing, 2021. +[12] Hu, J., Shen, L. and Sun, G., 2018. Squeeze-and-excitation networks. In Proceedings of the IEEE +conference on computer vision and pattern recognition (pp. 7132-7141). +[13] Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. "Imagenet classification with deep convolutional +neural networks." Advances in neural information processing systems 25 (2012). +[14] Huang L, Ruan S, Denoeux T. Covid-19 classification with deep neural network and belief functions. +InThe Fifth International Conference on Biological Information and Biomedical Engineering 2021 Jul 20 +(pp. 1-4). + diff --git a/P9E0T4oBgHgl3EQfkAHC/content/tmp_files/load_file.txt b/P9E0T4oBgHgl3EQfkAHC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d87d5013ba6284df11a452ccab4c67bca7809ed --- /dev/null +++ b/P9E0T4oBgHgl3EQfkAHC/content/tmp_files/load_file.txt @@ -0,0 +1,250 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf,len=249 +page_content="Deep Learning For Classification Of Chest X-Ray Images (Covid 19) Benbakreti Samir1, Said Mwanahija1, Benbakreti Soumia2, Umut Özkaya3 1 Specialty Department, Ecole Nationale des Télécommunications et des Technologies de l'Information et de la Communication (ENSTTIC), Oran, Algeria." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 2 Laboratoire des Mathématiques, University of Djillali Liabes, Sidi Bel Abbes, Algeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 3Electrical ans Electronic Engineering, Konya Technical University, Turkey ABSTRACT In medical practice, the contribution of information technology can be considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Most of these practices include the images that medical assistance uses to identify different pathologies of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' One of them is X-ray images which cover much of our work in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Chest x-rays have played an important role in Covid 19 identification and diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The Covid 19 virus has been declared a global pandemic since 2020 after the first case found in Wuhan China in December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Our goal in this project is to be able to classify different chest X-ray images containing Covid 19, viral pneumonia, lung opacity and normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' We used CNN architecture and different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The best result is obtained by the use of the ResNet 18 architecture with 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' We also note that The GPU execution time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge much faster than the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The time saving is very considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' With these results not only will solve the diagnosis time for patients, but will provide an interesting tool for practitioners, thus helping them in times of strong pandemic in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Keywords: Deep learning, Image classification, CNN, Covid-19, Chest Xray, Pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Introduction Computerized Tomography (CT) and X-ray scans are frequently used for chest imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' An X-ray is a scan of the body that looks for pneumonia, tumors, fractures, and lung infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' An upgraded X-ray machine called a CT scan can produce sharper images of bones, tissue, and organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Compared to CT, the X-ray approach is simpler, faster, and more affordable, but it is also more dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Doctors can visually diagnose viral bacterial infections, viruses like covid 19 [1], and other infections by examining chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The technique of visual diagnosis is typically unappealing, time-consuming, and inaccurate, because it can result in low accuracy and requires specialized human resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Coronavirus disease 2019 (COVID-19) is an infectious disease brought on by the coronavirus strain known as severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' It is a lung infection that is respiratory in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The root of the coronavirus word is Greek (κορώνη) which means "crown or halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='" It relates to the virus\'s appearance under an electron microscope, which resembles a royal crown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Because of this, coronavirus is also known as the crowned virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The purpose of this paper is therefore to provide a decision making tool that will lighten the burden on medical staff, especially during pandemic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Related work Since Corona was announced as a pandemic, different projects were carried out since 2020 to 2022 that it became among an interesting subject to learn from, some of the works related to image classification for different reasons are discussed in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The COVID-CT dataset of 2560 images was the database used in [3], 2214 of which were used for training and the remaining 246 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" By employing WOA to optimize the network's hyperparameters, the model used to train ResNet-50 became the WOANet model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' This last experiment looked at the accuracy of the classification using the suggested method on 246 CT scans, and found that 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='37% of them were categorized as COVID-19, while 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='18% were identified as non-COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The radiologists will be greatly assisted by this proposed WOANet in reducing the burden on the healthcare system and hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" In this study [4], patients' X-ray images are used to classify patients using CNN deep learning." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' One of the most powerful algorithms with generative and deterministic capabilities is the capsule network (CapsNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' However, compared to the basic CNN structures, this network has been relatively more responsive to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The dataset utilized was the NIH complete Chest X-rays [5] collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" VDSNet has a validation accuracy value of 73%, which is higher than the sample dataset's score of 70." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Using a dataset of 6432 images, the DLH COVID [6] model is distinct, trustworthy, and independently created without any input from the transfer learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The experimental findings from the prospective validation phase suggest that the DLH MODEL outperformed the majority of the pre-trained models since it distinguished COVID-19, pneumonia, and healthy/unhealthy patients from the image dataset with a promising accuracy of 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Dataset As seen in figure 1, a database of chest X-ray images for COVID-19 positive cases as well as images of normal and viral pneumonia was created in collaboration with medical professionals by a group of researchers from Qatar University, Doha, Qatar, and the University of Dhaka, Bangladesh, as well as their collaborators from Pakistan and Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' This dataset contains 3616 COVID-19 positive cases, 10,192 Normal, 6012 Lung Opacity (Non-COVID lung infection), and 1345 Viral Pneumonia images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The COVID-19 x-ray image database was created using different sources [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 1 CXR scans with four categories of pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' COVID Lung_Opacity Normal ViralPneumonia For training of datasets, we employed Matlab 2021 installed on computer with 64-bit operating system, windows 10 Pro, 24 GB of Random Access Memory (RAM), with an Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='40GHz and Graphical Processing Unit (GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Eighty percent of the datasets are used for training and 20% for testing (evaluating the model performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Proposed Model In this study, we analyzed the different techniques for image classification of COVID-19 using X-Ray radiographic images of the chest, then examined CNN’s architecture that is based on research on the visual cortex of the cat by Hubel and Wisiel [10], and different pre-trained models: AlexNet, ResNet18 and GoogleNet in order to see the variation of answers in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 2 The proposed model with different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The use of deep learning methods not only allows us to process a very large number of images but at the same time allows us to skip the feature extraction step, such a cumbersome step because it is done by hand crucially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Due to their capacity to extract features (see figure 2) and learn to distinguish between various classes, convolutional neural networks (CNNs) are the top DL tool that are widely employed in several fields of the healthcare system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=', positive and negative, infected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Transfer learning (TL) has made it simpler to quickly and accurately retrain neural networks on chosen datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Experiments and Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1 Experiment 1: Application of the CNN model The structure of our CNN includes a number of layers, as shown in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' CNN receives a CXR image with a size of 299 by 299 pixels as its input, and the rest of the architecture is mentioned in the table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Model: CNN/AlexNet/GoogleNet/ResNet 18 Fully Convolution Connected Pooling Output Input O O O 299 299 1 4 classes FeatureExtraction Classification Table 1 The architecture of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Name Type Description of output size Input layer Input data 299*299 Conv 1 Convolution +ReLU 32*32*8 S1 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Conv 2 Convolution +ReLU 64*64*3 S2 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Conv 3 Convolution +ReLU 128*128*5 S3 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Conv 4 Convolution +ReLU 256*256*5 S4 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Conv 5 Convolution +ReLU 512*512*5 S5 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Conv 6 Convolution +ReLU 1024*1024*5 S6 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Fc Fully connected 1 Fc (4) The trained parameters used in this model are in the options side where all the hyperparameters used were defined including the number of epochs used (1 or 5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' the mini batch (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='001 and frequency validation is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The given CNN was trained using different parameters to test the accuracy for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' We utilized the accuracy parameter to evaluate how well the trained models performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The percentage of correctly classified images over all the images is what is referred to as accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The following formula is utilized: 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 Table 2 The results of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The CNN model 1 epoch 5 epochs Accuracy 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='61% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='13% Time GPU execution 146 min 58s 703 min 16s The first top accuracy after training the model using one epoch provided us with 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='61% accuracy for our 4 classes classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' As training the model using only one epoch did not provide the best result, we had to increase the number of epochs and see the performance of our model and the results for our model gave us 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='13%, which is a lot better compared to our first experiment with one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2 Experiment 2: Application of the pretrained models AlexNet: With 5 convolutional layers and convolutional filter sizes of 3*3 and 2*2 for max pooling operation, AlexNet is an 8-layer convolutional neural network [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Fully connected layers are the final three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" The AlexNet model's standard input size is 227*227*3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' GoogleNet (Inception v3): A convolutional neural network with 50 layers in depth is called GoogleNet [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The program, titled "Going deeper with convolutions," was developed and taught by Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Up to 1000 objects can be classified using the pre-trained Inceptionv3 model with the ImageNet dataset [13] weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" This network's image input size was 299x299 pixels." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' ResNet18: A convolutional neural network with 18 layers in depth is called ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Deep Residual Learning for Image Recognition, as it is known, was developed and trained by Microsoft in 2015 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' To address the issue of vanishing gradient that may affect the weightage change in neural networks, ResNet architectures introduced the use of residual layers and skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' This made training easier and allowed neural networks to get much deeper with greater performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The network was trained on colored images with a resolution of 224x224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' In addition to the accuracy parameters, we estimated the time GPU execution for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The results obtained are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Table 3 The results of the pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Pretrained models AlexNet GoogleNet ResNet18 Accuracy 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='93% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='87% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% Time GPU execution 14 min 58s 41 min 34s 33 min 13s Confusion matrix is the common approach used for evaluation of model performance based on true positive (TP), true negative (TN), false positive (FP), and false negative (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The figure 3 represents the confusion matrix of the Resnet 18 model which gave the best result in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 3 The confusion matrix for the RestNet 18 model with the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' When the same dataset was used using the same hyper parameters the accuracy found was 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='93%, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='87% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1 % for AlexNet, GoogleNet and ResNet18 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Note that the pretrained models used only one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' We see that the results can be improved by using pretrained architectures, attaining an accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=" The increased classification rate attained by Resnet 18 can be attributed to the network's use of novel techniques to lessen over-fitting in its model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The first method involved artificially enlarging the dataset with the aid of a label-preserving transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' This involved extracting random patches (224x224 for ResNet 18) and training the network on them while varying the intensities of the RGB channels in the training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The result was the generation of image translations and horizontal reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The second strategy was "dropout," which involves removing neurons that do not participate in the forward pass or the backward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' As a result, the model is forced to learn more robust characteristics and decreases the complex co-adaptations of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The GPU execution time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge much faster than the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The time saving is very considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 707 COVID 6 6 0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='9% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% Lungopacity S 1060 LL 1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='3% Output Class Normal 8 136 1949 5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 1 0 3 263 ViralPneumonia 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='0% %00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='5% 981% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='8% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='9% 118% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='4% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='2% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='9% alPpr Target Class 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Conclusion This work aimed at developing a convolutional neural network (CNN) model that will help classify COVID-19 and non-COVID 19 disease such as viral pneumonia cases using chest X-ray images in the period caused by the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The model used in this work was CNN as well as pre-trained models including AlexNet, GoogleNet, and ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' The CNN gave a result with 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='13% accuracy for classifying the four classes after training 80% of the dataset and testing on 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' This motivated us not only to keep changing settings, but also to work on pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' In the latter, the pre-trained models were used on the same dataset but with just one epoch for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' And the results were 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='93%, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='87% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='1 % for AlexNet, GoogleNet and ResNet18 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' References [1] Kutlu, Yakup, and Yunus Camgözlü.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' "Detection of coronavirus disease (COVID-19) from X-ray images using deep convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='" Natural and Engineering Sciences 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' 1 (2021): 60-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' [2] Alakus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' and Turkoglu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Comparison of deep learning approaches to predict COVID-19 infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'} +page_content=' Chaos, Solitons & Fractals, 140, p.' metadata={'source': 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Motivated by this, we introduce a model of multi-population learning +that allows for heterogeneous beliefs within each population and where agents respond to their +beliefs via smooth fictitious play (SFP). We show that the system state — a probability distribution +over beliefs — evolves according to a system of partial differential equations. We establish the +convergence of SFP to Quantal Response Equilibria in different classes of games capturing both +network competition as well as network coordination. We also prove that the beliefs will eventually +homogenize in all network games. Although the initial belief heterogeneity disappears in the limit, +we show that it plays a crucial role for equilibrium selection in the case of coordination games as it +helps select highly desirable equilibria. Contrary, in the case of network competition, the resulting +limit behavior is independent of the initialization of beliefs, even when the underlying game has many +distinct Nash equilibria. +1 +Introduction +Smooth Fictitious play (SFP) and variants thereof are arguably amongst the most well-studied learning +models in AI and game theory [2, 3, 21, 22, 9, 19, 36, 37, 42, 18, 17]. SFP describes a belief-based learning +process: agents form beliefs about the play of opponents and update their beliefs based on observations. +Informally, an agent’s belief can be thought as reflecting how likely its opponents will play each strategy. +During game plays, each agent plays smoothed best responses to its beliefs. Much of the literature of +SFP is framed in the context of homogeneous beliefs models where all agents in a given role have the +same beliefs. This includes models with one agent in each player role [3, 2, 39] as well as models with a +single population but in which all agents have the same beliefs [21, 22]. SFP are known to converge in +large classes of homogeneous beliefs models (e.g., most 2-player games [9, 19, 3]). However, in the context +of heterogeneous beliefs, where agents in a population have different beliefs, SFP has been explored to a +less extent. +The study of heterogeneous beliefs (or more broadly speaking, population heterogeneity) is important +and practically relevant. From multi-agent system perspective, heterogeneous beliefs widely exist in +many applications, such as traffic management, online trading and video game playing. For example, +it is natural to expect that public opinions generally diverge on autonomous vehicles and that people +have different beliefs about the behaviors of taxi drivers vs non-professional drivers. From machine +learning perspective, recent empirical advances hint that injecting heterogeneity potentially accelerates +population-based training of neural networks and improves learning performance [25, 29, 44]. From game +theory perspective, considering heterogeneity of beliefs better explains results of some human experiments +[10, 11]. +Heterogeneous beliefs models of SFP are not entirely new. In the pioneering work [12], Fudenberg +and Takahashi examine the heterogeneity issue in 2-population settings by appealing to techniques from +the stochastic approximation theory. This approach, which is typical in the SFP literature, relates the +limit behavior of each individual to an ordinary differential equation (ODE) and has yielded significant +insights for many homogeneous beliefs models [3, 2, 19, 39]. However, this approach, as also noted by +Fudenberg and Takahashi, “does not provide very precise estimates of the effect of the initial condition of +the system.” Consider an example of a population of agents each can choose between two pure strategies +1 +arXiv:2301.04929v1 [cs.MA] 12 Jan 2023 + +s1 and s2. Let us imagine two cases: (i) every agents in the population share the same belief that their +opponents play a mixed strategy choosing s1 and s2 with equal probability 0.5, and (ii) half of the agents +believe that their opponents determinedly play the pure strategy s1 and the other half believe that +their opponents determinedly play the pure strategy s2. The stochastic approximation approach would +generally treat these two cases equally, providing little information about the heterogeneity in beliefs as +well as its consequential effects on the system evolution. This drives our motivating questions: +How does heterogeneous populations evolve under SFP? How much and under what conditions does +the heterogeneity in beliefs affect their long-term behaviors? +Model and Solutions. In this paper, we study the dynamics of SFP in general classes of multi- +population network games that allow for heterogeneous beliefs. In a multi-population network game, each +vertex of the network represents a population (continuum) of agents, and each edge represents a series +of 2-player subgames between two neighboring populations. Note that multi-population network games +include all the 2-population games considered in [12] and are representation of subclasses of real-world +systems where the graph structure is evident [?]. We consider that for a certain population, individual +agents form separate beliefs about each neighbor population and observe the mean strategy play of +that population. Taking a approach different from stochastic approximation, we define the system state +as a probability measure over the space of beliefs, which allows us to precisely examine the impact of +heterogeneous beliefs on system evolution. This probability measure changes over time in response to +agents’ learning. Thus, the main challenge is to analyze the evolution of the measure, which in general +requires the development of new techniques. +As a starting point, we establish a system of partial differential equations (PDEs) to track the +evolution of the measure in continuous time limit (Proposition 1). The PDEs that we derive are akin +to the continuity equations1 commonly encountered in physics and do not allow for a general solution. +Appealing to moment closure approximation [13], we circumvent the need of solving the PDEs and +directly analyze the dynamics of the mean and variance (Proposition 2 and Theorem 1). As one of our +key results, we prove that the variance of beliefs always decays quadratically fast with time in all network +games (Theorem 1). Put differently, eventually, beliefs will homogenize and the distribution of beliefs +will collapse to a single point, regardless of initial distributions of beliefs, 2-player subgames that agents +play, and the number of populations and strategies. This result is non-trivial and perhaps somewhat +counterintuitive. Afterall, one may find it more natural to expect that the distribution of beliefs would +converge to some distribution rather than a single point, as evidenced by recent studies on Q-learning +and Cross learning [23, 24, 27]. +Technically, the eventual belief homogenization has a significant implication — it informally hints +that the asymptotic system state of initially heterogeneous systems are likely to be the same as in +homogeneous systems. We show that the fixed point of SFP correspond to Quantual Response Equilibria +(QRE)2 in network games for both homogeneous and initially heterogeneous systems (Theorem 2). As +our main result, we establish the convergence of SFP to QRE in different classes of games capturing both +network competition as well as network coordination, independent of belief initialization. Specifically, +for competitive network games, we first prove via a Lyapunov argument that the SFP converges to a +unique QRE in homogeneous systems, even when the underlying game has many distinct Nash equilibria +(Theorem 3). Then, we show that this convergence result can be carried over to initially heterogeneous +systems (Theorem 4), by leveraging that the mean belief dynamics of initially heterogeneous systems is +asymptotically autonomous [31] with its limit dynamics being the belief dynamics of a homogeneous system +(Lemma 7). For coordination network games, we also prove the convergence to QRE for homogeneous +and initially heterogeneous systems, in which the underlying network has star structure (Theorem 5). +On the other hand, the eventual belief homogenization may lead to a misconception that belief +heterogeneity has little effect on system evolution. Using an example of 2-population stag hunt games, +we show that belief heterogeneity actually plays a crucial role in equilibrium selection, even though it +eventually vanishes. As shown in Figure 1, changing the variance of initial beliefs results in different limit +behaviors, even when the mean of initial beliefs remains unchanged; in particular, while a small variance +leads to the less desirable equilibrium pH, Hq, a large variance leads to the payoff dominant equilibrium +pS, Sq. Thus, in the case of network coordination, initial belief heterogeneity can help select the highly +desirable equilibrium and provides interesting insights to the seminal thorny problem of equilibrium +selection [26]. On the contrary, in the case of network competition, we prove (Theorems 3 and 4 on the +1The continuity equation is a PDE that describes the transport phenomena of some quantity (e.g., mass, energy, +momentum and other conserved quantities) in a physical system. +2QRE is a game theoretic solution concept under bounded rationality. By QRE, in this paper we refer to their canonical +form also referred to as logit equilibria or logit QRE in the literature [14]. +2 + +Figure 1: The system dynamics under the effects of different variances of initial beliefs (thin lines: +predictions of our PDE model, shaded wide lines: simulation results). ¯µ2S represents the mean belief +about population 2 and ¯x1S represents the mean probability of playing strategy S in population 1. Initially, +we set the mean beliefs ¯µ2S “ ¯µ1S “ 0.3 (details of the setup are summarized in the supplementary). +Given the same initial mean belief, different initial variances σ2pµ2Sq lead to the convergence to different +beliefs (the left panel) and even to different strategy choices (the right panel). In particular, a large initial +variance helps select the payoff dominant equilibrium pS, Sq in stag hunt games. +convergence to a unique QRE in competitive network games) as well as showcase experimentally that the +resulting limit behavior is independent of initialization of beliefs, even if the underlying game has many +distinct Nash equilibria. +Related Works. +SFP and its variants have recently attracted a lot of attention in AI research +[36, 37, 42, 18, 17]. There is a significant literature that analyze SFP in different models [3, 7, 21, 19], and +the paper that is most closely related to our work is [12]. Fudenberg and Takahashi [12] also examines the +heterogeneity issue and anticipate belief homogenization in the limit under 2-population settings. In this +paper, we consider multi-population network games, which is a generalization of their setting.3 Moreover, +our approach is more fundamental, as the PDEs that we derive can provide much richer information +about the system evolution and thus precisely estimates the temporal effects of heterogeneity, which is +generally intractable in [12]. Therefore, using our approach, we are able to show an interesting finding — +the initial heterogeneity plays a crucial role in equilibrium selection (Figure 1) — which unfortunately +cannot be shown using the approach in [12]. Last but not least, to our knowledge, our paper is the first +work that presents a systematic study of smooth fictitious play in general classes of network games. +On the other hand, networked multi-agent learning constitutes one of the current frontiers in AI and +ML research [43, 30, 16]. Recent theoretical advances on network games provide conditions for learning +behaviors to be not chaotic [6, 34], and investigate the convergence of Q-learning and continuous-time FP +in the case of network competitions [7, 28]. However, [7, 28] consider that there is only one agent on each +vertex, and hence their models are essentially for homogeneous systems. +Lahkar and Seymour [27] and Hu et al. [23, 24] also use the continuity equations as a tool to study +population heterogeneity in multi-agent systems where a single population of agents applies Cross learning +or Q-learning to play symmetric games. They either prove or numerically showcase that heterogeneity +generally persists. Our results complement these advances by showing that heterogeneity vanishes under +SFP and that heterogeneity helps select highly desirable equilibria. Moreover, methodologically, we +establish new proof techniques for the convergence of learning dynamics in heterogeneous systems by +leveraging seminal results (Lemmas 1 and 2) from the asymptotically autonomous dynamical system +literature, which may be of independent interest. +3The analysis presented in this paper covers all generic 2-population network games, all generic bipartite network games +where the game played on each edge is the same along all edges, and all weighted zero-sum games which do not require the +graph to be bipartite nor to have the same game played on each edge. +3 + +Small vs. Large Variance of Initial Population Beliefs +Mean Belief about the Others' Playing S +Mean Probability of Playing S +1S +0.8 +0.8 +0.6 +Mean Prob. +0.6 +)~ 0.023 +(μ2s) ~ 0.023 +0.A +α(μ2s) += 0.1 +0.4 +g(μ2s) += 0.1 +0.2 +0.2 +50 +100 +150 +200 +50 +100 +150 +200 +Time t +Time t2 +Preliminaries +Population Network Games. +A population network game (PNG) Γ “ pN, pV, Eq, pSi, ωiq@iPV , pAijqpi,jqPEq +consists of a multi-agent system N distributed over a graph pV, Eq, where V “ t1, ..., nu is the set of +vertices each represents a population (continuum) of agents, and E is the set of pairs, pi, jq, of population +i ‰ j P V . For each population i P V , agents of this population has a finite set Si of pure strategies (or +actions) with generic elements si P Si. Agents may also use mixed strategies (or choice distributions). +For an arbitrary agent k in population i, its mixed strategy is a vector xipkq P ∆i, where ∆i is the +simplex in R|Si| such that ř +siPSi xisipkq “ 1 and xisipkq ě 0, @si P Si. Each edge pi, jq P E defines +a series of two-player subgames between populations i and j, such that for a given time step, each +agent in population i is randomly paired up with another agent in population j to play a two-player +subgame. We denote the payoff matrices for agents of population i and j in these two-player subgames by +Aij P R|Si|ˆ|Sj| and Aji P R|Sj|ˆ|Si|, respectively. Note that at a given time step, each agent chooses a +(mixed or pure) strategy and plays that strategy in all two-player subgames. Let x “ pxi, txjupi,jqPEq be +a mixed strategy profile, where xi (or xj) denotes a generic mixed strategy in population i (or j). Given +the mixed strategy profile x, the expected payoff of using xi in the game Γ is +ripxq “ ripxi, txjupi,jqPEq :“ +ÿ +pi,jqPE +xJ +i Aijxj. +(1) +The game Γ is competitive (or weighted zero-sum), if there exist positive constants ω1, . . . , ωn such that +ÿ +iPV +ωiripxq “ +ÿ +pi,jqPE +` +ωixJ +i Aijxj ` ωjxJ +j Ajixi +˘ +“ 0, +@x P +ź +iPV +∆i. +(2) +On the other hand, Γ is a coordination network game, if for each edge pi, jq P E, the payoff matrices of +the two-player subgame satisfy Aij “ AJ +ji. +Smooth Fictitious Play. +SFP is a belief-based model for learning in games. In SFP, agents form +beliefs about the play of opponents and respond to the beliefs via smooth best responses. Given a +game Γ, consider an arbitrary agent k in a population i P V . Let Vi “ tj P V : pi, jq P Eu be the set +of neighbor populations. Agent k maintains a weight κi +jsjpkq for each opponent strategy sj P Sj of a +neighbor population j P Vi. Based on the weights, agent k forms a belief about the neighbor population +j, such that each opponent strategy sj is played with probability +µi +jsjpkq “ +κi +jsjpkq +ř +s1 +jPSj κi +js1 +jpkq. +(3) +Let µi +jpkq be the vector of beliefs with the sj-th element equals µi +jsjpkq. Agent k forms separate beliefs +for each neighbor population, and plays a smooth best response to the set of beliefs tµi +jpkqujPVi. Given a +game Γ, agent k’s expected payoff for using a pure strategy si P Si is +uisipkq “ ripesi, tµi +jpk, tqujPViq “ +ÿ +jPVi +eJ +siAijµi +jpkq +(4) +where esi is a unit vector where the si-th element is 1. The probability of playing strategy si is then +given by +xisipkq “ +exppβuisipkqq +ř +s1 +iPSi exppβuis1 +ipkqq +(5) +where β is a temperature (or the degree of rationality). We consider that agents observe the mean mixed +strategy of each neighbor population. As such, at a given time step t, agent k updates the weights for +each opponent strategy sj P Sj, j P Vi as follows: +κi +jsjpk, t ` 1q “ κi +jsjpk, tq ` ¯xjsjptq +(6) +where ¯xjsj is the mean probability of playing strategy sj in population j, i.e., ¯xjsj “ +1 +nj +ř +lPpopulation j xjsjplq +with the number of agents denoted by nj. +For simplicity, we assume the initial sum of weights +ř +sjPSj κi +jsjpk, 0q to be the same for every agent in the system N and denote this initial sum by λ. +Observe that Equation 6 can be rewritten as +pλ ` t ` 1qµi +jsjpk, t ` 1q “ pλ ` tqµi +jsjpk, tq ` ¯xjsjptq. +(7) +4 + +Hence, even though agent k directly updates the weights, its individual state can be characterized by the +set of beliefs tµi +jpkqujPVi. In the following, we usually drop the time index t and agent index k in the +bracket (depending on the context) for notational convenience. +3 +Belief Dynamics in Population Network Games +Observe that for an arbitrary agent k, its belief µi +jpkq is in the simplex ∆j “ tµi +jpkq P R|Sj|| ř +sjPSj µi +jsjpkq “ +1, µi +jsjpkq ě 0, @sj P Sju. We assume that the system state is characterized by a Borel probability measure +P defined on the state space ∆ “ ś +iPV ∆i. Given µi P ∆i, we write the marginal probability density func- +tion as ppµi, tq. Note that ppµi, tq is the density of agents having the belief µi about population i throughout +the system. Define µ “ tµiuiPV P ∆. Since agents maintain separate beliefs about different neighbor +populations, the joint probability density function ppµ, tq can be factorized, i.e., ppµ, tq “ ś +iPV ppµi, tq. +We make the following assumption for the initial marginal density functions. +Assumption 1. At time t “ 0, for each population i P V , the marginal density function ppµi, tq is +continuously differentiable and has zero mass at the boundary of the simplex ∆i. +This assumption is standard and common for a “nice” probability distribution. Under this mild +condition, we determine the evolution of the system state P with the following proposition, using the +techniques similar to those in [27, 23]. +Proposition 1 (Population Belief Dynamics). The continuous-time dynamics of the marginal density +function ppµi, tq for each population i P V is governed by a partial differential equation +´Bppµi, tq +Bt +“ ∇ ¨ +ˆ +ppµi, tq ¯xi ´ µi +λ ` t ` 1 +˙ +(8) +where ∇¨ is the divergence operator and ¯xi is the mean mixed strategy with each si-th element +¯xisi “ +ż +ś +jPVi ∆j +exp pβuisiq +ř +s1 +iPSi exp pβuis1 +iq +ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +(9) +where uisi “ ř +jPVi eJ +siAijµj. +For every marginal density function ppµi, tq, the total mass is always conserved (Corollary 1 of the +supplementary); moreover, the mass at the boundary of the simplex ∆i always remains zero, indicating +that agents’ beliefs will never go to extremes (Corollary 2 of the supplementary). +Generalizing the notion of a system state to a distribution over beliefs allows us to address a very +specific question — the impact of belief heterogeneity on system evolution. That said, partial differential +equations (Equation 8) are notoriously difficult to solve. Here we resort to the evolution of moments +based on the evolution of the distribution (Equation 8). In the following proposition, we show that the +characterization of belief heterogeneity is important, as the dynamics of the mean system state (or the +mean belief dynamics) is indeed affected by belief heterogeneity. +Proposition 2 (Mean Belief Dynamics). The dynamics of the mean belief ¯µi about each population +i P V is governed by a system of differential equations such that for each strategy si, +d¯µisi +dt +« fsiptµjujPViq ´ ¯µisi +λ ` t ` 1 +` +ř +jPVi +ř +sjPSj +B2fsiptµjujPViq +pBµjsj q2 +Varpµjsjq +2pλ ` t ` 1q +. +(10) +where fsiptµujPViq is the logit choice function (Equation 5) applied to strategy si P Si, and Varpµjsjq is +the variance of belief µjsj in the entire system. +In general, the mean belief dynamics is under the joint effects of the mean, variance, and infinitely +many higher moments of the belief distribution. To allow for more conclusive results, we apply the +moment closure approximation4 and assume the effects of the third and higher moments to be negligible. +Now, just for a moment, suppose that the system beliefs are homogeneous —- the beliefs of every +individuals are the same. Hence, the mean belief dynamics are effectively the belief dynamics of individuals. +The following proposition follows from Equation 7. +4Moment closure is a typical approximation method used to estimate moments of population models [13, 15, 32]. To use +moment closure, a level is chosen past which all cumulants are set to zero. The conventional choice of the level is 2, i.e., +setting the third and higher cumulants to be zero. +5 + +Proposition 3 (Belief Dynamics for Homogeneous Populations). For a homogeneous system, the +dynamics of the belief µi about each population i P V is governed by a system of differential equations +such that for each strategy si, +dµisi +dt +“ xisi ´ µisi +λ ` t ` 1 “ fsiptµjujPViq ´ µisi +λ ` t ` 1 +(11) +where µisi is the same for all agents in each neighbor population j P Vi. +Intuitively, the mean belief dynamics indicates the trend of beliefs in a system, and the variance of +beliefs indicates belief heterogeneity. Contrasting Propositions 2 and 3, it is clear that the variance of +belief (belief heterogeneity) plays a role in determining the mean belief dynamics (the trend of beliefs) for +heterogeneous systems. It is then natural to ask: how does the belief heterogeneity evolve over time? +How much does the belief heterogeneity affect the trend of beliefs? Our investigation to these questions +reveals an interesting finding — the variance of beliefs asymptotically tends to zero. +Theorem 1 (Quadratic Decay of the Variance of Population Beliefs). The dynamics of the +variance of beliefs µi about each population i P V is governed by a system of differential equations such +that for each strategy si, +dVarpµisiq +dt +“ ´2Varpµisiq +λ ` t ` 1 . +(12) +At given time t, Varpµisiq “ +´ +λ`1 +λ`t`1 +¯2 +σ2pµisiq, where σ2pµisiq is the initial variance. Thus, the variance +Varpµisiq decays to zero quadratically fast with time. +Such quadratic decay of the variance stands no matter what 2-player subgames agents play and what +initial conditions are. Put differently, the beliefs will eventually homogenize for all population network +games. This fact immediately implies the system state in the limit. +Corollary 1. As time t Ñ 8, the density function ppµi, tq for each population i P V evolves into a Dirac +delta function, and the variance of the choice distributions within each population i P V also goes to zero. +Note that while the choice distributions will homogenize within each population, they are not necessarily +the same across different populations. This is because the strategy choice of each population is in response +to its own set of neighbor populations (which are generally different). +4 +Convergence of Smooth Fictitious Play in Population Network +Games +The finding on belief homogenization is non-trivial and also technically important. One implication is +that the fixed points of systems with initially heterogeneous beliefs are the same as in systems with +homogeneous beliefs. Thus, it follows from the belief dynamics for homogeneous systems (Proposition 3) +that the fixed points of systems have the following property. +Theorem 2 (Fixed Points of System Dynamics). For any system that initially have homogeneous +or heterogeneous beliefs, the fixed points of the system dynamics is a pair pµ˚, x˚q that satisfy x˚ +i “ µ˚ +i +for each population i P V and are the solutions of the system of equations +x˚ +isi “ +exp +´ +β ř +jPVi eJ +siAijx˚ +j +¯ +ř +s1 +iPSi exp +´ +β ř +jPVi eJ +s1 +iAijx˚ +j +¯ +(13) +for every strategy si P Si and population i P V . Such fixed points always exist and coincide with the +Quantal Response Equilibria (QRE) [33] of the population network game Γ. +Note that the above theorem applies for all population network games. +We study the convergence of SFP to the QRE under the both cases of network competition and +network coordination. Due to space limits, in the following, we mainly focus on network competition and +present only the main result on network coordination. +6 + +4.1 +Network Competition +Consider a competitive population network game Γ. Note that in competitive network games, the Nash +equilibrium payoffs need not to be unique (which is in clear contrast to two-player settings), and it +generally allows for infinitely many Nash equilibria. In the following theorem, focusing on homogeneous +systems, we establish the convergence of the belief dynamics to a unique QRE, regardless of the number +of Nash equilibria in the underlying game. +Theorem 3 (Convergence in Homogeneous Network Competition). Given a competitive Γ, for +any system that has homogeneous beliefs, the belief dynamics (Equation 11) converges to a unique QRE +which is globally asymptotically stable. +Proof of Sketch. We proof this theorem by showing that the “distance” between xi and µi is strictly +decreasing until the QRE is reached. In particular, we measure the distance in terms of the perturbed +payoff and construct a strict Lyapunov function +L :“ +ÿ +iPV +ωi +“ +πi +` +xi, tµjujPVi +˘ +´ πi +` +µi, tµjujPVi +˘‰ +(14) +where ω1 . . . ωn are the positive weights given by Γ, and πi is a perturbed payoff function defined as +πi +` +xi, tµjujPVi +˘ :“ xJ +i +ř +jPVi Aijµj ´ 1 +β +ř +siPSi xisi lnpxisiq. +Next, we turn to systems with initially heterogeneous beliefs. Leveraging that the variance of beliefs +eventually goes to zero, we establish the following lemma. +Lemma 1. For a system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10) is +asymptotically autonomous [31] with the limit equation dµi +dt “ xi ´ µi, which after time-reparmeterization +is equivalent to the belief dynamics for homogeneous systems (Equation 11). +For ease of presentation, we follow the convention to denote the solution flows of an asymptotically +autonomous system and its limit equation by φ and Θ, respectively. Thieme [40] provides the following +seminal result that connects the limit behaviors of φ and Θ. +Lemma 2 (Thieme [40] Theorem 4.2). Given a metric space pX, dq. Assume that the equilibria of Θ +are isolated compact Θ-invariant subsets of X. The ω-Θ-limit set of any pre-compact Θ-orbit contains a +Θ-equilibrium. The point ps, xq, s ě t0, x P X, have a pre-compact φ-orbit. Then the following alternative +holds: 1) φpt, s, xq Ñ e, t Ñ 8, for some Θ-equilibrium e, and 2) the ω-φ-limit set of ps, xq contains +finitely many Θ-equilibria which are chained to each other in a cyclic way. +Combining the above results, we prove the convergence for initially heterogeneous systems. +Theorem 4 (Convergence in Initially Heterogeneous Network Competition). Given a compet- +itive Γ, for any system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10) +converges to a unique QRE. +The following corollary immediately follows as the result of belief homogenization. +Corollary 2. For any competitive Γ, under smooth fictitious play, the choice distributions and beliefs of +every individual converges to a unique QRE (given in Theorem 2), regardless of belief initialization and +the number of Nash equilibria in Γ. +4.2 +Network Coordination +We delegate most of the results on coordination network games to the supplementary, and summarize +only the main result here. +Theorem 5 (Convergence in Network Coordination with Star Structure). Given a coordination +Γ where the network structure consists of a single or disconnected multiple stars, each orbit of the belief +dynamics (Equation 11) for homogeneous systems as well as each orbit of the mean belief dynamics +(Equation 10) for initially heterogeneous systems converges to the set of QRE. +Note that this theorem applies to all 2-population coordination games, as network games with or +without star structure are essentially the same when there are only two vertices. We also remark that +pure or mixed Nash equilibria in coordination network games are complex; as reported in recent works +[5, 4, 1], finding a pure Nash equilibrium is PLS-complete. Hence, learning in the general case of network +coordination is difficult and generally requires some conditions for theoretical analysis [34, 35]. +7 + +H +S +H +(1, 1) +(2, 0) +S +(0, 2) +(4, 4) +Table 1: Stag Hunt. +Figure 2: Asymmetric Matching Pennies. +Figure 3: Belief heterogeneity helps select the payoff dominant equilibrium pS, Sq (yellow: the equilibrium +pS, Sq, blue: the equilibrium pH, Hq). As the variance of initial beliefs increases (from the left to right +panel), a larger range of initial mean beliefs will approximately reach the equilibrium pS, Sq in the limit. +For each panel, the initial variances of two populations σ2pµ1Hq and σ2pµ2Hq are the same. +5 +Experiments: +Equilibrium Selection in Population Network +Games +In this section, we complement our theory and present an empirical study of SFP in a two-population +coordination (stag hunt) game and a five-population zero-sum (asymmetric matching pennies) game. +Importantly, these two games both have multiple Nash equilibria, which naturally raises the problem of +equilibrium selection. +5.1 +Two-Population Stag Hunt Games +We have shown in Figure 1 (in the introduction) that given the same initial mean belief, changing the +variances of initial beliefs can result in different limit behaviors. In the following, we systematically study +the effect of initial belief heterogeneity by visualizing how it affects the regions of attraction to different +equilibria. +Game Description. We consider a two-population stag hunt game, where each player in populations +1 and 2 has two actions tH, Su. As shown in the payoff bi-matrices (Table 1), there are two pure strategy +Nash equilibria in this game: pH, Hq and pS, Sq. While pH, Hq is risk dominant, pS, Sq is indeed more +desirable as it is payoff dominant as well as Pareto optimal. +Results. In this game, population 1 forms beliefs about population 2 and vice versa. We denote +the initial mean beliefs by a pair p¯µ2H, ¯µ1Hq. We numerically solve the mean belief dynamics for a large +range of initial mean beliefs, given different variances of initial beliefs. In Figure 3, for each pair of initial +mean beliefs, we color the corresponding data point based on which QRE the system eventually converges +to. We observe that as the variance of initial beliefs increases (from the left to right panel), a larger range +of initial mean beliefs results in the convergence to the QRE that approximates the payoff dominant +equilibrium pS, Sq. Put differently, a higher degree of initial belief heterogeneity leads to a larger region +of attraction to pS, Sq. Hence, belief heterogeneity eventually vanishes though, it provides an approach to +equilibrium selection, as it helps select the highly desirable equilibrium. +5.2 +Five-Population Asymmetric Matching Pennies Games +We have shown in Corollary 2 that SFP converges to a unique QRE even if there are multiple Nash +equilibria in a competitive Γ. In the following, we corroborate this by providing empirical evidence in +8 + ++1 ++1 ++1 +1 +Population 1 +Population 2 +Population 3 +Population 4 +Population 5 +H +(L H) +{H, T} +[H, T} +Match +Match +Match +MatchBasins of Attraction of (S, S) and (H, H) under Different Variances of Initial Beliefs +g?(μ1H) += 0 +g2(μ1H) = 0.02 +g?(μH) += 0.05 +g2(μ1H) = 0.1 +(S,S) +j12H +H +2H +0.8 +0.8 +0.8 +0.8 +Initial +Initial +Initial +Initial +0.6 +0.6 +0.6 +0.6 +(H,H) +0.6 +0.8 +1 +0.6 +0.8 +1 +0.6 +0.8 +1 +0.6 +0.8 +1 +Initial jiH +Initial jiiH +Initial jiH +InitialjiiHFigure 4: With different belief initialization, SFP selects a unique equilibrium where all agents in +population 3 play strategy H with probability 0.5. We run 100 simulation runs for each initialization. The +thin lines represent the mean mixed strategy (the choice probability of H) and the shaded areas represent +the variance of the mixed strategies in the population. In the legends, B denotes Beta distribution; +the two Beta distributions correspond to the initial beliefs about the neighbor populations 2 and 4, +respectively. +agent-based simulations with different belief initialization (the details of simulations are summarized in +the supplementary). +Game Description. Consider a five-population asymmetric matching pennies game [28], where +the network structure is a line (depicted in Figure 2). Each agent has two actions tH, Tu. Agents in +populations 1 and 5 do not learn; they always play strategies H and T, respectively. For agents in +populations 2 to 4, they receive `1 if they match the strategy of the opponent in the next population, +and receive ´1 if they mismatch. On the contrary, they receive `1 if they mismatch the strategy of the +opponent in the previous population, and receive ´1 if they match. Hence, this game has infinitely many +Nash equilibria of the form: agents in populations 2 and 4 play strategy T, whereas agents in population +3 are indifferent between strategies H and T. +Results. In this game, agents in each population form two beliefs (one for the previous population +and one for the next population). We are mainly interested in the strategies of population 3, as the +Nash equilibria differ in the strategies in population 3. For validation, we vary population 3’s beliefs +about the neighbor populations 2 and 4, and fix population 3’s beliefs about the other populations. As +shown in Figure 4, given differential initialization of beliefs, agents in population 3 converge to the same +equilibrium where they all take strategy H with probability 0.5. Therefore, even when the underlying +zero-sum game has many Nash equilibria, SFP with different initial belief heterogeneity selects a unique +equilibria, addressing the problem of equilibrium selection. +6 +Conclusions +We study a heterogeneous beliefs model of SFP in network games. Representing the system state with a +distribution over beliefs, we prove that beliefs eventually become homogeneous in all network games. We +establish the convergence of SFP to Quantal Response Equilibria in general competitive network games +as well as coordination network games with star structure. We experimentally show that although the +initial belief heterogeneity vanishes in the limit, it plays a crucial role in equilibrium selection and helps +select highly desirable equilibria. +Appendix A: Corollaries and Proofs omitted in Section 3 +Proof of Proposition 1 +It follows from Equation 7 in the main paper that the change in µi +jpk, tq between two discrete time steps +is +µi +jpk, t ` 1q “ µi +jpk, tq ` +¯xjptq ´ µi +jpk, tq +λ ` t ` 1 +. +(15) +9 + +Probability of Playing H in Population 3 +Vary the inital Mean, Fix the initial Variance +Vary the initial Variance, Fix the initial Mean +B(5, 10), B(8, 2) +B(10, 20), B(16, 4) +B(5, 10), B(2, 8) +B(2.5, 5), B(4,1) +B(10, 5), B(8, 2) +B(50, 100),B(80,20) +0.5 +0 +0 +100 +200 +300 +400 +100 +200 +300 +400 +Time t +Time tLemma 3. Under Assumption 1 (in the main paper), for an arbitrary agent k in population i, its belief +µi +jpk, tq about a neighbor population j will never reach the extreme belief (i.e., the boundary of the simplex +∆i). +Proof. Assumption 1 ensures that ¯xjp0q is in the interior of the simplex ∆j. Moreover, the logit choice +function (Equation 5 in the main paper) also ensures that ¯xjptq stays in the interior of ∆j afterwards for +a finite temperature β. Hence, from Equation 15, one can see that µi +jpk, tq for every time step t will stay +in the interior of ∆j. +In the following, for notation convenience, we sometimes drop the agent index k and the time index +t depending on the context. Consider a population i. We rewrite the change in the beliefs about this +population as follows. +µipt ` 1q “ µiptq ` ¯xiptq ´ µiptq +λ ` t ` 1 +. +(16) +Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s. We +rewrite the above equation as +µipt ` δq “ µiptq ` δ ¯xiptq ´ µiptq +λ ` t ` 1 +. +(17) +Next, we consider a test function θpµiq. Define +Y “ Erθpµipt ` δqqs ´ Erθpµiptqqs +δ +. +(18) +Applying Taylor series for θpµipt ` δqq at µiptq, we obtain +θpµipt ` δqq “ θpµiptqq ` +δ +λ ` t ` 1Bµiθpµiq r¯xiptq ´ µiptqs +` +δ2 +2pλ ` t ` 1q2 r¯xiptq ´ µiptqsJ Hθpµiq r¯xiptq ´ µiptqs +` o +˜„ +δ ¯xiptq ´ µiptq +λ ` t ` 1 +ȷ2¸ +(19) +where H denotes the Hessian matrix. Hence, the expectation Erθpµipt ` δqqs is +Erθpµipt ` δqqs “ Erθpµiptqqs ` +δ +λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs +` +δ2 +2pλ ` t ` 1q2 E +“ +r¯xiptq ´ µiptqsJHθpµiq r¯xiptq ´ µiptqs +‰ +` +δ2 +2pλ ` t ` 1q2 Eropr¯xiptq ´ µiptqs2qs +(20) +Moving the term Erθpµiptqqs to the left hand side and dividing both sides by δ, we recover the quantity +Y , i.e., +Y “ +1 +λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs +` +δ +2pλ ` t ` 1q2 Err¯xiptq ´ µiptqsJHθpµiptqqr¯xiptq ´ µiptqs ` o +` +p¯xiptq ´ µiptqq2˘ +s +(21) +Taking the limit of Y with δ Ñ 0, the contribution of the second term on the right hand side vanishes, +yielding +lim +δÑ0 Y “ +1 +λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs +(22) +“ +1 +λ ` t ` 1 +ż +ppµiptq, tq +“ +Bµiθpµiptqqp¯xiptq ´ µiptqq +‰ +dµiptq. +(23) +Apply integration by parts. We obtain +lim +δÑ0 Y “ 0 ´ +1 +λ ` t ` 1 +ż +θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq +(24) +10 + +where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero as a result +of Lemma 1. On the other hand, according to the definition of Y , +lim +δÑ0 Y “ lim +δÑ0 +ż +θpµiptqqppµi, t ` δq ´ ppµi, tq +δ +dµi “ +ż +θpµiptqqBtppµi, tqdµi. +(25) +Therefore, we have the equality +ż +θpµiptqqBtppµi, tqdµi “ ´ +1 +λ ` t ` 1 +ż +θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq. +(26) +As θ is a test function, this leads to +Btppµi, tq “ ´ +1 +λ ` t ` 1∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs . +(27) +Rearranging the terms, we obtain Equation 8 in the main paper. By the definition of expectation given a +probability distribution, it is straightforward to obtain Equation 9 in the main paper. Q.E.D. +Remarks: The PDEs we derived are akin to the continuity equation commonly encountered in physics +in the study of conserved quantities.The continuity equation describes the transport phenomena (e.g., of +mass or energy) in a physical system. This renders a physical interpretation for our PDE model: under +SFP, the belief dynamics of a heterogeneous system is analogously the transport of the agent mass in the +simplex ∆ “ ś +iPV ∆i. +Corollaries of Proposition 1 +Corollary 3. For any population i P V , the system beliefs about this population never go to extremes. +Proof. This is a straightforward result of Lemma 1. +Corollary 4. For any population i P V , the total probability mass ppµi, tq always remains conserved. +Proof. Consider the time derivative of the total probability mass +d +dt +ż +ppµi, tqdµi. +(28) +Apply the Leibniz rule to interchange differentiation and integration, +d +dt +ż +ppµi, tqdµi “ +ż Bppµi, tq +Bt +dµi. +(29) +Substitute Bppµi,tq +Bt +with Equation 8 in the main paper, +d +dt +ż +ppµi, tqdµi +“ ´ +ż +∇ ¨ +ˆ +ppµi, tq ¯xi ´ µi +λ ` t ` 1 +˙ +dµi +(30) +“ ´ +ż +ÿ +siPSi +Bµisi +ˆ +ppµi, tq ¯xisi ´ µisi +λ ` t ` 1 +˙ +dµi +(31) +“ ´ +1 +λ ` t ` 1 +«ż +ÿ +siPSi +Bµisi ppµi, tq p¯xisi ´ µisiq dµi ` +ż +ppµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi +ff +(32) +Apply integration by parts, +ż +ÿ +siPSi +Bµisi ppµi, tq p¯xisi ´ µisiq dµi “ 0 ´ +ż +ppµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi. +(33) +where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero. Hence, the +terms within the bracket of Equation 32 cancel out, and +d +dt +ż +ppµi, tqdµi “ 0. +(34) +11 + +Proof of Proposition 2 +Lemma 4. The dynamics of the mean belief ¯µi about each population i P V is governed by a differential +equation +d¯µisi +dt +“ ¯xisi ´ ¯µisi +λ ` t ` 1 , +@si P Si. +(35) +Proof. The time derivative of the mean belief about strategy si is +d¯µisi +dt +“ d +dt +ż +µisippµi, tqdµi. +(36) +We apply the Leibniz rule to interchange differentiation and integration, and then substitute Bppµi,tq +Bt +with +Equation 8 in the main paper. +d +dt +ż +µisippµi, tqdµi +(37) +“ +ż +µisi +Bppµi, tq +Bt +dµi +(38) +“ ´ +ż +µisi∇ ¨ +ˆ +ppµi, tq ¯xi ´ µi +λ ` t ` 1 +˙ +dµi +(39) +“ ´ +ż +µisi +ÿ +siPSi +Bµisi +ˆ +ppµi, tq ¯xisi ´ µisi +λ ` t ` 1 +˙ +dµi +(40) +“ γ +«ż +µisi +ÿ +siPSi +` +Bµisi ppµi, tq +˘ +p¯xisi ´ µisiq dµi ` +ż +µisippµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi +ff +(41) +where γ :“ ´ +1 +λ`t`1. Apply integration by parts to the first term in Equation 41. +ż +µisi +ÿ +siPSi +` +Bµisi ppµi, tq +˘ +p¯xisi ´ µisiq dµi +“ ´ +ż +µisippµi, tq +» +– ÿ +s1 +iPSi +Bµis1 +i p¯xis1 +i ´ µis1 +iq +fi +fl ` ppµi, tqBµisi rµisip¯xisi ´ µisiqs dµi +(42) +where we have leveraged that the probability mass at the boundary remains zero. Hence, it follows from +Equation 41 that +d +dt +ż +µisippµi, tqdµi +(43) +“ ´γ +ż +µisippµi, tq +ÿ +s1 +iPSi +Bµis1 +i p¯xis1 +i ´ µis1 +iqdµi ´ γ +ż +ppµi, tqBµisi rµisip¯xisi ´ µisiqs dµi +` γ +ż +µisippµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi +(44) +“ γ +ż +ppµi, tq +“ +µisiBµisi p¯xisi ´ µisiq ´ Bµisi rµisip¯xisi ´ µisiqs +‰ +dµi +(45) +“ γ +ż +ppµi, tqµisidµi ´ +ż +ppµi, tq¯xisidµi +(46) +“ ¯xisi ´ ¯µisi +λ ` t ` 1 +(47) +We repeat the mean probability ¯xisi, which has been given in Equation 9 in the main paper, as follows: +¯xisi “ +ż +exp pβuisiq +ř +s1 +iPSi exp pβuis1 +iq +ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +(48) +12 + +where uisi “ ř +jPVi eJ +siAijµj. Define ¯µ :“ t¯µjujPVi and +fsiptµjujPViq :“ +exp pβ ř +jPVi eJ +siAijµjq +ř +s1 +iPSi exp pβ ř +jPVi eJ +s1 +iAijµjq. +(49) +Applying the Taylor expansion to approximate this function at the mean belief ¯µ, we have +fsiptµjujPViq « fsip¯µq ` ∇fsip¯µq ¨ pµ ´ ¯µq ` 1 +2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq ` Op||µ ´ ¯µ||3q +(50) +where H denotes the Hessian matrix. Hence, we can rewrite Equation 48 as +¯xisi “ +ż +fsiptµjujPViq +ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +(51) +« fsip¯µq ` +ż +∇fsip¯µq ¨ µ +ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +´ ∇fsip¯µq ¨ ¯µ +` +ż 1 +2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq +ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +` +ż +Op||µ ´ ¯µ||q3 ź +jPVi +ppµj, tq +˜ ź +jPVi +dµj +¸ +(52) +Observe that in Equation 52, the second and the third term can be canceled out. Moreover, for any +two neighbor populations j, k P Vi, the beliefs µj, µk about these two populations are separate and +independent. Hence, the covariance of these beliefs are zero. We apply the moment closure approximation +[32, 13] with the second order and obtain +¯xisi « fsip¯µq ` 1 +2 +ÿ +jPVi +ÿ +sjPSj +B2fsip¯µq +pBµjsjq2 Varpµjsjq. +(53) +Hence, substituting ¯xisi in Lemma 4 with the above approximation, we have the mean belief dynamics +d¯µisi +dt +« fsip¯µq ´ ¯µisi +λ ` t ` 1 +` +ř +jPVi +ř +sjPSj +B2fsip¯µq +pBµjsj q2 Varpµjsjq +2pλ ` t ` 1q +. +(54) +Q.E.D. +Remarks: the use of the moment closure approximation (considering only the first and the second +moments) is for obtaining more conclusive results. Strictly speaking, the mean belief dynamics also depend +on the third and higher moments. However, we observe in the experiments that these moments in general +have little effects on the mean belief dynamics. To be more specific, given the same initial mean beliefs, +while the variance of initial beliefs sometimes can change the limit behaviors of a system, we do not +observe similar phenomena for the third and higher moments. +Proof of Proposition 3 +Consider a population i. It follows from Equation 7 in the main paper that the change in the beliefs +about this population can be written as follows. +µipt ` 1q “ µiptq ` xiptq ´ µiptq +λ ` t ` 1 +. +(55) +Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s. We rewrite +the above equation as +µipt ` δq “ µiptq ` δ xiptq ´ µiptq +λ ` t ` 1 +. +(56) +Move the term µiptq to the right hand side and divide both sides by δ, +µipt ` δq ´ µiptq +δ +“ xiptq ´ µiptq +λ ` t ` 1 +. +(57) +13 + +Assume that the amount of time δ between two successive time steps goes to zero. we have +dµi +dt “ lim +δÑ0 +µipt ` δq ´ µiptq +δ +“ xiptq ´ µiptq +λ ` t ` 1 +. +(58) +Note that for continuous-time dynamics, we usually drop the time index in the bracket, yielding the belief +dynamics (Equation 11) in Proposition 3. Q.E.D. +Proof of Theorem 1 +Without loss of generality, we consider the variance of the belief µisi about strategy si of population i. +Note that +Varpµisiq “ Erpµisiq2s ´ p¯µisiq2. +(59) +Hence, we have +dVarpµisiq +dt +“ dErpµisiq2s +dt +´ 2¯µisi +d¯µisi +dt . +(60) +Consider the first term on the right hand side. We apply the Leibniz rule to interchange differentiation +and integration, and then substitute Bppµi,tq +Bt +with Equation 8 in the main paper. +dErpµisiq2s +dt +“ +ż +pµisiq2 Bppµi, tq +Bt +dµi +(61) +“ ´ +ż +pµisiq2∇ ¨ +ˆ +ppµi, tq ¯xi ´ µi +λ ` t ` 1 +˙ +dµi +(62) +“ ´ +ż +pµisiq2 ÿ +siPSi +Bµisi +ˆ +ppµi, tq ¯xisi ´ µisi +λ ` t ` 1 +˙ +dµi +(63) +“ γ +ż +pµisiq2 ÿ +siPSi +Bµisi ppµi, tq p¯xisi ´ µisiq dµi ` γ +ż +pµisiq2ppµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi +(64) +where γ :“ ´ +1 +λ`t`1. Applying integration by parts to the first term in Equation 64 yields +ż +pµisiq2 ÿ +siPSi +Bµisi ppµi, tq p¯xisi ´ µisiq dµi +“ ´ +ż +pµisiq2ppµi, tq +» +– ÿ +s1 +iPSi +Bµis1 +i p¯xis1 +i ´ µis1 +iq +fi +fl ` ppµi, tqBµisi +“ +pµisiq2p¯xisi ´ µisiq +‰ +dµi +(65) +where we have leveraged that the probability mass at the boundary remains zero. Combining the above +two equations, we obtain +dErpµisiq2s +dt +“ ´γ +ż +pµisiq2ppµi, tq +» +– ÿ +s1 +iPSi +Bµis1 +i p¯xis1 +i ´ µis1 +iq +fi +fl ` ppµi, tqBµisi +“ +pµisiq2p¯xisi ´ µisiq +‰ +dµi +` γ +ż +pµisiq2ppµi, tq +ÿ +siPSi +Bµisi p¯xisi ´ µisiq dµi +(66) +“ γ +ż “ +´ppµi, tqBµisi +“ +pµisiq2p¯xisi ´ µisiq +‰‰ +` pµisiq2ppµi, tqBµisi p¯xisi ´ µisiq dµi +(67) +“ γ +ż +2pµisiq2ppµi, tqdµi ´ γ +ż +2¯xisiµisippµi, tqdµi +(68) +“ ´2Erpµisiq2s ´ 2¯xisi ¯µisi +λ ` t ` 1 +. +(69) +14 + +Next, we consider the second term in Equation 60. By Lemma 4, we have +2¯µisi +d¯µisi +dt +“ 2¯µisip¯xisi ´ ¯µisiq +λ ` t ` 1 +. +(70) +Combining Equations 69 and 70, the dynamics of the variance is +dVarpµisiq +dt +“ ´2Erpµisiq2s ´ 2¯xisi ¯µisi +λ ` t ` 1 +´ 2¯µisip¯xisi ´ ¯µisiq +λ ` t ` 1 +(71) +“ 2p¯µisiq2 ´ 2Erpµisiq2s +λ ` t ` 1 +(72) +“ ´2Varpµisiq +λ ` t ` 1 . +(73) +Q.E.D. +Remarks: We believe that the rationale behind such a phenomenon is twofold: 1) agents apply smooth +fictitious play, and 2) agents respond to the mean strategy play of other populations rather than the +strategy play of some fixed agents. Regarding the former, we notice that under a similar setting, population +homogenization may not occur if agents apply other learning methods, e.g., Q-learning and Cross learning. +Regarding the latter, imagine that agents adjust their beliefs in response to the strategies of some fixed +agents. For example, consider two populations; one contains agents A and C, and the other one contains +agents B and D. Suppose that agents A and B form a fixed pair such that they adjust their beliefs only in +response to each other; the same applies to agents C and D. Belief homogenization may not happen. +Appendix B: Proofs omitted in Section 4.1 +Proof of Theorem 2 +Belief homogenization implies that the fixed points of systems with initially heterogeneous beliefs are the +same as in systems with homogeneous beliefs. Thus, we focus on homogeneous systems to analyze the +fixed points. It is straightforward to see that +dµi +dt “ xi ´ µi +λ ` t ` 1 “ 0 ùñ xi “ µi. +(74) +Denote the fixed points of the system dynamics, which satisfies the above equation, by px˚ +i , µ˚ +i q for each +population i. By the logit choice function (Equation 5 in the main paper), we have +x˚ +isi “ +exp pβuisiq +ř +s1 +iPSi exp pβuis1 +iq “ +exp pβ ř +jPVi eJ +siAijµ˚ +j q +ř +s1 +iPSi exp pβ ř +jPVi eJ +s1 +iAijµ˚ +j q. +(75) +Leveraging that x˚ +i “ µ˚ +i , @i P V at the fixed points, we can replace µ˚ +j with x˚ +j . Q.E.D. +Proof of Theorem 3 +Consider a population i. The set of neighbor populations is Vi, the set of beliefs about the neighbor +populations is tµjujPVi, and the choice distribution is xi. Given a population network game Γ, the +expected payoff is given by xJ +i +ř +pi,jqPE Aijµj. Define a perturbed payoff function +πi +` +xi, tµjujPVi +˘ :“ xJ +i +ÿ +jPVi +Aijµj ` vpxiq +(76) +where vpxiq “ ´ 1 +β +ř +siPSi xisi lnpxisiq. Under this form of vpxiq, the maximization of πi yields the choice +distribution xi from the logit choice function [8]. Based on this, we establish the following lemma. +Lemma 5. For a choice distribution xi of SFP in a population network game, +Bxiπi +` +xi, tµjujPVi +˘ +“ 0 +and +ÿ +jPVi +` +Aijµj +˘J “ ´Bxivpxiq. +(77) +Proof. This lemma immediately follows from the fact that the maximization of πi will yield the choice +distribution xi from the logit choice function [8]. +15 + +The belief dynamics of a homogeneous populations can be simplified after time-reparameterization. +Lemma 6. Given τ “ ln λ`t`1 +λ`1 , the belief dynamics of homogeneous systems (given in Equation 11 in +the main paper) is equivalent to +dµi +dτ “ xi ´ µi. +(78) +Proof. From τ “ ln λ`t`1 +λ`1 , we have +t “ pλ ` 1qpexp pτq ´ 1q. +(79) +By the chain rule, for each dimension si, +dµisi +dτ +“ dµisi +dt +dt +dτ +(80) +“ xisi ´ µisi +λ ` t ` 1 +d ppλ ` 1qpexp pτq ´ 1qq +dτ +(81) +“ +xisi ´ µisi +λ ` pλ ` 1qpexp pτq ´ 1q ` 1pλ ` 1q exp pτq +(82) +“ xisi ´ µisi. +(83) +Next, we define the Lyapunov function L as +L :“ +ÿ +iPV +ωiLi +s.t. +Li :“ πi +` +xi, tµjujPVi +˘ +´ πi +` +µi, tµjujPVi +˘ +. +(84) +where tωiuiPV is the set of positive weights defined in the weighted zero-sum Γ. The function L is +non-negative because for every i P V , xi maximizes the function πi. When for every i P V , xi “ µi, the +function L reaches the minimum value 0. +Rewrite L as +L “ +ÿ +iPV +« +ωiπi +` +xi, tµjujPVi +˘ +´ ωiµJ +i +ÿ +jPVi +Aijµj ´ ωivpµiq +ff +. +(85) +We observe that πi +` +xi, tµjujPVi +˘ +is convex in µj, j P Vi by Danskin’s theorem, and ´vpµiq is strictly +convex in µi. Moreover, by the weighted zero-sum property given in Equation 2 in the main paper, we +have +ÿ +iPV +˜ +ωiµJ +i +ÿ +jPVi +Aijµj +¸ +“ 0 +(86) +since µi P ∆i, µj P ∆j for every i, j P V. Therefore, the function L is a strictly convex function and attains +its minimum value 0 at a unique point xi “ µi, @i P V. +Consider the function Li. Its time derivative is +9Li “ Bxiπi +` +xi, tµjujPVi +˘ +9xi ` +ÿ +jPVi +” +Bµjπi +` +xi, tµjujPVi +˘ 9µj +ı +´ Bµiπi +` +µi, tµjujPVi +˘ 9µi ´ +ÿ +jPVi +” +Bµjπi +` +µi, tµjujPVi +˘ 9µj +ı +. +(87) +Note that the partial derivative Bxiπi equals 0 by Lemma 5. Thus, we can rewrite this as +9Li “ Bµiπi +` +µi, tµjujPVi +˘ 9µi ` +ÿ +jPVi +” +Bµjπi +` +xi, tµjujPVi +˘ +´ Bµjπi +` +µi, tµjujPVi +˘ı +9µj +(88) +“ ´ +« ÿ +jPVi +` +Aijµj +˘J ` Bµivpµiq +ff +pxi ´ µiq ` +ÿ +jPVi +` +xJ +i Aij ´ µJ +i Aij +˘ +pxj ´ µjq +(89) +“ rBxivpxiq ´ Bµivpµiqs pxi ´ µiq ` +ÿ +jPVi +` +xJ +i Aijxj ´ µJ +i Aijxj ´ xJ +i Aijµj ` µJ +i Aijµj +˘ +. +(90) +16 + +where from Equation 89 to 90, we apply Lemma 5 to substitute ř +jPVi +` +Aijµj +˘J with ´Bxivpxiq. Hence, +summing over all the populations, the time derivative of L is +9L “ +ÿ +iPV +ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq +` +ÿ +iPV +ÿ +jPVi +ωi +` +xJ +i Aijxj ´ µJ +i Aijxj ´ xJ +i Aijµj ` µJ +i Aijµj +˘ +. +(91) +The summation in the second line is equivalent to +ÿ +pi,jqPE +pωixJ +i Aijxj ` ωjxJ +j Ajixiq ´ pωiµJ +i Aijxj ` ωjxJ +j Ajiµiq +(92) +´ pωixJ +i Aijµj ` ωjµJ +j Ajixiq ` pωiµJ +i Aijµj ` ωjµJ +j Ajiµiq. +(93) +By the weighted zero-sum property given in Equation 2 in the main paper, this summation equals 0, +yielding +9L “ +ÿ +iPV +ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq. +(94) +Note that the function v is strictly concave such that its second derivative is negative definite. By this +property, 9L ď 0 with equality only if xi “ µi, @i P V , which corresponds to the QRE. Therefore, L is a +strict Lyapunov function, and the global asymptotic stability of the QRE follows. Q.E.D. +Remarks: Intuitively, the Lyapunov function defined above measures the distance between the QRE +and a given set of beliefs. The idea of measuring the distance in terms of entropy-regularized payoffs is +inspired from the seminal work [19]. However, different from the network games considered in this paper, +Hofbauer and Hopkins [19] consider SFP in two-player games. To our knowledge, so far there has been +no systematic study on SFP in network games. +Proof of Theorem 4 +The proof of Theorem 4 leverages the seminal results of the asymptotically autonomous dynamical system +[31, 40, 41] which conventionally is defined as follows. +Definition 1. A nonautonomous system of differential equations in Rn +x1 “ fpt, xq +(95) +is said to be asymptotically autonomous with limit equation +y1 “ gpyq, +(96) +if fpt, xq Ñ gpxq, t Ñ 8, where the convergence is uniform on each compact subset of Rn. Conventionally, +the solution flow of Eq. 95 is called the asymptotically autonomous semiflow (denoted by φ) and the +solution flow of Eq. 96 is called the limit semiflow (denoted by Θ). +Based on this definition, we establish Lemma 1 in the main paper, which is repeated as follows. +Lemma 7. For a system that initially has heterogeneous beliefs, the mean belief dynamics is asymptotically +autonomous [31] with the limit equation +dµi +dt “ xi ´ µi +(97) +which after time-reparameterization is equivalent to the belief dynamics for homogeneous systems. +Proof. We first time-reparameterize the mean belief dynamics of heterogeneous systems. Assume τ “ +17 + +Figure 5: Population network games where the underlying network consists of star structure. +ln λ`t`1 +λ`1 . By the chain rule and Equation 54, for each dimension si, +d¯µisi +dτ +“ d¯µisi +dt +dt +dτ +(98) +“ +» +—–fsip¯µq ´ ¯µisi +λ ` t ` 1 +` +ř +jPVi +ř +sjPSj +B2fsip¯µq +pBµjsj q2 Varpµjsjq +2pλ ` t ` 1q +fi +ffifl d ppλ ` 1qpexp pτq ´ 1qq +dτ +(99) +“ +fsip¯µq ´ ¯µisi ` 1 +2 +ř +jPVi +ř +sjPSj +B2fsip¯µq +pBµjsj q2 +´ +λ`1 +λ`t`1 +¯2 +σ2pµjsjq +λ ` pλ ` 1qpexp pτq ´ 1q ` 1 +pλ ` 1q exp pτq +(100) +“ fsip¯µq ´ ¯µisi ` 1 +2 +ÿ +jPVi +ÿ +sjPSj +B2fsip¯µq +pBµjsjq2 σ2pµjsjq exp p´2τq. +(101) +Observe that exp p´2τq decays to zero exponentially fast and that both σ2pµjsjq and +B2fsip¯µq +pBµjsj q2 are bounded +for every µ in the simplex ś +jPVi ∆j. Hence, Equation 101 converges locally and uniformly to the following +equation: +d¯µisi +dτ +“ fsip¯µq ´ ¯µisi. +(102) +Note that xisi “ fsip¯µq for homogeneous systems, and the above equation is algebraically equivalent to +Equation 97. Hence, by Definition 1, Equation 101 is asymptotically autonomous with the limit equation +being Equation 97. +By the above lemma, we can formally connect the limit behaviors of initially heterogeneous systems +and those of homogeneous systems. Recall that Theorem 3 in the main paper states that under SFP, +there is a unique rest point (QRE) for the belief dynamics in a weighted zero-sum network game Γ; +this excludes the case where there are finitely many equilibria that are chained to each other. Hence, +combining Lemma 2 in the main paper, we prove that the mean belief dynamics of initially heterogeneous +systems converges to a unique QRE. Q.E.D. +Appendix C: Results and Proofs omitted in Section 4.2 +For the case of network coordination, we consider networks that consist of a star or disconnected multiple +stars due to technical reasons. In Figure 1, we present examples of the considered network structure with +different numbers of nodes (populations). +In the following theorem, focusing on homogeneous systems, we establish the convergence of the belief +dynamics to the set of QRE. +Theorem 6 (Convergence in Homogeneous Network Coordination with Star Structure). +Given a coordination Γ where the network structure consists of a single or disconnected multiple stars, +each orbit of the belief dynamics for homogeneous systems converges to the set of QRE. +Proof. Consider a root population j of a star structure. Its set of leaf (neighbor) populations is Vj, the +set of beliefs about the leaf populations is tµiuiPVj, and the choice distribution is xj. Given the game Γ, +18 + +Five Populations +Five Populations +(Two Disconnected +Stars) +Three Populations +Two Populations +P2 +P1 +P3the expected payoff is xJ +j +ř +iPVj Ajiµi. Define a perturbed payoff function +πj +` +xj, tµiuiPVj +˘ :“ xJ +j +ÿ +iPVj +Ajiµi ` vpxjq +(103) +where vpxjq “ ´ 1 +β +ř +sjPSj xjsj lnpxjsjq. Under this form of vpxjq, the maximization of πj yields the choice +distribution xj from the logit choice function [8]. +Consider a leaf population i of the root population j. It has only one neighbor population, which +is population j. Thus, given the game Γ, the expected payoff is xJ +i Aijµj. Define a perturbed payoff +function +πi +` +xi, µj +˘ :“ xJ +i Aijµj ` vpxiq +(104) +where vpxiq “ ´ 1 +β +ř +siPSi xisi lnpxisiq. Similarly, the maximization of πi yields the choice distribution xi +from the logit choice function [8]. Based on this, we establish the following lemma. +Lemma 8. For choice distributions of SFP in a population network game with start structure, +Bxjπj +` +xj, tµiuiPVj +˘ +“ 0 +and +ÿ +iPVj +pAjiµiqJ “ ´Bxjvpxjq +if j is a root population, +(105) +Bxiπi +` +xi, µj +˘ +“ 0 +and +` +Aijµj +˘J “ ´Bxivpxiq +if i is a leaf population. +(106) +Proof. This lemma immediately follows from the fact that the maximization of πj and πi , respectively, +yield the choice distributions xj and xi from the logit choice function [8]. +For readability, we repeat the belief dynamics of a homogeneous population after time-reparameterization, +which has been proved in Lemma 4 in Appendix B, as follows: +dµi +dτ “ xi ´ µi. +(107) +Let R Ă V be the set of all root populations. We define +L :“ +ÿ +jPR +Lj +s.t. +Lj :“ µJ +j +ÿ +iPVj +Ajiµi ` vpµjq ` +ÿ +iPVj +vpµiq. +(108) +Consider the function Lj. Its time derivative 9Lj is +9Lj “ +» +–BµjpµJ +j +ÿ +iPVj +Ajiµiq 9µj ` +ÿ +iPVj +BµipµJ +j +ÿ +iPVj +Ajiµiq 9µi +fi +fl ` Bµjvpµjq 9µj ` +ÿ +iPVj +Bµivpµiq 9µi +(109) +“ +ÿ +iPVj +pAjiµiqJpxj ´ µjq ` +» +– ÿ +iPVj +µJ +j Ajipxi ´ µiq +fi +fl ` Bµjvpµjqpxj ´ µjq ` +ÿ +iPVj +Bµivpµiqpxi ´ µiq. (110) +Since Γ is a coordination game, we have +` +Aijµj +˘J “ µJ +j AJ +ij “ µJ +j Aji. Hence, applying Lemma 8, we can +substitute ř +iPVjpAjiµiqJ with ´v1pxjq, and µJ +j Aji with ´v1pxiq, yielding +9Lj “ ´Bxjvpxjqpxj ´ µjq ` +» +– ÿ +iPVj +p´Bxivpxiqqpxi ´ µiq +fi +fl ` Bµjvpµjqpxj ´ µjq ` +ÿ +iPVj +Bµivpµiqpxi ´ µiq +(111) +“ pBµjvpµjq ´ Bxjvpxjqqpxj ´ µjq ` +ÿ +iPVj +pBµivpµiq ´ Bxivpxiqqpxi ´ µiq +(112) +Note that the function v is strictly concave such that its second derivative is negative definite. By this +property, 9Lj ě 0 with equality only if xi “ µi, @i P Vj and xj “ µj. Thus, the time derivative of the +function L, i.e., 9L “ ř +jPR 9Lj ě 0 with equality only if xi “ µi, @i P Vj, xj “ µj, @j P R. +We generalize the convergence result to initially heterogeneous systems in the following theorem. +19 + +Theorem 7 (Convergence in Initially Heterogeneous Network Coordination with Star Struc- +ture). Given a coordination Γ where the network structure consists of a single or disconnected multiple +stars, each orbit of the mean belief dynamics for initially heterogeneous systems converges to the set of +QRE. +Proof. The proof technique is similar to that for initially heterogeneous competitive network games. By +Lemma 1 in the main paper, we show that the mean belief dynamics of initially heterogeneous systems is +asymptotically autonomous with the belief dynamics of homogeneous systems. Therefore, it follows from +Lemma 2 in the main paper that the convergence result for homogeneous systems can be carried over to +the initially heterogeneous systems. +Remarks: The convergence of SFP in coordination games and potential games has been established +under the 2-player settings [19] as well as some n-player settings [20, 39]. Our work differs from the +previous works in two aspects. First, our work allows for heterogeneous beliefs. Moreover, we consider that +agents maintain separate beliefs about other agents, while in the previous works agents do not distinguish +between other agents. Thus, even when the system beliefs are homogeneous, our setting is still different +from (and more complicated) than the previous settings. +Appendix D: Omitted Experimental Details +Numerical Method for the PDE model. +PDEs are notoriously difficult to solve, and only limited +types of PDEs allow analytic solutions. Hence, similar to previous research [23], we resort to numerical +method for PDEs; in particular, we consider the finite difference method [38]. +Agent-based Simulations. +The presented simulation results are averaged over 100 independent +simulation runs to smooth out the randomness. For each simulation run, there are 1, 000 agents in each +population. For each agent, the initial beliefs are sampled from the given initial probability distribution. +Detailed Experimental Setups for Figure 1. +In the case of small initial variance, the initial +beliefs µ1H and µ2H are distributed according to the distribution Betap280, 120q. On the contrary, +in the case of large initial variance, the initial beliefs µ1H and µ2H are distributed according to the +distribution Betap14, 6q. Thus, initially, the mean beliefs in these two cases are both ¯µ1H “ ¯µ2H “ 0.7 +and ¯µ1S “ ¯µ2S “ 0.3. In both cases, the initial sum of weights λ “ 10 and the temperature β “ 10. +Detailed Experimental Setups for Figure 3. +We visualize the regions of attraction of different +equilibria in stag hunt games by numerically solving the mean belief dynamics (Equation 10 in the main +paper). The initial variances have been given in the title of each panel. In all cases, the initial sum of +weights λ “ 0 and the temperature β “ 5. +Detailed Experimental Setups for Figure 4. +We let the initial beliefs about populations 1, 3 and 5 +remain unchanged across different cases, and vary the initial beliefs about populations 2 and 4. The initial +beliefs about populations 1, 3 and 5, denoted by µ1H, µ3H and µ5H, are distributed according to the +distributions Betap20, 10q, Betap6, 4q, and Betap10, 5q, respectively. The initial beliefs about populations +2 and 4 have been given in the legends of Figure 4. In all cases, the initial sum of weights λ “ 10 and the +temperature β “ 10. Note that µiT “ 1 ´ µiH for all populations i “ 1, 2, 3, 4, 5. +Source Code and Computing Resource. +We have attached the source code for reproducing our +main experiments. The Matlab script finitedifference.m numerically solves our PDE model presented +in Proposition 1 in the main paper. The Matlab script regionofattraction.m visualizes the region of +attraction of different equilibria in stag hunt games, which are presented in Figure 3. The Python scripts +simulation(staghunt).py and simulation(matchingpennies).py correspond to the agent-based simulations +in two-population stag hunt games and five-population asymmetric matching pennies games, respectively. +We use a laptop (CPU: AMD Ryzen 7 5800H) to run all the experiments. +20 + +References +[1] Yakov Babichenko and Aviad Rubinstein. Settling the complexity of nash equilibrium in congestion +games. 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Max- +imum entropy population based training for zero-shot human-ai coordination. +arXiv preprint +arXiv:2112.11701, 2021. +23 + diff --git a/QtE4T4oBgHgl3EQfKQz1/content/tmp_files/load_file.txt b/QtE4T4oBgHgl3EQfKQz1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70ed28bb572a861872d20e8bff85a73100f729ba --- /dev/null +++ b/QtE4T4oBgHgl3EQfKQz1/content/tmp_files/load_file.txt @@ -0,0 +1,741 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf,len=740 +page_content='Heterogeneous Beliefs and Multi-Population Learning in Network Games Shuyue Hu1, Harold Soh2, and Georgios Piliouras3 1Shanghai Artificial Intelligence Laboratory 2National University of Singapore 3Singapore University of Technology and Design Abstract The effect of population heterogeneity in multi-agent learning is practically relevant but remains far from being well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Motivated by this, we introduce a model of multi-population learning that allows for heterogeneous beliefs within each population and where agents respond to their beliefs via smooth fictitious play (SFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We show that the system state — a probability distribution over beliefs — evolves according to a system of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We establish the convergence of SFP to Quantal Response Equilibria in different classes of games capturing both network competition as well as network coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We also prove that the beliefs will eventually homogenize in all network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Although the initial belief heterogeneity disappears in the limit, we show that it plays a crucial role for equilibrium selection in the case of coordination games as it helps select highly desirable equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Contrary, in the case of network competition, the resulting limit behavior is independent of the initialization of beliefs, even when the underlying game has many distinct Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 1 Introduction Smooth Fictitious play (SFP) and variants thereof are arguably amongst the most well-studied learning models in AI and game theory [2, 3, 21, 22, 9, 19, 36, 37, 42, 18, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' SFP describes a belief-based learning process: agents form beliefs about the play of opponents and update their beliefs based on observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Informally, an agent’s belief can be thought as reflecting how likely its opponents will play each strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' During game plays, each agent plays smoothed best responses to its beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Much of the literature of SFP is framed in the context of homogeneous beliefs models where all agents in a given role have the same beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This includes models with one agent in each player role [3, 2, 39] as well as models with a single population but in which all agents have the same beliefs [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' SFP are known to converge in large classes of homogeneous beliefs models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', most 2-player games [9, 19, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' However, in the context of heterogeneous beliefs, where agents in a population have different beliefs, SFP has been explored to a less extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The study of heterogeneous beliefs (or more broadly speaking, population heterogeneity) is important and practically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' From multi-agent system perspective, heterogeneous beliefs widely exist in many applications, such as traffic management, online trading and video game playing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For example, it is natural to expect that public opinions generally diverge on autonomous vehicles and that people have different beliefs about the behaviors of taxi drivers vs non-professional drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' From machine learning perspective, recent empirical advances hint that injecting heterogeneity potentially accelerates population-based training of neural networks and improves learning performance [25, 29, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' From game theory perspective, considering heterogeneity of beliefs better explains results of some human experiments [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Heterogeneous beliefs models of SFP are not entirely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the pioneering work [12], Fudenberg and Takahashi examine the heterogeneity issue in 2-population settings by appealing to techniques from the stochastic approximation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This approach, which is typical in the SFP literature, relates the limit behavior of each individual to an ordinary differential equation (ODE) and has yielded significant insights for many homogeneous beliefs models [3, 2, 19, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' However, this approach, as also noted by Fudenberg and Takahashi, “does not provide very precise estimates of the effect of the initial condition of the system.” Consider an example of a population of agents each can choose between two pure strategies 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='04929v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='MA] 12 Jan 2023 s1 and s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Let us imagine two cases: (i) every agents in the population share the same belief that their opponents play a mixed strategy choosing s1 and s2 with equal probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='5, and (ii) half of the agents believe that their opponents determinedly play the pure strategy s1 and the other half believe that their opponents determinedly play the pure strategy s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The stochastic approximation approach would generally treat these two cases equally, providing little information about the heterogeneity in beliefs as well as its consequential effects on the system evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This drives our motivating questions: How does heterogeneous populations evolve under SFP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' How much and under what conditions does the heterogeneity in beliefs affect their long-term behaviors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Model and Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In this paper, we study the dynamics of SFP in general classes of multi- population network games that allow for heterogeneous beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In a multi-population network game, each vertex of the network represents a population (continuum) of agents, and each edge represents a series of 2-player subgames between two neighboring populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that multi-population network games include all the 2-population games considered in [12] and are representation of subclasses of real-world systems where the graph structure is evident [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We consider that for a certain population, individual agents form separate beliefs about each neighbor population and observe the mean strategy play of that population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Taking a approach different from stochastic approximation, we define the system state as a probability measure over the space of beliefs, which allows us to precisely examine the impact of heterogeneous beliefs on system evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This probability measure changes over time in response to agents’ learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, the main challenge is to analyze the evolution of the measure, which in general requires the development of new techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As a starting point, we establish a system of partial differential equations (PDEs) to track the evolution of the measure in continuous time limit (Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The PDEs that we derive are akin to the continuity equations1 commonly encountered in physics and do not allow for a general solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Appealing to moment closure approximation [13], we circumvent the need of solving the PDEs and directly analyze the dynamics of the mean and variance (Proposition 2 and Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As one of our key results, we prove that the variance of beliefs always decays quadratically fast with time in all network games (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Put differently, eventually, beliefs will homogenize and the distribution of beliefs will collapse to a single point, regardless of initial distributions of beliefs, 2-player subgames that agents play, and the number of populations and strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This result is non-trivial and perhaps somewhat counterintuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Afterall, one may find it more natural to expect that the distribution of beliefs would converge to some distribution rather than a single point, as evidenced by recent studies on Q-learning and Cross learning [23, 24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Technically, the eventual belief homogenization has a significant implication — it informally hints that the asymptotic system state of initially heterogeneous systems are likely to be the same as in homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We show that the fixed point of SFP correspond to Quantual Response Equilibria (QRE)2 in network games for both homogeneous and initially heterogeneous systems (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As our main result, we establish the convergence of SFP to QRE in different classes of games capturing both network competition as well as network coordination, independent of belief initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Specifically, for competitive network games, we first prove via a Lyapunov argument that the SFP converges to a unique QRE in homogeneous systems, even when the underlying game has many distinct Nash equilibria (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Then, we show that this convergence result can be carried over to initially heterogeneous systems (Theorem 4), by leveraging that the mean belief dynamics of initially heterogeneous systems is asymptotically autonomous [31] with its limit dynamics being the belief dynamics of a homogeneous system (Lemma 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For coordination network games, we also prove the convergence to QRE for homogeneous and initially heterogeneous systems, in which the underlying network has star structure (Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the other hand, the eventual belief homogenization may lead to a misconception that belief heterogeneity has little effect on system evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Using an example of 2-population stag hunt games, we show that belief heterogeneity actually plays a crucial role in equilibrium selection, even though it eventually vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As shown in Figure 1, changing the variance of initial beliefs results in different limit behaviors, even when the mean of initial beliefs remains unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' in particular, while a small variance leads to the less desirable equilibrium pH, Hq, a large variance leads to the payoff dominant equilibrium pS, Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, in the case of network coordination, initial belief heterogeneity can help select the highly desirable equilibrium and provides interesting insights to the seminal thorny problem of equilibrium selection [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the contrary, in the case of network competition, we prove (Theorems 3 and 4 on the 1The continuity equation is a PDE that describes the transport phenomena of some quantity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', mass, energy, momentum and other conserved quantities) in a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 2QRE is a game theoretic solution concept under bounded rationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By QRE, in this paper we refer to their canonical form also referred to as logit equilibria or logit QRE in the literature [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 2 Figure 1: The system dynamics under the effects of different variances of initial beliefs (thin lines: predictions of our PDE model, shaded wide lines: simulation results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' ¯µ2S represents the mean belief about population 2 and ¯x1S represents the mean probability of playing strategy S in population 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Initially, we set the mean beliefs ¯µ2S “ ¯µ1S “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='3 (details of the setup are summarized in the supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given the same initial mean belief, different initial variances σ2pµ2Sq lead to the convergence to different beliefs (the left panel) and even to different strategy choices (the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In particular, a large initial variance helps select the payoff dominant equilibrium pS, Sq in stag hunt games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' convergence to a unique QRE in competitive network games) as well as showcase experimentally that the resulting limit behavior is independent of initialization of beliefs, even if the underlying game has many distinct Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Related Works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' SFP and its variants have recently attracted a lot of attention in AI research [36, 37, 42, 18, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' There is a significant literature that analyze SFP in different models [3, 7, 21, 19], and the paper that is most closely related to our work is [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Fudenberg and Takahashi [12] also examines the heterogeneity issue and anticipate belief homogenization in the limit under 2-population settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In this paper, we consider multi-population network games, which is a generalization of their setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='3 Moreover, our approach is more fundamental, as the PDEs that we derive can provide much richer information about the system evolution and thus precisely estimates the temporal effects of heterogeneity, which is generally intractable in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Therefore, using our approach, we are able to show an interesting finding — the initial heterogeneity plays a crucial role in equilibrium selection (Figure 1) — which unfortunately cannot be shown using the approach in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Last but not least, to our knowledge, our paper is the first work that presents a systematic study of smooth fictitious play in general classes of network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the other hand, networked multi-agent learning constitutes one of the current frontiers in AI and ML research [43, 30, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Recent theoretical advances on network games provide conditions for learning behaviors to be not chaotic [6, 34], and investigate the convergence of Q-learning and continuous-time FP in the case of network competitions [7, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' However, [7, 28] consider that there is only one agent on each vertex, and hence their models are essentially for homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lahkar and Seymour [27] and Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' [23, 24] also use the continuity equations as a tool to study population heterogeneity in multi-agent systems where a single population of agents applies Cross learning or Q-learning to play symmetric games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' They either prove or numerically showcase that heterogeneity generally persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Our results complement these advances by showing that heterogeneity vanishes under SFP and that heterogeneity helps select highly desirable equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Moreover, methodologically, we establish new proof techniques for the convergence of learning dynamics in heterogeneous systems by leveraging seminal results (Lemmas 1 and 2) from the asymptotically autonomous dynamical system literature, which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 3The analysis presented in this paper covers all generic 2-population network games, all generic bipartite network games where the game played on each edge is the same along all edges, and all weighted zero-sum games which do not require the graph to be bipartite nor to have the same game played on each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 3 Small vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=" Large Variance of Initial Population Beliefs Mean Belief about the Others' Playing S Mean Probability of Playing S 1S 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 Mean Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 )~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='023 (μ2s) ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='A α(μ2s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='4 g(μ2s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2 50 100 150 200 50 100 150 200 Time t Time t2 Preliminaries Population Network Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' A population network game (PNG) Γ “ pN, pV, Eq, pSi, ωiq@iPV , pAijqpi,jqPEq consists of a multi-agent system N distributed over a graph pV, Eq, where V “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', nu is the set of vertices each represents a population (continuum) of agents, and E is the set of pairs, pi, jq, of population i ‰ j P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For each population i P V , agents of this population has a finite set Si of pure strategies (or actions) with generic elements si P Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Agents may also use mixed strategies (or choice distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For an arbitrary agent k in population i, its mixed strategy is a vector xipkq P ∆i, where ∆i is the simplex in R|Si| such that ř siPSi xisipkq “ 1 and xisipkq ě 0, @si P Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Each edge pi, jq P E defines a series of two-player subgames between populations i and j, such that for a given time step, each agent in population i is randomly paired up with another agent in population j to play a two-player subgame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We denote the payoff matrices for agents of population i and j in these two-player subgames by Aij P R|Si|ˆ|Sj| and Aji P R|Sj|ˆ|Si|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that at a given time step, each agent chooses a (mixed or pure) strategy and plays that strategy in all two-player subgames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Let x “ pxi, txjupi,jqPEq be a mixed strategy profile, where xi (or xj) denotes a generic mixed strategy in population i (or j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given the mixed strategy profile x, the expected payoff of using xi in the game Γ is ripxq “ ripxi, txjupi,jqPEq :“ ÿ pi,jqPE xJ i Aijxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (1) The game Γ is competitive (or weighted zero-sum), if there exist positive constants ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' , ωn such that ÿ iPV ωiripxq “ ÿ pi,jqPE ` ωixJ i Aijxj ` ωjxJ j Ajixi ˘ “ 0, @x P ź iPV ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (2) On the other hand, Γ is a coordination network game, if for each edge pi, jq P E, the payoff matrices of the two-player subgame satisfy Aij “ AJ ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Smooth Fictitious Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' SFP is a belief-based model for learning in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In SFP, agents form beliefs about the play of opponents and respond to the beliefs via smooth best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a game Γ, consider an arbitrary agent k in a population i P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Let Vi “ tj P V : pi, jq P Eu be the set of neighbor populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Agent k maintains a weight κi jsjpkq for each opponent strategy sj P Sj of a neighbor population j P Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Based on the weights, agent k forms a belief about the neighbor population j, such that each opponent strategy sj is played with probability µi jsjpkq “ κi jsjpkq ř s1 jPSj κi js1 jpkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (3) Let µi jpkq be the vector of beliefs with the sj-th element equals µi jsjpkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Agent k forms separate beliefs for each neighbor population, and plays a smooth best response to the set of beliefs tµi jpkqujPVi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a game Γ, agent k’s expected payoff for using a pure strategy si P Si is uisipkq “ ripesi, tµi jpk, tqujPViq “ ÿ jPVi eJ siAijµi jpkq (4) where esi is a unit vector where the si-th element is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The probability of playing strategy si is then given by xisipkq “ exppβuisipkqq ř s1 iPSi exppβuis1 ipkqq (5) where β is a temperature (or the degree of rationality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We consider that agents observe the mean mixed strategy of each neighbor population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As such, at a given time step t, agent k updates the weights for each opponent strategy sj P Sj, j P Vi as follows: κi jsjpk, t ` 1q “ κi jsjpk, tq ` ¯xjsjptq (6) where ¯xjsj is the mean probability of playing strategy sj in population j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', ¯xjsj “ 1 nj ř lPpopulation j xjsjplq with the number of agents denoted by nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For simplicity, we assume the initial sum of weights ř sjPSj κi jsjpk, 0q to be the same for every agent in the system N and denote this initial sum by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Observe that Equation 6 can be rewritten as pλ ` t ` 1qµi jsjpk, t ` 1q “ pλ ` tqµi jsjpk, tq ` ¯xjsjptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (7) 4 Hence, even though agent k directly updates the weights, its individual state can be characterized by the set of beliefs tµi jpkqujPVi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following, we usually drop the time index t and agent index k in the bracket (depending on the context) for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 3 Belief Dynamics in Population Network Games Observe that for an arbitrary agent k, its belief µi jpkq is in the simplex ∆j “ tµi jpkq P R|Sj|| ř sjPSj µi jsjpkq “ 1, µi jsjpkq ě 0, @sj P Sju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We assume that the system state is characterized by a Borel probability measure P defined on the state space ∆ “ ś iPV ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given µi P ∆i, we write the marginal probability density func- tion as ppµi, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that ppµi, tq is the density of agents having the belief µi about population i throughout the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define µ “ tµiuiPV P ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Since agents maintain separate beliefs about different neighbor populations, the joint probability density function ppµ, tq can be factorized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', ppµ, tq “ ś iPV ppµi, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We make the following assumption for the initial marginal density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' At time t “ 0, for each population i P V , the marginal density function ppµi, tq is continuously differentiable and has zero mass at the boundary of the simplex ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This assumption is standard and common for a “nice” probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Under this mild condition, we determine the evolution of the system state P with the following proposition, using the techniques similar to those in [27, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proposition 1 (Population Belief Dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The continuous-time dynamics of the marginal density function ppµi, tq for each population i P V is governed by a partial differential equation ´Bppµi, tq Bt “ ∇ ¨ ˆ ppµi, tq ¯xi ´ µi λ ` t ` 1 ˙ (8) where ∇¨ is the divergence operator and ¯xi is the mean mixed strategy with each si-th element ¯xisi “ ż ś jPVi ∆j exp pβuisiq ř s1 iPSi exp pβuis1 iq ź jPVi ppµj, tq ˜ ź jPVi dµj ¸ (9) where uisi “ ř jPVi eJ siAijµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For every marginal density function ppµi, tq, the total mass is always conserved (Corollary 1 of the supplementary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' moreover, the mass at the boundary of the simplex ∆i always remains zero, indicating that agents’ beliefs will never go to extremes (Corollary 2 of the supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Generalizing the notion of a system state to a distribution over beliefs allows us to address a very specific question — the impact of belief heterogeneity on system evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' That said, partial differential equations (Equation 8) are notoriously difficult to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Here we resort to the evolution of moments based on the evolution of the distribution (Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following proposition, we show that the characterization of belief heterogeneity is important, as the dynamics of the mean system state (or the mean belief dynamics) is indeed affected by belief heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proposition 2 (Mean Belief Dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The dynamics of the mean belief ¯µi about each population i P V is governed by a system of differential equations such that for each strategy si, d¯µisi dt « fsiptµjujPViq ´ ¯µisi λ ` t ` 1 ` ř jPVi ř sjPSj B2fsiptµjujPViq pBµjsj q2 Varpµjsjq 2pλ ` t ` 1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (10) where fsiptµujPViq is the logit choice function (Equation 5) applied to strategy si P Si, and Varpµjsjq is the variance of belief µjsj in the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In general, the mean belief dynamics is under the joint effects of the mean, variance, and infinitely many higher moments of the belief distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' To allow for more conclusive results, we apply the moment closure approximation4 and assume the effects of the third and higher moments to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Now, just for a moment, suppose that the system beliefs are homogeneous —- the beliefs of every individuals are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, the mean belief dynamics are effectively the belief dynamics of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The following proposition follows from Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 4Moment closure is a typical approximation method used to estimate moments of population models [13, 15, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' To use moment closure, a level is chosen past which all cumulants are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The conventional choice of the level is 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', setting the third and higher cumulants to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 5 Proposition 3 (Belief Dynamics for Homogeneous Populations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For a homogeneous system, the dynamics of the belief µi about each population i P V is governed by a system of differential equations such that for each strategy si, dµisi dt “ xisi ´ µisi λ ` t ` 1 “ fsiptµjujPViq ´ µisi λ ` t ` 1 (11) where µisi is the same for all agents in each neighbor population j P Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Intuitively, the mean belief dynamics indicates the trend of beliefs in a system, and the variance of beliefs indicates belief heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Contrasting Propositions 2 and 3, it is clear that the variance of belief (belief heterogeneity) plays a role in determining the mean belief dynamics (the trend of beliefs) for heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' It is then natural to ask: how does the belief heterogeneity evolve over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' How much does the belief heterogeneity affect the trend of beliefs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Our investigation to these questions reveals an interesting finding — the variance of beliefs asymptotically tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 1 (Quadratic Decay of the Variance of Population Beliefs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The dynamics of the variance of beliefs µi about each population i P V is governed by a system of differential equations such that for each strategy si, dVarpµisiq dt “ ´2Varpµisiq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (12) At given time t, Varpµisiq “ ´ λ`1 λ`t`1 ¯2 σ2pµisiq, where σ2pµisiq is the initial variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, the variance Varpµisiq decays to zero quadratically fast with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Such quadratic decay of the variance stands no matter what 2-player subgames agents play and what initial conditions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Put differently, the beliefs will eventually homogenize for all population network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This fact immediately implies the system state in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As time t Ñ 8, the density function ppµi, tq for each population i P V evolves into a Dirac delta function, and the variance of the choice distributions within each population i P V also goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that while the choice distributions will homogenize within each population, they are not necessarily the same across different populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This is because the strategy choice of each population is in response to its own set of neighbor populations (which are generally different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 4 Convergence of Smooth Fictitious Play in Population Network Games The finding on belief homogenization is non-trivial and also technically important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' One implication is that the fixed points of systems with initially heterogeneous beliefs are the same as in systems with homogeneous beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, it follows from the belief dynamics for homogeneous systems (Proposition 3) that the fixed points of systems have the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 2 (Fixed Points of System Dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For any system that initially have homogeneous or heterogeneous beliefs, the fixed points of the system dynamics is a pair pµ˚, x˚q that satisfy x˚ i “ µ˚ i for each population i P V and are the solutions of the system of equations x˚ isi “ exp ´ β ř jPVi eJ siAijx˚ j ¯ ř s1 iPSi exp ´ β ř jPVi eJ s1 iAijx˚ j ¯ (13) for every strategy si P Si and population i P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Such fixed points always exist and coincide with the Quantal Response Equilibria (QRE) [33] of the population network game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that the above theorem applies for all population network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We study the convergence of SFP to the QRE under the both cases of network competition and network coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Due to space limits, in the following, we mainly focus on network competition and present only the main result on network coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 Network Competition Consider a competitive population network game Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that in competitive network games, the Nash equilibrium payoffs need not to be unique (which is in clear contrast to two-player settings), and it generally allows for infinitely many Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following theorem, focusing on homogeneous systems, we establish the convergence of the belief dynamics to a unique QRE, regardless of the number of Nash equilibria in the underlying game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 3 (Convergence in Homogeneous Network Competition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a competitive Γ, for any system that has homogeneous beliefs, the belief dynamics (Equation 11) converges to a unique QRE which is globally asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof of Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We proof this theorem by showing that the “distance” between xi and µi is strictly decreasing until the QRE is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In particular, we measure the distance in terms of the perturbed payoff and construct a strict Lyapunov function L :“ ÿ iPV ωi “ πi ` xi, tµjujPVi ˘ ´ πi ` µi, tµjujPVi ˘‰ (14) where ω1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' ωn are the positive weights given by Γ, and πi is a perturbed payoff function defined as πi ` xi, tµjujPVi ˘ :“ xJ i ř jPVi Aijµj ´ 1 β ř siPSi xisi lnpxisiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Next, we turn to systems with initially heterogeneous beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Leveraging that the variance of beliefs eventually goes to zero, we establish the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For a system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10) is asymptotically autonomous [31] with the limit equation dµi dt “ xi ´ µi, which after time-reparmeterization is equivalent to the belief dynamics for homogeneous systems (Equation 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For ease of presentation, we follow the convention to denote the solution flows of an asymptotically autonomous system and its limit equation by φ and Θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thieme [40] provides the following seminal result that connects the limit behaviors of φ and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 2 (Thieme [40] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a metric space pX, dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Assume that the equilibria of Θ are isolated compact Θ-invariant subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The ω-Θ-limit set of any pre-compact Θ-orbit contains a Θ-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The point ps, xq, s ě t0, x P X, have a pre-compact φ-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Then the following alternative holds: 1) φpt, s, xq Ñ e, t Ñ 8, for some Θ-equilibrium e, and 2) the ω-φ-limit set of ps, xq contains finitely many Θ-equilibria which are chained to each other in a cyclic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Combining the above results, we prove the convergence for initially heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 4 (Convergence in Initially Heterogeneous Network Competition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a compet- itive Γ, for any system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10) converges to a unique QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The following corollary immediately follows as the result of belief homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For any competitive Γ, under smooth fictitious play, the choice distributions and beliefs of every individual converges to a unique QRE (given in Theorem 2), regardless of belief initialization and the number of Nash equilibria in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2 Network Coordination We delegate most of the results on coordination network games to the supplementary, and summarize only the main result here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 5 (Convergence in Network Coordination with Star Structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a coordination Γ where the network structure consists of a single or disconnected multiple stars, each orbit of the belief dynamics (Equation 11) for homogeneous systems as well as each orbit of the mean belief dynamics (Equation 10) for initially heterogeneous systems converges to the set of QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that this theorem applies to all 2-population coordination games, as network games with or without star structure are essentially the same when there are only two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We also remark that pure or mixed Nash equilibria in coordination network games are complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' as reported in recent works [5, 4, 1], finding a pure Nash equilibrium is PLS-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, learning in the general case of network coordination is difficult and generally requires some conditions for theoretical analysis [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 7 H S H (1, 1) (2, 0) S (0, 2) (4, 4) Table 1: Stag Hunt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Figure 2: Asymmetric Matching Pennies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Figure 3: Belief heterogeneity helps select the payoff dominant equilibrium pS, Sq (yellow: the equilibrium pS, Sq, blue: the equilibrium pH, Hq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As the variance of initial beliefs increases (from the left to right panel), a larger range of initial mean beliefs will approximately reach the equilibrium pS, Sq in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For each panel, the initial variances of two populations σ2pµ1Hq and σ2pµ2Hq are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 5 Experiments: Equilibrium Selection in Population Network Games In this section, we complement our theory and present an empirical study of SFP in a two-population coordination (stag hunt) game and a five-population zero-sum (asymmetric matching pennies) game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Importantly, these two games both have multiple Nash equilibria, which naturally raises the problem of equilibrium selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 Two-Population Stag Hunt Games We have shown in Figure 1 (in the introduction) that given the same initial mean belief, changing the variances of initial beliefs can result in different limit behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following, we systematically study the effect of initial belief heterogeneity by visualizing how it affects the regions of attraction to different equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Game Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We consider a two-population stag hunt game, where each player in populations 1 and 2 has two actions tH, Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As shown in the payoff bi-matrices (Table 1), there are two pure strategy Nash equilibria in this game: pH, Hq and pS, Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' While pH, Hq is risk dominant, pS, Sq is indeed more desirable as it is payoff dominant as well as Pareto optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In this game, population 1 forms beliefs about population 2 and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We denote the initial mean beliefs by a pair p¯µ2H, ¯µ1Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We numerically solve the mean belief dynamics for a large range of initial mean beliefs, given different variances of initial beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In Figure 3, for each pair of initial mean beliefs, we color the corresponding data point based on which QRE the system eventually converges to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We observe that as the variance of initial beliefs increases (from the left to right panel), a larger range of initial mean beliefs results in the convergence to the QRE that approximates the payoff dominant equilibrium pS, Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Put differently, a higher degree of initial belief heterogeneity leads to a larger region of attraction to pS, Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, belief heterogeneity eventually vanishes though, it provides an approach to equilibrium selection, as it helps select the highly desirable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2 Five-Population Asymmetric Matching Pennies Games We have shown in Corollary 2 that SFP converges to a unique QRE even if there are multiple Nash equilibria in a competitive Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following, we corroborate this by providing empirical evidence in 8 +1 +1 +1 1 Population 1 Population 2 Population 3 Population 4 Population 5 H (L H) {H, T} [H, T} Match Match Match MatchBasins of Attraction of (S, S) and (H, H) under Different Variances of Initial Beliefs g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (μ1H) = 0 g2(μ1H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='02 g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (μH) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='05 g2(μ1H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 (S,S) j12H H 2H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 Initial Initial Initial Initial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 (H,H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='8 1 Initial jiH Initial jiiH Initial jiH InitialjiiHFigure 4: With different belief initialization, SFP selects a unique equilibrium where all agents in population 3 play strategy H with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We run 100 simulation runs for each initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The thin lines represent the mean mixed strategy (the choice probability of H) and the shaded areas represent the variance of the mixed strategies in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the legends, B denotes Beta distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' the two Beta distributions correspond to the initial beliefs about the neighbor populations 2 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' agent-based simulations with different belief initialization (the details of simulations are summarized in the supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Game Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider a five-population asymmetric matching pennies game [28], where the network structure is a line (depicted in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Each agent has two actions tH, Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Agents in populations 1 and 5 do not learn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' they always play strategies H and T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For agents in populations 2 to 4, they receive `1 if they match the strategy of the opponent in the next population, and receive ´1 if they mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the contrary, they receive `1 if they mismatch the strategy of the opponent in the previous population, and receive ´1 if they match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, this game has infinitely many Nash equilibria of the form: agents in populations 2 and 4 play strategy T, whereas agents in population 3 are indifferent between strategies H and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In this game, agents in each population form two beliefs (one for the previous population and one for the next population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We are mainly interested in the strategies of population 3, as the Nash equilibria differ in the strategies in population 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For validation, we vary population 3’s beliefs about the neighbor populations 2 and 4, and fix population 3’s beliefs about the other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' As shown in Figure 4, given differential initialization of beliefs, agents in population 3 converge to the same equilibrium where they all take strategy H with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Therefore, even when the underlying zero-sum game has many Nash equilibria, SFP with different initial belief heterogeneity selects a unique equilibria, addressing the problem of equilibrium selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 6 Conclusions We study a heterogeneous beliefs model of SFP in network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Representing the system state with a distribution over beliefs, we prove that beliefs eventually become homogeneous in all network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We establish the convergence of SFP to Quantal Response Equilibria in general competitive network games as well as coordination network games with star structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We experimentally show that although the initial belief heterogeneity vanishes in the limit, it plays a crucial role in equilibrium selection and helps select highly desirable equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Appendix A: Corollaries and Proofs omitted in Section 3 Proof of Proposition 1 It follows from Equation 7 in the main paper that the change in µi jpk, tq between two discrete time steps is µi jpk, t ` 1q “ µi jpk, tq ` ¯xjptq ´ µi jpk, tq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (15) 9 Probability of Playing H in Population 3 Vary the inital Mean, Fix the initial Variance Vary the initial Variance, Fix the initial Mean B(5, 10), B(8, 2) B(10, 20), B(16, 4) B(5, 10), B(2, 8) B(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='5, 5), B(4,1) B(10, 5), B(8, 2) B(50, 100),B(80,20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='5 0 0 100 200 300 400 100 200 300 400 Time t Time tLemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Under Assumption 1 (in the main paper), for an arbitrary agent k in population i, its belief µi jpk, tq about a neighbor population j will never reach the extreme belief (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', the boundary of the simplex ∆i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Assumption 1 ensures that ¯xjp0q is in the interior of the simplex ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Moreover, the logit choice function (Equation 5 in the main paper) also ensures that ¯xjptq stays in the interior of ∆j afterwards for a finite temperature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, from Equation 15, one can see that µi jpk, tq for every time step t will stay in the interior of ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following, for notation convenience, we sometimes drop the agent index k and the time index t depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider a population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We rewrite the change in the beliefs about this population as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' µipt ` 1q “ µiptq ` ¯xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (16) Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We rewrite the above equation as µipt ` δq “ µiptq ` δ ¯xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (17) Next, we consider a test function θpµiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define Y “ Erθpµipt ` δqqs ´ Erθpµiptqqs δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (18) Applying Taylor series for θpµipt ` δqq at µiptq, we obtain θpµipt ` δqq “ θpµiptqq ` δ λ ` t ` 1Bµiθpµiq r¯xiptq ´ µiptqs ` δ2 2pλ ` t ` 1q2 r¯xiptq ´ µiptqsJ Hθpµiq r¯xiptq ´ µiptqs ` o ˜„ δ ¯xiptq ´ µiptq λ ` t ` 1 ȷ2¸ (19) where H denotes the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, the expectation Erθpµipt ` δqqs is Erθpµipt ` δqqs “ Erθpµiptqqs ` δ λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs ` δ2 2pλ ` t ` 1q2 E “ r¯xiptq ´ µiptqsJHθpµiq r¯xiptq ´ µiptqs ‰ ` δ2 2pλ ` t ` 1q2 Eropr¯xiptq ´ µiptqs2qs (20) Moving the term Erθpµiptqqs to the left hand side and dividing both sides by δ, we recover the quantity Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', Y “ 1 λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs ` δ 2pλ ` t ` 1q2 Err¯xiptq ´ µiptqsJHθpµiptqqr¯xiptq ´ µiptqs ` o ` p¯xiptq ´ µiptqq2˘ s (21) Taking the limit of Y with δ Ñ 0, the contribution of the second term on the right hand side vanishes, yielding lim δÑ0 Y “ 1 λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs (22) “ 1 λ ` t ` 1 ż ppµiptq, tq “ Bµiθpµiptqqp¯xiptq ´ µiptqq ‰ dµiptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (23) Apply integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We obtain lim δÑ0 Y “ 0 ´ 1 λ ` t ` 1 ż θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq (24) 10 where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero as a result of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the other hand, according to the definition of Y , lim δÑ0 Y “ lim δÑ0 ż θpµiptqqppµi, t ` δq ´ ppµi, tq δ dµi “ ż θpµiptqqBtppµi, tqdµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (25) Therefore, we have the equality ż θpµiptqqBtppµi, tqdµi “ ´ 1 λ ` t ` 1 ż θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (26) As θ is a test function, this leads to Btppµi, tq “ ´ 1 λ ` t ` 1∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (27) Rearranging the terms, we obtain Equation 8 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By the definition of expectation given a probability distribution, it is straightforward to obtain Equation 9 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Remarks: The PDEs we derived are akin to the continuity equation commonly encountered in physics in the study of conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='The continuity equation describes the transport phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', of mass or energy) in a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This renders a physical interpretation for our PDE model: under SFP, the belief dynamics of a heterogeneous system is analogously the transport of the agent mass in the simplex ∆ “ ś iPV ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Corollaries of Proposition 1 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For any population i P V , the system beliefs about this population never go to extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This is a straightforward result of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For any population i P V , the total probability mass ppµi, tq always remains conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider the time derivative of the total probability mass d dt ż ppµi, tqdµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (28) Apply the Leibniz rule to interchange differentiation and integration, d dt ż ppµi, tqdµi “ ż Bppµi, tq Bt dµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (29) Substitute Bppµi,tq Bt with Equation 8 in the main paper, d dt ż ppµi, tqdµi “ ´ ż ∇ ¨ ˆ ppµi, tq ¯xi ´ µi λ ` t ` 1 ˙ dµi (30) “ ´ ż ÿ siPSi Bµisi ˆ ppµi, tq ¯xisi ´ µisi λ ` t ` 1 ˙ dµi (31) “ ´ 1 λ ` t ` 1 «ż ÿ siPSi Bµisi ppµi, tq p¯xisi ´ µisiq dµi ` ż ppµi, tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi ff (32) Apply integration by parts, ż ÿ siPSi Bµisi ppµi, tq p¯xisi ´ µisiq dµi “ 0 ´ ż ppµi, tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (33) where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, the terms within the bracket of Equation 32 cancel out, and d dt ż ppµi, tqdµi “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (34) 11 Proof of Proposition 2 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The dynamics of the mean belief ¯µi about each population i P V is governed by a differential equation d¯µisi dt “ ¯xisi ´ ¯µisi λ ` t ` 1 , @si P Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The time derivative of the mean belief about strategy si is d¯µisi dt “ d dt ż µisippµi, tqdµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (36) We apply the Leibniz rule to interchange differentiation and integration, and then substitute Bppµi,tq Bt with Equation 8 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' d dt ż µisippµi, tqdµi (37) “ ż µisi Bppµi, tq Bt dµi (38) “ ´ ż µisi∇ ¨ ˆ ppµi, tq ¯xi ´ µi λ ` t ` 1 ˙ dµi (39) “ ´ ż µisi ÿ siPSi Bµisi ˆ ppµi, tq ¯xisi ´ µisi λ ` t ` 1 ˙ dµi (40) “ γ «ż µisi ÿ siPSi ` Bµisi ppµi, tq ˘ p¯xisi ´ µisiq dµi ` ż µisippµi, tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi ff (41) where γ :“ ´ 1 λ`t`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Apply integration by parts to the first term in Equation 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' ż µisi ÿ siPSi ` Bµisi ppµi, tq ˘ p¯xisi ´ µisiq dµi “ ´ ż µisippµi, tq » – ÿ s1 iPSi Bµis1 i p¯xis1 i ´ µis1 iq fi fl ` ppµi, tqBµisi rµisip¯xisi ´ µisiqs dµi (42) where we have leveraged that the probability mass at the boundary remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' it follows from Equation 41 that d dt ż µisippµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tqdµi (43) “ ´γ ż µisippµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tq ÿ s1 iPSi Bµis1 i p¯xis1 i ´ µis1 iqdµi ´ γ ż ppµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tqBµisi rµisip¯xisi ´ µisiqs dµi ` γ ż µisippµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi (44) “ γ ż ppµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tq “ µisiBµisi p¯xisi ´ µisiq ´ Bµisi rµisip¯xisi ´ µisiqs ‰ dµi (45) “ γ ż ppµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tqµisidµi ´ ż ppµi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tq¯xisidµi (46) “ ¯xisi ´ ¯µisi λ ` t ` 1 (47) We repeat the mean probability ¯xisi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' which has been given in Equation 9 in the main paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' as follows: ¯xisi “ ż exp pβuisiq ř s1 iPSi exp pβuis1 iq ź jPVi ppµj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' tq ˜ ź jPVi dµj ¸ (48) 12 where uisi “ ř jPVi eJ siAijµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define ¯µ :“ t¯µjujPVi and fsiptµjujPViq :“ exp pβ ř jPVi eJ siAijµjq ř s1 iPSi exp pβ ř jPVi eJ s1 iAijµjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (49) Applying the Taylor expansion to approximate this function at the mean belief ¯µ, we have fsiptµjujPViq « fsip¯µq ` ∇fsip¯µq ¨ pµ ´ ¯µq ` 1 2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq ` Op||µ ´ ¯µ||3q (50) where H denotes the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, we can rewrite Equation 48 as ¯xisi “ ż fsiptµjujPViq ź jPVi ppµj, tq ˜ ź jPVi dµj ¸ (51) « fsip¯µq ` ż ∇fsip¯µq ¨ µ ź jPVi ppµj, tq ˜ ź jPVi dµj ¸ ´ ∇fsip¯µq ¨ ¯µ ` ż 1 2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq ź jPVi ppµj, tq ˜ ź jPVi dµj ¸ ` ż Op||µ ´ ¯µ||q3 ź jPVi ppµj, tq ˜ ź jPVi dµj ¸ (52) Observe that in Equation 52, the second and the third term can be canceled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Moreover, for any two neighbor populations j, k P Vi, the beliefs µj, µk about these two populations are separate and independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, the covariance of these beliefs are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We apply the moment closure approximation [32, 13] with the second order and obtain ¯xisi « fsip¯µq ` 1 2 ÿ jPVi ÿ sjPSj B2fsip¯µq pBµjsjq2 Varpµjsjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (53) Hence, substituting ¯xisi in Lemma 4 with the above approximation, we have the mean belief dynamics d¯µisi dt « fsip¯µq ´ ¯µisi λ ` t ` 1 ` ř jPVi ř sjPSj B2fsip¯µq pBµjsj q2 Varpµjsjq 2pλ ` t ` 1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (54) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Remarks: the use of the moment closure approximation (considering only the first and the second moments) is for obtaining more conclusive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Strictly speaking, the mean belief dynamics also depend on the third and higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' However, we observe in the experiments that these moments in general have little effects on the mean belief dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' To be more specific, given the same initial mean beliefs, while the variance of initial beliefs sometimes can change the limit behaviors of a system, we do not observe similar phenomena for the third and higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof of Proposition 3 Consider a population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' It follows from Equation 7 in the main paper that the change in the beliefs about this population can be written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' µipt ` 1q “ µiptq ` xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (55) Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We rewrite the above equation as µipt ` δq “ µiptq ` δ xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (56) Move the term µiptq to the right hand side and divide both sides by δ, µipt ` δq ´ µiptq δ “ xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (57) 13 Assume that the amount of time δ between two successive time steps goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' we have dµi dt “ lim δÑ0 µipt ` δq ´ µiptq δ “ xiptq ´ µiptq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (58) Note that for continuous-time dynamics, we usually drop the time index in the bracket, yielding the belief dynamics (Equation 11) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof of Theorem 1 Without loss of generality, we consider the variance of the belief µisi about strategy si of population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that Varpµisiq “ Erpµisiq2s ´ p¯µisiq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (59) Hence, we have dVarpµisiq dt “ dErpµisiq2s dt ´ 2¯µisi d¯µisi dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (60) Consider the first term on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We apply the Leibniz rule to interchange differentiation and integration, and then substitute Bppµi,tq Bt with Equation 8 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' dErpµisiq2s dt “ ż pµisiq2 Bppµi, tq Bt dµi (61) “ ´ ż pµisiq2∇ ¨ ˆ ppµi, tq ¯xi ´ µi λ ` t ` 1 ˙ dµi (62) “ ´ ż pµisiq2 ÿ siPSi Bµisi ˆ ppµi, tq ¯xisi ´ µisi λ ` t ` 1 ˙ dµi (63) “ γ ż pµisiq2 ÿ siPSi Bµisi ppµi, tq p¯xisi ´ µisiq dµi ` γ ż pµisiq2ppµi, tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi (64) where γ :“ ´ 1 λ`t`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Applying integration by parts to the first term in Equation 64 yields ż pµisiq2 ÿ siPSi Bµisi ppµi, tq p¯xisi ´ µisiq dµi “ ´ ż pµisiq2ppµi, tq » – ÿ s1 iPSi Bµis1 i p¯xis1 i ´ µis1 iq fi fl ` ppµi, tqBµisi “ pµisiq2p¯xisi ´ µisiq ‰ dµi (65) where we have leveraged that the probability mass at the boundary remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Combining the above two equations, we obtain dErpµisiq2s dt “ ´γ ż pµisiq2ppµi, tq » – ÿ s1 iPSi Bµis1 i p¯xis1 i ´ µis1 iq fi fl ` ppµi, tqBµisi “ pµisiq2p¯xisi ´ µisiq ‰ dµi ` γ ż pµisiq2ppµi, tq ÿ siPSi Bµisi p¯xisi ´ µisiq dµi (66) “ γ ż “ ´ppµi, tqBµisi “ pµisiq2p¯xisi ´ µisiq ‰‰ ` pµisiq2ppµi, tqBµisi p¯xisi ´ µisiq dµi (67) “ γ ż 2pµisiq2ppµi, tqdµi ´ γ ż 2¯xisiµisippµi, tqdµi (68) “ ´2Erpµisiq2s ´ 2¯xisi ¯µisi λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (69) 14 Next, we consider the second term in Equation 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By Lemma 4, we have 2¯µisi d¯µisi dt “ 2¯µisip¯xisi ´ ¯µisiq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (70) Combining Equations 69 and 70, the dynamics of the variance is dVarpµisiq dt “ ´2Erpµisiq2s ´ 2¯xisi ¯µisi λ ` t ` 1 ´ 2¯µisip¯xisi ´ ¯µisiq λ ` t ` 1 (71) “ 2p¯µisiq2 ´ 2Erpµisiq2s λ ` t ` 1 (72) “ ´2Varpµisiq λ ` t ` 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (73) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Remarks: We believe that the rationale behind such a phenomenon is twofold: 1) agents apply smooth fictitious play, and 2) agents respond to the mean strategy play of other populations rather than the strategy play of some fixed agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Regarding the former, we notice that under a similar setting, population homogenization may not occur if agents apply other learning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', Q-learning and Cross learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Regarding the latter, imagine that agents adjust their beliefs in response to the strategies of some fixed agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For example, consider two populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' one contains agents A and C, and the other one contains agents B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Suppose that agents A and B form a fixed pair such that they adjust their beliefs only in response to each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' the same applies to agents C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Belief homogenization may not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Appendix B: Proofs omitted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='1 Proof of Theorem 2 Belief homogenization implies that the fixed points of systems with initially heterogeneous beliefs are the same as in systems with homogeneous beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, we focus on homogeneous systems to analyze the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' It is straightforward to see that dµi dt “ xi ´ µi λ ` t ` 1 “ 0 ùñ xi “ µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (74) Denote the fixed points of the system dynamics, which satisfies the above equation, by px˚ i , µ˚ i q for each population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By the logit choice function (Equation 5 in the main paper), we have x˚ isi “ exp pβuisiq ř s1 iPSi exp pβuis1 iq “ exp pβ ř jPVi eJ siAijµ˚ j q ř s1 iPSi exp pβ ř jPVi eJ s1 iAijµ˚ j q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (75) Leveraging that x˚ i “ µ˚ i , @i P V at the fixed points, we can replace µ˚ j with x˚ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof of Theorem 3 Consider a population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The set of neighbor populations is Vi, the set of beliefs about the neighbor populations is tµjujPVi, and the choice distribution is xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a population network game Γ, the expected payoff is given by xJ i ř pi,jqPE Aijµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define a perturbed payoff function πi ` xi, tµjujPVi ˘ :“ xJ i ÿ jPVi Aijµj ` vpxiq (76) where vpxiq “ ´ 1 β ř siPSi xisi lnpxisiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Under this form of vpxiq, the maximization of πi yields the choice distribution xi from the logit choice function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Based on this, we establish the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For a choice distribution xi of SFP in a population network game, Bxiπi ` xi, tµjujPVi ˘ “ 0 and ÿ jPVi ` Aijµj ˘J “ ´Bxivpxiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (77) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This lemma immediately follows from the fact that the maximization of πi will yield the choice distribution xi from the logit choice function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 15 The belief dynamics of a homogeneous populations can be simplified after time-reparameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given τ “ ln λ`t`1 λ`1 , the belief dynamics of homogeneous systems (given in Equation 11 in the main paper) is equivalent to dµi dτ “ xi ´ µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (78) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' From τ “ ln λ`t`1 λ`1 , we have t “ pλ ` 1qpexp pτq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (79) By the chain rule, for each dimension si, dµisi dτ “ dµisi dt dt dτ (80) “ xisi ´ µisi λ ` t ` 1 d ppλ ` 1qpexp pτq ´ 1qq dτ (81) “ xisi ´ µisi λ ` pλ ` 1qpexp pτq ´ 1q ` 1pλ ` 1q exp pτq (82) “ xisi ´ µisi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (83) Next, we define the Lyapunov function L as L :“ ÿ iPV ωiLi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Li :“ πi ` xi, tµjujPVi ˘ ´ πi ` µi, tµjujPVi ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (84) where tωiuiPV is the set of positive weights defined in the weighted zero-sum Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The function L is non-negative because for every i P V , xi maximizes the function πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' When for every i P V , xi “ µi, the function L reaches the minimum value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Rewrite L as L “ ÿ iPV « ωiπi ` xi, tµjujPVi ˘ ´ ωiµJ i ÿ jPVi Aijµj ´ ωivpµiq ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (85) We observe that πi ` xi, tµjujPVi ˘ is convex in µj, j P Vi by Danskin’s theorem, and ´vpµiq is strictly convex in µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Moreover, by the weighted zero-sum property given in Equation 2 in the main paper, we have ÿ iPV ˜ ωiµJ i ÿ jPVi Aijµj ¸ “ 0 (86) since µi P ∆i, µj P ∆j for every i, j P V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Therefore, the function L is a strictly convex function and attains its minimum value 0 at a unique point xi “ µi, @i P V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider the function Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Its time derivative is 9Li “ Bxiπi ` xi, tµjujPVi ˘ 9xi ` ÿ jPVi ” Bµjπi ` xi, tµjujPVi ˘ 9µj ı ´ Bµiπi ` µi, tµjujPVi ˘ 9µi ´ ÿ jPVi ” Bµjπi ` µi, tµjujPVi ˘ 9µj ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (87) Note that the partial derivative Bxiπi equals 0 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, we can rewrite this as 9Li “ Bµiπi ` µi, tµjujPVi ˘ 9µi ` ÿ jPVi ” Bµjπi ` xi, tµjujPVi ˘ ´ Bµjπi ` µi, tµjujPVi ˘ı 9µj (88) “ ´ « ÿ jPVi ` Aijµj ˘J ` Bµivpµiq ff pxi ´ µiq ` ÿ jPVi ` xJ i Aij ´ µJ i Aij ˘ pxj ´ µjq (89) “ rBxivpxiq ´ Bµivpµiqs pxi ´ µiq ` ÿ jPVi ` xJ i Aijxj ´ µJ i Aijxj ´ xJ i Aijµj ` µJ i Aijµj ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (90) 16 where from Equation 89 to 90, we apply Lemma 5 to substitute ř jPVi ` Aijµj ˘J with ´Bxivpxiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, summing over all the populations, the time derivative of L is 9L “ ÿ iPV ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq ` ÿ iPV ÿ jPVi ωi ` xJ i Aijxj ´ µJ i Aijxj ´ xJ i Aijµj ` µJ i Aijµj ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (91) The summation in the second line is equivalent to ÿ pi,jqPE pωixJ i Aijxj ` ωjxJ j Ajixiq ´ pωiµJ i Aijxj ` ωjxJ j Ajiµiq (92) ´ pωixJ i Aijµj ` ωjµJ j Ajixiq ` pωiµJ i Aijµj ` ωjµJ j Ajiµiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (93) By the weighted zero-sum property given in Equation 2 in the main paper, this summation equals 0, yielding 9L “ ÿ iPV ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (94) Note that the function v is strictly concave such that its second derivative is negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By this property, 9L ď 0 with equality only if xi “ µi, @i P V , which corresponds to the QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Therefore, L is a strict Lyapunov function, and the global asymptotic stability of the QRE follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Remarks: Intuitively, the Lyapunov function defined above measures the distance between the QRE and a given set of beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The idea of measuring the distance in terms of entropy-regularized payoffs is inspired from the seminal work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' However, different from the network games considered in this paper, Hofbauer and Hopkins [19] consider SFP in two-player games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' To our knowledge, so far there has been no systematic study on SFP in network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof of Theorem 4 The proof of Theorem 4 leverages the seminal results of the asymptotically autonomous dynamical system [31, 40, 41] which conventionally is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' A nonautonomous system of differential equations in Rn x1 “ fpt, xq (95) is said to be asymptotically autonomous with limit equation y1 “ gpyq, (96) if fpt, xq Ñ gpxq, t Ñ 8, where the convergence is uniform on each compact subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Conventionally, the solution flow of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 95 is called the asymptotically autonomous semiflow (denoted by φ) and the solution flow of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 96 is called the limit semiflow (denoted by Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Based on this definition, we establish Lemma 1 in the main paper, which is repeated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For a system that initially has heterogeneous beliefs, the mean belief dynamics is asymptotically autonomous [31] with the limit equation dµi dt “ xi ´ µi (97) which after time-reparameterization is equivalent to the belief dynamics for homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We first time-reparameterize the mean belief dynamics of heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Assume τ “ 17 Figure 5: Population network games where the underlying network consists of star structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' ln λ`t`1 λ`1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By the chain rule and Equation 54, for each dimension si, d¯µisi dτ “ d¯µisi dt dt dτ (98) “ » —–fsip¯µq ´ ¯µisi λ ` t ` 1 ` ř jPVi ř sjPSj B2fsip¯µq pBµjsj q2 Varpµjsjq 2pλ ` t ` 1q fi ffifl d ppλ ` 1qpexp pτq ´ 1qq dτ (99) “ fsip¯µq ´ ¯µisi ` 1 2 ř jPVi ř sjPSj B2fsip¯µq pBµjsj q2 ´ λ`1 λ`t`1 ¯2 σ2pµjsjq λ ` pλ ` 1qpexp pτq ´ 1q ` 1 pλ ` 1q exp pτq (100) “ fsip¯µq ´ ¯µisi ` 1 2 ÿ jPVi ÿ sjPSj B2fsip¯µq pBµjsjq2 σ2pµjsjq exp p´2τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (101) Observe that exp p´2τq decays to zero exponentially fast and that both σ2pµjsjq and B2fsip¯µq pBµjsj q2 are bounded for every µ in the simplex ś jPVi ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, Equation 101 converges locally and uniformly to the following equation: d¯µisi dτ “ fsip¯µq ´ ¯µisi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (102) Note that xisi “ fsip¯µq for homogeneous systems, and the above equation is algebraically equivalent to Equation 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, by Definition 1, Equation 101 is asymptotically autonomous with the limit equation being Equation 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By the above lemma, we can formally connect the limit behaviors of initially heterogeneous systems and those of homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Recall that Theorem 3 in the main paper states that under SFP, there is a unique rest point (QRE) for the belief dynamics in a weighted zero-sum network game Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' this excludes the case where there are finitely many equilibria that are chained to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, combining Lemma 2 in the main paper, we prove that the mean belief dynamics of initially heterogeneous systems converges to a unique QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Appendix C: Results and Proofs omitted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='2 For the case of network coordination, we consider networks that consist of a star or disconnected multiple stars due to technical reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In Figure 1, we present examples of the considered network structure with different numbers of nodes (populations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the following theorem, focusing on homogeneous systems, we establish the convergence of the belief dynamics to the set of QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Theorem 6 (Convergence in Homogeneous Network Coordination with Star Structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a coordination Γ where the network structure consists of a single or disconnected multiple stars, each orbit of the belief dynamics for homogeneous systems converges to the set of QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider a root population j of a star structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Its set of leaf (neighbor) populations is Vj, the set of beliefs about the leaf populations is tµiuiPVj, and the choice distribution is xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given the game Γ, 18 Five Populations Five Populations (Two Disconnected Stars) Three Populations Two Populations P2 P1 P3the expected payoff is xJ j ř iPVj Ajiµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define a perturbed payoff function πj ` xj, tµiuiPVj ˘ :“ xJ j ÿ iPVj Ajiµi ` vpxjq (103) where vpxjq “ ´ 1 β ř sjPSj xjsj lnpxjsjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Under this form of vpxjq, the maximization of πj yields the choice distribution xj from the logit choice function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Consider a leaf population i of the root population j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' It has only one neighbor population, which is population j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, given the game Γ, the expected payoff is xJ i Aijµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Define a perturbed payoff function πi ` xi, µj ˘ :“ xJ i Aijµj ` vpxiq (104) where vpxiq “ ´ 1 β ř siPSi xisi lnpxisiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Similarly, the maximization of πi yields the choice distribution xi from the logit choice function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Based on this, we establish the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For choice distributions of SFP in a population network game with start structure, Bxjπj ` xj, tµiuiPVj ˘ “ 0 and ÿ iPVj pAjiµiqJ “ ´Bxjvpxjq if j is a root population, (105) Bxiπi ` xi, µj ˘ “ 0 and ` Aijµj ˘J “ ´Bxivpxiq if i is a leaf population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (106) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' This lemma immediately follows from the fact that the maximization of πj and πi , respectively, yield the choice distributions xj and xi from the logit choice function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For readability, we repeat the belief dynamics of a homogeneous population after time-reparameterization, which has been proved in Lemma 4 in Appendix B, as follows: dµi dτ “ xi ´ µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (107) Let R Ă V be the set of all root populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We define L :“ ÿ jPR Lj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Lj :“ µJ j ÿ iPVj Ajiµi ` vpµjq ` ÿ iPVj vpµiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (108) Consider the function Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Its time derivative 9Lj is 9Lj “ » –BµjpµJ j ÿ iPVj Ajiµiq 9µj ` ÿ iPVj BµipµJ j ÿ iPVj Ajiµiq 9µi fi fl ` Bµjvpµjq 9µj ` ÿ iPVj Bµivpµiq 9µi (109) “ ÿ iPVj pAjiµiqJpxj ´ µjq ` » – ÿ iPVj µJ j Ajipxi ´ µiq fi fl ` Bµjvpµjqpxj ´ µjq ` ÿ iPVj Bµivpµiqpxi ´ µiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' (110) Since Γ is a coordination game, we have ` Aijµj ˘J “ µJ j AJ ij “ µJ j Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, applying Lemma 8, we can substitute ř iPVjpAjiµiqJ with ´v1pxjq, and µJ j Aji with ´v1pxiq, yielding 9Lj “ ´Bxjvpxjqpxj ´ µjq ` » – ÿ iPVj p´Bxivpxiqqpxi ´ µiq fi fl ` Bµjvpµjqpxj ´ µjq ` ÿ iPVj Bµivpµiqpxi ´ µiq (111) “ pBµjvpµjq ´ Bxjvpxjqqpxj ´ µjq ` ÿ iPVj pBµivpµiq ´ Bxivpxiqqpxi ´ µiq (112) Note that the function v is strictly concave such that its second derivative is negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By this property, 9Lj ě 0 with equality only if xi “ µi, @i P Vj and xj “ µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, the time derivative of the function L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=', 9L “ ř jPR 9Lj ě 0 with equality only if xi “ µi, @i P Vj, xj “ µj, @j P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We generalize the convergence result to initially heterogeneous systems in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 19 Theorem 7 (Convergence in Initially Heterogeneous Network Coordination with Star Struc- ture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Given a coordination Γ where the network structure consists of a single or disconnected multiple stars, each orbit of the mean belief dynamics for initially heterogeneous systems converges to the set of QRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The proof technique is similar to that for initially heterogeneous competitive network games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' By Lemma 1 in the main paper, we show that the mean belief dynamics of initially heterogeneous systems is asymptotically autonomous with the belief dynamics of homogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Therefore, it follows from Lemma 2 in the main paper that the convergence result for homogeneous systems can be carried over to the initially heterogeneous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Remarks: The convergence of SFP in coordination games and potential games has been established under the 2-player settings [19] as well as some n-player settings [20, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Our work differs from the previous works in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' First, our work allows for heterogeneous beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Moreover, we consider that agents maintain separate beliefs about other agents, while in the previous works agents do not distinguish between other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, even when the system beliefs are homogeneous, our setting is still different from (and more complicated) than the previous settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Appendix D: Omitted Experimental Details Numerical Method for the PDE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' PDEs are notoriously difficult to solve, and only limited types of PDEs allow analytic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Hence, similar to previous research [23], we resort to numerical method for PDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' in particular, we consider the finite difference method [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Agent-based Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The presented simulation results are averaged over 100 independent simulation runs to smooth out the randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For each simulation run, there are 1, 000 agents in each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' For each agent, the initial beliefs are sampled from the given initial probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Detailed Experimental Setups for Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In the case of small initial variance, the initial beliefs µ1H and µ2H are distributed according to the distribution Betap280, 120q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' On the contrary, in the case of large initial variance, the initial beliefs µ1H and µ2H are distributed according to the distribution Betap14, 6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Thus, initially, the mean beliefs in these two cases are both ¯µ1H “ ¯µ2H “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='7 and ¯µ1S “ ¯µ2S “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In both cases, the initial sum of weights λ “ 10 and the temperature β “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Detailed Experimental Setups for Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We visualize the regions of attraction of different equilibria in stag hunt games by numerically solving the mean belief dynamics (Equation 10 in the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The initial variances have been given in the title of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In all cases, the initial sum of weights λ “ 0 and the temperature β “ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Detailed Experimental Setups for Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We let the initial beliefs about populations 1, 3 and 5 remain unchanged across different cases, and vary the initial beliefs about populations 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The initial beliefs about populations 1, 3 and 5, denoted by µ1H, µ3H and µ5H, are distributed according to the distributions Betap20, 10q, Betap6, 4q, and Betap10, 5q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The initial beliefs about populations 2 and 4 have been given in the legends of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In all cases, the initial sum of weights λ “ 10 and the temperature β “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Note that µiT “ 1 ´ µiH for all populations i “ 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Source Code and Computing Resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We have attached the source code for reproducing our main experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The Matlab script finitedifference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='m numerically solves our PDE model presented in Proposition 1 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The Matlab script regionofattraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='m visualizes the region of attraction of different equilibria in stag hunt games, which are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' The Python scripts simulation(staghunt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='py and simulation(matchingpennies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content='py correspond to the agent-based simulations in two-population stag hunt games and five-population asymmetric matching pennies games, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' We use a laptop (CPU: AMD Ryzen 7 5800H) to run all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' 20 References [1] Yakov Babichenko and Aviad Rubinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' Settling the complexity of nash equilibrium in congestion games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1426–1437, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE4T4oBgHgl3EQfKQz1/content/2301.04929v1.pdf'} +page_content=' [2] 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The files +are stored at the servers such that any I colluding servers can not obtain any information about the files, while any +I + L servers can together recover all files. In addition, the system are designed to satisfy: (a) Each user’s demand +must be satisfied upon receiving the signals from any I + L servers; (b) The content of the library must be kept +secure from a wiretapper who obtains all the signals from the servers; (c) Any subset of users together with all the +servers can not obtain any information about the demands of the remaining users. A coded scheme is derived to +satisfy the above constraints by incoporating the idea of secret sharing and key superposition into the framework +of Placement Delivery Array (PDA), originally proposed to characterize the single-server coded caching system +without any security or privacy constraints, where the memory size at each user, the storage size at each server and +the total communication load over all servers are characterized by the PDA parameters. It is shown that the PDAs +describing the original Maddah-Ali and Niesen’s coded caching scheme result in an achievable memory-storage- +communication region, such that the storage size and the communication load are optimal to within a constant +multiplicative gap, except for the small memory regime when the number of files is smaller than the number of +users. +Index Terms +Cache, Secret Sharing, Placement delivery array, Robust decoding, Scalar linear function retrieval, Security, +Private +I. INTRODUCTION +Coded caching is a technique to reduce the peak-time communication load by jointly designing the +pre-stored cache contents and communication signals, which was introduced by Maddah-Ali and Niesen +(MAN) [1]. The model consists of a single server holding a set of files that connects to multiple cache- +aided users through a shared-link. The system operates at two phases: In the placement phase, the users +fill their own caches without knowing their future demands; In the delivery phase, knowing the users’ +demands, the server satisfies them by transmitting coded signals to them. For a system with N files and +K users, the MAN scheme achieves the optimal load-memory tradeoff among all uncoded placement +schemes when N ≥ K [2], and for N < K after removing some redundant transmissions [3]. Recently, it +was showed that allowing the users to demand arbitrary linear combinations of the files does not increase +the load compared to the case single file retrieval, at least under uncoded placement [4]. One important +problem in MAN scheme is that, the number of packets each file needs to be partitioned into (called +subpacketization) grows exponentially with the number of users K [5]. In [6], Yan et al proposed an +combination object called Placement Delivery Array (PDA) to characterize a subclass of coded caching +schemes with uncoded placement, and proposed two constructions of PDA. +Content security and demand privacy are critical aspects of practical systems. In [7], the content of the +library must be protected against an external wiretapper who obtains the signals transmitted during the +delivery phase. This was achieved by pre-storinging security keys into user caches for the part of the files +The authors are with the Information Coding & Transmission Key Lab of Sichuan Province, CSNMT Int. Coop. Res. Centre (MoST), +Southwest Jiaotong University, Chengdu 611756, China. (email: qifayan@swjtu.edu.cn, xhutang@swjtu.edu.cn, zzc@swjtu.edu.cn) +arXiv:2301.08711v1 [cs.IT] 20 Jan 2023 + +that were not cached in the MAN scheme. The security keys are stored in a structured way such that +each user can decode all the multicast signals it needs to decode. In [8], [9], users’ demand privacy are +guarranteed by adding virtual users, so that the real users can not distinguish if the demands are from real +or virtual users. Another technique to guarantee privacy is privacy key proposed in [10], which imposes +that any subset of colluding users must not obtain any information about the demands of other users. The +key idea in [10] is that, each user also privately caches some random linear combination of the parts of the +files that were not cached in the MAN scheme, which was called privacy keys. The queries are generated +by adding the coefficients used to generate the privacy keys to the real demands, so that the users can +decode the linear combination of files with the queries and further decode their own demands with the +privacy keys. In [11], a key superposition scheme was proposed to guarantee both content security against +a wiretapper and demand privacy against colluding users simultaneously. The idea is superposing (i.e., +sum together) the security keys and privacy keys. It was showed that the load-memory tradeoff in this +case is the same as in the setup with only content security guarantees. The idea of key superposition +was incorporated into the framework of Placement Delivery Array (PDA), with the advantage that low +subpacketization schemes can be obtained directly from existing PDA constructions, such as the ones +in [6], [12]–[15]. +The above progresses are obtained under a basic setup with a single server and multiple users. Another +interesting setup involves distributed systems with multiple servers and a single user. Demand privacy +for such networks were known as Private Information Retrieval (PIR) [16]. The capacity of PIR has been +characterized in [17] for single file retrieval, in [18] for scalar linear function retrieval, and in [19] for +single file retrieval and colluding servers. PIR with a cache-aided user was investigated in [20]–[23]. +Coding across the distributed servers with Maximum Distance Separable (MDS) codes are useful, since +it saves storage while allowing node failures or erasures, which commonly happen in such systems [24]. +The capacity of PIR from MDS-coded servers was charactered in [25], while the schemes achieving the +smallest download rate with almost optimal sub-packetization are proposed in [26], [27]. In addition to +signal security, in distributed systems, another type of security is blind server, where the servers might be +untrusted and required to be blind to the data. In 1979, Shamir proposes a technique that allows multiple +servers to jointly storing a message while keeping the message secret from any subset of servers with +cardinality not exceeding some threshold [28]. The techique is called secret sharing, which is well used in +many distributed systems where the contents need to be kept blind from servers, e.g., the private polynomial +computation from coded servers [29], and multi-user blind private information retrieval [30]. For multiple- +server-multiple-user systems, the techniques from coded caching and PIR were combined in [31], [32] +to guarantee server-side privacy. In [33], [34], the technique of key superposition is incoprated into the +MDS-coded server system under PDA framework, which guarantees demand privacy against both servers +and colluding users, signal security against an wiretapper, and robust decoding against server failures. As +far as we known, the blind server constraint is not investigated in multiple-server-multiple-user networks. +In this paper, we aim to design coded caching schemes with blind servers, and all the other constraints +in [34] still hold. That is, in addition to demand privacy, signal security and robust decoding, the design of +the distributed storage at the servers need to guarrantee that the content of the files are blind to any subset +of I servers, and the whole files can be recovered from the contents at any I + L servers. We refer the +model as Robust, Secure and Private Scalar Linear Function Retrieval from Blind Servers (RSP-LFRB). +Unlike the assumption that the files are stored in the MDS coded form across the servers, the model here +allows to flexibly design the contents stored at the servers, and the storage size at each server is identified +as a permance measure. Our main contributions for the proposed RSP-LFRB model are: +1) We propose a procedure to obtain a RSP-LFRB scheme from a given PDA. The advantage of +establishing a PDA framework is that RSP-LFRB schemes with low subpacketizations can be easily +obtained from various existing PDA constructions [6], [12]–[15]. The key idea is to incoporate the +secret sharing scheme into the PDA framework, and using the key superpositions in a similar way +as in the single server system [11]. The memory size at each user, the storage size at the servers, +and the communication load is characterized with the PDA parameters. + +Fig. 1: System model +2) Following the proposed procedure, an achievable region of memory-storage-communication1 triples +based on the PDAs that describe the original MAN scheme in [1] (MAN-PDAs) are derived. +Moreover, the achievable storage size is optimal within a constant multiplicative gap 2, and the +achievable communication load is optimal within a constant multiplicative gap 12, except for the +small memory regime when the number of files is smaller than the number of users. +The rest of this paper is organized as follows. Section II gives the formal system model description. +Section III reviews the PDA framework and gives an illustrative example. Section IV summarizes our main +results, where the proof details are deferred to Sections V and VI. Section VII presents some numerical +results, and Section VIII concludes the paper. +Notations: In this paper, N+ denotes the set of positive integers; Fq and Fn +q denote the finite field of +cardinality q, for some prime power q, and the n-dimensional vector space over Fq, respectively. For two +integers m, n such that m ≤ n, we use [m : n] to denote the set of the first positive integers {m, . . . , n}; +[1 : n] is also denoted by [n] for short. We use XA to denote the tuple composed of {Xi : i ∈ A} for +some integer set A, where the elements are ordered increasingly, e.g., X[3] = (X1, X2, X3). For variables +with two or more indices, e.g., Xi,j, we use XA,B to denote the tuple {Xi,j : i ∈ A, j ∈ B}, where the +elements are listed in lexicographical order, e.g. X[3],[2] = (X1,1, X1,2, X2,1, X2,2, X3,1, X3,2). +II. SYSTEM MODEL +Let N, K, H, I, L be positive integers satisfying I + L ≤ H. A (N, K, H, I, L) system, illustrated in +Fig. 1, consists of a file library of N files, H servers (denoted by 1, . . . , H), where each server is connected +to K users (denoted by 1, . . . , K) via a dedicated shared-link. The N files are mutually independent and +uniformly distributed over FB +q for some prime power q, where B denotes the file length, that is, +H(W1) = . . . = H(WN) = B, +H(W1, . . . , WN) = H(W1) + . . . + H(WN), +where the entropy function H(·) is calculated with logorithm q. For any vector a = (a1, . . . , aN) ∈ FN +q , +we use Wa to denote the linear combination of the files with coefficient vector a: +Wa := +� +n∈[N] +an · Wn. +(1) +The system generates the stored content of server h (denoted by Zh) from the N files and a random +variable U from some alphabet U with an encoding function χh : FNB +q +× U �→ F⌊TB⌋ +q +, i.e., +Zh = χh(W[N], U) ∈ F⌊TB⌋ +q +, +∀ h ∈ [H], +(2) +where T is the storage size at each server. The storage contents at the servers need to satisfy the following +properties: +1To distinguish the storage capacity between users and servers, in this paper, we will refer “memory” and “storage” for the user side and +server side respectively. + +Files +0.... +N +Random Variable +U +Encoder +Colluding +Blind Servers +2 +X1 +X2 +XH +Links +Wiretapper +Users & Caches +2 +3 +K1) I-Security: Any I servers should be oblivious to the files W[N]: +I(W[N]; ZI) = 0, +∀ I ⊆ [H], |I| = I. +(3a) +2) (I + L)-Robust Recovery: The whole files W[N] can be recovered from any I + L servers: +H(W[N]| ZJ ) = 0, +∀ J ⊆ [H], |J | = I + L. +(3b) +The system operates in two phases as follows. +Placement Phase: Each user k ∈ [K] generates some random variable Pk from some finite alphabet +Pk and cache some content Ck as a function of Pk, U and the file library W[N] using some function +ϕk : Pk × U × FNB +q +�→ F⌊MB⌋ +q +, i.e., +Ck := ϕk(Pk, U, W[N]) ∈ F⌊MB⌋ +q +, ∀ k ∈ [K], +where M is the memory size of each user. The encoding functions ϕ1, . . . , ϕK are known by the servers, +but the randomness P1, . . . , PK are kept private by the corresponding users. +Delivery Phase: Each user k ∈ [K] generates a demand dk = (dk,1, . . . , dk,N)⊤ ∈ FN +q , meaning that +it is interested in retrieving the linear combination of the files Wdk. The random variables d1, . . . , dK, +P1, . . . , PK, W1, . . . , WN, U are independent, i.e., +H(d[K], W[N], P[K], U) = +� +k∈[K] +H(dk) + +� +n∈[N] +H(Wn) + +� +k∈[K] +H(Pk) + H(U). +User k ∈ [K] sends a query Qk,h of length ℓk,h to server h for each h ∈ [H], where the queries +Qk,[H] := (Qk,1, . . . , Qk,H) are generated from some query function κk,h : FN +q × Pk �→ F +ℓk,h +q +, i.e., +Qk,h := κk,h(dk, Pk) ∈ F +ℓk,h +q +, ∀ h ∈ [H], +(4) +Upon receiving the queries from all the users, server h ∈ [H] creates a signal Xh as +Xh := φh(Zh, Q[K],h) ∈ F⌊RhB⌋ +q +, ∀h ∈ [H], +(5) +for some encoding function φh : F⌊TB⌋ +q +× F +� +k∈[K] ℓk,h +q +�→ F⌊RhB⌋ +q +. The quantity Rh, h ∈ [H], is referred to +as the communication load of server h. The (total) communication load of the system is defined as +R := +� +h∈[H] +Rh. +The following conditions must hold for all demands d1, . . . , dK ∈ FN +q : +1) Robust Decoding: Each user k can retrieval its demanded linear combination of the files Wdk from +any (I + L)-subset of signals and the content in its memory, i.e., for each k ∈ [K]: +H(Wdk | XAk, dk, Ck) = 0, +∀ Ak ⊆ [H] : |Ak| = I + L. +(6a) +2) Signal Security: A wiretapper, who is not a user and observes all the delivery signals, can not obtain +any information about the contents of the library files: +I(W[N]; X[H]) = 0. +(6b) +3) Demand Privacy: Any subset of users together with the servers can not jointly learn any information +on the demands of the other users, regardless of the file realizations: +I(d[K]\S; CS, dS, Q[K],[H], Z[H] | W[N]) = 0, +∀ S ⊆ [K]. +(6c) +Definition 1. A Memory-Storage-Communication (MSC) triple (M, T, R) ∈ [1, N] × R+ × R+ is said to + +be B-achievable if, for any ϵ > 0, there exists a scheme satisfying all the conditions in (3) and (6) with +memory size less than M + ϵ, storage size less than T + ϵ, and load less than R + ϵ with file-length B. +The MSC region of the system is defined as +G = {(M, T, R) : (M, T, R) is achievable.}. +The main objective of this paper is to characterize the MSC region G. For simplicity, we call a valid +scheme satsify the constraints (3) and (6) a Robust, Secure, Private Scalar Linear Function Retrieval from +Blind Servers (RSP-LFRB) scheme. For a given scheme, we are also interested in its subpacketization +level, which is defined as the number of packets each file has to be partitioned into in order to implement +the scheme. Throughout this paper, we consider the case N ≥ 2, since demand privacy is impossible for +N = 1 (i.e., there is only one possible file to be demanded). +Remark 1 (Privacy & Security Against the Wiretapper). Notice that, by (6c), +I(d[K]; X[H] | W[N]) +≤ I(d[K]; X[H], Q[K],[H], Z[H] | W[N]) += I(d[K]; Q[K],[H], Z[H] | W[N]) +(7) += 0, +where (7) follows from (5). Together with (6b), it holds that +I(W[N], d[K]; X[H]) = 0, +that is, the wiretapper having access to X[H] in fact can not obtain any information on both the files and +the demands of the users. +Remark 2 (Minimum Memory Size). It was proved in [7] that, in order to guarantee the conditions in (6a) +and (6b) simultaneously, the memory size M has to be no less than one. Thus the MSC region is defined +for M ∈ [1, N]. +III. PDAS AND A TOY EXAMPLE +We will construct our RSP-LFRB scheme for any given PDA [6], which was introduced to reduce the +subpacketization in coded caching in the single server system without any security or privacy constraints. +In this section, we first review the definition of PDA, and then give an example to highlight the key ideas +in the design of our RSP-LFRB scheme. The general construction will be given in Section V. +A. Placement Delivery Array +Definition 2 (PDA [6]). For given K, F ∈ N+ and Z, S ∈ N, an F ×K array A = [aj,k], j ∈ [F], k ∈ [K], +composed of Z specific symbols “∗” in each column and some ordinary symbols 1, . . . , S, each occurring +at least once, is called a (K, F, Z, S) PDA, if, for any two distinct entries aj,k and aj′,k′, we have +aj,k = aj′,k′ = s, for some ordinary symbol s ∈ [S] only if +a) j ̸= j′, k ̸= k′, i.e., they lie in distinct rows and distinct columns; and +b) aj,k′ = aj′,k = ∗, i.e., the corresponding 2 × 2 sub-array formed by rows j, j′ and columns k, k′ +must be of the following form +� +s +∗ +∗ +s +� +or +� +∗ +s +s +∗ +� +. +It was showed in [6] that, with a given (K, F, Z, S) PDA, there exists an associated coded caching +scheme in the single server system without any security or privacy constraint. The parameter K is the +number of users, F is the number of packets each file is split into (i.e., subpacketization), Z is the number +of uncoded packets from each file stored at each user, and S is the number of coded multicast signals. + +In our model, each file will first split into L equal-size subfiles, and those implications will be used on +the subfiles. +B. A Toy RSP-LFRB Example from PDAs +We derive here a RSP-LFRB scheme associated to the (K, F, Z, S) = (3, 3, 1, 3) PDA +A = +� +� +∗ +1 +2 +1 +∗ +3 +2 +3 +∗ +� +� +(8) +for an (N, K, H, I, L) = (4, 3, 4, 1, 2) distributed system. +Let the four files be W1, W2, W3, W4 ∈ FB +q . Firstly, each file Wn, n ∈ [4] is split into L = 2 equal-size +subfiles, +Wn = (Wn,1, Wn,2), +(9) +and each subfile Wn,l, l ∈ [2] is further split into F = 3 equal-size packets: +Wn,l = (Wn,l,1, Wn,l,2, Wn,l,3). +(10) +Generate NI = 4 uniform and independent random variables {Jn : n ∈ [4]} of the same size with a +subfile, and SL = 6 uniform and independent random variables {Vl,s : l ∈ [2], s ∈ [3]} of the same size +with a packet. Then construct two groups of polynomials: +� +Wn(x) = Wn,1 + Wn,2x + Jnx2, +∀ n ∈ [4], +(11a) +�Vs(x) = V1,s + V2,sx, +∀ s ∈ [3]. +(11b) +Notice that, in accordance with (10), for each n ∈ [3], � +Wn(x) and Jn can be decomposed into 3 equal-size +components, i.e., � +Wn(x) = (� +Wn,1(x), � +Wn,2(x), � +Wn,3(x)) and Jn = (Jn,1, Jn,2, Jn,3), where +� +Wn,j(x) = Wn,1,j + Wn,2,jx + Jn,jx2, +j ∈ [3]. +(12) +Let α1, α2, α3, α4 be H = 4 distinct non-zero elements in Fq. The system operates as follows: +Server Storage Design: Each server h ∈ [4] stores the evaluations of the polynomials in (11): +Zh = {� +Wn(αh) : n ∈ [4]} ∪ {�Vs(αh) : s ∈ [3]}. +(13) +Placement Phase: Each user k ∈ [3] generates a random vector pk = (pk,1, pk,2, pk,3, pk,4)⊤ ∈ F4 +q. +The cache content of the user k is composed of pk and the (un)coded packets in the corresponding column +in Table I. The packets W[4],[2],j are associated to the j-th row of A in (8) and user k is associated to the +TABLE I: The cached contents of users† according to A in (8). +Row +User 1 +User 2 +User 3 +1 +W[4],[2],1 +Wp2,[2],1 ⊕ V[2],1 +Wp3,[2],1 ⊕ V[2],2 +2 +Wp1,[2],2 ⊕ V[2],1 +W[4],[2],2 +Wp3,[2],2 ⊕ V[2],3 +3 +Wp1,[2],3 ⊕ V[2],2 +Wp2,[2],3 ⊕ V[2],3 +W[4],[2],3 +† In addition, each user k ∈ [3] caches pk. +k-th column of A. The packets in the j-th row of Table I of user k are created according to the entry +aj,k of A in (8): if aj,k = ∗, user k caches NL = 8 uncoded packets W[4],[2],j, otherwise it caches L = 2 +coded packets Wpk,[2],j ⊕ V[2],aj,k. +Delivery Phase: Assume that user 1, 2, 3 demands the linear combination Wd1, Wd2 and Wd3, +respectively, where d1, d2, d3 ∈ F4 +q. Each user k ∈ [3] sends qk = pk ⊕ dk to all the servers as queries. +Upon receiving the query vectors q[3], each server h ∈ [4] sends a signal Xh to the users, where Xh is + +composed of the query vectors q[3] and S = 3 coded packets �Y[3](αh), which are associated to the ordinary +symbols s = 1, 2, 3 respectively: +s = a1,2 = a2,1 = 1 : +�Y1(αh) = �V1(αh) + � +Wq1,2(αh) + � +Wq2,1(αh), +s = a1,3 = a3,1 = 2 : +�Y2(αh) = �V2(αh) + � +Wq1,3(αh) + � +Wq3,1(αh), +s = a2,3 = a3,2 = 3 : +�Y3(αh) = �V3(αh) + � +Wq2,3(αh) + � +Wq3,2(αh), +where a polynomial � +Wa,j(x) is defined for each a ∈ F4 +q and j ∈ [3] as: +� +Wa,j(x) ≜ a1� +W1,j(x) + a2� +W2,j(x) + a3� +W3,j(x) + a4� +W4,j(x). +Robust Decoding: Let’s take s = 1 to show how the users utilize the signals to decode. Consider +the polynomial +�Y1(x) = �V1(x) + � +Wq1,2(x) + � +Wq2,1(x) += �V1(x) + +4 +� +n=1 +q1,n� +Wn,2(x) + +4 +� +n=1 +q2,n� +Wn,1(x) += (V1,1 + Wq1,1,2 + Wq2,1,1) + (V2,1 + Wq1,2,2 + Wq2,2,1)x + (Jq1,2 + Jq2,1)x2. +Notice that, the signal �Y1(αh) sent by server h is the evaluation of �Y1(x), which is a polynomial of degree +2, thus each user can obtain �Y1(x) by interpolation with any 3 signals, and thus decode the coefficients. +Table II lists all the coefficients that a user can decode from the signals of any 3 servers. +TABLE II: The signals a user can decode from the signals of any 3 servers† according to A. +s +Subfiles 1 +Subfiles 2 +Noises +1 +V1,1 + Wq1,1,2 + Wq2,1,1 +V2,1 + Wq1,2,2 + Wq2,2,1 +Jq1,2 + Jq2,1 +2 +V1,2 + Wq1,1,3 + Wq3,1,1 +V2,2 + Wq1,2,3 + Wq3,2,1 +Jq1,3 + Jq3,1 +3 +V1,3 + Wq2,1,3 + Wq3,1,2 +V2,3 + Wq2,2,3 + Wq3,2,2 +Jq2,3 + Jq3,2 +† In addition, each server h ∈ [4] transmits the query vectors q[3]. +Upon obtaining the signals in Table II, each user k ∈ [3] can proceed with the decoding process for +each subfile l ∈ [2] as in [10]. For example, for subfile 1, as a1,2 = a2,1 = 1, user 1 can decode Wd1,1,2 +and user 2 can decode Wd2,1,1 from the signal +V1,1 + Wq1,1,2 + Wq2,1,1 += Wd1,1,2 + (Wp1,1,2 + V1,1) + Wq2,1,1 +(14a) += Wd2,1,1 + (Wp2,1,1 + V1,1) + Wq1,1,2, +(14b) +since +• user 1 has cached the superposition key Wp1,1,2 + V1,1 and can compute Wq2,1,1 its cached subfiles +W[4],1,1 and the query vector q2, thus can decode Wd1,1,2 by (14a); +• user 2 has cached the superposition key Wp2,1,1 + V1,1 and can compute Wq1,1,2 with its cached +subfiles W[4],1,2 and the query vector q1, and thus can decode Wd2,1,1 by (14b). +One can verify that each user k ∈ [3] can decode all the remaining packets Wdk,[2],[3]\{k} from its stored +contents, the signals in Table II and the query vectors q[3]. +Security and Privacy Conditions: In addition to the robust decoding, the other conditions in (3) +and (6) are also guaranteed: In fact, the secret sharing techique [28] to encode files with polynomials in +(11a) guarantees the I-Security, and (I + L)-Robust Recovery; Signal Securiy and Demand Privacy are +guaranteed since each signal or demand is accompanied by a key of random and uniformly distributed +vector. + +Performance: Recall that each packet is of length B +6 . Each user caches 12 packets and 1 vector in +F4 +q, whose length does not scale with B. Thus the needed memory is M = 12 × 1 +6 = 2 files. Each server +stores N = 4 subfiles, each of length B +2 , and 3 coded packets, each of length B +6 , thus the storage size +is T = 4 × 1 +2 + 3 × 1 +6 = +5 +2. Each of the 4 servers transmits 3 packets and 3 vectors in F4 +q, thus the +achieved load is R = 4 × 3 × 1 +6 = 2 files. Hence, the scheme achieves the memory-storage-load triple +(M, R) = +� +2, 5 +2, 2 +� +. Since each file is split into 6 equal-size packets, the subpacketization level is LF = 6. +IV. MAIN RESULTS +A. PDA based RSP-LFRB Schemes +With any given PDA, we will construct an associated RSP-LFRB scheme. We will prove the following +theorem by presenting and analyzing the construction in Section V. +Theorem 1. For any (N, K, H, I, L) system and a given (K, F, Z, S) PDA A, there exists an associated +RSP-LFRB scheme that achieves the MSC triple +� +MA, TA, RA +� += +� +1 + Z +F (N − 1), 1 +L +� +N + S +F +� +, HS +LF +� +. +(15) +with subpacketization LF. +Remark 3 (Comparison with MDS Coded Server Systems). A previous work in [34] has extended the key +superposition scheme in [11] to MDS coded servers. Unlike the previous work, which assumes that the +files are stored at the servers in the form of MDS code, the design of the storage at the servers is a part of +the RSP-LFRB scheme, and we also take the storage size at each server into account as a cost measure. +Moreover, this work further inserts an additional constraint that any I servers are blind to the file library. +This is achieved by inserting random variables in the coded subfiles with secret sharing technique [28]. It +can be observed that, when I = 0, the strategy degrades to the setup in [34], where the MDS generator +matrix is given by +G = +� +��� +1 +1 +. . . +1 +α1 +α2 +. . . +αH +... +... +... +... +αL−1 +1 +αL−1 +2 +. . . +αL−1 +H +� +��� . +(16) +The proposed scheme indicates that the techniques of coded caching, key superpositions can be integrated +with the secret sharing to provide robustness and protections for data security and demand privacy. +The advantage of establishing a PDA framework is that, all the existing PDAs can be used to derive +the RSP-LFRB scheme. As the subpacketization of the PDA based RSP-LFRB scheme is LF, which is +proportional to F, PDAs with small number of rows result low subpacketization schemes. Among all +PDA constructions, the MAN-PDAs that describe the Maddah-Ali and Niesen’s coded caching scheme +[1] is of very important, which we will mainly discuss in the following. +B. Optimality of MAN-PDA based RSP-LFRB Schemes +For any integer t ∈ [0 : K], define the set +Ωt ≜ {T ⊆ [K] : |T | = t}. +(17) +It was proved in [6] that the following construction is a (K, +�K +t +� +, +�K−1 +t−1 +� +, +� K +t+1 +� +) PDA, which describes the +MAN scheme in [1] and will be referred to as MAN-PDA in the following. + +Fig. 2: Achievable MSC Region, N = 50, K = 10, H = 20, L = 10, I ≤ 10 +Definition 3 (MAN-PDA). Fix any integer t ∈ [0 : K], denote the set Ωt = {Tj : j ∈ [ +�K +t +� +]}. Also, choose +an arbitrary bijective function κt+1 from Ωt+1 to the set +�� K +t+1 +�� +. Then, define the array At = [aj,k] as +aj,k ≜ +� +∗, +if k ∈ Tj +κt+1({k} ∪ Tj), +if k /∈ Tj . +(18) +Example 1 (A MAN-PDA). Consider K = 4, t = 2, let T1 = {1, 2}, T2 = {1, 3}, T3 = {1, 4}, T4 = +{2, 3}, T5 = {2, 4} and T6 = {3, 4}. Let κ3 be the lexicographic order of a subset of size 3 in Ω3, +e.g., κ3({1, 2, 3}) = 1, κ3({1, 2, 4}) = 2 and κ3({1, 3, 4}) = 3 and κ3({2, 3, 4}) = 4. The corresponding +(4, 6, 3, 4) PDA is given by +A2 = +� +������ +∗ +∗ +1 +2 +∗ +1 +∗ +3 +∗ +2 +3 +∗ +1 +∗ +∗ +4 +2 +∗ +4 +∗ +3 +4 +∗ +∗ +� +������ +. +The following theorem summarizes the performance of MAN-PDA and its optimality. +Theorem 2. Let (M, T(M), R(M)) be the curve formed by connecting the points +(Mt, Tt, Rt) = +� +1 + t(N − 1) +K +, 1 +L +� +N + K − t +t + 1 +� +, H(K − t) +L(t + 1) +� +, +t ∈ [0 : K] +(19) +sequentially, then the region +{(M, T, R) : T ≥ T(M), R ≥ R(M)} +(20) +is achievable in an (N, K, H, I, L) RSPB-LFR system, where the point (Mt, Tt, Rt) can be achieved with +subpacketization L +�K +t +� +. Moreover, the optimal storage size T ∗(M) and load R∗(M) satisfies +T(M) +T ∗(M) ≤ 2, +R(M) +R∗(M) ≤ 12, +(21) +if K ≤ N or K > N, M ≥ 2. +As illustrated in Fig. 2, the curve (M, T(M), R(M)) is composed of K line segments. Then the + +- +- +1 +-R + Load +10 +0 +6 +5.9 +5.8 +5.7 +5.6 +5.5 +5.4 +5.3 +5.2 +Storage Size T +5.1 +5 +50M +I(M2, T2, R2) +(M3, T3, R3) +(M4, T4, R4) +0 +(M5, T5, R5) +10 +(M6, T6, R6) +20 +(M, TR) +30 +Ms,T8, Rs) +Tg, Rg)40 +Memory Size M +R10)(Mo, To, Ro) +20boundary is formed by the above surfaces starting from the curve and parallel to the T- and R-axes. +Obviously, the curve (M, T(M), R(M)), M ∈ [1, N] forms the all Pareto-optimal points of the set, thus, +we only need to verify the achievability of the curve. We only need to prove the achievability of the curve +(M, T(M), R(M)). In fact, the achievability of the point (Mt, Tt, Rt) directly follows from Theorem 1 +and the fact that At in Definition 3 is a (K, +�K +t +� +, +�K−1 +t−1 +� +, +� K +t+1 +� +) MAN-PDA [6]. Moreover, for general +M ∈ [1, N], M lies in [Mt−1, Mt] for some t ∈ [K], the point (M, T(M), R(M)) can be achieved by +memory-sharing [1] between the points (Mt−1, Tt−1, Rt−1) and (Mt, Tt, Rt). Thus, the achievability of +the region (20) follows directly Theorem 1 and the construction of MAN-PDA. The optimality in proved +by deriving a lower bound for the T(M) and R(M) respectively, where the details are presented in +Section VI. +Notice that, the storage size +T(M) = N +L + 1 +H R(M). +(22) +In fact, from scheme description in Section V, N +L is the storage size used to store the encoded version +of the files, while +1 +H R(M) is the size of the encoded version of security keys used to protect the signal +from the wiretapper, which is the same with the signal size transmitted by each server. +Notice that the achievable region does not depend on the paramenter I. This is because that the +communication load is reflects the load over all servers. From the user’s view, the load to each user +needs to be larger than +� +1 + I +L +�� +1 − t +K +� +(Remark 5 in Section V-B). Thus, although the parameter I does +not influence the total communication load transmitted by the servers (as long as I +L ≤ H), it increases +the amount of signals that each user need to decode. We will present some numerical results to illustrate +the influence of the parameters I and L in Section VII-B. +Remark 4 (On Unbounded Regime). In the regime K > N, 1 ≤ M < 2, the gap is unbounded. The main +problem in small memory regime for the single server model when K > N is that, if security keys are +used [7], [11], for the point M = 1 the best know achievable communication load is K, while the best +known converse is N. Thus, it seems that the larger communication load when K > N is mainly caused +by the security condition; closing the gap in small memory regime when K > N is an open problem in +the single server system with signal security constraint constraint setup [7], [11], and scalar linear function +retrieval from MDS coded servers with signal security constraint setups [34]. +V. PDA BASED RSPB-LFR SCHEME +In this section, we prove Theorem 1 by deriving a RSP-LFRB scheme for an (N, K, H, I, L) system +from any given (K, F, Z, S) PDA A = [aj,k]F×K. +A. PDA Based Scheme +Based on A, each file Wn is partitioned into LF equal-size packets as follows: +• Each file Wn is first patitioned into L subfiles +Wn = (Wn,1, . . . , Wn,L), +∀ n ∈ [N], +(23) +where Wn,l ∈ F +B +L +q is the l-th subfile of Wn for all l ∈ [L]. +• Each subfile Wn,l is further partitioned into F packets +Wn,l = (Wn,l,1, . . . , Wn,l,F), +∀ n ∈ [N], l ∈ [L], +(24) +where Wn,l,j ∈ F +B +LF +q +is the j-th packet of the subfile Wn,l for all j ∈ [F]. +The system generates NI random subfiles, denoted by J1,1, J1,2, . . . , JN,I, and LS random packets, +denoted by V1,1, V1,2, . . . , VL,S, where the random subfiles Jn,i are uniformally distributed over F +B +L +q , and + +the random packets Vl,s are uniformally distributed over F +B +LF +q +. That is, the random variable U is generated +as +U = (J1,1, . . . , JN,I, V1,1, . . . , VL,S) ∼ Unif +� +F +B(NIF +S) +LF +q +� +. +(25) +Similar to (24), Jn,i is partitioned into F equal-size random packets: +Jn,i=(Jn,i,1, . . . , Jn,i,F), +∀ n ∈ [N], i ∈ [I], +(26) +where Jn,i,j ∈ F +B +L +q is the j-th packet of the random subfile Jn,i. +Define two set of polynomials +� +Wn(x) = +L +� +l=1 +Wn,lxl−1 + +I +� +i=1 +Jn,ixL+i−1, +∀ n ∈ [N], +(27) +�Vs(x) = +L +� +l=1 +Vl,sxl−1, +∀ s ∈ [S]. +(28) +In accordance to the partitions in (24) and (26), each of the polynomials in (27) can be decomposed into +F components, i.e., +� +Wn(x) = +�� +Wn,1(x), . . . , � +Wn,F(x) +� +, ∀ n ∈ [N] +(29) +where +� +Wn,j(x) = +L +� +l=1 +Wn,l,jxl−1 + +I +� +i=1 +Jn,i,jxL+i−1, +∀ j ∈ [F]. +(30) +In the following, for a given a = (a1, . . . , aN) ∈ FN +q , we will adopt similar notations as in (1) to denote +the linear combinations of {� +Wn(x) : n ∈ [N]}, {� +Wn,j(x) : n ∈ [N]} and {Jn,i,j : n ∈ [N]} with coefficient +vector a, respectively: +� +Wa(x) = +� +n∈[N] +an · � +Wn(x), +(31) +� +Wa,j(x) = +� +n∈[N] +an · � +Wn,j(x), +∀ j ∈ [F], +(32) +Ja,i,j = +� +n∈[N] +an · Jn,i,j, +∀ i ∈ [I], j ∈ [F]. +(33) +Notice that, by (29), � +Wa(x) is composed of F components, i.e., � +Wa(x) = +�� +Wa,1(x), � +Wa,2, . . . , � +Wa,F +� +. +Server Storage Design: Let α1, . . . , αH be H distinct nonzero elements in Fq. For each h ∈ [H], the +system evaluates the values of the polynomials � +Wn(x) and �Vs(x) at αh, and stores them at server h, that +is, +Zh = +�� +Wn(αh) : n ∈ [N] +� +∪ +��Vs(αh) : s ∈ [S] +� +. +(34) +Notice that, by (29), each coded subfile � +Wn(αh) is composed of F equal-size packets, which are the +evaluations of the polynomials in (30) at αh: +� +Wn(αh) = +�� +Wn,1(αh), ..., � +Wn,F(αh) +� +, +∀ n ∈ [N]. +(35) + +Placement Phase: Each user k ∈ [K] locally generates a random vector pk uniformaly over FN +q , and +constructs its local cache Ck as +Ck ={pk} +(36a) +∪{Wn,l,j : n ∈ [N], l ∈ [L], j ∈ [F], aj,k = ∗} +(36b) +∪{Wpk,l,j + Vl,aj,k : l ∈ [L], j ∈ [F], aj,k ̸= ∗}. +(36c) +Delivery Phase: Assume that user k ∈ [K] demands Wdk, for some dk ∈ FN +q . Then user k ∈ [K] +sends query qk = dk + pk to all the servers, i.e., the queries Qk,[H] are constructed as +Qk,h = qk = dk + pk, +∀ h ∈ [H]. +(37) +Upon receiving the queries Q[K],h = q[K], each server h ∈ [H] constructs S coded signals, one for each +s ∈ [S]: +�Ys(αh) := �Vs(αh) + +� +(u,v)∈[F]×[K] +au,v=s +� +Wqv,u(αh). +(38) +Then each server h ∈ [H] sends the signal +Xh = +� +q[K], �Yh,[S](αh) +� +(39) +to the users. +B. Constraint Condition Verifications +In this subsection, we prove that, the constraints in (3) and (6) are satisfied in the PDA based RSPB-LFR +scheme. +I-Security: Notice that, by the secret-sharing arguments [28], for any I ⊆ [H] such that |I| ≤ I, +I +� +W[N]; +�� +W[N](αh) : h ∈ I +�� += 0. +(40) +Therefore, +I +� +W[N]; ZI +� += I +� +W[N]; {� +W[N](αh)}h∈I, {V[S](αh)}h∈I +� += I +� +W[N]; {� +W[N](αh)}h∈I +� ++ I +� +W[N]; {V[S](αh)}h∈I|{� +W[N](αh)}h∈I +� += 0, +where the last equality follows from (40) and the fact that the random variables V[L],[S] are generated +independently and uniformly over F +B +LF +q +, and independent of W[N] and {� +W[N](αh)}h∈I, so that the ran- +dom variables {V[S](αh)}h∈I are distributed independently and uniformly over F +B +LF +q +and independent of +� +W[N], {� +W[N](αh)}h∈I +� +. +(I + L)-Recovery: Notice that, each polynomial � +Wn(x) is of degree I + L − 1. As a result, with +values at any I + L distinct points, the polynomial can be recovered by interpolation. Then by (34), the +(I + L)-Recovery condition is straghtforward. +Robust Decoding: We need to show that for each user k ∈ [K], with any J ⊆ [H] such that +|J | = I + L, with the signals {Xh}h∈J and the cache content Ck, user k can decode its demanded scalar +linear function Wdk, i.e., all the packets Wdk,[L],[F]. +For each i ∈ [F] such that ai,k = ∗, by (36b), user k ∈ [K] has stored all the packets W[N],[L],i, thus it +can directly compute the packets Wdk,l,i for each l ∈ [L]. + +Now, consider any i ∈ [F] such that ai,k ̸= ∗. Fixed a subset J ⊆ [H] such that |J | = I + L. Let +s ≜ ai,k, consider the polynomial +�Ys(x) = �Vs(x) + +� +(u,v)∈[F]×[K] +au,v=s +� +Wqv,u(x) += +L +� +l=1 +Vl,sxl−1 + +� +(u,v)∈[F]×[K] +au,v=s +N +� +n=1 +qv,n · � +Wn,u(x) += +L +� +l=1 +Vl,sxl−1 + +� +(u,v)∈[F]×[K] +au,v=s +N +� +n=1 +qv,n · +� +L +� +l=1 +Wn,l,uxl−1 + +I +� +i=1 +Jn,i,uxL+i−1 +� += +L +� +l=1 +� +Vl,s + +� +(u,v)∈[F]×[K] +au,v=s +Wqv,l,u +� +xl−1 + +I +� +i=1 +� +� +(u,v)∈[F]×[K] +au,v=s +Jqv,i,u +� +xL+i−1 +(41) += +L +� +l=1 +Yl,sxl−1 + +I +� +i=1 +�Ji,sxL+i−1, +(42) +where Yl,s and �Ji,s are defined as +Yl,s ≜ Vl,s + +� +(u,v)∈[F]×[K] +au,v=s +Wqv,l,u, +∀ l ∈ [L] +(43a) +�Ji,s ≜ +� +(u,v)∈[F]×[K] +au,v=s +Jqv,i,u, +∀ i ∈ [I]. +(43b) +Notice that, the signal �Ys(αh) sent by server h is the value of �Ys(x) evaluated at the point αh. As �Ys(x) is +a polynomial of degree I + L − 1, user k can decode the coefficients in (43) by interpolation with signals +from any I + L servers. Moreover, since ai,k = s, for each l ∈ [L], the signal Yl,s in (43a) can be written +as +Yl,s = Vl,s + Wqk,l,i + +� +(u,v)∈[F]×[K] +au,v=s,(u,v)̸=(i,k) +Wqv,l,u +(44a) += Wdk,l,i + (Vl,ai,k + Wpk,l,i) ++ +� +(u,v)∈[F]×[K] +au,v=s=ai,k,(u,v)̸=(i,k) +Wqv,l,u, +(44b) +where (44b) follows from qk = pk + dk. Therefore, user k ∈ [K] can decode Wdk,l,i from the the signal +Yl,s by canceling the remaining terms since +1) the coded packet Vl,ai,k + Wpk,l,i is cached by user k by (36c); +2) for each (u, v) ∈ [F] × [K] such that au,v = s and (u, v) ̸= (i, k), since ai,k = au,v = s, by the +definition of PDA, i ̸= u, v ̸= k and ai,v = au,k = ∗. Thus, user k ∈ [K] stores all the packets +W[N],[L],u. Hence, user k can compute Wqv,l,u for each l ∈ [L]. +Remark 5 (Communication Load to Each User). From the decoding process, each user can decode L +packets from I + L coded packets. Since each user need to decode L(F − Z) packets in total, each user +need to collect (I + L)(F − Z) coded packets in total. Thus, the communication load to each user is +(I+L)(F−Z) +LF += +� +1 + I +L +�� +1 − Z +F +� +. + +Signal Security: We have +I(W[N]; X[H]) += I +� +W[N]; q[K], {�Yh,[S](αh)}h∈[H] +� +≤ I +� +W[N]; q[K], Y[L],[S], J[N],[I], {�Yh,[S](αh)}h∈[H] +� += I +� +W[N]; q[K], Y[L],[S], J[N],[I] +� +(45a) += I +� +W[N]; q[K], J[N],[I] +� ++ I +� +W[N]; Y[L],[S]|q[K], J[N],[I] +� += 0, +(45b) +where (45a) holds because {�Yh,[S](αh)}h∈[H] can be evaluated from q[K], Y[L],[S], J[N],[I]; and (45b) follows +since the variables Y[L],[S] are of form (44a), where the variables V[L],[S] are independently and uniformly +distributed over F +B +LF +q +. +Demand Privacy: Notice that, for any S ⊆ [K], +I(d[K]\S; CS, dS, Q[K],[H], Z[H] | W[N]) += I(d[K]\S; CS, dS, q[K], Z[H] | W[N]) +(46a) += I(d[K]\S; CS, dS, qS, Z[H] | W[N]) + I(d[K]\S; q[K]\S | CS, dS, qS, Z[H], W[N]) +(46b) += 0, +(46c) +where (46a) follows from (37); and (46c) follows since the demands d[K]\S are independent of the random +variables CS, dS, q[K], Z[H], W[N], and qk = pk + dk for each k ∈ [K]\S where the random variables +p[K]\S are independently and uniformly distributed over FN +q . +C. Performance +By (23) and (24), each file is split into LF equal-size packets, each of length +B +LF symbols, thus the +subpacketization is LF. Denote the achieved memory-storage-load triple by (MA, TA, RA)., then +Memory Size: For each user k ∈ [K], by the cached content in (36), for each i ∈ [F] such that +ai,k = ∗, there are LN associated packets cached by the user, one from each file (see (36b)). For each +i ∈ [F] such that ai,k ̸= ∗, there are L associated coded packet cached at the user (see (36c)). In addition, +the pk in (36a) can be stored with N symbols. Recall that, each column of a (K, F, Z, S) PDA has Z +“ ∗ ”s and F − Z ordinary symbols, thus, the achieved memory size is +MA = inf +B∈N+ +1 +B +� +(Z · LN + (F − Z) · L) B +LF + N +� += F + Z (N − 1) +F +. +Storage Size: By (34), since each coded subfile � +Wn(αh) is of B +L symbols, and each �Vs(αh) is of +B +LF +symbols, the storage size is +TA = 1 +B +�B +L · N + B +LF · S +� += N +L + S +LF . +(47) +Communication Load: By (39), each server h ∈ [H] sends S coded packets Yh,[S], each of +B +LF +symbols, and the coefficient vectors q[K] can be sent in KN symbols, thus the achieved load is +RA = inf +B∈N+ +1 +B +� +H · S · B +LF + H · KN +� += HS +LF . +VI. OPTIMALITY OF MAN-PDA BASED RSP-LFRB SCHEME +In the following subsections, we separately derive the lower bounds in (21). + +Let r(M) be the lower convex envelope of the following points: +�� +1 + t(N − 1) +K +, K − t +t + 1 +� +: t = 0, 1, . . . , K +� +. +(48) +Notice that, r(M) is the achievable communication load with memory size M for the key superpostion +scheme under signal security and demand privacy in the single server setup in [11]. Moreover, r(M) has +the following relationship with T(M) and R(M): +T(M) = 1 +L +� +N + r(M) +� +, +R(M) = H +L · r(M). +(49) +We will use the following upper bound for r(M), which was proved in [11, Lemma 4]: +r(M) ≤ N − M +M − 1 , +∀ M ∈ (1, N]. +(50) +In the following, we prove the bounds in (21) by deriving a lower bound for T ∗(M) and R∗(M) +respectively. +A. Upper bound for +T(M) +T ∗(M) +For any M ∈ [1, N], the optimal storage size +T ∗(M) ≥ N +L . +(51) +In fact, +NB = H(W[N]) += I(W[N]; Z[I+L]) + H(W[N]|Z[I+L]) += I(W[N]; Z[I+L]) +(52a) += I(W[N]; Z[I]) + I(W[N]; Z[I+1:I+L]|Z[I]) += I(W[N]; Z[I+1:I+L]|Z[I]) +(52b) +≤ H(Z[I+1:I+L]|Z[I]) +≤ LT ∗(M)B, +(52c) +where (52a) and (52b) follows from the constraints (3a) and (3b), respectively. +Therefore, +1) for any K and M ≥ 2, +T(M) +T ∗(M) ≤ +1 +L(N + r(M)) +N +L +(53a) += 1 + 1 +N · r(M) += 1 + 1 +N · N − M +M − 1 +(53b) +≤ 1 + N − 2 +N +(53c) +≤ 2, +where (53a) and (53b) follows from(51) and (50), respectively, and (53c) follows from the fact +M ≥ 2. + +2) for K ≤ N, since r(M) is upper bounded by K, +T(M) +T ∗(M) ≤ (N + K)/L +N/L += 1 + K +N +(54) +≤ 2. +(55) +B. Upper bound for +R(M) +R∗(M) +We first derive a lower bound for R∗(M) in Lemma 1. +Lemma 1. For any M ∈ [1, N], the optimal load R∗(M) has the following lower bound: +R∗(M) ≥ +max +u∈[min{⌊ N +2 ⌋,K}] +H +L · u(N − u + 1 − uM) +N +. +(56) +Proof: Consider the case where each user demands a file, and denote Dk the index of the file demanded +by user k for all k ∈ [K]. Denote the signal of server h ∈ [H] under the demands (D1, . . . , DK) = +(d1, . . . , dK) by X(d1,...,dK),h. For each u ∈ [min{⌊ N +2 ⌋, K}], consider the first u caches C1, · · · , Cu. For +each r ∈ [⌊ N +u ⌋] and h ∈ [H], denote +Xr,h ≜ X((r−1)u+1,(r−1)u+2,...,ru,1,1,...,1),h. +(57) +Let L and I be disjoint subsets of [H] with cardinality L and I, respectively. Denote � +Nu = u +� N +u ⌋, then +the files {Wn : n = 1, 2, . . . , � +Nu} can be decoded with the signals {Xr,h : r ∈ [⌊ N +u ⌋], h ∈ L ∪ I} and the +caches {C1, . . . , Cu}. Notice that � +Nu ≥ u( N +u − u−1 +u ) = N − u + 1, therefore +(N − u + 1)B +≤ � +NuB += I(W[ � +Nu]; X[⌊ N +u ⌋],L∪I, C[u]) += I(W[ � +Nu]; X[⌊ N +u ⌋],I) + I(W[ � +Nu]; X[⌊ N +u ⌋],L, C[u] | X[⌊ N +u ⌋],I) +≤ I(W[N]; X[⌊ N +u ⌋],I, Q[K],I, ZI) + +� +h∈L +⌊ N +u ⌋ +� +r=1 +H(Xr,h) + +u +� +k=1 +H(Cu) +≤ I(W[N]; Q[K],I, ZI) + +� +h∈L +⌊ N +u ⌋ +� +r=1 +H(Xr,h) + uMB +(58a) += I(W[N]; ZI) + +� +h∈L +⌊ N +u ⌋ +� +r=1 +H(Xr,h) + uMB +(58b) +≤ +� +h∈L +⌊ N +u ⌋ +� +r=1 +H(Xr,h) + uMB, +(58c) +where (58a) follows since the signals X[⌊ N +u ⌋],I are determined by ZI and Q[K],I by (5); (58b) follows since +the queries Q[K],I are independent of (W[N], ZI) by (4), and (58c) follows from the I-Security constraint +(3a). +Sum the inequalities in (58c) over all subsets L ⊂ [H] of cardinality L, +�H +L +� +(N − u + 1)B + +≤ +� +L⊂[H],|L|=L +� +h∈L +⌊ N +u ⌋ +� +r=1 +H(Xr,h) + +�H +L +� +uMB +≤ +⌊ N +u ⌋ +� +r=1 +H +� +h=1 +� +L⊂[H],|L|=L +H(Xr,h) · 1(h ∈ L) + +�H +L +� +uMB += +�H − 1 +L − 1 +� ⌊ N +u ⌋ +� +r=1 +H +� +h=1 +H(Xr,h) + +�H +L +� +uMB +≤ +�H − 1 +L − 1 +��N +u +� +R∗(M)B + +�H +L +� +uMB +≤ +�H − 1 +L − 1 +�N +u R∗(M)B + +�H +L +� +uMB. +Therefore, +R∗(M) ≥ H +L · u(N − u + 1 − uM) +N +, +∀ u ∈ +� +min +��N +2 +� +, K +�� +. +(59) +Then the proof is completed. +Now for each u ∈ +� +min{⌊ N +2 ⌋, K} +� +, define +Lu(M) ≜ H +L · u(N − u + 1 − uM) +N +, +∀ M ∈ [0, N]. +(60) +Then Lemma 1 indicates that for each u ∈ +� +min{⌊ N +2 ⌋, K} +� +, +R∗(M) ≥ Lu(M), +∀ M ∈ [1, N]. +(61) +For any M ∈ [0, N], define +f(M) ≜ +H +4LN · (N − M)(N + M + 2) +M + 1 +. +(62) +Notice that f(M) is convex in M. The following lemma shows that f(M) is a lower bound of R∗(M) +on +� +max{2, N−2K +2K+1 }, N +� +. +Lemma 2. For any M ∈ [max{2, N−2K +2K+1 }, N +� +, +R∗(M) ≥ f(M). +(63) +Proof: Consider the interval +� +max +� N−2⌊N/2⌋ +2⌊N/2⌋+1 , N−2K +2K+1 +� +, N +� +, which can be split into min{⌊ N +2 ⌋, K} +intervals: +� +max +�N − 2⌊N/2⌋ +2⌊N/2⌋ + 1 , N − 2K +2K + 1 +� +, N +� += +min{⌊ N +2 ⌋,K} +� +u=1 +�N − 2u +2u + 1 , N − 2u + 2 +2u − 1 +� +. +(64) +Notice that, by the fact ⌊ N +2 ⌋ ≥ +N−1 +2 , +N−2⌊N/2⌋ +2⌊N/2⌋+1 ≤ +1 +N ≤ 2, the interval +� +max +� N−2⌊N/2⌋ +2⌊N/2⌋+1 , N−2K +2K+1 +� +, N +� +encloses [max{2, N−2K +2K+1 }, N +� +as its sub-interval. Therefore, for any M ∈ [max{2, N−2K +2K+1 }, N +� +, there exists +u ∈ [min{⌊ N +2 ⌋, K}] such that M ∈ +� N−2u +2u+1 , N−2u+2 +2u−1 +� +. It is easy to verify: +Lu +�N − 2u +2u + 1 +� += f +�N − 2u +2u + 1 +� += H(N + 1) +LN +· u(u + 1) +2u + 1 , +(65) +Lu +�N − 2u + 2 +2u − 1 +� += f +�N − 2u + 2 +2u − 1 +� += H(N + 1) +LN +· u(u − 1) +2u − 1 . +(66) + +That is Lu(M) and f(M) concides on the end points of the interval +� N−2u +2u+1 , N−2u+2 +2u−1 +� +. Since f(M) is +convex on M, we concludes that +Lu(x) ≥ f(x), +∀ x ∈ +�N − 2u +2u + 1 , N − 2u + 2 +2u − 1 +� +. +(67) +Therefore, with (61) and (67), we concludes (63), which holds for all M ∈ +� +max +� +2, N−2K +2K+1 +� +, N +� +. +Therefore, +1) for any K, M ≥ max +� +2, N−2K +2K+1 +� +, by (49), (50) and Lemma 2, +R(M) +R∗(M) +≤ +H +L · r(M) +f(M) +≤ +H +L · N−M +M−1 +H +4NL · (N−M)(N+M+2) +M+1 += 4N · +M + 1 +(M − 1)(N + M + 2) += 4N · +� +1 +N + M + 2 + +2 +(M − 1)(N + M + 2) +� +≤ 12 · +N +N + 4 +(68) +< 12, +where in (68), we used the fact M ≥ 2. This completes the proof of second inequalilty in (21). +2) for K ≤ N and M ≤ max +� +2, N−2K +2K+1 +� +, +R(M) ≤ HK +L , +(69) +and since R∗(M) must be non-increasing with M, +R∗(M) ≥ R∗� +max +� +2, N − 2K +2K + 1 +�� +(70) +a) if N ≥ 6K + 2, max +� +2, N−2K +2K+1 +� += N−2K +2K+1 , by (63), +R∗� +max +� +2, N − 2K +2K + 1 +�� +≥ f +�N − 2K +2K + 1 +� +(71) += +H +4LN · +� +N − N−2K +2K+1 +�� +N + N−2K +2K+1 + 2) +N−2K +2K+1 + 1 +(72) += +H +4LN · +� +N + 1 − N+1 +2K+1 +�� +N + 1 + N+1 +2K+1 +� +N+1 +2K+1 +(73) += H +4L · N + 1 +N +· 2K · +� +1 + +1 +2K + 1 +� +(74) +≥ HK +2L , +(75) +thus by (69), (70) and (75), +R(M) +R∗(M) ≤ 2 ≤ 12. +(76) + +b) if N < 6K + 2 and N ≥ 4, max +� +2, N−2K +2K+1 +� += 2, by (63), +R∗� +max +� +2, N − 2K +2K + 1 +�� +≥ f(2) +(77) += +H +4LN · (N − 2)(N + 4) +3 +. +(78) +thus by (69), (70) and (78), +R(M) +R∗(M) ≤ +12KN +N 2 + 2N − 8 +(79) +≤ +12N 2 +N 2 + 2N − 8 +(80) +≤ 12, +(81) +where the last step follows from N ≥ 4. +c) if N < 6K + 2 and N ≤ 3, it is easy to obtain +R(M) ≤ H +L · K +� +1 − M − 1 +N − 1 +� +, +(82) +and by Lemma 1, +R∗(M) ≥ H +L +� +1 − M +N +� +. +(83) +thus, +R(M) +R∗(M) ≤ KN +N − 1 ≤ +N 2 +N − 1 ≤ 4.5 ≤ 12. +(84) +From the above arguments, we conclude that, for all K, N except for K > N and 1 ≤ M ≤ max +� +2, N−2K +2K+1 +� +, +R(M) +R∗(M) ≤ 12. +(85) +Notice that, when K > N, max +� +2, N−2K +2K+1 +� += 2. Hence, we proved (85) for K ≤ N or K > N, M ≥ 2. +VII. NUMERICAL RESULTS +In this section, we present some numerical results to illustrate the comparison of the achievable MSC +region and its lower bounds, and the influences of parameters I and L. +A. Achievable MSC Region & Lower Bounds +By (20), as the boundaries of the region are parallel to T- and R- axes, as illustrated in Fig. 2, it is +sufficient to plot the projections to the M-T plane and M-R plane, i.e., the functions T(M) and R(M). +In Fig. 3, we plot T(M) and R(M) for a system with H = 20, L = 10 to illustrate the following two +regimes: +a) K ≤ N: See Fig. 3(a) and 3(b); +b) K > N: See Fig. 3(c) and 3(d). +For comparison, the corresponding bound (see (51) or (56)) is also plotted. We choose (N, K) = (100, 10) +and (10, 100) for the above two regimes, respectively. From the figures, we have the following observations: +1) For the regime K ≤ N, the storage size T(M) is close to the constant lower bound N +L over all +M ∈ [1, N]. This is because T(M) achieves its maximum at T(1) = K+N +L , which is dominated +by +N +L . In fact, the bound in (54) indicates that the bound can be tighted to 1.1 for the chosen + +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +10 +10.1 +10.2 +10.3 +10.4 +10.5 +10.6 +10.7 +10.8 +10.9 +11 +(a) T(M) for N = 100, K = 10 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +(b) R(M) for N = 100, K = 10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +(c) T(M) for N = 10, K = 100 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +(d) R(M) for N = 10, K = 100 +Fig. 3: Storage size T(M) and communication load R(M) for H = 20 servers with L = 10. +parameters. The communication R(M) is close to its lower bound over all M ∈ [1, N]. For the +chosen parameters, (76) indicates that the bound can be tighted to 2 for 1 ≤ M ≤ N−2K +2K+1 . +2) For the regime K > N, the storage size T(M) departs far from the constant bound N +L for M close +to 1, because T(1) = K+N +L +is dominated by K +L . But it dramatically decreases with M, since the +coded version of the superposition key sizes are the same with the communication load of each +server, which dramatically decreases with M due to the coded multi-cast opportunities provided by +coded caching. Similarly, the communication load R(M) scales with K at M = 1, and decreases +dramatically around M = 1. In particular, as stated in Theorem 2, when M ≥ 2, both T(M) and +R(M) are close to their own lower bounds. +3) In both regimes, T(M) and R(M) have similar trends in M, since they are linear to each other, +see (22). +In summary, the numerical results are consistent with Remark 4: the only un-bounded regime is K > N +and 1 ≤ M < 2. +B. Influence of Parameters I and L +As mentioned in Section IV-B, the number of colluding servers that the system can resist does not affect +the total communication load, but I has to satisfy I + L ≤ H. The cost of resisting the colluding servers +is that each user has to gather more signals to decode if I is larger. The analysis in Remark 5 indicates +that the communication load to each user has to reached (1 + I +L)(1 − t +K +� +for the point (Mt, Tt, Rt). +To see how the parameter L influences the total communication load, we plot the total communication +load as a function of L for a system with H = 20, L = 10, N = 50, K = 10 and M = 25.5 (or +equivalently t = 5) in Fig. 4(a). It is clear that, the larger L, the total communication load is smaller. The +L is inversely proportional to the total communication load by (19). In fact, the parameter L determines +the size of each subfile by (23), and each server is responsible for transmiting one encoded subfile over all + +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +(a) Total communication load R, I ≤ H − L +(b) Communication load to each user Ru +Fig. 4: Influence of parameters L and I to total communication load R and the communication load to each user Ru for a system with +N = 50, K = 10, M = 25.5. +files.With larger L, the size of each encoded subfile is smaller, and thus results in smaller communication +load. +To see how I and L jointly influence the communication load to each user, we plot Ru = (1+ I +L)(1− t +K +� +for the above system. It can be seen that, with given I, Ru decreases with L, but with given L, Ru increases +linearly with I. Notice that, with the parameters I and L, the system can resist I colluding servers and +H − I − L non-responding (or un-connected) servers. Notice that, each user need signals from at least +I + L servers, thus, with larger I + L, the connectivity requirement for the topology to implement the +scheme is higher. +In summary, larger L decreases both total communication load and the load to each user, but larger +I only increases the communication load to users, and I + L determines the threshold of the number of +servers that a user need to access. +VIII. CONCLUSION +A RSP-LFRB scheme is proposed within PDA framework, where the techniques of coded caching, +secret sharing, key superpositons are integrated such that the signal security, users’ demand privacy, and +the blindness to colluding servers are simultaneously guaranteed. The storage size and the communication +load of MAN-PDA based RSP-LFRB scheme are showed to be to within a multiplicative gap of at most +2 and 12 from optimal in all regimes, except for small memory regime with less files than users. +REFERENCES +[1] M. A. Maddah-Ali, and U. 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Sel. +Areas Commun., vol. 40, no. 3, pp. 968-981, Mar. 2022. + diff --git a/StFAT4oBgHgl3EQf1x6P/content/tmp_files/load_file.txt b/StFAT4oBgHgl3EQf1x6P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb98b67171fe754ecc21027f958ccf1e048b6d7a --- /dev/null +++ b/StFAT4oBgHgl3EQf1x6P/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf,len=1008 +page_content='Robust, Private and Secure Cache-Aided Scalar Linear Function Retrieval from Blind Servers Qifa Yan Member, IEEE,, Xiaohu Tang, Senior Member, IEEE, and Zhengchun Zhou, Member, IEEE Abstract This work investigates a system composed of multiple distributed servers and users, where each user is equiped with a local cache, and aims to retrieve a scalar linear function of the files of a library from the servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The files are stored at the servers such that any I colluding servers can not obtain any information about the files, while any I + L servers can together recover all files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In addition, the system are designed to satisfy: (a) Each user’s demand must be satisfied upon receiving the signals from any I + L servers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (b) The content of the library must be kept secure from a wiretapper who obtains all the signals from the servers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (c) Any subset of users together with all the servers can not obtain any information about the demands of the remaining users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A coded scheme is derived to satisfy the above constraints by incoporating the idea of secret sharing and key superposition into the framework of Placement Delivery Array (PDA), originally proposed to characterize the single-server coded caching system without any security or privacy constraints, where the memory size at each user, the storage size at each server and the total communication load over all servers are characterized by the PDA parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It is shown that the PDAs describing the original Maddah-Ali and Niesen’s coded caching scheme result in an achievable memory-storage- communication region, such that the storage size and the communication load are optimal to within a constant multiplicative gap, except for the small memory regime when the number of files is smaller than the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Index Terms Cache, Secret Sharing, Placement delivery array, Robust decoding, Scalar linear function retrieval, Security, Private I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' INTRODUCTION Coded caching is a technique to reduce the peak-time communication load by jointly designing the pre-stored cache contents and communication signals, which was introduced by Maddah-Ali and Niesen (MAN) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The model consists of a single server holding a set of files that connects to multiple cache- aided users through a shared-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The system operates at two phases: In the placement phase, the users fill their own caches without knowing their future demands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In the delivery phase, knowing the users’ demands, the server satisfies them by transmitting coded signals to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For a system with N files and K users, the MAN scheme achieves the optimal load-memory tradeoff among all uncoded placement schemes when N ≥ K [2], and for N < K after removing some redundant transmissions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Recently, it was showed that allowing the users to demand arbitrary linear combinations of the files does not increase the load compared to the case single file retrieval, at least under uncoded placement [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' One important problem in MAN scheme is that, the number of packets each file needs to be partitioned into (called subpacketization) grows exponentially with the number of users K [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In [6], Yan et al proposed an combination object called Placement Delivery Array (PDA) to characterize a subclass of coded caching schemes with uncoded placement, and proposed two constructions of PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Content security and demand privacy are critical aspects of practical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In [7], the content of the library must be protected against an external wiretapper who obtains the signals transmitted during the delivery phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' This was achieved by pre-storinging security keys into user caches for the part of the files The authors are with the Information Coding & Transmission Key Lab of Sichuan Province, CSNMT Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Coop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Centre (MoST), Southwest Jiaotong University, Chengdu 611756, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (email: qifayan@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='cn, xhutang@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='cn, zzc@swjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='cn) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='08711v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='IT] 20 Jan 2023 that were not cached in the MAN scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The security keys are stored in a structured way such that each user can decode all the multicast signals it needs to decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In [8], [9], users’ demand privacy are guarranteed by adding virtual users, so that the real users can not distinguish if the demands are from real or virtual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Another technique to guarantee privacy is privacy key proposed in [10], which imposes that any subset of colluding users must not obtain any information about the demands of other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The key idea in [10] is that, each user also privately caches some random linear combination of the parts of the files that were not cached in the MAN scheme, which was called privacy keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The queries are generated by adding the coefficients used to generate the privacy keys to the real demands, so that the users can decode the linear combination of files with the queries and further decode their own demands with the privacy keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In [11], a key superposition scheme was proposed to guarantee both content security against a wiretapper and demand privacy against colluding users simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The idea is superposing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', sum together) the security keys and privacy keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It was showed that the load-memory tradeoff in this case is the same as in the setup with only content security guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The idea of key superposition was incorporated into the framework of Placement Delivery Array (PDA), with the advantage that low subpacketization schemes can be obtained directly from existing PDA constructions, such as the ones in [6], [12]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The above progresses are obtained under a basic setup with a single server and multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Another interesting setup involves distributed systems with multiple servers and a single user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Demand privacy for such networks were known as Private Information Retrieval (PIR) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The capacity of PIR has been characterized in [17] for single file retrieval, in [18] for scalar linear function retrieval, and in [19] for single file retrieval and colluding servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' PIR with a cache-aided user was investigated in [20]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Coding across the distributed servers with Maximum Distance Separable (MDS) codes are useful, since it saves storage while allowing node failures or erasures, which commonly happen in such systems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The capacity of PIR from MDS-coded servers was charactered in [25], while the schemes achieving the smallest download rate with almost optimal sub-packetization are proposed in [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In addition to signal security, in distributed systems, another type of security is blind server, where the servers might be untrusted and required to be blind to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In 1979, Shamir proposes a technique that allows multiple servers to jointly storing a message while keeping the message secret from any subset of servers with cardinality not exceeding some threshold [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The techique is called secret sharing, which is well used in many distributed systems where the contents need to be kept blind from servers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', the private polynomial computation from coded servers [29], and multi-user blind private information retrieval [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For multiple- server-multiple-user systems, the techniques from coded caching and PIR were combined in [31], [32] to guarantee server-side privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In [33], [34], the technique of key superposition is incoprated into the MDS-coded server system under PDA framework, which guarantees demand privacy against both servers and colluding users, signal security against an wiretapper, and robust decoding against server failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' As far as we known, the blind server constraint is not investigated in multiple-server-multiple-user networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In this paper, we aim to design coded caching schemes with blind servers, and all the other constraints in [34] still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' That is, in addition to demand privacy, signal security and robust decoding, the design of the distributed storage at the servers need to guarrantee that the content of the files are blind to any subset of I servers, and the whole files can be recovered from the contents at any I + L servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We refer the model as Robust, Secure and Private Scalar Linear Function Retrieval from Blind Servers (RSP-LFRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Unlike the assumption that the files are stored in the MDS coded form across the servers, the model here allows to flexibly design the contents stored at the servers, and the storage size at each server is identified as a permance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Our main contributions for the proposed RSP-LFRB model are: 1) We propose a procedure to obtain a RSP-LFRB scheme from a given PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The advantage of establishing a PDA framework is that RSP-LFRB schemes with low subpacketizations can be easily obtained from various existing PDA constructions [6], [12]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The key idea is to incoporate the secret sharing scheme into the PDA framework, and using the key superpositions in a similar way as in the single server system [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The memory size at each user, the storage size at the servers, and the communication load is characterized with the PDA parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 1: System model 2) Following the proposed procedure, an achievable region of memory-storage-communication1 triples based on the PDAs that describe the original MAN scheme in [1] (MAN-PDAs) are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, the achievable storage size is optimal within a constant multiplicative gap 2, and the achievable communication load is optimal within a constant multiplicative gap 12, except for the small memory regime when the number of files is smaller than the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Section II gives the formal system model description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Section III reviews the PDA framework and gives an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Section IV summarizes our main results, where the proof details are deferred to Sections V and VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Section VII presents some numerical results, and Section VIII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notations: In this paper, N+ denotes the set of positive integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Fq and Fn q denote the finite field of cardinality q, for some prime power q, and the n-dimensional vector space over Fq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For two integers m, n such that m ≤ n, we use [m : n] to denote the set of the first positive integers {m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' [1 : n] is also denoted by [n] for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We use XA to denote the tuple composed of {Xi : i ∈ A} for some integer set A, where the elements are ordered increasingly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', X[3] = (X1, X2, X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For variables with two or more indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', Xi,j, we use XA,B to denote the tuple {Xi,j : i ∈ A, j ∈ B}, where the elements are listed in lexicographical order, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[3],[2] = (X1,1, X1,2, X2,1, X2,2, X3,1, X3,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' SYSTEM MODEL Let N, K, H, I, L be positive integers satisfying I + L ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A (N, K, H, I, L) system, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 1, consists of a file library of N files, H servers (denoted by 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , H), where each server is connected to K users (denoted by 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , K) via a dedicated shared-link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The N files are mutually independent and uniformly distributed over FB q for some prime power q, where B denotes the file length, that is, H(W1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' = H(WN) = B, H(W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , WN) = H(W1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' + H(WN), where the entropy function H(·) is calculated with logorithm q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For any vector a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , aN) ∈ FN q , we use Wa to denote the linear combination of the files with coefficient vector a: Wa := � n∈[N] an · Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (1) The system generates the stored content of server h (denoted by Zh) from the N files and a random variable U from some alphabet U with an encoding function χh : FNB q × U �→ F⌊TB⌋ q , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', Zh = χh(W[N], U) ∈ F⌊TB⌋ q , ∀ h ∈ [H], (2) where T is the storage size at each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The storage contents at the servers need to satisfy the following properties: 1To distinguish the storage capacity between users and servers, in this paper, we will refer “memory” and “storage” for the user side and server side respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Files 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='. N Random Variable U Encoder Colluding Blind Servers 2 X1 X2 XH Links Wiretapper Users & Caches 2 3 K1) I-Security: Any I servers should be oblivious to the files W[N]: I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' ZI) = 0, ∀ I ⊆ [H], |I| = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (3a) 2) (I + L)-Robust Recovery: The whole files W[N] can be recovered from any I + L servers: H(W[N]| ZJ ) = 0, ∀ J ⊆ [H], |J | = I + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (3b) The system operates in two phases as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Placement Phase: Each user k ∈ [K] generates some random variable Pk from some finite alphabet Pk and cache some content Ck as a function of Pk, U and the file library W[N] using some function ϕk : Pk × U × FNB q �→ F⌊MB⌋ q , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', Ck := ϕk(Pk, U, W[N]) ∈ F⌊MB⌋ q , ∀ k ∈ [K], where M is the memory size of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The encoding functions ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , ϕK are known by the servers, but the randomness P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , PK are kept private by the corresponding users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Delivery Phase: Each user k ∈ [K] generates a demand dk = (dk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , dk,N)⊤ ∈ FN q , meaning that it is interested in retrieving the linear combination of the files Wdk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The random variables d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , dK, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , PK, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , WN, U are independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', H(d[K], W[N], P[K], U) = � k∈[K] H(dk) + � n∈[N] H(Wn) + � k∈[K] H(Pk) + H(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' User k ∈ [K] sends a query Qk,h of length ℓk,h to server h for each h ∈ [H], where the queries Qk,[H] := (Qk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , Qk,H) are generated from some query function κk,h : FN q × Pk �→ F ℓk,h q , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', Qk,h := κk,h(dk, Pk) ∈ F ℓk,h q , ∀ h ∈ [H], (4) Upon receiving the queries from all the users, server h ∈ [H] creates a signal Xh as Xh := φh(Zh, Q[K],h) ∈ F⌊RhB⌋ q , ∀h ∈ [H], (5) for some encoding function φh : F⌊TB⌋ q × F � k∈[K] ℓk,h q �→ F⌊RhB⌋ q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The quantity Rh, h ∈ [H], is referred to as the communication load of server h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The (total) communication load of the system is defined as R := � h∈[H] Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The following conditions must hold for all demands d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , dK ∈ FN q : 1) Robust Decoding: Each user k can retrieval its demanded linear combination of the files Wdk from any (I + L)-subset of signals and the content in its memory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', for each k ∈ [K]: H(Wdk | XAk, dk, Ck) = 0, ∀ Ak ⊆ [H] : |Ak| = I + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (6a) 2) Signal Security: A wiretapper, who is not a user and observes all the delivery signals, can not obtain any information about the contents of the library files: I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[H]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (6b) 3) Demand Privacy: Any subset of users together with the servers can not jointly learn any information on the demands of the other users, regardless of the file realizations: I(d[K]\\S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' CS, dS, Q[K],[H], Z[H] | W[N]) = 0, ∀ S ⊆ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (6c) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A Memory-Storage-Communication (MSC) triple (M, T, R) ∈ [1, N] × R+ × R+ is said to be B-achievable if, for any ϵ > 0, there exists a scheme satisfying all the conditions in (3) and (6) with memory size less than M + ϵ, storage size less than T + ϵ, and load less than R + ϵ with file-length B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The MSC region of the system is defined as G = {(M, T, R) : (M, T, R) is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The main objective of this paper is to characterize the MSC region G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For simplicity, we call a valid scheme satsify the constraints (3) and (6) a Robust, Secure, Private Scalar Linear Function Retrieval from Blind Servers (RSP-LFRB) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For a given scheme, we are also interested in its subpacketization level, which is defined as the number of packets each file has to be partitioned into in order to implement the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Throughout this paper, we consider the case N ≥ 2, since demand privacy is impossible for N = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', there is only one possible file to be demanded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Remark 1 (Privacy & Security Against the Wiretapper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that, by (6c), I(d[K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[H] | W[N]) ≤ I(d[K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[H], Q[K],[H], Z[H] | W[N]) = I(d[K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Q[K],[H], Z[H] | W[N]) (7) = 0, where (7) follows from (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Together with (6b), it holds that I(W[N], d[K];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[H]) = 0, that is, the wiretapper having access to X[H] in fact can not obtain any information on both the files and the demands of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Remark 2 (Minimum Memory Size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It was proved in [7] that, in order to guarantee the conditions in (6a) and (6b) simultaneously, the memory size M has to be no less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus the MSC region is defined for M ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' PDAS AND A TOY EXAMPLE We will construct our RSP-LFRB scheme for any given PDA [6], which was introduced to reduce the subpacketization in coded caching in the single server system without any security or privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In this section, we first review the definition of PDA, and then give an example to highlight the key ideas in the design of our RSP-LFRB scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The general construction will be given in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Placement Delivery Array Definition 2 (PDA [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For given K, F ∈ N+ and Z, S ∈ N, an F ×K array A = [aj,k], j ∈ [F], k ∈ [K], composed of Z specific symbols “∗” in each column and some ordinary symbols 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , S, each occurring at least once, is called a (K, F, Z, S) PDA, if, for any two distinct entries aj,k and aj′,k′, we have aj,k = aj′,k′ = s, for some ordinary symbol s ∈ [S] only if a) j ̸= j′, k ̸= k′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', they lie in distinct rows and distinct columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' and b) aj,k′ = aj′,k = ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', the corresponding 2 × 2 sub-array formed by rows j, j′ and columns k, k′ must be of the following form � s ∗ ∗ s � or � ∗ s s ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It was showed in [6] that, with a given (K, F, Z, S) PDA, there exists an associated coded caching scheme in the single server system without any security or privacy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The parameter K is the number of users, F is the number of packets each file is split into (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', subpacketization), Z is the number of uncoded packets from each file stored at each user, and S is the number of coded multicast signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In our model, each file will first split into L equal-size subfiles, and those implications will be used on the subfiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A Toy RSP-LFRB Example from PDAs We derive here a RSP-LFRB scheme associated to the (K, F, Z, S) = (3, 3, 1, 3) PDA A = � � ∗ 1 2 1 ∗ 3 2 3 ∗ � � (8) for an (N, K, H, I, L) = (4, 3, 4, 1, 2) distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Let the four files be W1, W2, W3, W4 ∈ FB q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Firstly, each file Wn, n ∈ [4] is split into L = 2 equal-size subfiles, Wn = (Wn,1, Wn,2), (9) and each subfile Wn,l, l ∈ [2] is further split into F = 3 equal-size packets: Wn,l = (Wn,l,1, Wn,l,2, Wn,l,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (10) Generate NI = 4 uniform and independent random variables {Jn : n ∈ [4]} of the same size with a subfile, and SL = 6 uniform and independent random variables {Vl,s : l ∈ [2], s ∈ [3]} of the same size with a packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Then construct two groups of polynomials: � Wn(x) = Wn,1 + Wn,2x + Jnx2, ∀ n ∈ [4], (11a) �Vs(x) = V1,s + V2,sx, ∀ s ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (11b) Notice that, in accordance with (10), for each n ∈ [3], � Wn(x) and Jn can be decomposed into 3 equal-size components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', � Wn(x) = (� Wn,1(x), � Wn,2(x), � Wn,3(x)) and Jn = (Jn,1, Jn,2, Jn,3), where � Wn,j(x) = Wn,1,j + Wn,2,jx + Jn,jx2, j ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (12) Let α1, α2, α3, α4 be H = 4 distinct non-zero elements in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The system operates as follows: Server Storage Design: Each server h ∈ [4] stores the evaluations of the polynomials in (11): Zh = {� Wn(αh) : n ∈ [4]} ∪ {�Vs(αh) : s ∈ [3]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (13) Placement Phase: Each user k ∈ [3] generates a random vector pk = (pk,1, pk,2, pk,3, pk,4)⊤ ∈ F4 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The cache content of the user k is composed of pk and the (un)coded packets in the corresponding column in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The packets W[4],[2],j are associated to the j-th row of A in (8) and user k is associated to the TABLE I: The cached contents of users† according to A in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Row User 1 User 2 User 3 1 W[4],[2],1 Wp2,[2],1 ⊕ V[2],1 Wp3,[2],1 ⊕ V[2],2 2 Wp1,[2],2 ⊕ V[2],1 W[4],[2],2 Wp3,[2],2 ⊕ V[2],3 3 Wp1,[2],3 ⊕ V[2],2 Wp2,[2],3 ⊕ V[2],3 W[4],[2],3 † In addition, each user k ∈ [3] caches pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' k-th column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The packets in the j-th row of Table I of user k are created according to the entry aj,k of A in (8): if aj,k = ∗, user k caches NL = 8 uncoded packets W[4],[2],j, otherwise it caches L = 2 coded packets Wpk,[2],j ⊕ V[2],aj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Delivery Phase: Assume that user 1, 2, 3 demands the linear combination Wd1, Wd2 and Wd3, respectively, where d1, d2, d3 ∈ F4 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Each user k ∈ [3] sends qk = pk ⊕ dk to all the servers as queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Upon receiving the query vectors q[3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' each server h ∈ [4] sends a signal Xh to the users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' where Xh is composed of the query vectors q[3] and S = 3 coded packets �Y[3](αh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' which are associated to the ordinary symbols s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3 respectively: s = a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2 = a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1 = 1 : �Y1(αh) = �V1(αh) + � Wq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2(αh) + � Wq2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1(αh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' s = a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3 = a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1 = 2 : �Y2(αh) = �V2(αh) + � Wq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3(αh) + � Wq3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1(αh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' s = a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3 = a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2 = 3 : �Y3(αh) = �V3(αh) + � Wq2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3(αh) + � Wq3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2(αh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' where a polynomial � Wa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x) is defined for each a ∈ F4 q and j ∈ [3] as: � Wa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x) ≜ a1� W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x) + a2� W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x) + a3� W3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x) + a4� W4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Robust Decoding: Let’s take s = 1 to show how the users utilize the signals to decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Consider the polynomial �Y1(x) = �V1(x) + � Wq1,2(x) + � Wq2,1(x) = �V1(x) + 4 � n=1 q1,n� Wn,2(x) + 4 � n=1 q2,n� Wn,1(x) = (V1,1 + Wq1,1,2 + Wq2,1,1) + (V2,1 + Wq1,2,2 + Wq2,2,1)x + (Jq1,2 + Jq2,1)x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that, the signal �Y1(αh) sent by server h is the evaluation of �Y1(x), which is a polynomial of degree 2, thus each user can obtain �Y1(x) by interpolation with any 3 signals, and thus decode the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Table II lists all the coefficients that a user can decode from the signals of any 3 servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' TABLE II: The signals a user can decode from the signals of any 3 servers† according to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' s Subfiles 1 Subfiles 2 Noises 1 V1,1 + Wq1,1,2 + Wq2,1,1 V2,1 + Wq1,2,2 + Wq2,2,1 Jq1,2 + Jq2,1 2 V1,2 + Wq1,1,3 + Wq3,1,1 V2,2 + Wq1,2,3 + Wq3,2,1 Jq1,3 + Jq3,1 3 V1,3 + Wq2,1,3 + Wq3,1,2 V2,3 + Wq2,2,3 + Wq3,2,2 Jq2,3 + Jq3,2 † In addition, each server h ∈ [4] transmits the query vectors q[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Upon obtaining the signals in Table II, each user k ∈ [3] can proceed with the decoding process for each subfile l ∈ [2] as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For example, for subfile 1, as a1,2 = a2,1 = 1, user 1 can decode Wd1,1,2 and user 2 can decode Wd2,1,1 from the signal V1,1 + Wq1,1,2 + Wq2,1,1 = Wd1,1,2 + (Wp1,1,2 + V1,1) + Wq2,1,1 (14a) = Wd2,1,1 + (Wp2,1,1 + V1,1) + Wq1,1,2, (14b) since user 1 has cached the superposition key Wp1,1,2 + V1,1 and can compute Wq2,1,1 its cached subfiles W[4],1,1 and the query vector q2, thus can decode Wd1,1,2 by (14a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' user 2 has cached the superposition key Wp2,1,1 + V1,1 and can compute Wq1,1,2 with its cached subfiles W[4],1,2 and the query vector q1, and thus can decode Wd2,1,1 by (14b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' One can verify that each user k ∈ [3] can decode all the remaining packets Wdk,[2],[3]\\{k} from its stored contents, the signals in Table II and the query vectors q[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Security and Privacy Conditions: In addition to the robust decoding, the other conditions in (3) and (6) are also guaranteed: In fact, the secret sharing techique [28] to encode files with polynomials in (11a) guarantees the I-Security, and (I + L)-Robust Recovery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Signal Securiy and Demand Privacy are guaranteed since each signal or demand is accompanied by a key of random and uniformly distributed vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Performance: Recall that each packet is of length B 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Each user caches 12 packets and 1 vector in F4 q, whose length does not scale with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus the needed memory is M = 12 × 1 6 = 2 files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Each server stores N = 4 subfiles, each of length B 2 , and 3 coded packets, each of length B 6 , thus the storage size is T = 4 × 1 2 + 3 × 1 6 = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Each of the 4 servers transmits 3 packets and 3 vectors in F4 q, thus the achieved load is R = 4 × 3 × 1 6 = 2 files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Hence, the scheme achieves the memory-storage-load triple (M, R) = � 2, 5 2, 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Since each file is split into 6 equal-size packets, the subpacketization level is LF = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' MAIN RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' PDA based RSP-LFRB Schemes With any given PDA, we will construct an associated RSP-LFRB scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We will prove the following theorem by presenting and analyzing the construction in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For any (N, K, H, I, L) system and a given (K, F, Z, S) PDA A, there exists an associated RSP-LFRB scheme that achieves the MSC triple � MA, TA, RA � = � 1 + Z F (N − 1), 1 L � N + S F � , HS LF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (15) with subpacketization LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Remark 3 (Comparison with MDS Coded Server Systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A previous work in [34] has extended the key superposition scheme in [11] to MDS coded servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Unlike the previous work, which assumes that the files are stored at the servers in the form of MDS code, the design of the storage at the servers is a part of the RSP-LFRB scheme, and we also take the storage size at each server into account as a cost measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, this work further inserts an additional constraint that any I servers are blind to the file library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' This is achieved by inserting random variables in the coded subfiles with secret sharing technique [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It can be observed that, when I = 0, the strategy degrades to the setup in [34], where the MDS generator matrix is given by G = � ��� 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 1 α1 α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' αH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' αL−1 1 αL−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' αL−1 H � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (16) The proposed scheme indicates that the techniques of coded caching, key superpositions can be integrated with the secret sharing to provide robustness and protections for data security and demand privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The advantage of establishing a PDA framework is that, all the existing PDAs can be used to derive the RSP-LFRB scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' As the subpacketization of the PDA based RSP-LFRB scheme is LF, which is proportional to F, PDAs with small number of rows result low subpacketization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Among all PDA constructions, the MAN-PDAs that describe the Maddah-Ali and Niesen’s coded caching scheme [1] is of very important, which we will mainly discuss in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Optimality of MAN-PDA based RSP-LFRB Schemes For any integer t ∈ [0 : K], define the set Ωt ≜ {T ⊆ [K] : |T | = t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (17) It was proved in [6] that the following construction is a (K, �K t � , �K−1 t−1 � , � K t+1 � ) PDA, which describes the MAN scheme in [1] and will be referred to as MAN-PDA in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2: Achievable MSC Region, N = 50, K = 10, H = 20, L = 10, I ≤ 10 Definition 3 (MAN-PDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Fix any integer t ∈ [0 : K], denote the set Ωt = {Tj : j ∈ [ �K t � ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Also, choose an arbitrary bijective function κt+1 from Ωt+1 to the set �� K t+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Then, define the array At = [aj,k] as aj,k ≜ � ∗, if k ∈ Tj κt+1({k} ∪ Tj), if k /∈ Tj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (18) Example 1 (A MAN-PDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Consider K = 4, t = 2, let T1 = {1, 2}, T2 = {1, 3}, T3 = {1, 4}, T4 = {2, 3}, T5 = {2, 4} and T6 = {3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Let κ3 be the lexicographic order of a subset of size 3 in Ω3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', κ3({1, 2, 3}) = 1, κ3({1, 2, 4}) = 2 and κ3({1, 3, 4}) = 3 and κ3({2, 3, 4}) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The corresponding (4, 6, 3, 4) PDA is given by A2 = � ������ ∗ ∗ 1 2 ∗ 1 ∗ 3 ∗ 2 3 ∗ 1 ∗ ∗ 4 2 ∗ 4 ∗ 3 4 ∗ ∗ � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The following theorem summarizes the performance of MAN-PDA and its optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Let (M, T(M), R(M)) be the curve formed by connecting the points (Mt, Tt, Rt) = � 1 + t(N − 1) K , 1 L � N + K − t t + 1 � , H(K − t) L(t + 1) � , t ∈ [0 : K] (19) sequentially, then the region {(M, T, R) : T ≥ T(M), R ≥ R(M)} (20) is achievable in an (N, K, H, I, L) RSPB-LFR system, where the point (Mt, Tt, Rt) can be achieved with subpacketization L �K t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, the optimal storage size T ∗(M) and load R∗(M) satisfies T(M) T ∗(M) ≤ 2, R(M) R∗(M) ≤ 12, (21) if K ≤ N or K > N, M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2, the curve (M, T(M), R(M)) is composed of K line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Then the 1 R Load 10 0 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2 Storage Size T 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1 5 50M I(M2, T2, R2) (M3, T3, R3) (M4, T4, R4) 0 (M5, T5, R5) 10 (M6, T6, R6) 20 (M, TR) 30 Ms,T8, Rs) Tg, Rg)40 Memory Size M R10)(Mo, To, Ro) 20boundary is formed by the above surfaces starting from the curve and parallel to the T- and R-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Obviously, the curve (M, T(M), R(M)), M ∈ [1, N] forms the all Pareto-optimal points of the set, thus, we only need to verify the achievability of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We only need to prove the achievability of the curve (M, T(M), R(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In fact, the achievability of the point (Mt, Tt, Rt) directly follows from Theorem 1 and the fact that At in Definition 3 is a (K, �K t � , �K−1 t−1 � , � K t+1 � ) MAN-PDA [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, for general M ∈ [1, N], M lies in [Mt−1, Mt] for some t ∈ [K], the point (M, T(M), R(M)) can be achieved by memory-sharing [1] between the points (Mt−1, Tt−1, Rt−1) and (Mt, Tt, Rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus, the achievability of the region (20) follows directly Theorem 1 and the construction of MAN-PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The optimality in proved by deriving a lower bound for the T(M) and R(M) respectively, where the details are presented in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that, the storage size T(M) = N L + 1 H R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (22) In fact, from scheme description in Section V, N L is the storage size used to store the encoded version of the files, while 1 H R(M) is the size of the encoded version of security keys used to protect the signal from the wiretapper, which is the same with the signal size transmitted by each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that the achievable region does not depend on the paramenter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' This is because that the communication load is reflects the load over all servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' From the user’s view, the load to each user needs to be larger than � 1 + I L �� 1 − t K � (Remark 5 in Section V-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus, although the parameter I does not influence the total communication load transmitted by the servers (as long as I +L ≤ H), it increases the amount of signals that each user need to decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We will present some numerical results to illustrate the influence of the parameters I and L in Section VII-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Remark 4 (On Unbounded Regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In the regime K > N, 1 ≤ M < 2, the gap is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The main problem in small memory regime for the single server model when K > N is that, if security keys are used [7], [11], for the point M = 1 the best know achievable communication load is K, while the best known converse is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus, it seems that the larger communication load when K > N is mainly caused by the security condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' closing the gap in small memory regime when K > N is an open problem in the single server system with signal security constraint constraint setup [7], [11], and scalar linear function retrieval from MDS coded servers with signal security constraint setups [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' PDA BASED RSPB-LFR SCHEME In this section, we prove Theorem 1 by deriving a RSP-LFRB scheme for an (N, K, H, I, L) system from any given (K, F, Z, S) PDA A = [aj,k]F×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' PDA Based Scheme Based on A, each file Wn is partitioned into LF equal-size packets as follows: Each file Wn is first patitioned into L subfiles Wn = (Wn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , Wn,L), ∀ n ∈ [N], (23) where Wn,l ∈ F B L q is the l-th subfile of Wn for all l ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Each subfile Wn,l is further partitioned into F packets Wn,l = (Wn,l,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , Wn,l,F), ∀ n ∈ [N], l ∈ [L], (24) where Wn,l,j ∈ F B LF q is the j-th packet of the subfile Wn,l for all j ∈ [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The system generates NI random subfiles, denoted by J1,1, J1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , JN,I, and LS random packets, denoted by V1,1, V1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , VL,S, where the random subfiles Jn,i are uniformally distributed over F B L q , and the random packets Vl,s are uniformally distributed over F B LF q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' That is, the random variable U is generated as U = (J1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , JN,I, V1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , VL,S) ∼ Unif � F B(NIF +S) LF q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (25) Similar to (24), Jn,i is partitioned into F equal-size random packets: Jn,i=(Jn,i,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , Jn,i,F), ∀ n ∈ [N], i ∈ [I], (26) where Jn,i,j ∈ F B L q is the j-th packet of the random subfile Jn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Define two set of polynomials � Wn(x) = L � l=1 Wn,lxl−1 + I � i=1 Jn,ixL+i−1, ∀ n ∈ [N], (27) �Vs(x) = L � l=1 Vl,sxl−1, ∀ s ∈ [S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (28) In accordance to the partitions in (24) and (26), each of the polynomials in (27) can be decomposed into F components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', � Wn(x) = �� Wn,1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , � Wn,F(x) � , ∀ n ∈ [N] (29) where � Wn,j(x) = L � l=1 Wn,l,jxl−1 + I � i=1 Jn,i,jxL+i−1, ∀ j ∈ [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (30) In the following, for a given a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , aN) ∈ FN q , we will adopt similar notations as in (1) to denote the linear combinations of {� Wn(x) : n ∈ [N]}, {� Wn,j(x) : n ∈ [N]} and {Jn,i,j : n ∈ [N]} with coefficient vector a, respectively: � Wa(x) = � n∈[N] an · � Wn(x), (31) � Wa,j(x) = � n∈[N] an · � Wn,j(x), ∀ j ∈ [F], (32) Ja,i,j = � n∈[N] an · Jn,i,j, ∀ i ∈ [I], j ∈ [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (33) Notice that, by (29), � Wa(x) is composed of F components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', � Wa(x) = �� Wa,1(x), � Wa,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , � Wa,F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Server Storage Design: Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , αH be H distinct nonzero elements in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For each h ∈ [H], the system evaluates the values of the polynomials � Wn(x) and �Vs(x) at αh, and stores them at server h, that is, Zh = �� Wn(αh) : n ∈ [N] � ∪ ��Vs(αh) : s ∈ [S] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (34) Notice that, by (29), each coded subfile � Wn(αh) is composed of F equal-size packets, which are the evaluations of the polynomials in (30) at αh: � Wn(αh) = �� Wn,1(αh), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', � Wn,F(αh) � , ∀ n ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (35) Placement Phase: Each user k ∈ [K] locally generates a random vector pk uniformaly over FN q , and constructs its local cache Ck as Ck ={pk} (36a) ∪{Wn,l,j : n ∈ [N], l ∈ [L], j ∈ [F], aj,k = ∗} (36b) ∪{Wpk,l,j + Vl,aj,k : l ∈ [L], j ∈ [F], aj,k ̸= ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (36c) Delivery Phase: Assume that user k ∈ [K] demands Wdk, for some dk ∈ FN q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Then user k ∈ [K] sends query qk = dk + pk to all the servers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', the queries Qk,[H] are constructed as Qk,h = qk = dk + pk, ∀ h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (37) Upon receiving the queries Q[K],h = q[K], each server h ∈ [H] constructs S coded signals, one for each s ∈ [S]: �Ys(αh) := �Vs(αh) + � (u,v)∈[F]×[K] au,v=s � Wqv,u(αh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (38) Then each server h ∈ [H] sends the signal Xh = � q[K], �Yh,[S](αh) � (39) to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Constraint Condition Verifications In this subsection, we prove that, the constraints in (3) and (6) are satisfied in the PDA based RSPB-LFR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' I-Security: Notice that, by the secret-sharing arguments [28], for any I ⊆ [H] such that |I| ≤ I, I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' �� W[N](αh) : h ∈ I �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (40) Therefore, I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' ZI � = I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' {� W[N](αh)}h∈I, {V[S](αh)}h∈I � = I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' {� W[N](αh)}h∈I � + I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' {V[S](αh)}h∈I|{� W[N](αh)}h∈I � = 0, where the last equality follows from (40) and the fact that the random variables V[L],[S] are generated independently and uniformly over F B LF q , and independent of W[N] and {� W[N](αh)}h∈I, so that the ran- dom variables {V[S](αh)}h∈I are distributed independently and uniformly over F B LF q and independent of � W[N], {� W[N](αh)}h∈I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (I + L)-Recovery: Notice that, each polynomial � Wn(x) is of degree I + L − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' As a result, with values at any I + L distinct points, the polynomial can be recovered by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Then by (34), the (I + L)-Recovery condition is straghtforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Robust Decoding: We need to show that for each user k ∈ [K], with any J ⊆ [H] such that |J | = I + L, with the signals {Xh}h∈J and the cache content Ck, user k can decode its demanded scalar linear function Wdk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', all the packets Wdk,[L],[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For each i ∈ [F] such that ai,k = ∗, by (36b), user k ∈ [K] has stored all the packets W[N],[L],i, thus it can directly compute the packets Wdk,l,i for each l ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Now, consider any i ∈ [F] such that ai,k ̸= ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Fixed a subset J ⊆ [H] such that |J | = I + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Let s ≜ ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' consider the polynomial �Ys(x) = �Vs(x) + � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s � Wqv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u(x) = L � l=1 Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='sxl−1 + � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s N � n=1 qv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='n · � Wn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u(x) = L � l=1 Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='sxl−1 + � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s N � n=1 qv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='n · � L � l=1 Wn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='uxl−1 + I � i=1 Jn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='uxL+i−1 � = L � l=1 � Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s + � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s Wqv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u � xl−1 + I � i=1 � � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s Jqv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u � xL+i−1 (41) = L � l=1 Yl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='sxl−1 + I � i=1 �Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='sxL+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (42) where Yl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s and �Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s are defined as Yl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s ≜ Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s + � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s Wqv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' ∀ l ∈ [L] (43a) �Ji,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='s ≜ � (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v)∈[F]×[K] au,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='v=s Jqv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' ∀ i ∈ [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (43b) Notice that, the signal �Ys(αh) sent by server h is the value of �Ys(x) evaluated at the point αh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' As �Ys(x) is a polynomial of degree I + L − 1, user k can decode the coefficients in (43) by interpolation with signals from any I + L servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, since ai,k = s, for each l ∈ [L], the signal Yl,s in (43a) can be written as Yl,s = Vl,s + Wqk,l,i + � (u,v)∈[F]×[K] au,v=s,(u,v)̸=(i,k) Wqv,l,u (44a) = Wdk,l,i + (Vl,ai,k + Wpk,l,i) + � (u,v)∈[F]×[K] au,v=s=ai,k,(u,v)̸=(i,k) Wqv,l,u, (44b) where (44b) follows from qk = pk + dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Therefore, user k ∈ [K] can decode Wdk,l,i from the the signal Yl,s by canceling the remaining terms since 1) the coded packet Vl,ai,k + Wpk,l,i is cached by user k by (36c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2) for each (u, v) ∈ [F] × [K] such that au,v = s and (u, v) ̸= (i, k), since ai,k = au,v = s, by the definition of PDA, i ̸= u, v ̸= k and ai,v = au,k = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus, user k ∈ [K] stores all the packets W[N],[L],u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Hence, user k can compute Wqv,l,u for each l ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Remark 5 (Communication Load to Each User).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' From the decoding process, each user can decode L packets from I + L coded packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Since each user need to decode L(F − Z) packets in total, each user need to collect (I + L)(F − Z) coded packets in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Thus, the communication load to each user is (I+L)(F−Z) LF = � 1 + I L �� 1 − Z F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Signal Security: We have I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[H]) = I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' q[K], {�Yh,[S](αh)}h∈[H] � ≤ I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' q[K], Y[L],[S], J[N],[I], {�Yh,[S](αh)}h∈[H] � = I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' q[K], Y[L],[S], J[N],[I] � (45a) = I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' q[K], J[N],[I] � + I � W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Y[L],[S]|q[K], J[N],[I] � = 0, (45b) where (45a) holds because {�Yh,[S](αh)}h∈[H] can be evaluated from q[K], Y[L],[S], J[N],[I];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' and (45b) follows since the variables Y[L],[S] are of form (44a), where the variables V[L],[S] are independently and uniformly distributed over F B LF q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Demand Privacy: Notice that, for any S ⊆ [K], I(d[K]\\S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' CS, dS, Q[K],[H], Z[H] | W[N]) = I(d[K]\\S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' CS, dS, q[K], Z[H] | W[N]) (46a) = I(d[K]\\S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' CS, dS, qS, Z[H] | W[N]) + I(d[K]\\S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' q[K]\\S | CS, dS, qS, Z[H], W[N]) (46b) = 0, (46c) where (46a) follows from (37);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' and (46c) follows since the demands d[K]\\S are independent of the random variables CS, dS, q[K], Z[H], W[N], and qk = pk + dk for each k ∈ [K]\\S where the random variables p[K]\\S are independently and uniformly distributed over FN q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Performance By (23) and (24), each file is split into LF equal-size packets, each of length B LF symbols, thus the subpacketization is LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Denote the achieved memory-storage-load triple by (MA, TA, RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', then Memory Size: For each user k ∈ [K], by the cached content in (36), for each i ∈ [F] such that ai,k = ∗, there are LN associated packets cached by the user, one from each file (see (36b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For each i ∈ [F] such that ai,k ̸= ∗, there are L associated coded packet cached at the user (see (36c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In addition, the pk in (36a) can be stored with N symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Recall that, each column of a (K, F, Z, S) PDA has Z “ ∗ ”s and F − Z ordinary symbols, thus, the achieved memory size is MA = inf B∈N+ 1 B � (Z · LN + (F − Z) · L) B LF + N � = F + Z (N − 1) F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Storage Size: By (34), since each coded subfile � Wn(αh) is of B L symbols, and each �Vs(αh) is of B LF symbols, the storage size is TA = 1 B �B L · N + B LF · S � = N L + S LF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (47) Communication Load: By (39), each server h ∈ [H] sends S coded packets Yh,[S], each of B LF symbols, and the coefficient vectors q[K] can be sent in KN symbols, thus the achieved load is RA = inf B∈N+ 1 B � H · S · B LF + H · KN � = HS LF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' OPTIMALITY OF MAN-PDA BASED RSP-LFRB SCHEME In the following subsections, we separately derive the lower bounds in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Let r(M) be the lower convex envelope of the following points: �� 1 + t(N − 1) K , K − t t + 1 � : t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (48) Notice that, r(M) is the achievable communication load with memory size M for the key superpostion scheme under signal security and demand privacy in the single server setup in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Moreover, r(M) has the following relationship with T(M) and R(M): T(M) = 1 L � N + r(M) � , R(M) = H L · r(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (49) We will use the following upper bound for r(M), which was proved in [11, Lemma 4]: r(M) ≤ N − M M − 1 , ∀ M ∈ (1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (50) In the following, we prove the bounds in (21) by deriving a lower bound for T ∗(M) and R∗(M) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Upper bound for T(M) T ∗(M) For any M ∈ [1, N], the optimal storage size T ∗(M) ≥ N L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (51) In fact, NB = H(W[N]) = I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Z[I+L]) + H(W[N]|Z[I+L]) = I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Z[I+L]) (52a) = I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Z[I]) + I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Z[I+1:I+L]|Z[I]) = I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Z[I+1:I+L]|Z[I]) (52b) ≤ H(Z[I+1:I+L]|Z[I]) ≤ LT ∗(M)B, (52c) where (52a) and (52b) follows from the constraints (3a) and (3b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Therefore, 1) for any K and M ≥ 2, T(M) T ∗(M) ≤ 1 L(N + r(M)) N L (53a) = 1 + 1 N · r(M) = 1 + 1 N · N − M M − 1 (53b) ≤ 1 + N − 2 N (53c) ≤ 2, where (53a) and (53b) follows from(51) and (50), respectively, and (53c) follows from the fact M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2) for K ≤ N, since r(M) is upper bounded by K, T(M) T ∗(M) ≤ (N + K)/L N/L = 1 + K N (54) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (55) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Upper bound for R(M) R∗(M) We first derive a lower bound for R∗(M) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For any M ∈ [1, N], the optimal load R∗(M) has the following lower bound: R∗(M) ≥ max u∈[min{⌊ N 2 ⌋,K}] H L · u(N − u + 1 − uM) N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (56) Proof: Consider the case where each user demands a file, and denote Dk the index of the file demanded by user k for all k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Denote the signal of server h ∈ [H] under the demands (D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , DK) = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , dK) by X(d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=',dK),h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For each u ∈ [min{⌊ N 2 ⌋, K}], consider the first u caches C1, · · · , Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For each r ∈ [⌊ N u ⌋] and h ∈ [H], denote Xr,h ≜ X((r−1)u+1,(r−1)u+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=',ru,1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=',1),h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (57) Let L and I be disjoint subsets of [H] with cardinality L and I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Denote � Nu = u � N u ⌋, then the files {Wn : n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , � Nu} can be decoded with the signals {Xr,h : r ∈ [⌊ N u ⌋], h ∈ L ∪ I} and the caches {C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' , Cu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that � Nu ≥ u( N u − u−1 u ) = N − u + 1, therefore (N − u + 1)B ≤ � NuB = I(W[ � Nu];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[⌊ N u ⌋],L∪I, C[u]) = I(W[ � Nu];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[⌊ N u ⌋],I) + I(W[ � Nu];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[⌊ N u ⌋],L, C[u] | X[⌊ N u ⌋],I) ≤ I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' X[⌊ N u ⌋],I, Q[K],I, ZI) + � h∈L ⌊ N u ⌋ � r=1 H(Xr,h) + u � k=1 H(Cu) ≤ I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Q[K],I, ZI) + � h∈L ⌊ N u ⌋ � r=1 H(Xr,h) + uMB (58a) = I(W[N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' ZI) + � h∈L ⌊ N u ⌋ � r=1 H(Xr,h) + uMB (58b) ≤ � h∈L ⌊ N u ⌋ � r=1 H(Xr,h) + uMB, (58c) where (58a) follows since the signals X[⌊ N u ⌋],I are determined by ZI and Q[K],I by (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (58b) follows since the queries Q[K],I are independent of (W[N], ZI) by (4), and (58c) follows from the I-Security constraint (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Sum the inequalities in (58c) over all subsets L ⊂ [H] of cardinality L, �H L � (N − u + 1)B ≤ � L⊂[H],|L|=L � h∈L ⌊ N u ⌋ � r=1 H(Xr,h) + �H L � uMB ≤ ⌊ N u ⌋ � r=1 H � h=1 � L⊂[H],|L|=L H(Xr,h) · 1(h ∈ L) + �H L � uMB = �H − 1 L − 1 � ⌊ N u ⌋ � r=1 H � h=1 H(Xr,h) + �H L � uMB ≤ �H − 1 L − 1 ��N u � R∗(M)B + �H L � uMB ≤ �H − 1 L − 1 �N u R∗(M)B + �H L � uMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Therefore, R∗(M) ≥ H L · u(N − u + 1 − uM) N , ∀ u ∈ � min ��N 2 � , K �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (59) Then the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Now for each u ∈ � min{⌊ N 2 ⌋, K} � , define Lu(M) ≜ H L · u(N − u + 1 − uM) N , ∀ M ∈ [0, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (60) Then Lemma 1 indicates that for each u ∈ � min{⌊ N 2 ⌋, K} � , R∗(M) ≥ Lu(M), ∀ M ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (61) For any M ∈ [0, N], define f(M) ≜ H 4LN · (N − M)(N + M + 2) M + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (62) Notice that f(M) is convex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The following lemma shows that f(M) is a lower bound of R∗(M) on � max{2, N−2K 2K+1 }, N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For any M ∈ [max{2, N−2K 2K+1 }, N � , R∗(M) ≥ f(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (63) Proof: Consider the interval � max � N−2⌊N/2⌋ 2⌊N/2⌋+1 , N−2K 2K+1 � , N � , which can be split into min{⌊ N 2 ⌋, K} intervals: � max �N − 2⌊N/2⌋ 2⌊N/2⌋ + 1 , N − 2K 2K + 1 � , N � = min{⌊ N 2 ⌋,K} � u=1 �N − 2u 2u + 1 , N − 2u + 2 2u − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (64) Notice that, by the fact ⌊ N 2 ⌋ ≥ N−1 2 , N−2⌊N/2⌋ 2⌊N/2⌋+1 ≤ 1 N ≤ 2, the interval � max � N−2⌊N/2⌋ 2⌊N/2⌋+1 , N−2K 2K+1 � , N � encloses [max{2, N−2K 2K+1 }, N � as its sub-interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Therefore, for any M ∈ [max{2, N−2K 2K+1 }, N � , there exists u ∈ [min{⌊ N 2 ⌋, K}] such that M ∈ � N−2u 2u+1 , N−2u+2 2u−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It is easy to verify: Lu �N − 2u 2u + 1 � = f �N − 2u 2u + 1 � = H(N + 1) LN u(u + 1) 2u + 1 , (65) Lu �N − 2u + 2 2u − 1 � = f �N − 2u + 2 2u − 1 � = H(N + 1) LN u(u − 1) 2u − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (66) That is Lu(M) and f(M) concides on the end points of the interval � N−2u 2u+1 , N−2u+2 2u−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Since f(M) is convex on M, we concludes that Lu(x) ≥ f(x), ∀ x ∈ �N − 2u 2u + 1 , N − 2u + 2 2u − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (67) Therefore, with (61) and (67), we concludes (63), which holds for all M ∈ � max � 2, N−2K 2K+1 � , N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Therefore, 1) for any K, M ≥ max � 2, N−2K 2K+1 � , by (49), (50) and Lemma 2, R(M) R∗(M) ≤ H L · r(M) f(M) ≤ H L · N−M M−1 H 4NL · (N−M)(N+M+2) M+1 = 4N · M + 1 (M − 1)(N + M + 2) = 4N · � 1 N + M + 2 + 2 (M − 1)(N + M + 2) � ≤ 12 · N N + 4 (68) < 12, where in (68), we used the fact M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' This completes the proof of second inequalilty in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2) for K ≤ N and M ≤ max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' N−2K 2K+1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' R(M) ≤ HK L ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (69) and since R∗(M) must be non-increasing with M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' R∗(M) ≥ R∗� max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' N − 2K 2K + 1 �� (70) a) if N ≥ 6K + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' N−2K 2K+1 � = N−2K 2K+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' by (63),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' R∗� max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' N − 2K 2K + 1 �� ≥ f �N − 2K 2K + 1 � (71) = H 4LN · � N − N−2K 2K+1 �� N + N−2K 2K+1 + 2) N−2K 2K+1 + 1 (72) = H 4LN · � N + 1 − N+1 2K+1 �� N + 1 + N+1 2K+1 � N+1 2K+1 (73) = H 4L · N + 1 N 2K · � 1 + 1 2K + 1 � (74) ≥ HK 2L ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (75) thus by (69),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (70) and (75),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' R(M) R∗(M) ≤ 2 ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (76) b) if N < 6K + 2 and N ≥ 4, max � 2, N−2K 2K+1 � = 2, by (63), R∗� max � 2, N − 2K 2K + 1 �� ≥ f(2) (77) = H 4LN · (N − 2)(N + 4) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (78) thus by (69), (70) and (78), R(M) R∗(M) ≤ 12KN N 2 + 2N − 8 (79) ≤ 12N 2 N 2 + 2N − 8 (80) ≤ 12, (81) where the last step follows from N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' c) if N < 6K + 2 and N ≤ 3, it is easy to obtain R(M) ≤ H L · K � 1 − M − 1 N − 1 � , (82) and by Lemma 1, R∗(M) ≥ H L � 1 − M N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (83) thus, R(M) R∗(M) ≤ KN N − 1 ≤ N 2 N − 1 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='5 ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (84) From the above arguments, we conclude that, for all K, N except for K > N and 1 ≤ M ≤ max � 2, N−2K 2K+1 � , R(M) R∗(M) ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' (85) Notice that, when K > N, max � 2, N−2K 2K+1 � = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Hence, we proved (85) for K ≤ N or K > N, M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we present some numerical results to illustrate the comparison of the achievable MSC region and its lower bounds, and the influences of parameters I and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Achievable MSC Region & Lower Bounds By (20), as the boundaries of the region are parallel to T- and R- axes, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2, it is sufficient to plot the projections to the M-T plane and M-R plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=', the functions T(M) and R(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3, we plot T(M) and R(M) for a system with H = 20, L = 10 to illustrate the following two regimes: a) K ≤ N: See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3(a) and 3(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' b) K > N: See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3(c) and 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For comparison, the corresponding bound (see (51) or (56)) is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' We choose (N, K) = (100, 10) and (10, 100) for the above two regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' From the figures, we have the following observations: 1) For the regime K ≤ N, the storage size T(M) is close to the constant lower bound N L over all M ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' This is because T(M) achieves its maximum at T(1) = K+N L , which is dominated by N L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In fact, the bound in (54) indicates that the bound can be tighted to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1 for the chosen 0 10 20 30 40 50 60 70 80 90 100 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='9 11 (a) T(M) for N = 100, K = 10 0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 12 14 16 18 20 (b) R(M) for N = 100, K = 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 (c) T(M) for N = 10, K = 100 0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 120 140 160 180 200 (d) R(M) for N = 10, K = 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3: Storage size T(M) and communication load R(M) for H = 20 servers with L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The communication R(M) is close to its lower bound over all M ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' For the chosen parameters, (76) indicates that the bound can be tighted to 2 for 1 ≤ M ≤ N−2K 2K+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2) For the regime K > N, the storage size T(M) departs far from the constant bound N L for M close to 1, because T(1) = K+N L is dominated by K L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' But it dramatically decreases with M, since the coded version of the superposition key sizes are the same with the communication load of each server, which dramatically decreases with M due to the coded multi-cast opportunities provided by coded caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Similarly, the communication load R(M) scales with K at M = 1, and decreases dramatically around M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In particular, as stated in Theorem 2, when M ≥ 2, both T(M) and R(M) are close to their own lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 3) In both regimes, T(M) and R(M) have similar trends in M, since they are linear to each other, see (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In summary, the numerical results are consistent with Remark 4: the only un-bounded regime is K > N and 1 ≤ M < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Influence of Parameters I and L As mentioned in Section IV-B, the number of colluding servers that the system can resist does not affect the total communication load, but I has to satisfy I + L ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The cost of resisting the colluding servers is that each user has to gather more signals to decode if I is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The analysis in Remark 5 indicates that the communication load to each user has to reached (1 + I L)(1 − t K � for the point (Mt, Tt, Rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' To see how the parameter L influences the total communication load, we plot the total communication load as a function of L for a system with H = 20, L = 10, N = 50, K = 10 and M = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='5 (or equivalently t = 5) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It is clear that, the larger L, the total communication load is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The L is inversely proportional to the total communication load by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In fact, the parameter L determines the size of each subfile by (23), and each server is responsible for transmiting one encoded subfile over all 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 (a) Total communication load R, I ≤ H − L (b) Communication load to each user Ru Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 4: Influence of parameters L and I to total communication load R and the communication load to each user Ru for a system with N = 50, K = 10, M = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content='With larger L, the size of each encoded subfile is smaller, and thus results in smaller communication load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' To see how I and L jointly influence the communication load to each user, we plot Ru = (1+ I L)(1− t K � for the above system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' It can be seen that, with given I, Ru decreases with L, but with given L, Ru increases linearly with I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that, with the parameters I and L, the system can resist I colluding servers and H − I − L non-responding (or un-connected) servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Notice that, each user need signals from at least I + L servers, thus, with larger I + L, the connectivity requirement for the topology to implement the scheme is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' In summary, larger L decreases both total communication load and the load to each user, but larger I only increases the communication load to users, and I + L determines the threshold of the number of servers that a user need to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' CONCLUSION A RSP-LFRB scheme is proposed within PDA framework, where the techniques of coded caching, secret sharing, key superpositons are integrated such that the signal security, users’ demand privacy, and the blindness to colluding servers are simultaneously guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' The storage size and the communication load of MAN-PDA based RSP-LFRB scheme are showed to be to within a multiplicative gap of at most 2 and 12 from optimal in all regimes, except for small memory regime with less files than users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Forensics Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 10,no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 355–370, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' [8] K.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Caire,“A new design of cache-aided multiuser private information retrieval with uncoded prefetching,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' Theory (ISIT), Melbourne, Australia, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StFAT4oBgHgl3EQf1x6P/content/2301.08711v1.pdf'} +page_content=' 2021.' 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a/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/2301.02014v1.pdf.txt b/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/2301.02014v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9153d445d85345b0808bdb9070be2479fbbc37ad --- /dev/null +++ b/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/2301.02014v1.pdf.txt @@ -0,0 +1,1122 @@ +𝐂⃗ Sequential Optimization Numbers Group +Zile Hui, 51174500096@stu.ecnu.edu.cn +ABSTRACT. We define C sequential optimization numbers, where C is a k+1-tuple vector. We prove +that the unsigned Stirling numbers of first kind are (0,1) sequential optimization numbers. Many +achievements of the Stirling numbers of first kind can be transformed into the properties of C +sequential optimization numbers. We give some examples such as the recurrence formula and an +instance of C sequential optimization numbers. We also extend some properties such as an upper +bounder of them. +KEYWORDS: C Sequential optimization numbers, Stirling numbers of first kind, k-dimensional +sequential optimization numbers, Upper bounder. +1 INTRODUCTION +Stirling numbers were introduced by the Scottish mathematician James Stirling in his famous treatise +and were subsequently rediscovered in various forms by numerous authors [Bla16][Sti30]. Stirling +numbers of the first kind, denoted by s(𝑛, 𝑚), were studied in a large number of works and have +various properties [Bla16]. Recurrence formula is one of the basic ways to define the unsigned +Stirling numbers of first kind. There is no known closed-form expression so far for the unsigned +Stirling numbers of first kind and asymptotic formulas have been studied by numerous authors +[Wil93][Tem93][AD17]. C Sequential optimization numbers are defined based on optimization set. +We give some properties of C sequential optimization numbers and some of them are fundamental +properties of Stirling numbers when C is (0,1). We mainly refer to the definitions and proofs in one +article [Hui22]. +2 SEQUENTIAL OPTIMIZATION NUMBERS +We introduce the optimization set in Definition 2.1 and define the C sequential optimization numbers +in Definition 2.2. Then, we give an expression which is not closed-form, a recurrence formula, some +properties and applications of C sequential optimization numbers. At the end of this chapter, we give +an upper bound of sequential optimization numbers and some results derived from this upper bound. +Definition 2.1. Let 𝒙(𝑥1, 𝑥2, … , 𝑥𝑘), 𝒚(𝑦1, 𝑦2, … , 𝑦𝑘) be k-dimensional vectors and 𝑹(𝑅1, 𝑅2, … , 𝑅𝑘) be k- +dimensional relation vector. If for all 𝑖 = 1,2, … , 𝑘, ( 𝑥𝑖, 𝑦𝑖) ∈ 𝑹𝒊, then x is said to be related to y by R, +denoted by (𝒙, 𝒚) ∈R. Let U be a set of vectors, R be a relation vector and 𝐴 ⊆ 𝑈, if for ∀𝒖 ∈ 𝑈, ∃𝒂 ∈ 𝐴 +imply (𝒂, 𝒖) ∈ 𝑹 or 𝒂 = 𝒖, then A is said to be a majorization set of U by R. If 𝐵 is a majorization set of +U by R and |𝐵| is a minimum of all the cardinalities of majorization sets, then 𝐵 is said to be an +optimization set of U by R, denoted by 𝐵𝑜𝑹 +⊆ 𝑈, and |𝐵| is said to be the weight of U by R, denoted by +𝑊𝑜𝑹(𝑈). +We can find similar selection strategies of optimization set in many existing studies +[Mou79][Xue00][ARY21][Hui22]. It can be proved that this strategy can include all possible optimal +terms and exclude impossible optimal terms in the iteration of the algorithm [Hui22]. +Definition 2.2. Let 𝒂𝟏(1, 𝑎11, 𝑎12, … 𝑎1𝑘), 𝒂𝟐(2, 𝑎21, 𝑎22, … 𝑎2𝑘),…, 𝒂𝒏(𝑛, 𝑎𝑛1, 𝑎𝑛2, … 𝑎𝑛𝑘) be n k+1- +dimensional +vectors, +𝑹(𝑅1, 𝑅2, … , 𝑅𝑘) = (<, <, … , <) + be +k-dimensional +relation +vector, +C=(𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘), 𝑈 = {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏 } and 𝑆 ⊆ 𝑈 . For all 𝑗 = 0,1, … , 𝑘 , 𝑐𝑗 ∈ {0,1}. For all 𝑖 = +1,2, … , 𝑛 and 𝑗 = 1,2, … , 𝑘 , 𝒃𝒊𝒋 = (𝑖, 𝑎𝑖𝑗) , 𝑈𝑗 = {𝒃𝟏𝒋, 𝒃𝟐𝒋, … , 𝒃𝒏𝒋 }, 𝑎1𝑗, 𝑎2𝑗, … 𝑎𝑛𝑗 are 1,2, … , 𝑛 , + +2 +respectively. 𝑆𝑗 is an optimization set of 𝑈𝑗 by (<, 𝑅𝑗). For a certain 𝑖 and all 𝑗 = 1,2, … , 𝑘, the total +number of 𝒃𝒊𝒋 ∈ 𝑆𝑗 is l, if 𝑐𝑙 = 1 , then 𝒂𝒊 ∈ 𝑆 , otherwise, 𝒂𝒊 ∉ 𝑆 , 𝑆 is said to be an C sequential +optimization set of U by R, denoted by 𝑆𝑂𝑹, 𝑪 +⊆ +𝑈. |𝑆| is said to be sequential optimization weight of U by R, +denoted by 𝑊𝑂𝑹,𝑪(𝑈). The numbers of ways that 𝑊𝑂𝑹,𝑪(𝑈) = 𝑚 are said to be C sequential optimization +numbers, denoted by 𝑂𝑪(𝑛, 𝑚). +In Definition 2.2, 𝑪 = (𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘). We define 𝑩𝑗 = (𝑏𝑗,0, 𝑏𝑗,1, 𝑏𝑗,2, … , 𝑏𝑗,𝑘), 𝑪′ = (1,1, … ,1) − 𝑪 +and 𝐹𝑗(𝑪) = 𝑩𝑗 ∙ 𝑪𝑇, where 𝑏𝑗,𝑝 = 𝐶𝑘 +𝑝 +1 +(𝑗−1)𝑝, 𝑝 = 0,1, … , 𝑘 and 𝑗 = 2,3, … , 𝑛. In this paper, we define +∏ +𝐹𝑗(𝑪) +𝑚 +𝑗=𝑚+1 += 1 and ∑ +𝑓(𝑝) +𝑚 +𝑝=𝑚+1 += 0 to make the expressions concise, where 𝑓(𝑝) is a function of +𝑝. We give some properties about 𝐹𝑗(𝑪). +(a). If 2 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑛, + 𝐹𝑗2(𝑪) ≤ 𝐹𝑗1(𝑪) (1) +Proof. If 2 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑛, +𝑏𝑗2,𝑝 +𝑏𝑗1,𝑝 += (𝑗1 − 1)𝑝 +(𝑗2 − 1)𝑝 ≤ 1 +𝑏𝑗2,𝑝 ≤ 𝑏𝑗1,𝑝 + 𝐹𝑗2(𝑪) ≤ 𝐹𝑗1(𝑪) +(b). + ∏ +𝐹𝑗(𝑪) +𝑛 +𝑗=2 +≤ 𝑛𝑘 (2) +Proof. + ∏ 𝐹𝑗(𝑪) +𝑛 +𝑗=2 +≤ ∏ 𝐹𝑗([1,1, … ,1]𝑇) +𝑛 +𝑗=2 + + = ∏ ∑ 𝑏𝑗,𝑝 +𝑘 +𝑝=0 +𝑛 +𝑗=2 + + = ∏ ∑ 𝐶𝑘 +𝑝 +1 +(𝑗 − 1)𝑝 +𝑘 +𝑝=0 +𝑛 +𝑗=2 + + = ∏ (1 + +1 +𝑗 − 1) +𝑘 +𝑛 +𝑗=2 + + = 𝑛𝑘 +(c). If 𝑐0 = 1, + 𝐹𝑗(𝑪) ≥ 1 (3) +Proof. + 𝐹𝑗(𝑪) = 𝑩 ∙ 𝑪𝑇 ≥ 𝑐0𝑏𝑗,0 = 1 +Theorem 2.1. For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, +𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = {(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 +, +1 ≤ 𝑚 ≤ 𝑛 +0, +𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 + +where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of m-1 elements in set {2,3, … , 𝑛} and +{𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}. + +3 +Proof of Theorem 2.1. In the scenario of Definition 2.2, 𝑪 = (𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘) and 𝑆𝑂𝑹, 𝑪 +⊆ +𝑈. 𝑔(𝑆) +denotes the numbers of ways where S is a set consisting of specific elements. +First, we discuss the case where 𝑪 = (0,1). We can get 𝑆 is optimization set. +For 𝑚 = 1, the number of positions of 𝒂𝟐~𝒂𝒏 is (𝑛 − 1)! as shown in Figure 1.a. So 𝑂(0,1)(𝑛, 1) = +(𝑛 − 1)!. + +(a) + +(b) + +(c) +Figure 1: Three ways to optimization set. +For 𝑚 = 2,3, … , 𝑛, |𝑈| = 𝑛 and 𝑆1 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓, 𝒂𝒋}, we use mathematical induction to prove +that 𝑔(𝑆1) = +(𝑛−1)! +(𝑝−1)(𝑞−1)∙…∙(𝑟−1)(𝑗−1). +Basis step: we prove 𝑔(𝑆2) = +(𝑛−1)! +𝑗−1 , where 𝑆2 = {𝒂𝟏, 𝒂𝒋}. As shown in Figure 1.b, 𝑎11 = 𝑖, 𝑎𝑗1 = 1, +where 𝑖 = 2,3, … , 𝑛, 𝑗 = 2,3, … , 𝑛 and 𝑖 + 𝑗 ≤ 𝑛 + 2. First, place row 2~(𝑖 − 1) and the number of +positions is +(𝑛−𝑗)! +(𝑛−𝑖−𝑗+2)!. Second, place row (𝑖 + 1)~𝑛 and the number of positions is (𝑛 − 𝑖)!. Total +number is +(𝑛−𝑖)!(𝑛−𝑗)! +(𝑛−𝑖−𝑗+2)! . +𝑔(𝑆2) = ∑ (𝑛 − 𝑖)! (𝑛 − 𝑗)! +(𝑛 − 𝑖 − 𝑗 + 2)! +𝑛+2−𝑗 +𝑖=2 + + = (𝑛 − 𝑗)! +𝑗 − 1 +∑ (𝑛 − 𝑖)! [(𝑛 − 𝑖 + 1) − (𝑛 − 𝑖 − 𝑗 + 2)] +(𝑛 − 𝑖 − 𝑗 + 2)! +𝑛+2−𝑗 +𝑖=2 + + = (𝑛 − 𝑗)! +𝑗 − 1 +∑ [ (𝑛 − 𝑖 + 1)! +(𝑛 − 𝑖 − 𝑗 + 2)! − (𝑛 − 𝑖)! (𝑛 − 𝑖 − 𝑗 + 2) +(𝑛 − 𝑖 − 𝑗 + 2)! +] +𝑛+2−𝑗 +𝑖=2 + +In the brackets, the right side of the minus is equal to 0 when 𝑖 = 𝑛 + 2 − 𝑗. The right side of the +minus with 𝑖 = 𝑡 is equal to the left side of the minus with 𝑖 = 𝑡 + 1, where 𝑡 = 2,3, … , 𝑛 + 1 − 𝑗. So, +𝑔(𝑆2) = (𝑛 − 𝑗)! +𝑗 − 1 ∙ (𝑛 − 1)! +(𝑛 − 𝑗)! + = (𝑛 − 1)! +𝑗 − 1 + +n +ai +nn +ai +aj +nn +al +ap +aq +aj +n4 +Inductive step: assume 𝑔(𝑆3) = +(𝑛−1)! +(𝑝−1)(𝑞−1)∙…∙(𝑟−1), where 𝑆3 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓}. Then, we show +𝑔(𝑆1) = +(𝑛−1)! +(𝑝−1)(𝑞−1)∙…∙(𝑟−1)(𝑗−1) where 𝑆1 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓, 𝒂𝒋} and 1 < 𝑝 < 𝑞 < 𝑟 < 𝑗 ≤ 𝑛 . In +Figure 1.c, 𝑎𝑟1 = 𝑖, 𝑎𝑗1 = 1, where 𝑖 = 2,3, … , 𝑛 − 1, 𝑗 = 3,4, … , 𝑛 and 𝑖 + 𝑗 ≤ 𝑛 + 2. First, place row +2~(𝑖 − 1) and the number of positions is +(𝑛−𝑗)! +(𝑛−𝑖−𝑗+2)!. We remove all the rows and columns where the +points in row 1~(𝑖 − 1) are. In the rest of the figure, |𝑈’| = 𝑛 − 𝑖 + 1, 𝑆3 +′ = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓} and the +number of ways is +(𝑛−𝑖)! +(𝑝−1)(𝑞−1)∙…∙(𝑟−1). So, +𝑔(𝑆1) = ∑ +(𝑛 − 𝑖)! (𝑛 − 𝑗)! +(𝑝 − 1)(𝑞 − 1) ∙ … ∙ (𝑟 − 1)(𝑛 − 𝑖 − 𝑗 + 2)! +𝑛+2−𝑗 +𝑖=2 + + = +(𝑛 − 1)! +(𝑝 − 1)(𝑞 − 1) ∙ … ∙ (𝑟 − 1)(𝑗 − 1) +So, for 𝑪 = (0,1), change 𝑆3 to 𝑆1 by add 𝒂𝒋 and we can get 𝑔(𝑆1) = +1 +𝑗−1 𝑔(𝑆3), where 𝑗 = 2,3, … , 𝑛. +Then, for 𝑘 > 1, we discuss the relationship between g(𝑆1) and g(𝑆3). In 𝒂𝟏~𝒂𝒏, let dimension 1, +{1, 2, … 𝑛} and dimension w+1, {𝑎1𝑤, 𝑎2𝑤, … , 𝑎𝑛𝑤} form k groups of n 2-dimensional vectors +(1, 𝑎1𝑤), ( 2, 𝑎2𝑤), … (n, 𝑎𝑛𝑤) and 𝑈𝑤 = {(1, 𝑎1𝑤), ( 2, 𝑎2𝑤), … (n, 𝑎𝑛𝑤)} , where 𝑤 = 1,2, … , 𝑘 . Let +𝑆𝑤𝑂𝑹, (0,1) +⊆ +𝑈𝑤 and we can get the relationship between 𝑆𝑤 and 𝑔(𝑆𝑤) is the same as the +relationship between 𝑆 and 𝑔(𝑆) when 𝑪 = (0,1). +We define 𝑆3,0 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓} and there are 0 (𝑗, 𝑎𝑗𝑤) in all 𝑆3,0 +𝑤 . In k+1-dimensional vector, +to change 𝑆3,0 to 𝑆1 by add 𝒂𝒋, we need to change 𝑙 𝑆3,0 +𝑤 to 𝑆1 +𝑤 by add (𝑗, 𝑎𝑗𝑤) if 𝑐𝑙 = 1. If 𝑐𝑙 = 0, we +change 𝑙 𝑆3,0 +𝑤 to 𝑆1 +𝑤 by add (𝑗, 𝑎𝑗𝑤) to change 𝑆3,0 to 𝑆3. We can get +𝑔(𝑆1) = 𝑩 ∙ 𝑪𝑇𝑔(𝑆3,0) +𝑔(𝑆3) = 𝑩 ∙ 𝑪′𝑇𝑔(𝑆3,0) +and +𝑔(𝑆1) = 𝑩 ∙ 𝑪𝑇 +𝑩 ∙ 𝑪′𝑇 𝑔(𝑆3) = 𝐹𝑗(𝑪) +𝐹𝑗(𝑪′) 𝑔(𝑆3) +We also get 𝒂𝟏 in all 𝑆𝑤. So, +(a). If 𝑐𝑘 = 1, +𝑂𝑪(𝑛, 𝑚) = +{ + + + + + + + + +(𝑛 − 1)!𝑘 ∑ ∏ 𝐹𝑗′𝑖(𝑪′) +𝑛−1 +𝑖=1 +, +𝑚 = 1 +(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 +, +2 ≤ 𝑚 ≤ 𝑛 − 1 +(𝑛 − 1)!𝑘 ∑ ∏ 𝐹𝑗𝑖(𝑪) +𝑛−1 +𝑖=1 +, +𝑚 = 𝑛 +0, +𝑚 = 0 or 𝑚 > 𝑛 + +where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of 𝑚 − 1 elements in set {2,3, … , 𝑛} and +{𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}. +(b). If 𝑐𝑘 = 0, + +5 + 𝑂𝑪(𝑛, 𝑚) = +{ + + + + + + + + +(𝑛 − 1)!𝑘 ∑ ∏ 𝐹𝑗′𝑖(𝑪′) +𝑛−1 +𝑖=1 +, +𝑚 = 0 +(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚−1 +𝑖=1 +, +1 ≤ 𝑚 ≤ 𝑛 − 2 +(𝑛 − 1)!𝑘 ∑ ∏ 𝐹𝑗𝑖(𝑪) +𝑛−1 +𝑖=1 +, +𝑚 = 𝑛 − 1 +0, +𝑚 = −1 or 𝑚 > 𝑛 − 1 + +where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚 are all combinations consisting of 𝑚 elements in set {2,3, … , 𝑛} and +{𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚−1} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚}. +To sum up, for all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, +𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = {(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 +, +1 ≤ 𝑚 ≤ 𝑛 +0, +𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 + +where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of 𝑚 − 1 elements in set {2,3, … , 𝑛} and +{𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}. +Theorem 2.2. For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ≥ 𝑐𝑘 − 1, the recurrence formula of C sequential optimization +numbers is +𝑂𝑪(𝑛 + 1, 𝑚 + 1) = 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) + 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) +and the boundary condition is +𝑂𝒄(𝑛, 𝑚) = {0, +𝑚 = 𝑐𝑘 − 1 𝑜𝑟 𝑚 > 𝑐𝑘 − 1 + 𝑛 +1, +𝑚 = 𝑐𝑘, 𝑛 = 1 + +Proof of Theorem 2.2. In Definition 2.2, for 𝑐𝑘 = 1 , we divide the number of ways that +𝑂𝑘(𝑛 + 1, 𝑚 + 1) denotes into two parts, which are with and without 𝒂𝒏+𝟏, where 𝑛 > 2 and 𝑚 = +2,3, … , 𝑛 − 1. +First, ways with 𝒂𝒏+𝟏 and 𝑚 vectors from {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏}. let 𝑗𝑚 be 𝑛 + 1, 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 be all +combinations consisting of m-1 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − +{𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}. +𝑛!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 += 𝑛!𝑘 𝐹𝑛+1(𝑪) ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 + += 𝑛𝑘𝐹𝑛+1(𝑪)(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 + += 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) +Then, ways without 𝒂𝒏+𝟏 and m vectors from {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏}. let 𝑗′𝑛−𝑚 be 𝑛 + 1, 𝑗1, 𝑗2, ⋯ , 𝑗𝑚 are all +combinations consisting of m elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚−1} = {2,3, … , 𝑛} − +{𝑗1, 𝑗2, ⋯ , 𝑗𝑚}. +𝑛!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚 +𝑖=1 += 𝑛!𝑘 𝐹𝑛+1(𝑪′) ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑚−1 +𝑖=1 + + +6 += 𝑛𝑘𝐹𝑛+1(𝑪′)(𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑚 +𝑖=1 +∏ 𝐹𝑗′𝑖(𝑪′)) +𝑛−𝑚−1 +𝑖=1 + += 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) +To sum up, + 𝑂𝑪(𝑛 + 1, 𝑚 + 1) = 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) + 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) +We can get the boundary condition according to Theorem 2.1. + 𝑂𝒄(𝑛, 𝑚) = {0, +𝑚 = 𝑐𝑘 − 1 𝑜𝑟 𝑚 > 𝑐𝑘 − 1 + 𝑛 +1, +𝑚 = 𝑐𝑘, 𝑛 = 1 + +We can do the same thing for 𝑐𝑘 = 0. We can prove that the recurrence formula works for all 𝑘, 𝑛 ∈ +𝑁+ and 𝑚 ≥ 𝑐𝑘 − 1. +We give some properties and applications of C sequential optimization numbers below and we +prove part of them in this paper. +Lemma 2.1 For 𝑐𝑘 − 1 ≤ 𝑚 ≤ 𝑛 + 𝑐𝑘, +𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) +For 0 ≤ 𝑚 ≤ 𝑛 + 1, + 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = 0 = 𝑂𝑪′(𝑛, 𝑛 + 1 − 𝑚 + 𝑐′𝑘 − 1). +Proof of Lemma 2.1. For 1 ≤ 𝑡 ≤ 𝑛, + 𝑂𝑪(𝑛, 𝑡 + 𝑐𝑘 − 1) = (𝑛 − 1)!𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) +𝑡−1 +𝑖=1 +∏ 𝐹𝑗′ +𝑖(𝑪′)) +𝑛−𝑡 +𝑖=1 + + = (𝑛 − 1)!𝑘 ∑( +∏ +𝐹𝑗′ +𝑖(𝑪′) +𝑛−𝑡+1−1 +𝑖=1 +∏ +𝐹𝑗𝑖(𝑪) +𝑛−(𝑛−𝑡+1) +𝑖=1 +) + = 𝑂𝑪′(𝑛, 𝑛 − 𝑡 + 1 + 𝑐′𝑘 − 1) + = 𝑂𝑪′(𝑛, 𝑛 − 𝑡 + 𝑐′𝑘) +For 𝑡 = 0 and 𝑡 = 𝑛 + 1, + 𝑂𝑪(𝑛, 𝑡 + 𝑐𝑘 − 1) = 0 = 𝑂𝑪′(𝑛, 𝑛 + 1 − 𝑡 + 𝑐′𝑘 − 1) +Let 𝑚 = 𝑡 + 𝑐𝑘 − 1, + 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) +where 𝑐𝑘 − 1 ≤ 𝑚 ≤ 𝑛 + 𝑐𝑘. +Lemma 2.2. +∑ 𝑂𝑪(𝑛, 𝑚) +𝑛 +𝑚=0 += 𝑛!𝑘 +Lemma 2.3. For 𝑪 = (0,1), 𝑂𝑪(𝑛, 𝑚) = 𝑠𝑢(𝑛, 𝑚), where 𝑠𝑢(𝑛, 𝑚) are the unsigned Stirling numbers of +first kind. +Lemma 2.4. Let 𝑪 = (0,1,1, … ,1) be a k+1-tuple vector, 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑘(𝑛, 𝑚), where 𝑂𝑘(𝑛, 𝑚) is k- +dimensional sequential optimization numbers [Hui22]. +Lemma 2.5. For all 𝑘, 𝑛 ∈ 𝑁+, we define 𝑥𝑪 +1↑ = 𝑥, +𝑥𝑪 +𝑛↑ = 𝑥[𝐹2(𝑪)𝑥 + 𝐹2(𝑪′)][2𝑘𝐹3(𝑪)𝑥 + 2𝑘𝐹3(𝑪′)]⋯ [(𝑛 − 1)𝑘𝐹𝑛(𝑪)𝑥 + (𝑛 − 1)𝑘𝐹𝑛(𝑪′)] +where 𝑛 ≥ 2 and 𝑂𝑪 +𝑢(𝑛, 𝑚) = 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). Then, we can get + +7 +𝑥𝑪 +𝑛↑ = ∑ 𝑂𝑪 +𝑢(𝑛, 𝑚) +𝑛 +𝑚=0 +𝑥𝑚 +and the zero points are 𝑥 = 0 and 𝑥 = − +𝐹𝑚(𝑪′) +𝐹𝑚(𝑪) , where 𝑚 = 2,3, , … , 𝑛. We call 𝑂𝑪 +𝑢(𝑛, 𝑚) unsigned 𝑪 +sequential optimization numbers. +For all 𝑘, 𝑛 ∈ 𝑁+, we define 𝑥𝑪 +1↓ = 𝑥, +𝑥𝑪 +𝑛↓ = 𝑥[𝐹2(𝑪)𝑥 − 𝐹2(𝑪′)][2𝑘𝐹3(𝑪)𝑥 − 2𝑘𝐹3(𝑪′)]⋯ [(𝑛 − 1)𝑘𝐹𝑛(𝑪)𝑥 − (𝑛 − 1)𝑘𝐹𝑛(𝑪′)] +where 𝑛 ≥ 2 and 𝑂𝑪 +𝑠(𝑛, 𝑚) = (−1)𝑛+𝑚𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). Then, we can get +𝑥𝑪 +𝑛↓ = ∑ 𝑂𝑪 +𝑠(𝑛, 𝑚) +𝑛 +𝑚=0 +𝑥𝑚 +and the zero points are 𝑥 = 0 and 𝑥 = +𝐹𝑚(𝑪′) +𝐹𝑚(𝑪) , where 𝑚 = 2,3, , … , 𝑛 . We call 𝑂𝑪 +𝑠(𝑛, 𝑚) signed 𝑪 +sequential optimization numbers. +Instance 2.1. C=(𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘). There are n boards and the sequence of them are fixed. Each board is +painted one color and divided into k smaller boards. For all 𝑖 = 1,2, … , 𝑘, the 𝑖th smaller boards of each +board form a group and their height are 1,2, … , 𝑛 respectively. In {𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘}, 𝑐𝑝1, 𝑐𝑝2, … , 𝑐𝑝𝑞 are all +elements which are equal to 1. Following the direction of the arrow, the number of ways that m colors can +be seen 𝑝1 or 𝑝2 or…or 𝑝𝑞 times is 𝑂𝑪(𝑛, 𝑚). + We call it k-dimensional color boards problem. An example is shown in Figure 2.a and the number +of colors that can be seen is shown in Figure 2.b. + +(a) + +(b) +Figure 2: An example of k-dimensional color boards problem. +Proof of Instance 2.1. Label as the 𝑖th board follow direction of the arrow. Label the height of the +smaller boards in same color from left to right as ℎ𝑖1, ℎ𝑖2, … ℎ𝑖𝑘 . Let 𝑹 = (>, >, … , >) be k- +dimensional relation vector, 𝒉𝒊 = (𝑖, ℎ𝑖1, ℎ𝑖2, … ℎ𝑖𝑘) and 𝑎𝑖𝑗 = 𝑛 + 1 − ℎ𝑖𝑗 , where 𝑖 = 1,2, … , 𝑛 and +𝑗 = 1,2, … , 𝑘. When 𝑪 = (0,1), as shown in Figure 3, we can get same optimization set in Figure 1.c +follow direction of the line of sight. When 𝑘 ≥ 1, 𝑗th color is seen 𝑙 times means that change 𝑙 𝑆3,0 +𝑤 to +𝑆1 +𝑤 by add (𝑗, 𝑎𝑗𝑤) in proof of Theorem 2.1. First color is seen 𝑘 time and 𝒂𝟏 in all 𝑆𝑤 in proof of Theorem +2.1. So, we can conclude that the number of ways that 𝑚 colors can be seen 𝑝1 or 𝑝2 or…or 𝑝𝑞 times is +𝑂𝑪(𝑛, 𝑚). + +8 + +Figure 3: Relationship between smaller color boards group and optimization set. +The properties of C dimensional sequential optimization numbers above are basic properties of +Stirling numbers of first kind when C is (0,1). Since Stirling numbers of first kind have been widely +studied and a large number of achievements have been made, many of them can be transformed into +the properties of C dimensional sequential optimization numbers and many valuable results can be +obtained. +The properties of C dimensional sequential optimization numbers below are extended by +properties of k-dimensional sequential optimization numbers [Hui22]. Theorem 2.3 is an upper +bounder of C dimensional sequential optimization numbers. Theorem 2.4 and Lemma 2.6 show C +dimensional sequential optimization numbers are almost concentrated in particular interval. +Theorem 2.5 shows the upper ratio of upper bounder to themselves. +Theorem 2.3. For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, we define + 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) = { +(𝑛 − 1)!𝑘 +(𝑚 − 1)! (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛−𝑚 +𝑖=1 +, +1 ≤ 𝑚 ≤ 𝑛 +0, +𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 + +where 𝑯𝑛 = (ℎ0, ℎ1, … , ℎ𝑘) and ℎ𝑝 = 𝐶𝑘 +𝑝 ∑ +1 +𝑗𝑝 +𝑛−1 +𝑗=1 += ∑ +𝐶𝑘 +𝑝 +1 +(𝑗−1)𝑝 +𝑛 +𝑗=2 += ∑ +𝑏𝑗,𝑝 +𝑛 +𝑗=2 +. +We +can +get +𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). +We can get a formula below. + 𝑯𝑛 ∙ 𝑪𝑇 = ∑ ℎ𝑝𝑐𝑝 +𝑘 +𝑝=1 += ∑ ∑ 𝑏𝑗,𝑝 +𝑛 +𝑗=2 +𝑐𝑝 +𝑘 +𝑝=1 += ∑ ∑ 𝑏𝑗,𝑝 +𝑘 +𝑝=1 +𝑐𝑝 +𝑛 +𝑗=2 += ∑ 𝐹𝑗(𝑪) +𝑛 +𝑗=2 + (4) +Proof of Theorem 2.3. We prove it using Theorem 2.2. First, we prove the boundary condition. For +all 𝑘, 𝑛 ∈ 𝑁+, +𝑂𝐂𝑚𝑎𝑥(𝑛, m + 𝑐𝑘 − 1) = {(𝑛 − 1)!𝑘 ∏ 𝐹𝑖+1(𝑪′) +𝑛−1 +𝑖=1 +, +𝑚 = 1 +0, +𝑚 = 0 𝑜𝑟 𝑚 > 𝑛 + +So, the boundary condition satisfy 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). +Then, we prove the inequality for 2 ≤ 𝑚 ≤ 𝑛 by mathematical induction. +Basis step: For 𝑛 = 2, according to formula (2), + 𝑂𝑪𝑚𝑎𝑥(2,2 + 𝑐𝑘 − 1) = 𝑯2 ∙ 𝑪𝑇 = 𝐹2(𝑪) = 𝑂𝑪(2,2 + 𝑐𝑘 − 1) 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (4) + +n+1 +n +ai +ap +lineof sight +aq +ar +aj +n9 +So, for 𝑛 = 2, 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). +Inductive step: For 𝑛 ≥ 2 and 2 ≤ 𝑚 ≤ 𝑛, we assume 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). +Then, we prove 𝑂𝑪𝑚𝑎𝑥(𝑛 + 1, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛 + 1, 𝑚 + 𝑐𝑘 − 1). + 𝑂𝑪𝑚𝑎𝑥(𝑛 + 1, m + 𝑐𝑘 − 1) = +𝑛!𝑘 +(𝑚 − 1)! (𝑯𝑛+1 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + += +𝑛!𝑘 +(𝑚 − 1)! [𝑯𝑛 ∙ 𝑪𝑇 + 𝐹𝑛+1(𝑪)]𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (4) +≥ +𝑛!𝑘 +(𝑚 − 1)! 𝐶𝑚−1 +0 +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 ++ +𝑛!𝑘 +(𝑚 − 1)! 𝐶𝑚−1 +1 +𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + += 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 1)! +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 ++ 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 2)! +𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + += 𝐹𝑛+2−m(𝑪′) 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 1)! +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛−m +𝑖=1 ++ 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 2)! +𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + +≥ 𝐹𝑛+1(𝑪′) 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 1)! +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛−m +𝑖=1 ++ 𝑛𝑘(𝑛 − 1)!𝑘 +(𝑚 − 2)! +𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) +𝑛+1−m +𝑖=1 + 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (1) += 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) + 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 − 1 + 𝑐𝑘 − 1) +≥ 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) + 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚 − 1 + 𝑐𝑘 − 1) += 𝑂𝑪(𝑛 + 1, m + 𝑐𝑘 − 1) +We can do same thing for 𝑚 = 𝑛 + 1. To sum up, for all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ N, 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − +1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1). +Theorem 2.4. For all 𝑛 ≥ 2, 𝑘 ≥ 1 and 𝑐0 = 0,𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1)] = +𝑂𝑪(𝑛,𝑚+𝑐𝑘−1) +𝑛!𝑘 + be the probability +of 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) and 𝑀 = 𝑒𝑘𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 +𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + + 𝑀1, where 𝑀1 is a positive +integer. We can get +𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑒−𝑀1 +Proof of Theorem 2.4. Let 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] = +𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚+𝑐𝑘−1) +𝑛!𝑘 + be the probability of the +upper bound of C sequential optimization numbers. For 𝑐0 = 0 and 1 ≤ 𝑚 ≤ 𝑛 − 1, + +10 +𝑃𝑟 [𝑂𝑘𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] = 𝑂𝑘𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘) +𝑛𝑘 + += +1 +𝑛𝑘𝑚! (𝑯𝑛 ∙ 𝑪𝑇)𝑚 ∏ 𝐹𝑖+1(𝑪′) +𝑛−m−1 +𝑖=1 + +≤ +1 +𝑛𝑘𝑚! (𝑯𝑛 ∙ 𝑪𝑇)𝑚(𝑛 − 𝑚)𝑘 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (2) +≤ 1 +𝑚! (𝑯𝑛 ∙ 𝑪𝑇)𝑚 +and +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚 + 𝑐𝑘)] +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚 + 𝑐𝑘 − 1)] = +𝑯𝑛 ∙ 𝑪𝑇 +𝑚𝐹𝑛−𝑚+1(𝑪′) + ≤ 𝑯𝑛 ∙ 𝑪𝑇 +𝑚 + 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (3) +Since ℎ𝑝 = ∑ +𝐶𝑘 +𝑝 +1 +(𝑗−1)𝑝 +𝑛 +𝑗=2 +, we can get ℎ0 = 𝑛 − 1, ℎ1 = 𝑘 ∑ +1 +𝑗−1 +𝑛 +𝑗=2 +≤ 𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] and for 𝑝 ≥ 2, +ℎ𝑝 ≤ 𝐶𝑘 +𝑝 ∑ +1 +(𝑗−1)2 +𝑛 +𝑗=2 +≤ +𝜋2 +6 𝐶𝑘 +𝑝. We know 𝑚! > √2𝜋𝑚( +𝑚 +𝑒 )𝑚 (Stirling's approximation). +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] ≤ +𝑐1 𝐶𝑘 +1[log(𝑛 − 1) + 1] + 𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 +𝑚 + +When 𝑚 ≥ 𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 +𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 +, +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] ≤ 1 +𝑒 +and +𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] = 1 +𝑚! (𝑯𝑛 ∙ 𝑪𝑇)𝑚 +≤ +{𝑐1 𝐶𝑘 +1[log(𝑛 − 1) + 1] + 𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 +}𝑚 +√2𝜋𝑚(𝑚 +𝑒 )𝑚 + + = +1 +√2𝜋𝑚 +[ +𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 +𝑚 +]𝑚 + ≤ 𝑒−1 +When 𝑀 = 𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 +𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + + 𝑀1, where 𝑀1 is a positive integer. +𝑃𝑟 [𝑂𝑪(𝑛, 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑒−𝑀1 +So, + +11 +𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] + = +∑ 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑖)] +𝑛 +𝑖=𝑀+1 + + ≤ 𝑒−𝑀1 +Lemma 2.6 If 𝑐0 = 1, we can get 𝑐′0 = 0 and 𝑃 𝑟[𝑂𝑪(𝑛, 𝑚 > 𝑛 − 𝑀 + 𝑐𝑘)] = 𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − +1)] ≤ 𝑒−𝑀1, where 𝑀 = 𝑒𝑘𝑐′1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 +𝜋2 +6 ∑ +𝑐′𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + + 𝑀1 and 𝑀1 is a positive integer. +Proof of Lemma 2.6. 𝑐′0 = 1 − 𝑐0 = 0. +𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − 1)] ≤ 𝑒−𝑀1 + 𝑃 𝑟[𝑂𝑪(𝑛, 𝑚 > 𝑛 + 1 − 𝑀 + 𝑐𝑘 − 1)] = 𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − 1)] ≤ 𝑒−𝑀1 𝐿𝑒𝑚𝑚𝑎 2.1 +where 𝑀 = 𝑒𝑘𝑐′1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 +𝜋2 +6 ∑ +𝑐′𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + + 𝑀1 and 𝑀1 is a positive integer. +Theorem 2.5. For all 𝑘 ≥ 1, 𝑛 ≥ 2 and 0 ≤ 𝑚 ≤ 𝑛, 𝑂𝑪(𝑛, 𝑚) ≤ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑛 − 𝑚), +∑ +𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=0 +∑ +𝑂𝑪(𝑛, 𝑚) +𝑛 +𝑚=0 +≤ 𝑒𝜆 +and +∑ +𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=0 +∑ +𝑂𝑪(𝑛, 𝑚) +𝑛 +𝑚=0 +≤ 𝑒𝜆′ +where 𝜆 = 𝑐0(𝑛 − 1) + 𝑐1𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] + +𝜋2 +6 ∑ +𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + and 𝜆′ = 𝑐′0(𝑛 − 1) + +𝑐′1𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] + +𝜋2 +6 ∑ +𝑐′𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 +.In particular, for 𝑠𝑢(𝑛, 𝑚) = 𝑂(0,1)(𝑛, 𝑚), +∑ +𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=0 +∑ +𝑠𝑢(𝑛, 𝑚) +𝑛 +𝑚=0 +≤ (𝑛 − 1) +𝑛 +𝑒 +∑ +1 +𝑗−1 +𝑛 +𝑗=2 +−𝑙𝑜𝑔 (𝑛−1) ≤ 𝑒𝛾 +where 𝛾 is Euler-Mascheroni constant and 𝑒𝛾 ≤ 1.7811. +Proof of Theorem 2.5. + 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) ≤ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑛 − 𝑚) 𝐿𝑒𝑚𝑚𝑎 2.2 +Let +𝜆 = 𝑯𝑛 ∙ 𝑪𝑇 +≤ 𝑐0(𝑛 − 1) + 𝑐1𝑘[log(𝑛 − 1) + 1] + 𝜋2 +6 ∑ 𝑐𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + +and +𝜆′ = 𝑯𝑛 ∙ 𝑪′𝑇 + ≤ 𝑐′0(𝑛 − 1) + 𝑐′1𝑘[log(𝑛 − 1) + 1] + 𝜋2 +6 ∑ 𝑐′𝑝𝐶𝑘 +𝑝 +𝑘 +𝑝=2 + +then, + +12 +∑ +𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=0 +∑ +𝑂𝑪(𝑛, 𝑖) +𝑛 +𝑚=0 += 1 +𝑛!𝑘 ∑ (𝑛 − 1)!𝑘 +(𝑚 − 1)! +𝑛 +𝑚=1 +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛−m +𝑖=1 + + ≤ 1 +𝑛𝑘 ∑ +𝜆𝑖−1 +(𝑚 − 1)! +𝑛 +𝑚=1 +(𝑛 − 𝑚 + 1)𝑘 + ≤ ∑ +𝜆𝑖−1 +(𝑚 − 1)! +𝑛 +𝑚=1 + + ≤ 𝑒𝜆 (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) +and +∑ +𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=0 +∑ +𝑂𝑪(𝑛, 𝑚) +𝑛 +𝑚=0 += 1 +𝑛!𝑘 ∑ (𝑛 − 1)!𝑘 +(𝑚 − 1)! +𝑛 +𝑚=1 +(𝑯𝑛 ∙ 𝑪′𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪) +𝑛−m +𝑖=1 + +≤ 1 +𝑛𝑘 ∑ +𝜆′𝑖−1 +(𝑚 − 1)! +𝑛 +𝑚=1 +(𝑛 − 𝑚 + 1)𝑘 +≤ ∑ +𝜆′𝑖−1 +(𝑚 − 1)! +𝑛 +𝑚=1 + +≤ 𝑒𝜆′ (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) +In particular, for 𝑠(𝑛, 𝑚) = 𝑂(0,1)(𝑛, 𝑚), +∑ +𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=1 +∑ +𝑠𝑢(𝑛, 𝑚) +𝑛 +𝑚=1 += 1 +𝑛! ∑ (𝑛 − 1)! +(𝑚 − 1)! +𝑛 +𝑚=1 +(𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) +𝑛−m +𝑖=1 + +≤ 1 +𝑛 ∑ +𝜆𝑖−1 +(𝑚 − 1)! +𝑛 +𝑚=1 + +≤ 𝑒𝜆 +𝑛 (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) += 1 +𝑛 𝑒 +∑ +1 +𝑗−1 +𝑛 +𝑗=2 + +Let +𝑎𝑛 = 1 +𝑛 𝑒 +∑ +1 +𝑗−1 +𝑛 +𝑗=2 + +and +𝑓(𝑥) = +𝑥 +𝑥 + 1 𝑒 +1 +𝑥 +where 𝑛 > 0 and 𝑥 > 0. + +13 + 𝑎𝑛+1 +𝑎𝑛 += +𝑛 +𝑛 + 1 𝑒 +1 +𝑛 +𝑓′(𝑥) = − +1 +𝑥(𝑥 + 1)2 𝑒 +1 +𝑥 < 0 +lim +𝑥→+∞ 𝑓(𝑥) = 1 +So, 𝑓(𝑥) > 1, +𝑎𝑛+1 +𝑎𝑛 > 1 and +∑ +𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) +𝑛 +𝑚=1 +∑ +𝑠𝑢(𝑛, 𝑚) +𝑛 +𝑚=1 +≤ lim +𝑛→+∞ +1 +𝑛 𝑒 +∑ +1 +𝑗−1 +𝑛 +𝑗=2 + += lim +𝑛→+∞ +(𝑛 − 1) +𝑛 +𝑒 +∑ +1 +𝑗−1 +𝑛 +𝑗=2 +−𝑙𝑜𝑔 (𝑛−1) += 𝑒𝛾 +where 𝛾 is Euler-Mascheroni constant and 𝑒𝛾 ≤ 1.7811. +3 CONCLUSION +In the article [Hui22], the proof of the Stirling numbers of first kind is more basic, which led to the +discovery of C sequential optimization numbers. We deal with many inequalities succinctly and +roughly and some of them still have much room for improvement. For the k+1-tuple vector C, there +are 2𝑘 kinds of sequences and some of them can have special properties. +REFERENCES +[AD17] +R. Arratia and S. DeSalvo. 2017. Completely Effective Error Bounds for Stirling Numbers +of the First and Second Kinds via Poisson Approximation. Annals of Combinatorics, +21(1),1-24. https://doi.org/10.1007/s00026-017-0339-z +[ARY21] +N. Alon, K. Rudow and L. Yariv. 2021. Dominance Solvability in Random Games (May +2021). http://dx.doi.org/10.2139/ssrn.3850992 +[Bla16] +I. V. Blagouchine. 2016. Two series expansions for the logarithm of the gamma function +involving Stirling numbers and containing only rational coefficients for certain +arguments related to 𝜋−1 . Journal of Mathematical Analysis and Applications, Vol. +442(October 2016), 404-434. https://doi.org/10.1016/j.jmaa.2016.04.032 +[Hui22] +Z.Hui. Sequential Optimization Numbers and Conjecture about Edge-Symmetry and +Weight-Symmetry +Shortest +Weight-Constrained +Path +(Jun +2022). +https://doi.org/10.48550/arXiv.2206.07052 +[Mou79] +H. Moulin. 1979. Dominance solvable voting schemes. Econometric, Vol. 47, No. +6(November 1979), 1337-1351. https://doi.org/10.2307/1914004 +[Sti30] +J. Stirling. 1730. Methodus differentialis, sive Tractatus de summatione et interpolatione +serierum infinitarum. +[Tem93] +N. M. Temme. 1993. Asymptotic Estimates of Stirling Numbers. Studies in Applied +Mathematics, 89(3),233-243. https://doi.org/10.1002/sapm1993893233 +[Wil93] +H. S. Wilf. 1993. The asymptotic behavior of the Stirling numbers of the first kind. Journal +of Combinatorial Theory, Series A 64,344-349. https://doi.org/10.1016/0097- +3165(93)90103-F +[Xue00] +G. Xue. 2000. Primal dual algorithms for computing weight-constrained shortest paths + +14 +and weight- constrained minimum spanning trees. Conference Proceeding of the 2000 +IEEE International Performance, Computing and Communications Conference (February +2000), 271-277. http://dx.doi.org/10.1109/PCCC.2000.830328 + diff --git a/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/load_file.txt b/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cfe43cae6961a87358e95efa6c5065fb67d4d06 --- /dev/null +++ b/TNA0T4oBgHgl3EQfEP8t/content/tmp_files/load_file.txt @@ -0,0 +1,382 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf,len=381 +page_content='𝐂⃗ Sequential Optimization Numbers Group Zile Hui, 51174500096@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='cn ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We define C sequential optimization numbers, where C is a k+1-tuple vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We prove that the unsigned Stirling numbers of first kind are (0,1) sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Many achievements of the Stirling numbers of first kind can be transformed into the properties of C sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We give some examples such as the recurrence formula and an instance of C sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We also extend some properties such as an upper bounder of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' KEYWORDS: C Sequential optimization numbers, Stirling numbers of first kind, k-dimensional sequential optimization numbers, Upper bounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 1 INTRODUCTION Stirling numbers were introduced by the Scottish mathematician James Stirling in his famous treatise and were subsequently rediscovered in various forms by numerous authors [Bla16][Sti30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Stirling numbers of the first kind, denoted by s(𝑛, 𝑚), were studied in a large number of works and have various properties [Bla16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Recurrence formula is one of the basic ways to define the unsigned Stirling numbers of first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' There is no known closed-form expression so far for the unsigned Stirling numbers of first kind and asymptotic formulas have been studied by numerous authors [Wil93][Tem93][AD17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' C Sequential optimization numbers are defined based on optimization set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We give some properties of C sequential optimization numbers and some of them are fundamental properties of Stirling numbers when C is (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We mainly refer to the definitions and proofs in one article [Hui22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 2 SEQUENTIAL OPTIMIZATION NUMBERS We introduce the optimization set in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1 and define the C sequential optimization numbers in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we give an expression which is not closed-form, a recurrence formula, some properties and applications of C sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' At the end of this chapter, we give an upper bound of sequential optimization numbers and some results derived from this upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝒙(𝑥1, 𝑥2, … , 𝑥𝑘), 𝒚(𝑦1, 𝑦2, … , 𝑦𝑘) be k-dimensional vectors and 𝑹(𝑅1, 𝑅2, … , 𝑅𝑘) be k- dimensional relation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If for all 𝑖 = 1,2, … , 𝑘, ( 𝑥𝑖, 𝑦𝑖) ∈ 𝑹𝒊, then x is said to be related to y by R, denoted by (𝒙, 𝒚) ∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let U be a set of vectors, R be a relation vector and 𝐴 ⊆ 𝑈, if for ∀𝒖 ∈ 𝑈, ∃𝒂 ∈ 𝐴 imply (𝒂, 𝒖) ∈ 𝑹 or 𝒂 = 𝒖, then A is said to be a majorization set of U by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 𝐵 is a majorization set of U by R and |𝐵| is a minimum of all the cardinalities of majorization sets, then 𝐵 is said to be an optimization set of U by R, denoted by 𝐵𝑜𝑹 ⊆ 𝑈, and |𝐵| is said to be the weight of U by R, denoted by 𝑊𝑜𝑹(𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can find similar selection strategies of optimization set in many existing studies [Mou79][Xue00][ARY21][Hui22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' It can be proved that this strategy can include all possible optimal terms and exclude impossible optimal terms in the iteration of the algorithm [Hui22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝒂𝟏(1, 𝑎11, 𝑎12, … 𝑎1𝑘), 𝒂𝟐(2, 𝑎21, 𝑎22, … 𝑎2𝑘),…, 𝒂𝒏(𝑛, 𝑎𝑛1, 𝑎𝑛2, … 𝑎𝑛𝑘) be n k+1- dimensional vectors, 𝑹(𝑅1, 𝑅2, … , 𝑅𝑘) = (<, <, … , <) be k-dimensional relation vector, C=(𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘), 𝑈 = {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏 } and 𝑆 ⊆ 𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑗 = 0,1, … , 𝑘 , 𝑐𝑗 ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑖 = 1,2, … , 𝑛 and 𝑗 = 1,2, … , 𝑘 , 𝒃𝒊𝒋 = (𝑖, 𝑎𝑖𝑗) , 𝑈𝑗 = {𝒃𝟏𝒋, 𝒃𝟐𝒋, … , 𝒃𝒏𝒋 }, 𝑎1𝑗, 𝑎2𝑗, … 𝑎𝑛𝑗 are 1,2, … , 𝑛 , 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑆𝑗 is an optimization set of 𝑈𝑗 by (<, 𝑅𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For a certain 𝑖 and all 𝑗 = 1,2, … , 𝑘, the total number of 𝒃𝒊𝒋 ∈ 𝑆𝑗 is l, if 𝑐𝑙 = 1 , then 𝒂𝒊 ∈ 𝑆 , otherwise, 𝒂𝒊 ∉ 𝑆 , 𝑆 is said to be an C sequential optimization set of U by R, denoted by 𝑆𝑂𝑹, 𝑪 ⊆ 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' |𝑆| is said to be sequential optimization weight of U by R, denoted by 𝑊𝑂𝑹,𝑪(𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' The numbers of ways that 𝑊𝑂𝑹,𝑪(𝑈) = 𝑚 are said to be C sequential optimization numbers, denoted by 𝑂𝑪(𝑛, 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2, 𝑪 = (𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We define 𝑩𝑗 = (𝑏𝑗,0, 𝑏𝑗,1, 𝑏𝑗,2, … , 𝑏𝑗,𝑘), 𝑪′ = (1,1, … ,1) − 𝑪 and 𝐹𝑗(𝑪) = 𝑩𝑗 ∙ 𝑪𝑇, where 𝑏𝑗,𝑝 = 𝐶𝑘 𝑝 1 (𝑗−1)𝑝, 𝑝 = 0,1, … , 𝑘 and 𝑗 = 2,3, … , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In this paper, we define ∏ 𝐹𝑗(𝑪) 𝑚 𝑗=𝑚+1 = 1 and ∑ 𝑓(𝑝) 𝑚 𝑝=𝑚+1 = 0 to make the expressions concise, where 𝑓(𝑝) is a function of 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We give some properties about 𝐹𝑗(𝑪).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 2 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑛, 𝐹𝑗2(𝑪) ≤ 𝐹𝑗1(𝑪) (1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 2 ≤ 𝑗1 ≤ 𝑗2 ≤ 𝑛, 𝑏𝑗2,𝑝 𝑏𝑗1,𝑝 = (𝑗1 − 1)𝑝 (𝑗2 − 1)𝑝 ≤ 1 𝑏𝑗2,𝑝 ≤ 𝑏𝑗1,𝑝 𝐹𝑗2(𝑪) ≤ 𝐹𝑗1(𝑪) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' ∏ 𝐹𝑗(𝑪) 𝑛 𝑗=2 ≤ 𝑛𝑘 (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' ∏ 𝐹𝑗(𝑪) 𝑛 𝑗=2 ≤ ∏ 𝐹𝑗([1,1, … ,1]𝑇) 𝑛 𝑗=2 = ∏ ∑ 𝑏𝑗,𝑝 𝑘 𝑝=0 𝑛 𝑗=2 = ∏ ∑ 𝐶𝑘 𝑝 1 (𝑗 − 1)𝑝 𝑘 𝑝=0 𝑛 𝑗=2 = ∏ (1 + 1 𝑗 − 1) 𝑘 𝑛 𝑗=2 = 𝑛𝑘 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 𝑐0 = 1, 𝐹𝑗(𝑪) ≥ 1 (3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐹𝑗(𝑪) = 𝑩 ∙ 𝑪𝑇 ≥ 𝑐0𝑏𝑗,0 = 1 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = {(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 , 1 ≤ 𝑚 ≤ 𝑛 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of m-1 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 3 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In the scenario of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2, 𝑪 = (𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘) and 𝑆𝑂𝑹, 𝑪 ⊆ 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑔(𝑆) denotes the numbers of ways where S is a set consisting of specific elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First, we discuss the case where 𝑪 = (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can get 𝑆 is optimization set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For 𝑚 = 1, the number of positions of 𝒂𝟐~𝒂𝒏 is (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' So 𝑂(0,1)(𝑛, 1) = (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='. (a) (b) (c) Figure 1: Three ways to optimization set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For 𝑚 = 2,3, … , 𝑛, |𝑈| = 𝑛 and 𝑆1 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓, 𝒂𝒋}, we use mathematical induction to prove that 𝑔(𝑆1) = (𝑛−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝−1)(𝑞−1)∙…∙(𝑟−1)(𝑗−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Basis step: we prove 𝑔(𝑆2) = (𝑛−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑗−1 , where 𝑆2 = {𝒂𝟏, 𝒂𝒋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' As shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='b, 𝑎11 = 𝑖, 𝑎𝑗1 = 1, where 𝑖 = 2,3, … , 𝑛, 𝑗 = 2,3, … , 𝑛 and 𝑖 + 𝑗 ≤ 𝑛 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First, place row 2~(𝑖 − 1) and the number of positions is (𝑛−𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛−𝑖−𝑗+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='. Second, place row (𝑖 + 1)~𝑛 and the number of positions is (𝑛 − 𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='. Total number is (𝑛−𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛−𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛−𝑖−𝑗+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑔(𝑆2) = ∑ (𝑛 − 𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑖 − 𝑗 + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛+2−𝑗 𝑖=2 = (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑗 − 1 ∑ (𝑛 − 𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' [(𝑛 − 𝑖 + 1) − (𝑛 − 𝑖 − 𝑗 + 2)] (𝑛 − 𝑖 − 𝑗 + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛+2−𝑗 𝑖=2 = (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑗 − 1 ∑ [ (𝑛 − 𝑖 + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑖 − 𝑗 + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' − (𝑛 − 𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑖 − 𝑗 + 2) (𝑛 − 𝑖 − 𝑗 + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' ] 𝑛+2−𝑗 𝑖=2 In the brackets, the right side of the minus is equal to 0 when 𝑖 = 𝑛 + 2 − 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' The right side of the minus with 𝑖 = 𝑡 is equal to the left side of the minus with 𝑖 = 𝑡 + 1, where 𝑡 = 2,3, … , 𝑛 + 1 − 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' So, 𝑔(𝑆2) = (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑗 − 1 ∙ (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' = (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑗 − 1 n ai nn ai aj nn al ap aq aj n4 Inductive step: assume 𝑔(𝑆3) = (𝑛−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝−1)(𝑞−1)∙…∙(𝑟−1), where 𝑆3 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we show 𝑔(𝑆1) = (𝑛−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝−1)(𝑞−1)∙…∙(𝑟−1)(𝑗−1) where 𝑆1 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓, 𝒂𝒋} and 1 < 𝑝 < 𝑞 < 𝑟 < 𝑗 ≤ 𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='c, 𝑎𝑟1 = 𝑖, 𝑎𝑗1 = 1, where 𝑖 = 2,3, … , 𝑛 − 1, 𝑗 = 3,4, … , 𝑛 and 𝑖 + 𝑗 ≤ 𝑛 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First, place row 2~(𝑖 − 1) and the number of positions is (𝑛−𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛−𝑖−𝑗+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='. We remove all the rows and columns where the points in row 1~(𝑖 − 1) are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In the rest of the figure, |𝑈’| = 𝑛 − 𝑖 + 1, 𝑆3 ′ = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓} and the number of ways is (𝑛−𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝−1)(𝑞−1)∙…∙(𝑟−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' So, 𝑔(𝑆1) = ∑ (𝑛 − 𝑖)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑛 − 𝑗)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝 − 1)(𝑞 − 1) ∙ … ∙ (𝑟 − 1)(𝑛 − 𝑖 − 𝑗 + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛+2−𝑗 𝑖=2 = (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑝 − 1)(𝑞 − 1) ∙ … ∙ (𝑟 − 1)(𝑗 − 1) So, for 𝑪 = (0,1), change 𝑆3 to 𝑆1 by add 𝒂𝒋 and we can get 𝑔(𝑆1) = 1 𝑗−1 𝑔(𝑆3), where 𝑗 = 2,3, … , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, for 𝑘 > 1, we discuss the relationship between g(𝑆1) and g(𝑆3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In 𝒂𝟏~𝒂𝒏, let dimension 1, {1, 2, … 𝑛} and dimension w+1, {𝑎1𝑤, 𝑎2𝑤, … , 𝑎𝑛𝑤} form k groups of n 2-dimensional vectors (1, 𝑎1𝑤), ( 2, 𝑎2𝑤), … (n, 𝑎𝑛𝑤) and 𝑈𝑤 = {(1, 𝑎1𝑤), ( 2, 𝑎2𝑤), … (n, 𝑎𝑛𝑤)} , where 𝑤 = 1,2, … , 𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝑆𝑤𝑂𝑹, (0,1) ⊆ 𝑈𝑤 and we can get the relationship between 𝑆𝑤 and 𝑔(𝑆𝑤) is the same as the relationship between 𝑆 and 𝑔(𝑆) when 𝑪 = (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We define 𝑆3,0 = {𝒂𝟏, 𝒂𝒑, 𝒂𝒒, … , 𝒂𝒓} and there are 0 (𝑗, 𝑎𝑗𝑤) in all 𝑆3,0 𝑤 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In k+1-dimensional vector, to change 𝑆3,0 to 𝑆1 by add 𝒂𝒋, we need to change 𝑙 𝑆3,0 𝑤 to 𝑆1 𝑤 by add (𝑗, 𝑎𝑗𝑤) if 𝑐𝑙 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 𝑐𝑙 = 0, we change 𝑙 𝑆3,0 𝑤 to 𝑆1 𝑤 by add (𝑗, 𝑎𝑗𝑤) to change 𝑆3,0 to 𝑆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can get 𝑔(𝑆1) = 𝑩 ∙ 𝑪𝑇𝑔(𝑆3,0) 𝑔(𝑆3) = 𝑩 ∙ 𝑪′𝑇𝑔(𝑆3,0) and 𝑔(𝑆1) = 𝑩 ∙ 𝑪𝑇 𝑩 ∙ 𝑪′𝑇 𝑔(𝑆3) = 𝐹𝑗(𝑪) 𝐹𝑗(𝑪′) 𝑔(𝑆3) We also get 𝒂𝟏 in all 𝑆𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' So, (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 𝑐𝑘 = 1, 𝑂𝑪(𝑛, 𝑚) = { (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ ∏ 𝐹𝑗′𝑖(𝑪′) 𝑛−1 𝑖=1 , 𝑚 = 1 (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 , 2 ≤ 𝑚 ≤ 𝑛 − 1 (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ ∏ 𝐹𝑗𝑖(𝑪) 𝑛−1 𝑖=1 , 𝑚 = 𝑛 0, 𝑚 = 0 or 𝑚 > 𝑛 where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of 𝑚 − 1 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' If 𝑐𝑘 = 0, 5 𝑂𝑪(𝑛, 𝑚) = { (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ ∏ 𝐹𝑗′𝑖(𝑪′) 𝑛−1 𝑖=1 , 𝑚 = 0 (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚−1 𝑖=1 , 1 ≤ 𝑚 ≤ 𝑛 − 2 (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ ∏ 𝐹𝑗𝑖(𝑪) 𝑛−1 𝑖=1 , 𝑚 = 𝑛 − 1 0, 𝑚 = −1 or 𝑚 > 𝑛 − 1 where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚 are all combinations consisting of 𝑚 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚−1} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' To sum up, for all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = {(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 , 1 ≤ 𝑚 ≤ 𝑛 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 are all combinations consisting of 𝑚 − 1 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ≥ 𝑐𝑘 − 1, the recurrence formula of C sequential optimization numbers is 𝑂𝑪(𝑛 + 1, 𝑚 + 1) = 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) + 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) and the boundary condition is 𝑂𝒄(𝑛, 𝑚) = {0, 𝑚 = 𝑐𝑘 − 1 𝑜𝑟 𝑚 > 𝑐𝑘 − 1 + 𝑛 1, 𝑚 = 𝑐𝑘, 𝑛 = 1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2, for 𝑐𝑘 = 1 , we divide the number of ways that 𝑂𝑘(𝑛 + 1, 𝑚 + 1) denotes into two parts, which are with and without 𝒂𝒏+𝟏, where 𝑛 > 2 and 𝑚 = 2,3, … , 𝑛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First, ways with 𝒂𝒏+𝟏 and 𝑚 vectors from {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' let 𝑗𝑚 be 𝑛 + 1, 𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1 be all combinations consisting of m-1 elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 = 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 𝐹𝑛+1(𝑪) ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 = 𝑛𝑘𝐹𝑛+1(𝑪)(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 = 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) Then, ways without 𝒂𝒏+𝟏 and m vectors from {𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' let 𝑗′𝑛−𝑚 be 𝑛 + 1, 𝑗1, 𝑗2, ⋯ , 𝑗𝑚 are all combinations consisting of m elements in set {2,3, … , 𝑛} and {𝑗′1, 𝑗′2, ⋯ , 𝑗′𝑛−𝑚−1} = {2,3, … , 𝑛} − {𝑗1, 𝑗2, ⋯ , 𝑗𝑚}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚 𝑖=1 = 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 𝐹𝑛+1(𝑪′) ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑚−1 𝑖=1 6 = 𝑛𝑘𝐹𝑛+1(𝑪′)(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑚 𝑖=1 ∏ 𝐹𝑗′𝑖(𝑪′)) 𝑛−𝑚−1 𝑖=1 = 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) To sum up, 𝑂𝑪(𝑛 + 1, 𝑚 + 1) = 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚) + 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 1) We can get the boundary condition according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑂𝒄(𝑛, 𝑚) = {0, 𝑚 = 𝑐𝑘 − 1 𝑜𝑟 𝑚 > 𝑐𝑘 − 1 + 𝑛 1, 𝑚 = 𝑐𝑘, 𝑛 = 1 We can do the same thing for 𝑐𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can prove that the recurrence formula works for all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ≥ 𝑐𝑘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We give some properties and applications of C sequential optimization numbers below and we prove part of them in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1 For 𝑐𝑘 − 1 ≤ 𝑚 ≤ 𝑛 + 𝑐𝑘, 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) For 0 ≤ 𝑚 ≤ 𝑛 + 1, 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) = 0 = 𝑂𝑪′(𝑛, 𝑛 + 1 − 𝑚 + 𝑐′𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For 1 ≤ 𝑡 ≤ 𝑛, 𝑂𝑪(𝑛, 𝑡 + 𝑐𝑘 − 1) = (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑(∏ 𝐹𝑗𝑖(𝑪) 𝑡−1 𝑖=1 ∏ 𝐹𝑗′ 𝑖(𝑪′)) 𝑛−𝑡 𝑖=1 = (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑( ∏ 𝐹𝑗′ 𝑖(𝑪′) 𝑛−𝑡+1−1 𝑖=1 ∏ 𝐹𝑗𝑖(𝑪) 𝑛−(𝑛−𝑡+1) 𝑖=1 ) = 𝑂𝑪′(𝑛, 𝑛 − 𝑡 + 1 + 𝑐′𝑘 − 1) = 𝑂𝑪′(𝑛, 𝑛 − 𝑡 + 𝑐′𝑘) For 𝑡 = 0 and 𝑡 = 𝑛 + 1, 𝑂𝑪(𝑛, 𝑡 + 𝑐𝑘 − 1) = 0 = 𝑂𝑪′(𝑛, 𝑛 + 1 − 𝑡 + 𝑐′𝑘 − 1) Let 𝑚 = 𝑡 + 𝑐𝑘 − 1, 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) where 𝑐𝑘 − 1 ≤ 𝑚 ≤ 𝑛 + 𝑐𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' ∑ 𝑂𝑪(𝑛, 𝑚) 𝑛 𝑚=0 = 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For 𝑪 = (0,1), 𝑂𝑪(𝑛, 𝑚) = 𝑠𝑢(𝑛, 𝑚), where 𝑠𝑢(𝑛, 𝑚) are the unsigned Stirling numbers of first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝑪 = (0,1,1, … ,1) be a k+1-tuple vector, 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑘(𝑛, 𝑚), where 𝑂𝑘(𝑛, 𝑚) is k- dimensional sequential optimization numbers [Hui22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+, we define 𝑥𝑪 1↑ = 𝑥, 𝑥𝑪 𝑛↑ = 𝑥[𝐹2(𝑪)𝑥 + 𝐹2(𝑪′)][2𝑘𝐹3(𝑪)𝑥 + 2𝑘𝐹3(𝑪′)]⋯ [(𝑛 − 1)𝑘𝐹𝑛(𝑪)𝑥 + (𝑛 − 1)𝑘𝐹𝑛(𝑪′)] where 𝑛 ≥ 2 and 𝑂𝑪 𝑢(𝑛, 𝑚) = 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we can get 7 𝑥𝑪 𝑛↑ = ∑ 𝑂𝑪 𝑢(𝑛, 𝑚) 𝑛 𝑚=0 𝑥𝑚 and the zero points are 𝑥 = 0 and 𝑥 = − 𝐹𝑚(𝑪′) 𝐹𝑚(𝑪) , where 𝑚 = 2,3, , … , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We call 𝑂𝑪 𝑢(𝑛, 𝑚) unsigned 𝑪 sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+, we define 𝑥𝑪 1↓ = 𝑥, 𝑥𝑪 𝑛↓ = 𝑥[𝐹2(𝑪)𝑥 − 𝐹2(𝑪′)][2𝑘𝐹3(𝑪)𝑥 − 2𝑘𝐹3(𝑪′)]⋯ [(𝑛 − 1)𝑘𝐹𝑛(𝑪)𝑥 − (𝑛 − 1)𝑘𝐹𝑛(𝑪′)] where 𝑛 ≥ 2 and 𝑂𝑪 𝑠(𝑛, 𝑚) = (−1)𝑛+𝑚𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we can get 𝑥𝑪 𝑛↓ = ∑ 𝑂𝑪 𝑠(𝑛, 𝑚) 𝑛 𝑚=0 𝑥𝑚 and the zero points are 𝑥 = 0 and 𝑥 = 𝐹𝑚(𝑪′) 𝐹𝑚(𝑪) , where 𝑚 = 2,3, , … , 𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We call 𝑂𝑪 𝑠(𝑛, 𝑚) signed 𝑪 sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Instance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' C=(𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' There are n boards and the sequence of them are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Each board is painted one color and divided into k smaller boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑖 = 1,2, … , 𝑘, the 𝑖th smaller boards of each board form a group and their height are 1,2, … , 𝑛 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' In {𝑐0, 𝑐1, 𝑐2, … , 𝑐𝑘}, 𝑐𝑝1, 𝑐𝑝2, … , 𝑐𝑝𝑞 are all elements which are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Following the direction of the arrow, the number of ways that m colors can be seen 𝑝1 or 𝑝2 or…or 𝑝𝑞 times is 𝑂𝑪(𝑛, 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We call it k-dimensional color boards problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' An example is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='a and the number of colors that can be seen is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (a) (b) Figure 2: An example of k-dimensional color boards problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Proof of Instance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Label as the 𝑖th board follow direction of the arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Label the height of the smaller boards in same color from left to right as ℎ𝑖1, ℎ𝑖2, … ℎ𝑖𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝑹 = (>, >, … , >) be k- dimensional relation vector, 𝒉𝒊 = (𝑖, ℎ𝑖1, ℎ𝑖2, … ℎ𝑖𝑘) and 𝑎𝑖𝑗 = 𝑛 + 1 − ℎ𝑖𝑗 , where 𝑖 = 1,2, … , 𝑛 and 𝑗 = 1,2, … , 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' When 𝑪 = (0,1), as shown in Figure 3, we can get same optimization set in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='c follow direction of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' When 𝑘 ≥ 1, 𝑗th color is seen 𝑙 times means that change 𝑙 𝑆3,0 𝑤 to 𝑆1 𝑤 by add (𝑗, 𝑎𝑗𝑤) in proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First color is seen 𝑘 time and 𝒂𝟏 in all 𝑆𝑤 in proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' So, we can conclude that the number of ways that 𝑚 colors can be seen 𝑝1 or 𝑝2 or…or 𝑝𝑞 times is 𝑂𝑪(𝑛, 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 8 Figure 3: Relationship between smaller color boards group and optimization set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' The properties of C dimensional sequential optimization numbers above are basic properties of Stirling numbers of first kind when C is (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Since Stirling numbers of first kind have been widely studied and a large number of achievements have been made, many of them can be transformed into the properties of C dimensional sequential optimization numbers and many valuable results can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' The properties of C dimensional sequential optimization numbers below are extended by properties of k-dimensional sequential optimization numbers [Hui22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='3 is an upper bounder of C dimensional sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='6 show C dimensional sequential optimization numbers are almost concentrated in particular interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='5 shows the upper ratio of upper bounder to themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ 𝑁, we define 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) = { (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛−𝑚 𝑖=1 , 1 ≤ 𝑚 ≤ 𝑛 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where 𝑯𝑛 = (ℎ0, ℎ1, … , ℎ𝑘) and ℎ𝑝 = 𝐶𝑘 𝑝 ∑ 1 𝑗𝑝 𝑛−1 𝑗=1 = ∑ 𝐶𝑘 𝑝 1 (𝑗−1)𝑝 𝑛 𝑗=2 = ∑ 𝑏𝑗,𝑝 𝑛 𝑗=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can get 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can get a formula below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑯𝑛 ∙ 𝑪𝑇 = ∑ ℎ𝑝𝑐𝑝 𝑘 𝑝=1 = ∑ ∑ 𝑏𝑗,𝑝 𝑛 𝑗=2 𝑐𝑝 𝑘 𝑝=1 = ∑ ∑ 𝑏𝑗,𝑝 𝑘 𝑝=1 𝑐𝑝 𝑛 𝑗=2 = ∑ 𝐹𝑗(𝑪) 𝑛 𝑗=2 (4) Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We prove it using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' First, we prove the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘, 𝑛 ∈ 𝑁+, 𝑂𝐂𝑚𝑎𝑥(𝑛, m + 𝑐𝑘 − 1) = {(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∏ 𝐹𝑖+1(𝑪′) 𝑛−1 𝑖=1 , 𝑚 = 1 0, 𝑚 = 0 𝑜𝑟 𝑚 > 𝑛 So, the boundary condition satisfy 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we prove the inequality for 2 ≤ 𝑚 ≤ 𝑛 by mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Basis step: For 𝑛 = 2, according to formula (2), 𝑂𝑪𝑚𝑎𝑥(2,2 + 𝑐𝑘 − 1) = 𝑯2 ∙ 𝑪𝑇 = 𝐹2(𝑪) = 𝑂𝑪(2,2 + 𝑐𝑘 − 1) 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (4) n+1 n ai ap lineof sight aq ar aj n9 So, for 𝑛 = 2, 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Inductive step: For 𝑛 ≥ 2 and 2 ≤ 𝑚 ≤ 𝑛, we assume 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Then, we prove 𝑂𝑪𝑚𝑎𝑥(𝑛 + 1, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛 + 1, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑂𝑪𝑚𝑎𝑥(𝑛 + 1, m + 𝑐𝑘 − 1) = 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛+1 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 = 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' [𝑯𝑛 ∙ 𝑪𝑇 + 𝐹𝑛+1(𝑪)]𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (4) ≥ 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐶𝑚−1 0 (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 + 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐶𝑚−1 1 𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 = 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 + 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 = 𝐹𝑛+2−m(𝑪′) 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛−m 𝑖=1 + 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 ≥ 𝐹𝑛+1(𝑪′) 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛−m 𝑖=1 + 𝑛𝑘(𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝐹𝑛+1(𝑪)(𝑯𝑛 ∙ 𝑪𝑇)𝑚−2 ∏ 𝐹𝑖+1(𝑪′) 𝑛+1−m 𝑖=1 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (1) = 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) + 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 − 1 + 𝑐𝑘 − 1) ≥ 𝑛𝑘𝐹𝑛+1(𝑪′)𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) + 𝑛𝑘𝐹𝑛+1(𝑪)𝑂𝑪(𝑛, 𝑚 − 1 + 𝑐𝑘 − 1) = 𝑂𝑪(𝑛 + 1, m + 𝑐𝑘 − 1) We can do same thing for 𝑚 = 𝑛 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' To sum up, for all 𝑘, 𝑛 ∈ 𝑁+ and 𝑚 ∈ N, 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1) ≥ 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑛 ≥ 2, 𝑘 ≥ 1 and 𝑐0 = 0,𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1)] = 𝑂𝑪(𝑛,𝑚+𝑐𝑘−1) 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 be the probability of 𝑂𝑪(𝑛, 𝑚 + 𝑐𝑘 − 1) and 𝑀 = \uf0e9𝑒𝑘𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 \uf0f9 + 𝑀1, where 𝑀1 is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We can get 𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑒−𝑀1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Let 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] = 𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚+𝑐𝑘−1) 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 be the probability of the upper bound of C sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For 𝑐0 = 0 and 1 ≤ 𝑚 ≤ 𝑛 − 1, 10 𝑃𝑟 [𝑂𝑘𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] = 𝑂𝑘𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘) 𝑛𝑘 = 1 𝑛𝑘𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚 ∏ 𝐹𝑖+1(𝑪′) 𝑛−m−1 𝑖=1 ≤ 1 𝑛𝑘𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚(𝑛 − 𝑚)𝑘 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (2) ≤ 1 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚 and 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚 + 𝑐𝑘)] 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛,𝑚 + 𝑐𝑘 − 1)] = 𝑯𝑛 ∙ 𝑪𝑇 𝑚𝐹𝑛−𝑚+1(𝑪′) ≤ 𝑯𝑛 ∙ 𝑪𝑇 𝑚 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 (3) Since ℎ𝑝 = ∑ 𝐶𝑘 𝑝 1 (𝑗−1)𝑝 𝑛 𝑗=2 , we can get ℎ0 = 𝑛 − 1, ℎ1 = 𝑘 ∑ 1 𝑗−1 𝑛 𝑗=2 ≤ 𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] and for 𝑝 ≥ 2, ℎ𝑝 ≤ 𝐶𝑘 𝑝 ∑ 1 (𝑗−1)2 𝑛 𝑗=2 ≤ 𝜋2 6 𝐶𝑘 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We know 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=" > √2𝜋𝑚( 𝑚 𝑒 )𝑚 (Stirling's approximation)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] ≤ 𝑐1 𝐶𝑘 1[log(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 𝑚 When 𝑚 ≥ \uf0e9𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 \uf0f9, 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘 − 1)] ≤ 1 𝑒 and 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 + 𝑐𝑘)] = 1 𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑯𝑛 ∙ 𝑪𝑇)𝑚 ≤ {𝑐1 𝐶𝑘 1[log(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 }𝑚 √2𝜋𝑚(𝑚 𝑒 )𝑚 = 1 √2𝜋𝑚 [ 𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 𝑚 ]𝑚 ≤ 𝑒−1 When 𝑀 = \uf0e9𝑒k𝑐1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 \uf0f9 + 𝑀1, where 𝑀1 is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑃𝑟 [𝑂𝑪(𝑛, 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑒−𝑀1 So, 11 𝑃𝑟 [𝑂𝑪(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] ≤ 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚 > 𝑀 + 𝑐𝑘 − 1)] = ∑ 𝑃𝑟 [𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑖)] 𝑛 𝑖=𝑀+1 ≤ 𝑒−𝑀1 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='6 If 𝑐0 = 1, we can get 𝑐′0 = 0 and 𝑃 𝑟[𝑂𝑪(𝑛, 𝑚 > 𝑛 − 𝑀 + 𝑐𝑘)] = 𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − 1)] ≤ 𝑒−𝑀1, where 𝑀 = \uf0e9𝑒𝑘𝑐′1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐′𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 \uf0f9 + 𝑀1 and 𝑀1 is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑐′0 = 1 − 𝑐0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − 1)] ≤ 𝑒−𝑀1 𝑃 𝑟[𝑂𝑪(𝑛, 𝑚 > 𝑛 + 1 − 𝑀 + 𝑐𝑘 − 1)] = 𝑃 𝑟[𝑂𝑪′(𝑛, 𝑚 < 𝑀 + 𝑐′𝑘 − 1)] ≤ 𝑒−𝑀1 𝐿𝑒𝑚𝑚𝑎 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1 where 𝑀 = \uf0e9𝑒𝑘𝑐′1[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝑒 𝜋2 6 ∑ 𝑐′𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 \uf0f9 + 𝑀1 and 𝑀1 is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For all 𝑘 ≥ 1, 𝑛 ≥ 2 and 0 ≤ 𝑚 ≤ 𝑛, 𝑂𝑪(𝑛, 𝑚) ≤ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑛 − 𝑚), ∑ 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=0 ∑ 𝑂𝑪(𝑛, 𝑚) 𝑛 𝑚=0 ≤ 𝑒𝜆 and ∑ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=0 ∑ 𝑂𝑪(𝑛, 𝑚) 𝑛 𝑚=0 ≤ 𝑒𝜆′ where 𝜆 = 𝑐0(𝑛 − 1) + 𝑐1𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 and 𝜆′ = 𝑐′0(𝑛 − 1) + 𝑐′1𝑘[𝑙𝑜𝑔(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐′𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='In particular, for 𝑠𝑢(𝑛, 𝑚) = 𝑂(0,1)(𝑛, 𝑚), ∑ 𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=0 ∑ 𝑠𝑢(𝑛, 𝑚) 𝑛 𝑚=0 ≤ (𝑛 − 1) 𝑛 𝑒 ∑ 1 𝑗−1 𝑛 𝑗=2 −𝑙𝑜𝑔 (𝑛−1) ≤ 𝑒𝛾 where 𝛾 is Euler-Mascheroni constant and 𝑒𝛾 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='7811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑂𝑪(𝑛, 𝑚) = 𝑂𝑪′(𝑛, 𝑛 − 𝑚) ≤ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑛 − 𝑚) 𝐿𝑒𝑚𝑚𝑎 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2 Let 𝜆 = 𝑯𝑛 ∙ 𝑪𝑇 ≤ 𝑐0(𝑛 − 1) + 𝑐1𝑘[log(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 and 𝜆′ = 𝑯𝑛 ∙ 𝑪′𝑇 ≤ 𝑐′0(𝑛 − 1) + 𝑐′1𝑘[log(𝑛 − 1) + 1] + 𝜋2 6 ∑ 𝑐′𝑝𝐶𝑘 𝑝 𝑘 𝑝=2 then, 12 ∑ 𝑂𝑪𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=0 ∑ 𝑂𝑪(𝑛, 𝑖) 𝑛 𝑚=0 = 1 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛−m 𝑖=1 ≤ 1 𝑛𝑘 ∑ 𝜆𝑖−1 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 (𝑛 − 𝑚 + 1)𝑘 ≤ ∑ 𝜆𝑖−1 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 ≤ 𝑒𝜆 (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) and ∑ 𝑂𝑪′𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=0 ∑ 𝑂𝑪(𝑛, 𝑚) 𝑛 𝑚=0 = 1 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 ∑ (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='𝑘 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 (𝑯𝑛 ∙ 𝑪′𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪) 𝑛−m 𝑖=1 ≤ 1 𝑛𝑘 ∑ 𝜆′𝑖−1 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 (𝑛 − 𝑚 + 1)𝑘 ≤ ∑ 𝜆′𝑖−1 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 ≤ 𝑒𝜆′ (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) In particular, for 𝑠(𝑛, 𝑚) = 𝑂(0,1)(𝑛, 𝑚), ∑ 𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=1 ∑ 𝑠𝑢(𝑛, 𝑚) 𝑛 𝑚=1 = 1 𝑛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' ∑ (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 (𝑯𝑛 ∙ 𝑪𝑇)𝑚−1 ∏ 𝐹𝑖+1(𝑪′) 𝑛−m 𝑖=1 ≤ 1 𝑛 ∑ 𝜆𝑖−1 (𝑚 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 𝑛 𝑚=1 ≤ 𝑒𝜆 𝑛 (𝑇𝑎𝑦𝑙𝑜𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚) = 1 𝑛 𝑒 ∑ 1 𝑗−1 𝑛 𝑗=2 Let 𝑎𝑛 = 1 𝑛 𝑒 ∑ 1 𝑗−1 𝑛 𝑗=2 and 𝑓(𝑥) = 𝑥 𝑥 + 1 𝑒 1 𝑥 where 𝑛 > 0 and 𝑥 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 13 𝑎𝑛+1 𝑎𝑛 = 𝑛 𝑛 + 1 𝑒 1 𝑛 𝑓′(𝑥) = − 1 𝑥(𝑥 + 1)2 𝑒 1 𝑥 < 0 lim 𝑥→+∞ 𝑓(𝑥) = 1 So, 𝑓(𝑥) > 1, 𝑎𝑛+1 𝑎𝑛 > 1 and ∑ 𝑠𝑢𝑚𝑎𝑥(𝑛, 𝑚) 𝑛 𝑚=1 ∑ 𝑠𝑢(𝑛, 𝑚) 𝑛 𝑚=1 ≤ lim 𝑛→+∞ 1 𝑛 𝑒 ∑ 1 𝑗−1 𝑛 𝑗=2 = lim 𝑛→+∞ (𝑛 − 1) 𝑛 𝑒 ∑ 1 𝑗−1 𝑛 𝑗=2 −𝑙𝑜𝑔 (𝑛−1) = 𝑒𝛾 where 𝛾 is Euler-Mascheroni constant and 𝑒𝛾 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='7811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 3 CONCLUSION In the article [Hui22], the proof of the Stirling numbers of first kind is more basic, which led to the discovery of C sequential optimization numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' We deal with many inequalities succinctly and roughly and some of them still have much room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' For the k+1-tuple vector C, there are 2𝑘 kinds of sequences and some of them can have special properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' REFERENCES [AD17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Arratia and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='3850992 [Bla16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Blagouchine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Primal dual algorithms for computing weight-constrained shortest paths 14 and weight- constrained minimum spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' Conference Proceeding of the 2000 IEEE International Performance, Computing and Communications Conference (February 2000), 271-277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='1109/PCCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} +page_content='830328' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNA0T4oBgHgl3EQfEP8t/content/2301.02014v1.pdf'} diff --git a/TNAzT4oBgHgl3EQfJftl/vector_store/index.faiss b/TNAzT4oBgHgl3EQfJftl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0e92b868fe4c3e34c2c25eba74116772434c2fcb --- /dev/null +++ b/TNAzT4oBgHgl3EQfJftl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:01aeb83002bbbd7a07dd53b9cd7eaa8f68e20ba62969f5354dbb909c573e8c04 +size 3801133 diff --git 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100644 index 0000000000000000000000000000000000000000..70e215201cb0b485791dcf5ce7719be1e1b0b3b8 --- /dev/null +++ b/TtE4T4oBgHgl3EQfmQ2m/content/tmp_files/2301.05167v1.pdf.txt @@ -0,0 +1,4189 @@ +arXiv:2301.05167v1 [cs.GT] 12 Jan 2023 +On the Optimal Fixed-Price Mechanism in Bilateral Trade +Yang Cai* +Yale University, USA +yang.cai@yale.edu +Jinzhao Wu +Yale University, USA +jinzhao.wu@yale.edu +January 13, 2023 +Abstract +We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer +and a seller, trade an indivisible item. The seminal result of Myerson and Satterthwaite [29] shows +that no incentive compatible and budget balanced (i.e., the mechanism does not run a deficit) mech- +anism can achieve the optimal social welfare in bilateral trade. Motivated by this impossibility result, +we focus on approximating the optimal social welfare. We consider arguably the simplest form of +mechanisms – the fixed-price mechanisms, where the designer offers trade at a fixed price to the +seller and buyer. Besides the simple form, fixed-price mechanisms are also the only dominant strat- +egy incentive compatible and budget balanced mechanisms in bilateral trade [23]. +We obtain improved approximation ratios of fixed-price mechanisms in both (i) the full infor- +mation setting, where the designer knows the value distributions of both the seller and buyer; and +(ii) the limited information settings. In the full information setting, we show that the optimal fixed- +price mechanism can achieve at least 0.72 of the optimal welfare, and no fixed-price mechanism can +achieve more than 0.7381 of the optimal welfare. Prior to our result the state of the art approximation +ratio was 1 − 1 +e + 0.0001 ≈ 0.632 [24]. We further consider two limited information settings. In the +first one, the designer is only given the mean of the buyer’s value (or the mean of the seller’s value). +We show that with such minimal information, one can already design a fixed-price mechanism that +achieves 0.65 of the optimal social welfare, which surpasses the previous state of the art ratio in the +full information setting. In the second limited information setting, we assume that the designer has +access to finitely many samples from the value distributions. Recent results show that one can al- +ready obtain a constant factor approximation to the optimal welfare using a single sample from the +seller’s distribution [3, 16, 24]. Our goal is to understand what approximation ratios are possible if +the designer has more than one but still finitely many samples. This is usually a technically more +challenging regime and requires tools different from the single-sample analysis. We propose a new +family of sample-based fixed-price mechanisms that we refer to as the order statistic mechanisms and +provide a complete characterization of their approximation ratios for any fixed number of samples. +Using the characterization, we provide the optimal approximation ratios obtainable by order statis- +tic mechanism for small sample sizes (no more than 10 samples) and observe that they significantly +outperform the single sample mechanism. +*Yang Cai is supported by a Sloan Foundation Research Fellowship and the National Science Foundation Award CCF-1942583 +(CAREER). Part of this work was done while the author was visiting the Simons Institute for the Theory of Computing. + +1 +Introduction +We study a fundamental problem in mechanism design – maximizing social welfare in bilateral trade, +in which two agents, a seller and a buyer, trade an indivisible item. More specifically, we consider the +Bayesian setting where the seller’s private value S for the item that is drawn from distribution FS, and +the buyer’s private value B for the item is drawn from distribution FB. The social welfare is therefore +defined as EB,S [S +(B −S)· x(B,S)], where x(B,S) denotes the probability that the trade happens when +the seller’s value is S and the buyer’s value is B. +Surprisingly, exactly maximizing the social welfare in bilateral trade is impossible. The seminal re- +sult by Myerson and Satterthwaite [29] shows that no mechanism can simultaneously be (i) incentive +compatible (to the buyer and the seller), (ii) budget balanced, i.e., the mechanism does not run a deficit, +and (iii) maximizes the social welfare. For example, the VCG mechanism is incentive compatible and +maximizes the social welfare but is not budget balanced in general. Motivated by this impossibility re- +sult, our goal is to design incentive compatible and budget balanced mechanisms to approximate the +optimal welfare. We focus on the fixed-price mechanisms, in which the designer offers trade at a fixed +price to the seller and buyer. It is also known that fixed-price mechanisms are the only dominant strategy +incentive compatible and budge balance mechanisms in bilateral trade [23]. +1.1 +Our Contributions +We make progress on this problem on multiple fronts. We first consider the full information setting, +where the designer knows both FS and FB. We show how to use a factor revealing min-max program to +improve the approximation ratio of achievable by a fixed-price mechanism. +Contribution 1: For any FS,FB, there exists a fixed-price mechanism whose welfare is at least +0.72 ·OPT, where OPT = ES,B[max{S,B}] is the optimal welfare. Moreover, there exists a FS and +FB such that no fixed-price mechanism can attain welfare more than 0.7381 ·OPT. The formal +statement of our result can be found in Theorem 3.1. +We also have a “constant time” algorithm for computing the fixed-price mechanism that achieves the +welfare guarantee above. More specifically, we construct a collection of numbers p1,··· ,pn in [0,1], so +that for any FS, FB, our algorithm chooses the best price in the set +� +p1 ·OPT,··· ,pn ·OPT +� +. Clearly, the +approximation ratio will be better when we increase n. We show that when n = 16, our algorithm already +computes a fixed-price mechanism that has welfare at least 0.72·OPT. +Our result significantly improves on the state-of-the-art approximation 1− 1 +e +0.0001 ≈ 0.6322 [24]. +Our new hardness result also strengthens the previous best bound of 0.7385 [25]. Our upper and lower +bounds are obtained by considering two discretized variants of an infinite-dimensional min-max opti- +mization problem defined in Section 3.1. We show in Lemma 3.4 that, in the limit when the discretization +accuracy approaches 0, the upper bound and lower bound obtainable by our method will converge to the +optimal approximation ratio. Of course, the factor-revealing program become more expensive to solve +with finer discretization. Our upper and lower bounds are derived using the finest discretization that we +can computationally solve, but one could further close the gap with more computational resources. +Fixed-price mechanism based on only E[B] or E[S]. +Our first result requires the designer to know both +FS and FB.1 However, information about the underlying distributions of the agents’ values is often scarce +in practice, thus it is more desirable to design approximately optimal mechanisms using limited prior +information. Our second contribution concerns the case where the designer does not have the full infor- +mation of the underlying distributions but only knows the mean of FS or FB. +1Our first result uses FS and FB in two places: (1) to compute OPT and (2) to identify the best price in the set. +1 + +Contribution 2: Given access to E[B] or E[S], we can design a randomized fixed-price mechanism +whose welfare is at least 0.65·OPT. See Theorem 4.1 for details. +Note that the ratio of 0.65 already exceeds the previous state-of-the-art approximation ratio in the +full information setting. [5, 24] consider the setting where only FS is known to the designer and show +that a quantile mechanism (Mechanism 1), i.e., a fixed-price mechanism that chooses the trading price +according to a distribution of quantiles of the seller’s distribution, can obtain at least 1− 1 +e ≈ 0.6321 frac- +tion of the optimal welfare. [24] further shows that no quantile mechanism can obtain more than 1− 1 +e +fraction of the optimal welfare in the worst case. This result is sometimes interpreted as saying no mech- +anism can obtain an approximation ratio better than 1− 1 +e with only information about the seller’s value +distribution. Our second result shows that there is a strictly better way to use the information about +seller’s value distribution. Indeed, with minimal information about FS, i.e., its mean E[S], one can design +a fixed-price mechanism that strictly outperforms the optimal quantile mechanism that requires full +knowledge of FS. Moreover, the quantile mechanism is asymmetric and only defined when we know the +seller’s value distribution. We show in Theorem D.1 that this is unavoidable, as no quantile mechanism +over buyer’s value distribution can guarantee a constant fraction of the optimal welfare.2 In contrast, our +second result holds when the designer only knows the mean of the buyer’s value distribution FB. +Mechanism 1: +Quantile mechanism. +1 Input: A distribution Q over the interval [0,1]; +2 Randomly choose a quantile x ∈ [0,1] according to Q; +3 Output the x-quantile of the seller’s distribution as the price. Let FS be the seller’s distribution +and F −1 +S (·) be seller’s quantile function mapping any quantile to its corresponding value. The +quantile mechanism outputs F −1 +S (x) as the price; +Fixed-price mechanism using finitely many samples. +Finally, we consider a different limited infor- +mation model and initiate the study of approximating the optimal social welfare in using finitely many +samples. Namely, we are given a finite and limited number of samples, e.g., 3 or 5 samples, and the goal +is to design the best mechanism possible using these samples. It is important to distinguish this set- +ting from the more standard large sample setting, where the goal is to determine the number of samples +needed to design a (1−ε)-optimal mechanism (or optimal in a certain mechanism class) grows as a func- +tion of 1 +ε and other parameters of the mechanism design environment. The sample complexity in large +sample settings is usually stated using the big-O notation and ignores the accompanying constant. As a +result, these bounds are often vacuous when apply to the small sample regime, where there are only a +small finite number of samples available. +Contribution 3: We introduce a new family of mechanisms – order statistic mechanisms (Mecha- +nism 2); and provide an exact characterization of the optimal order statistic mechanisms for any +fixed number of samples (Theorem 5.1 and Theorem 5.2). Using our characterization, we can +compute the optimal approximation ratio obtainable for any sample size. +Recent results show that one can already obtain a constant factor approximation to the optimal wel- +fare using a single sample from the seller’s distribution [3, 16, 24]. However, techniques from these pa- +pers are tailored for the single sample setting and are difficult to generalize to even the case when two +samples are available. We provide a rich family of mechanisms that is well-defined for any number of +samples and characterize their performance. Using the characterization, we manage to optimize within +2This asymmetry is due to the asymmetry of the initial allocation – the item is owned by the seller. +2 + +this family of mechanisms for any fixed number of samples. +Mechanism 2: +Order statistic mechanism with N samples. +1 Input: A distribution P over [N]; +2 Randomly choose a number i ∈ [N] according to the distribution P; +3 Given N samples from the seller, select the i-th smallest sample as the price, which is the i-th +order statistic of these samples; +By numerically computing the optimal approximation ratios of order statistic mechanisms, we ob- +serve that the optimal order statistic mechanism with a small number of samples is usually sufficient +to significantly boost the approximation ratio. For example, in the symmetric setting, i.e., FS = FB, five +samples is sufficient to obtain an approximation ratio that is within 1% of the optimal ratio achievable by +any fixed-price mechanism; in the asymmetric setting, i.e., FS ̸= FB, the approximation ratio improves +from 1/2 to 0.578 when the sample size increases from one to three. Another natural mechanism is the +empirical risk minimization (ERM) mechanism, where one selects a price to maximize the social welfare +w.r.t. the empirical distribution. We compare the performance of the optimal order statistic mechanism +with ERM for sample size N = 2,3,5,10 in the symmetric setting. In all cases, the order statistic mecha- +nism substantially outperform the ERM. See Table 1 and 2 for our computed ratios in the symmetric and +asymmetric cases respectively. +Our analysis of the order statistic mechanisms builds on an interesting connection between the or- +der statistic mechanisms and the quantile mechanisms, that is, any order statistic mechanism is also a +quantile mechanism. Note that the i-th order statistic over N samples drawn uniformly and indepen- +dently from [0,1] has density f i +N(x) = N +�N−1 +i−1 +� +· xi−1 ·(1− x)N−i . Suppose we use the i-th order statistic as +the price, then it is equivalent to the quantile mechanism who selects a quantile corresponding to the +density function f i +N(·). More generally, if we choose the i-th order statistic with probability Pi, then the +order statistic mechanism is equivalent to the quantile mechanism that chooses the quantile according +to the density function q(·) = �N +i=1Pi f i +N(·). With this connection, we can focus on quantile mechanisms, +and we characterize the approximation ratio of any quantile mechanism as the solution of a minimiza- +tion problem (Lemma 5.3). By applying this characterization for quantile mechanisms to order statistic +mechanisms, we show that for any fixed sample size N, the ratio of the optimal order statistic mechanism +is exactly the solution of a max-min optimization problem. Despite that the optimization problem seems +intractable in general, we manage to solve it with sufficient numerical accuracy for N ≤ 10. Although we +only study approximating social welfare in bilateral trade in this paper, we believe this perspective of +viewing sample-based mechanisms through the lens of quantile mechanisms is novel and has broader +applications, especially in the small sample regime where the designer only has access to finitely many +samples. +1.2 +Related Work +Gains from Trade Maximization in Two-Sided Markets. +Another important objective in two-sided mar- +kets is the gains from trade (GFT), which measures the increment of the welfare after the trade. Note +that [29] also implies that optimal GFT is not achievable in bilateral trade. There has been increasing +interest from the algorithmic mechanism design community to study the approximability of the optimal +GFT [6, 8, 12, 2, 3, 10, 14]. It will be interesting to study the optimal approximation ratio obtainable for +GFT maximization in both the full information and the limited information settings. +Sample-Based Mechanism Design. +Sample-based mechanism design has become a central topic in +algorithmic mechanism design as it provides an alternative model that weakens the classical but some- +times unrealistic Bayesian assumption. The results in this direction can be roughly partition into two +groups: (1) Large sample results, where the goal is to determine the number of samples needed to design +3 + +a 1 − ε-optimal mechanism (or optimal in a certain mechanism class) as a function of 1 +ε and other pa- +rameters of the mechanism design environment, e.g., [17, 11, 26, 21, 31, 28, 27, 9, 7] or (2) Single sample +results, where the goal is to determine the optimal approximation ratio obtainable using a single sample, +e.g.,[18, 15, 19, 24, 20, 16]. Our result does not fit in to either of the groups. In particular, we study the +regime where the designer has a small fixed number of samples, as a result, the machinery developed for +large number of samples or a single sample does not apply to our setting. A recent line of works focus on +the same regime as ours but for the monopolist pricing problem [4, 13, 1]. Due to the different nature of +the studied problems, their techniques also do not apply here. +2 +Preliminaries +Bilateral Trade. +We study the bilateral trade problem. In this setting, there are two agents, a buyer +and a seller, trade a single indivisible item. The seller owns the item and values it at S while the buyer +values the item at B. Both S and B are non-negative and unknown to us but they are respectively drawn +from distributions FS and FB independently. We assume that FS and FB are continous distributions. +Actually, such assumption is w.l.o.g. and we discuss the reduction from distributions with point masses +to continous ones in Appendix A. +Fixed-price Mechanism. +We consider fixed-price mechanisms, which offer a price p to trade the item. +The trade happens if and only if both the seller and the buyer accept the price, i.e., B ≥ p ≥ S. As shown +by [23], fixed-price mechanism is the only dominant-strategy incentive-compatibility mechanism. In +this paper, we consider (possibly randomized) fixed-price mechanisms. We abuse notation and use +M(FS,FB) or M(I) where I = (FS,FB) to denote the distribution of prices p selected by mechanism +M on instance I = (FS,FB). +Welfare and Approximation Ratio. +We consider the objective of social welfare in this paper. For an +instance I = (FS,FB), the optimal welfare is defined as: +OPT-W (I) = +E +S∼FS,B∼FB +[max(S,B)] +Similarly, for a fixed-price mechanism M, the expected welfare on instance I can be written as: +W (M,I) = +E +S∼FS ,B∼FB +p∼M(FS ,FB ) +� +S + +1[S ≤ p ≤ B]·(B −S) +� +Our goal is to maximize the approximiation ratio. That is, find some mechanism M maximize the +following ratio. +min +I =(FS,FB) +W (M,I) +OPT-W (I) +Quantile Function. +Suppose F(·) is the c.d.f. of a distribution, and we define F −1(·) as the quantile func- +tion mapping the quantile to its corresponding value in this distribution. That is, F −1(x) = inf{y | F(y) = x}. +3 +A Near-Optimal Mechanism in the Full Information Setting +In this section, we show a near-optimal fixed-price mechanism when given the full information of the +buyer and the seller. +4 + +Theorem 3.1. There exists a DSIC, individually rational, budget balanced mechanism that achieves at +least 0.72 fraction of the optimal welfare for any instance I = (FS,FB). Moreover, no such mechanism has +an approximation ratio better than 0.7381. +To prove this, we first identify the best fixed-price mechanism when given the instance I = (FS,FB). +Then, the approximation ratio is determined by the mechanism’s performance on the worst-case in- +stance. Such a worst-case instance could be characterized by an infinite dimensional quadratically con- +strained quadratic program (QCQP). However, the infinite dimensional program is hard to solve directly. +Instead, we use two finite programs that can be solved numerically to upper bound and lower bound the +infinite dimensional program. Additionally, we show that the optimal solutions of these two programs +converge to the optimal solution of the infinite dimensional program as the number of variables tends +to infinity. +3.1 +Characterizing the Optimal Mechanism +We first characterize the optimal fixed-price mechanism via an infinite dimensional QCQP. Given any +instance I = (FS,FB), we could assume that OPT-W (I) = 1 without loss of generality since we can al- +ways scale the instance so that this is true. The optimal fixed-price mechanism corresponds to choosing +a price p ∈ argmaxp W (I,p). The following program captures the worst-case instance for fixed-price +mechanisms. +The Optimization Problem FullOp +min +µ,ν,r +r +s.t. +µ,ν are probability measures defined on R≥0 +(1) +OPT-W (I) +def= +� +R≥0 +� +R≥0 +max(x, y)ν(d y)µ(dx) ≥ 1 +W (I,t) +def += +� +R≥0 +x µ(dx)+ +� +R≥0 +� +R≥0 +(y − x)· +1[x ≤ t ≤ y]ν(d y)µ(dx) ≤ r +∀t ≥ 0 +Lemma 3.1. The value of the optimal solution of FullOp is the tight worst-case approximation ratio +achievable by a fixed-price mechanism. +The proof of Lemma 3.1 is postponed to Appendix B.1. Since it is difficult to directly solve an infinite +dimensional program like FullOp, we approximate FullOp from both above and below by constructing +two families of finite programs which provide an upper bound and a lower bound respectively. +3.2 +Factor Revealing Program for the Approximation Ratio under Full Information +We show that the approximation ratio of the optimal fixed price mechanism is at least 0.72, which signif- +icantly improves the previous state of the art bound of 1−1/e +ε with ε ≈ 10−4. Our approach is to find +a fixed-price mechanism whose performance under the worst distribution is maximized. This is exactly +captured by the optimization problem FullOp. However, it is an infinite-dimensional program. In this +section, we consider a discretized version of FullOp. More specifically, we assume that OPT-W (I) = 1, +and we restrict the mechanism to only choose price from a finite set P = {p1,p2,...,pk}. What we manage +to show is that the optimal value of the optimization problem LowerOp is indeed a lower bound on the +maximum approximation ratio one can obtain using prices from P for instance I. We establish the fol- +lowing two crucial properties: (i) For any I = (FS,FB) satisfying OPT-W (I) = 1, we can carefully round +5 + +FS and FB to two discrete distributions supported on P, where {s1,...,sn} and {b1,...,bn} can be viewed +as the corresponding “probability mass function” for the discretized distributions of the seller and the +buyer.3 Importantly, {s1,...,sn} and {b1,...,bn} satisfy inequalities (2) - (5). (ii) For any price pt, the wel- +fare from the corresponding fixed-price mechanism under I is at least the welfare under the rounded +distributions �n +i=1 si pi + �t−1 +i=1 +�n +j=t+1 sib j(p j − pi). Therefore, if we choose r to be maxt∈[n] W (I,pt), +{s1,...,sn}, {b1,...,bn}, and r form a feasible solution of LowerOp, which implies that the optimal value +of LowerOp is no greater than the constructed r. As the rounded distribution needs to satisfy a sequence +of constraints (especially constraint (5)), the procedure we use to round FS and FB is subtle and does not +simply round things up or down. See Appendix B.2 for details. +The Optimization Problem LowerOp +min +s1,s2···,sn +b1,b2,···,bn,r +r +s.t. +si,bi ≥ 0 +∀i ∈ [n] +(2) +n� +i=1 +si ≥ 1 +and +n� +i=1 +bi ≥ 1 +(3) +n� +i=1 +si ≤ 1+ 1 +pn +and +n� +i=1 +bi ≤ 1+ 1 +pn +(4) +n� +i=1 +n� +j=1 +sib j max(pi,p j) ≥ 1 +(5) +n� +i=1 +si pi + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi) ≤ r +∀t ∈ [n] +(6) +Lemma 3.2. For any 0 = p1 < p2 < ··· < pn, let r ∗ be the optimal value of LowerOp. Suppose M is the +mechanism that chooses the best price from the set +� +p1·E[max(S,B)],p2·E[max(S,B)],··· ,pn·E[max(S,B)] +� +to maximize the welfare. The welfare obtained by M is at least r ∗ ·OPT. +We defer the proof of the lemma to Appendix B.2. +3.3 +Hardness Result under Full Information +In this section, our goal is to find a threshold and an instance such that no fixed-price mechanism has an +approximation ratio better than the threshold on this instance. We focus on discrete distributions and +consider an instance I = (FS,FB) where FS is a discrete distribution supported on {p1+ε,p2+ε,··· ,pn + +ε}, and FB is a discrete distributions supported on P = {p1,p2,··· ,pn}. For such instance, the optimal +price must also lie in the set {pi +ε}i∈[n], as choosing a price x where pi +ε ≤ x < pi+1+ε is equivalent to +choosing a price of pi +ε . Therefore, any valid solution for the optimization problem below corresponds +to a hard instance. +Lemma 3.3. For any valid solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r) of UpperOp (defined in Appendix B.3) +satisfying r = maxt∈[n] +�n +i=1 si pi +�t +i=1 +�n +j=t+1 sib j(p j −pi) and ε > 0, there exists an instance I = (FS,FB) +such that no fixed-price mechanism can achieve more than (r +ε)-fraction of the optimal welfare on this +instance. +3For technical reasons, {s1,...,sn} and {b1,...,bn} do not exactly correspond to probability mass functions, but viewing them +as the probability mass functions gives the right intuition. +6 + +The proof of Lemma 3.3 is deferred to Appendix B.3. +Proof of Theorem 3.1. With Lemma 3.2 and Lemma 3.3, we are now ready to prove Theorem 3.1. For the +numerical results, our anonymous GitHub repository(https://github.com/BilateralTradeAnonymou +s/On-the-Optimal-Fixed-Price-Mechanism-in-Bilateral-Trade) provides all the certificates and +codes and also carefully explains all the details. +For the lower bound, we choose n = 16. Using Gurobi [22], we obtain a lower bound of 0.72 for +the optimization problem LowerOp for a carefully chosen set of price {p1,p2,··· ,pn}.4 Therefore, by +Lemma 3.2, there exists a 0.72-approximate fixed-price mechanism. +Things become much easier for the upper bound since we only need to find a feasible solution in- +stead of proving a lower bound of the optimal value. We choose n = 100 and numerically solve UpperOp +with a specific support {p1,p2,··· ,pn} and find a feasible solution that satisfies the constraints in Lemma 3.3 +where r ≤ 0.7381. Together with Lemma 3.3, we then find a hard instance such that no fixed-price mech- +anism attains a 0.7381-approximation of the optimal welfare. Please check our GitHub repository for the +detailed specification of the distributions. +Finally, we would like to point out that the optimal value obtained by LowerOp and UpperOp will +converge to the optimal value as the discretization accuracy tends to 0. +Lemma 3.4. Let r ∗ be the optimal value of FullOp, i.e. the optimal approximation ratio. For any ε > 0, +there exists two sets numbers 0 = p1 < p2 < ··· < pn and {p′ +1,p′ +2,··· ,p′ +n′} such that the optimal value of +LowerOp with respect to {p1,p2,··· ,pn} is at least r ∗−ε and the optimal value of UpperOp w.r.t. {p′ +1,p′ +2,··· ,p′ +n′} +is at most r ∗ +ε. +The proof of Lemma 3.4 is deferred to Appendix B.4 +4 +Breaking 1−1/e with Limited Information +We consider the limited information setting where we only knows the mean of the seller or the buyer. [24] +shows that any mechanism that only uses quantile information from the seller can not achieve a better +performance of 1−1/e. However, we observe that even with minimal information of FS such as its mean +E[S] (or similarly E[B]), we can break the 1−1/e barrier. We again provide a factor revealing program for +this setting. Although it looks similar to LowerOp, there is some subtle differences in how we discretize a +continuous distribution. See Appendix C for details. +Theorem 4.1. Consider the following fixed-price mechanism: Given E[B] (or E[S]), it randomly pick a +number x ∼ P according to a distribution P, and selects x · E[B] (or x · E[S]) as the price. There exists a +distribution PS for the seller and a distribution PB for the buyer such that the corresponding fixed-price +mechanism achieves at least 0.65·OPT welfare. +The high level idea is as follows. Lemma B.1 shows us how to discretize a continuous distribution +so that E[max(S,B)] increases and E[W (I,pt)] decreases. In other words, the discretization worsens the +instance. Therefore, we could use a similar technique to derive a lower bound of the approximation ratio. +The complete proof of Theorem 4.1 is in Appendix C +4We choose {p1,p2,··· ,p16} to be {0.0,0.1,0.19,0.27,0.315,0.355,0.395,0.44,0.485,0.535,0.595,0.665,0.74,0.875,1.195,1000.0} +to derive the 0.72. These numbers are chosen heuristically to provide good coverage between 0.3 to 0.5, which is the region +with concentration of probability mass in some bad instances we encounter. +7 + +5 +Fixed-Price Mechanism with Different Numbers of Samples +In this section, we consider the limited information setting where we only have sample access to the +distributions. We focus on order statistic mechanisms which is defined in Section 1.1 and our results +cover different number of samples for both symmetric and general instances. In the small sample regime, +we are able to characterize the optimal order statistic mechanism with any fixed number of samples. +When the number of samples goes to infinity, we show that the optimal quantile mechanism can be +approximated by order statistics mechanism as closely as desired and also obtain an upper bound on +the sample complexity. Finally, recall that we assume the distributions for the seller and the buyer are +continuous. See Appendix A for details. +5.1 +Order Statistic Mechanisms +To start with, we briefly discuss these two families of mechanisms that is used in the sample setting and +give high level ideas on how to design the order statistic mechanisms. Order statistic mechanisms will +be used when we only have samples from the distribution and quantile mechanisms will help us analyze +the performance of order statistic mechanisms. Actually, we will point out that quantile mechanisms +and order statistic mechanisms are equivalent in some sense. +5.1.1 +Connection Between Two Mechanisms +Next we aim to show the connection between these two mechanisms. Such observations give us insights +on designing mechanisms with small or large number of samples. +The order statistic mechanism is a special kind of quantile mechanisms +First, we can see that the +following two operations are equivalent: +• Draw a sample from distribution F. +• Uniformly sample a quantile x from [0,1], and use F −1(x) as the sample. +Now suppose f i +N(x) = N +�N−1 +i−1 +� +· xi−1 ·(1− x)N−i be the p.d.f. of the i-th order statistic over N samples +drawn uniformly and independently from [0,1] and let Pi to be Prx∼P[x = i] for any distribution P over +[N]. Using similar ideas above, it can be proved that any order statistic mechanism P is equivalent to a +quantile mechanism Q with probability density function +q(x) = +N� +i=1 +Pi f i +N(x) +Therefore, we can analyze the approximation ratio of quantile mechanism Q instead of order statistic +mechanism P. If we are able to compute the approximation ratio of any quantile mechanism Q, it fol- +lows that we can also characterize the optimal order statistic mechanism exactly. When the number of +samples are small, we can have a fine-grained analysis of the order statistic mechanisms and use these +limited samples carefully. Section 5.2 actually follow such intuitions to characterize the best possible +order statistic mechanism. +Quantile mechanisms can be approximated by order statistic mechanisms within any small error +Our goal is that for any quantile mechanism Q with p.d.f. q(x), we need to find some integer N and +a distribution P over [N], such that +q(x) ≈ +N� +i=1 +Pi f i +N(x) +8 + +Since �N +i=1Pi f i +N(x) is a polynomial of degree N − 1, this could be done for any continuous q(x) on +[0,1] since the Weierstrass approximation theorem states that every continuous function defined on a +closed interval can be uniformly approximated as closely as desired by a polynomial function. What’s +more interesting is that {f i +N (x)}N +i=1 are Bernstein basis polynomials and there are a series of work show- +ing that (stochastic) Bernstein polynomials can efficiently and uniformly approximate to any continous +function. Therefore, we can have an asymptotic analysis of the order statistic mechanism. What’s more, +such observation also shows that we have a block-box transformation from any quantile mechanism to +mechanisms only using samples. Section 5.3 uses such techniques and ideas. +5.2 +Small Sample Regime +In this section, we characterize the optimal order statistic mechanisms with any fixed number of sam- +ples for both symmetric and general instances. We first show that, in any setting, if we are able to give +a tight analysis of the quantile mechanism, we could directly characterize the optimal order statistic +mechanism with any fixed number of samples via an optimization problem. In the next, we show a tight +analysis of the quantile mechanism on both symmetric and general instances, and thus we obtain the +characterization of the optimal order statistic mechanism. +Recall that an order statistics mechanism with N samples randomly choose a number i ∈ [N] ac- +cording to a previously defined distribution P and select the i-th smallest sample as the price, and a +quantile mechanism randomly choose a quantile x ∈ [0,1] from a determined distribution Q and choose +the x-quantile, i.e. F −1 +S (x), as the price. Since every quantile mechanism and order statistic mechanism +is determined by the previously defined distribution, we abuse the notation and use distribution P over +[N] denote its corresponding order statistic mechanism and distribution Q over [0,1] denote its corre- +sponding quantile mechanism. +Lemma 5.1. Suppose C : ∆([0,1]) �→ R maps every quantile mechanism P to its exact approximation ratio. +Let P (Q) be the corresponding quantile mechanism of the order statisticmechanismQ. Fixing the number +of samples N, the optimal order statistic mechanism with N samples Q∗ +N is characterized by the following +optimization problem: +Q∗ +N = arg max +Q∈∆N +C (P (Q)) +where ∆([0,1]) is the set of all distributions over [0,1], i.e. the set of all quantile mechanisms, and ∆N is the +set of all distributions over [N], i.e. the set of all order statistic mechanisms with N samples. +The proof of Lemma 5.1 is quite straightforward and thus is postponed to Appendix D.2. +5.2.1 +Symmetric Instances +Now we study the case when the distributions are symmetric, i.e., FS = FB, which means that the seller’s +value S and the buyer’s value B are drawn from the same distribution. For simplification, we will use F +to refer to their distributions in this setting. +In order to find out the optimal order statistic mechanism, we need to first give a tight analysis of the +quantile mechanism. +Lemma 5.2. For any quantile mechanism for symmetric instance with distribution Q over [0,1], the ap- +proximation ratio is exactly +inf +x∈[0,1) +� +[0,x] t(1− x)dQ(t)+ +� +(x,1](1− t)x dQ(t)+(1− x) +1− x2 +where Q(t) is the cumulative distribution function of distributionQ. +9 + +Therefore, combining Lemma 5.1 and Lemma 5.2, we could characterize the optimal order statistic +mechanism via an optimization problem. +Theorem 5.1. The optimal order statistic mechanism with N samples for symmetric instances is the solu- +tion to the following optimization problem: +P∗ +N = +max +P1,P2,···,PN ≥0 +�N +i=1 Pi =1 +inf +x∈[0,1) +�x +0 p(t)t(1− x)dt + +�1 +x p(t)(1− t)x dt +(1− x) +1− x2 +where p(t) = �N +i=1Pi f i +N(x) and f i +N(x) = N +�N−1 +i−1 +� +· xi−1 ·(1− x)N−i is the p.d.f. of the i-th order statistic over +N samples drawn uniformly and independently from [0,1]. +The proof of Lemma 5.2 and Theorem 5.1 is in Appendix D.3 and D.4. +It turns out the optimization above is computationally tractable when N is not too large. We solve +the optimization problem and find out the optimal order statistic mechanism numerally with different +numbers of samples N. +To compare with the order statistic mechanisms, we also consider the most natural sample-based +mechanism – the Empirical Risk Minimization mechanism (ERM). We first provide the formal definition +below. +Definition 5.1 (Empirical Risk Minimization Mechanism). Given N samples X1, X2,..., XN drawn from +F, define ˜F be the empirical distribution of these N samples. That is to say, ˜F is the distribution with c.d.f. +˜F(x) satisfying: +˜F(x) = 1 +N +N +� +i=1 +1[x ≥ Xi] +The Empirical Risk Minimization mechanism (ERM) is the mechanism that computes the optimal +price according the empirical distribution ˜F. In particular, for N samples X1, X2,..., XN, +ERM(X1, X2,..., XN) = argmax +p +E +S∼ ˜F,B∼ ˜F +[S +(B −S)· +1[B ≥ p ≥ S]] +If there are multiple prices p that maximize the expected welfare, the ERM mechanism may select any +of them. +For N = 1,2,3,5,10, we compute the approximation ratio of order statistic mechanisms and also show +the upper bound of ERM. The results are listed below. To prove the upper bound, we use a counter +example in [24] and show that ERM has a bad performance on this instance. We defer the complete +proof of the upper bound of ERM to Appendix D.5 and the details of numerical results to Appendix E.1. +#Samples +Order Statistics Mechanism +ERM +1 +0.75 +0.5 +2 +0.821 +≤ 0.67 +3 +0.822 +≤ 0.75 +5 +0.847 +≤ 0.76 +10 +0.852 +≤ 0.80 +∞ +2+ +� +2 +4 +≈ 0.8536 +/ +Table 1: Approximation Ratios with Different Number of Samples +in the Symmetric Setting. +10 + +5.2.2 +General Instances +Now we consider the general setting, where the buyer’s distribution may be different from the seller’s. +Recall that we only consider mechanisms over seller’s information since there is no constant quantile or +order statistic mechanism over seller’s information. Using similar ideas, we first show a tight analysis re- +garding quantile mechanisms, which would guide us to discover the optimal order statistic mechanism. +Lemma 5.3 (Theorem 4.1 of [5]). For any quantile mechanism Q (over seller’s distribution) with cumula- +tive distribution function Q, its approximation ratio is exactly +min +x∈[0,1] +� +[0,x] +t dQ(t)+1− x +Similarly, combining Lemma 5.1 and Lemma 5.3, we are able to charaterize the optimal order statistic +mechanism over with N samples from seller’s distribution by an optimization problem: +Theorem 5.2. The optimal order statistic mechanism with N samples for symmetric instances is the solu- +tion to the following optimization problem: +P∗ +N = +max +P1,P2,···,PN ≥0 +�N +i=1 Pi =1 +min +x∈[0,1] +� +[0,x] +t · p(t)dt +1− x, +where p(x) = �N +i=1Pi · f i +N (x) and f i +N (x) = N +�N−1 +i−1 +� +·xi−1·(1−x)N−i is the p.d.f. of the i-th order statistic over +N samples drawn uniformly and independently from [0,1]. +The proof of Lemma 5.3 and Theorem 5.2 is deferred to Appendix D.6 and D.7. +Similarly, such optimization problem is easy to solve when the number of samples N is not to large. +We solve the optimization problem numerically for N = 1,2,3,5,10. Note that we do not compare our +mechanism to the Empirical Risk Minimization mechanism in the general setting. This is because we +only have sample access to the seller’s distribution, and the ERM can not be implemented without the +buyer’s samples. The details of numerical results is defered to Appendix E.2. +#Samples +Order Statistic Mechanism +1 +0.5 +2 +0.531 +3 +0.578 +5 +0.601 +10 +0.615 +∞ +1− 1 +e ≈ 0.632 +Table 2: Approximation Ratios with Different Number of Samples +In the General Setting. +5.3 +Asymptotic Analysis: From Quantile to Order Statistics +In this section, we turn to the case when the number of samples tends to infinity. As we show in sec- +tion 5.1.1, we could approximate any quantile mechanisms by order statistic mechanisms within any +small error. Using such ideas, we provide a "black-box" reduction that allows us to convert any quantile +mechanism with continuous probability density function q(x) to order statistic mechanism with N sam- +ples. Here N is usually a polynomial of 1 +ε, as long as the probability density function is not too crazy. We +now formally write it down. +11 + +Lemma 5.4. Let C : ∆([0,1]) �→ R be a function that maps every quantile mechanism Q with continuous +probability density function to its approximation ratio such that for any quantile mechanism Q1 with +p.d.f. q1(x) and quantile mechanism Q2 with p.d.f. q2(x), it holds that +C (Q1)−C (Q2) ≥ −c · +��q1 − q2 +�� +∞ +where c is a constant. +Now let Q be any quantile mechanism with continuous probability density function q(x). Define M as +maxx∈[0,1] q(x). For any ε > 0, suppose n is a positive integer satisfying that +ω +� +1 +� +n −1 +� +≤ ε/100 +(7) +2n exp + +− +ε2 +8ω2 +� +1 +� +n−1 +� + + ≤ ε +(8) +exp +� +− ε2n +48M3 +� +≤ ε/2 +(9) +where w(h) = sup x,y∈[0,1] +|x−y|≤h +��q(x)− q(y) +��. Then, there exists an order statistic mechanism with n samples that +achieves an approximation ratio of C (Q)−c ·ε. +The high level idea of the proof is as follows. Since we know that probability density functions of order +statistics form Bernstein basis polynomials, we could approximate the p.d.f. of the quantile mechanism +q(x) within any small error. Inequality (7), (8) and (9) actually help us to get an order statistic mechanism +whose corresponding distribution of quantile is close to the desired quantile mechanism Q. Finally, +by the property of C , we know that their approximation ratio is also close. The proof is postponed to +Appendix D.8. +Finally, we show that we could apply lemma 5.4 to both the symmetric and general settings and +convert the optimal quantile mechanism to order statistic mechanism within a error of at most ε using +poly +� 1 +ε +� +samples. We leave the details of such applications to Appendix D.9. +References +[1] Amine Allouah, Achraf Bahamou, and Omar Besbes. 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In +Advances in Neural Information Processing Systems, pages 5352–5359, 2017. +A +Tie Breaking +For distribution D with point masses, the following reduction will convert it to continuous one. We will +overload the notation of D and think of it as a bivariate distribution with the first coordinate drawn from +the previous single-variate distribution D and the second tie-breaker coordinate drawn independently +and uniformly from [0,1]. And (X1,t1) > (X2,t2) if and only if either X1 > X2, or X1 = X2 and t1 > t2. Since +14 + +the tie-breaker coordinate is continuous, the probability of having (X1,t1) = (X2,t2) for any two values +during a run of any mechanism is zero. Therefore we could define the c.d.f. of D as +FD(X ,t) = +Pr +(Y,u)∼(D,U[0,1])[(Y ,u) < (X ,t)] +Remind the second coordinate is only used to break ties, and it does not affect the calculation of +welfare. After including the additional random variable, we can see that D has been converted into a +continuous distribution since its second coordinate is continuous. +B +Missing Proofs in Section 3 +B.1 +Proof of Lemma 3.1 +We first show the proof of Lemma 3.1. +The approximation ratio of the optimal fixed-price mechanism could be written as +min +I =(FS,FB)max +p∈R +W (I,p) +OPT-W (I). +We first show that for any instance I = (FS,FB), there is a valid solution (u,v,r) such that r = maxp∈R +W (I ,p) +OPT-W (I ). We could first simply scale the instance by +1 +OPT-W (I ) to I ′ = (F ′ +S,F ′ +B) where OPT-W (I ′) = 1. +Such scaling means that W (I,OPT-W (I)· p) = OPT-W (I)·W (I ′,p) for all p ∈ R≥0. This implies that +max +p∈R +W (I,p) +OPT-W (I) = max +p∈R W (I ′,p). +Therefore, let u and v be the probability measures of F ′ +S and F ′ +B and r be maxp∈R W (I ′,p). It is +easy to verify that (u,v,r) is a valid solution. Let r ∗ be the optimal value of FullOp, this implies that +r ∗ ≤ maxp∈R +W (I ,p) +OPT-W (I ) for any instance I = (FS,FB). Taking the minimum over all possible I, we then +get that +r ∗ ≤ +min +I =(FS,FB)max +p∈R +W (I,p) +OPT-W (I) +(10) +Next, let (u∗,v∗,r ∗) be the optimal solution of FullOp. Since u∗,v∗ are both probability measures, +let F ∗ +S ,F ∗ +B be the corresponding distributions of u and v and I ∗ = (F ∗ +S ,F ∗ +B) be the instance. Now by the +constraint of FullOp, we know that OPT-W (I ∗) ≥ 1. Besides, (u∗,v∗,r ∗) is an optimal solution implies +that r ∗ = maxp∈R W (I ∗,p). Therefore, +r ∗ = max +p∈R W (I ∗,p) ≥ max +p∈R +W (I ∗,p) +OPT-W (I ∗) ≥ +min +I =(FS,FB)max +p∈R +W (I,p) +OPT-W (I) +(11) +Now combining inequality (10) and (11), it follows that +r ∗ = +min +I =(FS,FB)max +p∈R +W (I,p) +OPT-W (I) +which completes the proof. +15 + +B.2 +Proof of Lemma 3.2 +Before we give the proof of Lemma 3.2, we first show prove a lemma that helps us discretize a continuous +distribution. +Lemma B.1. For any instance I = (FS,FB), and 0 = p1 < p2 < ··· < pn, there exists a set of numbers +{si}i∈[n],{bi}i∈[n] satisfying the following equations. +si,bi ≥ 0 +∀i ∈ [n] +(12) +1 ≤ +n� +i=1 +si ≤ 1+ E[S] +pn +1 ≤ +n� +i=1 +bi ≤ 1+ E[B] +pn +(13) +n−1 +� +i=1 +si ≤ 1 +n−1 +� +i=1 +bi ≤ 1 +(14) +E +S∼FS +[S] = +n� +i=1 +sipi +(15) +E +B∼FB +[B] = +n� +j=1 +b j p j +(16) +OPT-W (I) +def= +E +S∼FS +B∼FB +[max(S,B)] ≤ +n� +i=1 +n� +j=1 +sib j ·max(pi,p j) +(17) +W (I,pt) +def= +E +S∼FS +[S]+ E +S∼FS +B∼FB +� +(B −S)· +1[S ≤ pt ≤ B] +� +≥ +n� +i=1 +sipi + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi) +∀t ∈ [n] +(18) +Proof. We construct (s1,··· ,sn,b1,··· ,bn) as follows. For the seller, define +qs,i = Pr +S∼FS +[pi ≤ S < pi+1] and Es,i = +E +S∼FS +[S · +1[pi ≤ S < pi+1]], ∀i ∈ [n], +where we assume that pn+1 = +∞. It is clear from the definition that qs,i · pi ≤ Es,i ≤ qs,i · pi+1. +Therefore, for any i ∈ [n −1], there exists non-negative numbers si,LEFT and si+1,RIGHT such that +si,LEFT + si+1,RIGHT = qs,i and si,LEFT · pi + si+1,RIGHT · pi+1 = Es,i. +(19) +We further define sn,LEFT as Es,n/pn and s1,RIGHT = 0. Now set si = si,LEFT + si,RIGHT for all i ∈ [n]. For +the buyer, we define {bi}i∈[n] similarly. +We now verify that ({si}i∈[n],{bi}i∈[n]) satisfies the properties above. The non-negativity of si and bi is +immediately derived from si,LEFT,si,RIGHT ≥ 0. From our definition, it is clear that �n +i=1 qs,i = 1, therefore +n� +i=1 +si = +n� +i=1 +si,LEFT + si,RIGHT ≥ +n� +i=1 +qs,i = 1, +and +n−1 +� +i=1 +si = +n−1 +� +i=1 +si,LEFT + si,RIGHT ≤ +n� +i=1 +qs,i = 1. +We could also see that +n� +i=1 +si = +n� +i=1 +si,LEFT + si,RIGHT ≤ +n� +i=1 +qs,i + sn,LEFT = 1+ Es,n +pn +≤ 1+ E[S] +pn +. +16 + +For the expectations, it holds that +n� +i=1 +si pi = +n� +i=1 +� +si,LEFT + si,RIGHT +� +· pi = +n� +i=1 +Es,i = E[S] +By symmetry, similar inequalities also holds for for {bi}i∈[n]. So far, we have verified that properties +(12), (13) , (14), (15) and (16) are satisfied. It only remains to show that (17) and(18) holds. +For any i ̸= j ∈ [n], w.l.o.g. we can assume that i < j. We could see that +E +S∼FS +B∼FB +� +max(S,B)· +1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] +� += +E +S∼FS +B∼FB +� +B · +1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] +� += +E +B∼FB +� +B · +1[B ∈ [p j,p j+1)] +� +· Pr +S∼FS +� +S ∈ [pi,pi+1) +� += Eb,j · qs,i += (b j,LEFT · p j +b j+1,RIGHT · p j+1)·(si,LEFT + si+1,RIGHT) += si,LEFT ·b j,LEFTmax(pi,p j)+ si+1,RIGHT·b j,LEFTmax(pi+1,p j) ++ si,LEFT ·b j+1,RIGHTmax(pi,p j+1)+ si+1,RIGHT·b j+1,RIGHTmax(pi+1,p j+1) +(20) +The second equality is due to the independence between S and B. +Now consider the case when i = j ≤ n −1. For any x, y ∈ [pi,pi+1], we have +max(x, y) ≤ max(pi, y)· pi+1 − x +pi+1 − pi ++max(pi+1, y)· +x − pi +pi+1 − pi +, +as pi+1−x +pi+1−pi + +x−pi +pi+1−pi = 1 and pi+1−x +pi+1−pi · pi + +x−pi +pi+1−pi · pi+1 = x. +Based on the inequality above, for any fixed y ∈ [pi,pi+1), we have +E +S∼FS +� +max(S, y)· +1[S ∈ [pi,pi+1)] +� +≤ E +S∼FS +�� +max(pi, y)· pi+1 −S +pi+1 − pi ++max(pi+1, y)· +S − pi +pi+1 − pi +� +1[S ∈ [pi,pi+1)] +� +=max(pi, y) E +S∼FS +� pi+1 −S +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� ++max(pi+1, y) E +S∼FS +� S − pi +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� +=y · si,LEFT + pi+1 · si+1,RIGHT +The last equality is because of the following identities: +E +S∼FS +� pi+1 −S +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� ++ E +S∼FS +� S − pi +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� += Pr[S ∈ [pi,pi+1)]] +pi · E +S∼FS +� pi+1 −S +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� ++ pi+1 · E +S∼FS +� S − pi +pi+1 − pi +· +1[S ∈ [pi,pi+1)] +� += E[S · +1[S ∈ [pi,pi+1)]]. +Hence, we can conclude that +� +ES∼FS +� pi+1−S +pi+1−pi · +1[S ∈ [pi,pi+1)] +� +,ES∼FS +� S−pi +pi+1−pi · +1[S ∈ [pi,pi+1)] +�� +is the +unique solution to (19). Thus, these two numbers respectively equal to si,LEFT and si+1,RIGHT. +17 + +Due to the inequality above, we have +E +B∼FB +� +E +S∼FS +� +max(S,B)· +1[S ∈ [pi,pi+1)] +� +· +1[B ∈ [pi,pi+1)] +� +≤ +E +B∼FB +�� +B · si,LEFT + pi+1 · si+1,RIGHT +� +· +1[B ∈ [pi,pi+1)] +� +=bi,LEFTsi,LEFT · pi +bi+1,RIGHTsi,LEFT · pi+1 +bi,LEFTsi+1,RIGHT· pi+1 +bi+1,RIGHTsi+1,RIGHT· pi+1 +(21) +The last special case is when i = j = n. +E +S∼FS +B∼FB +� +max(S,B)· +1[S ≥ pn] 1[B ≥ pn] +� +≤ E +S∼FS +B∼FB +� +BS/pn · +1[S ≥ pn] 1[B ≥ pn] +� += pn · +� +E +S∼FS +� +S · +1[S ≥ pn] +� +/pn +� +· +� +E +S∼FB +� +B · +1[B ≥ pn] +� +/pn +� += sn,LEFT ·bn,LEFT · pn. +(22) +Combining inequality (20), (21) and (22), we have +n� +i=1 +n� +j=1 +sib j ·max(pi,p j) ≥ +n� +i=1 +n� +j=1 +E +S∼FS +B∼FB +� +max(S,B)1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] +� += +E +S∼FS +B∼FB +[max(S,B)], +so inequality (17) is satisfied. +Finally, we are only left to show that property (18) holds. For any t ∈ [n], it follows that +W (I,pt) = +E +S∼FS +[S]+ +E +S∼FS +B∼FB +� +(B −S)· +1[S ≤ pt ≤ B] +� +≥ +E +S∼FS +[S]+ +E +S∼FS +B∼FB +� +(B −S)· +1[S < pt ≤ B] +� += +n� +i=1 +Es,i + +E +B∼FB +[B · +1[B ≥ pt]]· Pr +S∼FS +[S < pt]− E +S∼FS +[S · +1[S < pt]]· Pr +B∼FB +[B ≥ pt] +≥ +n� +i=1 +si · pi + +� +n� +j=t+1 +bj · p j +bt,LEFT · pt +� +· +� +t−1 +� +i=1 +si + st,RIGHT +� +− +� +t−1 +� +i=1 +si · pi + st,RIGHT · pt +� +· +� +n� +j=t+1 +bj +bt,LEFT +� += +n� +i=1 +si · pi + +t−1 +� +i=1 +n� +j=t+1 +si bj (p j − pi)+ +n� +j=t+1 +bj · st,RIGHT ·(p j − pt)+ +t−1 +� +i=1 +si ·bt,LEFT ·(pt − pi) +≥ +n� +i=1 +si · pi + +t−1 +� +i=1 +n� +j=t+1 +si bj (p j − pi). +(23) +where the second inequality follows from the fact that +Pr[B ≥ pt] = +n−1 +� +j=t +(b j,LEFT +b j+1,RIGHT)+Pr[B ≥ pn]≤ +n−1 +� +j=t +(b j,LEFT +b j+1,RIGHT)+bn,LEFT ≤ +n� +j=t+1 +b j +bt,LEFT +Therefore, we could see that inequality (18) holds. This finishes our proof. +18 + +With the lemma above, we are ready to give the proof of Lemma 3.2. +Consider the following fixed-price mechanism: Given any instance I = (FS,FB), we first compute the +optimal welfare of the instance. Suppose OPT-W (I) = c, we choose the fixed price p∗ from {cp1,··· ,cpn} +to maximizes the welfare, i.e., p∗ ∈ argmaxp∈{cp1,···,cpn}W (I,p). In the following, we show that this mech- +anism is an r ∗-approximation to the optimal welfare. +Note that the approximation ratio of our mechanism is independent of c.5 To keep our analysis +clean, we first assume that the instance I = (FS,FB) has optimal welfare 1. The approximation ratio of +our mechanism could be written as +min +I =(FS,FB ) +OPT-W (I )=1 +max +p∈{p1,p2,···,pn}W (I,p). +Next, we argue that for any instance I = (FS,FB) satisfying OPT-W (I) = 1, there exists a valid solu- +tion (s1,··· ,sn,b1,··· ,bn,r) of LowerOp such that r ≤ maxp∈{p1,p2,···,pn} W (I,p). This immediately implies +that r ∗ is a lower bound of the approximation ratio. +Given an instance I = (FS,FB) s.t. OPT-W (I) = 1, the solution (s1,··· ,sn,b1,··· ,bn,r) is constructed +as follows. Let (s1,s2,··· ,sn,b1,b2,··· ,bn) be the set of numbers that satisfies all the properties stated in +Lemma B.1. Let r be +max +t∈[n] +n� +i=1 +si pi + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi). +We first verify that ({si}i∈[n],{bi}i∈[n],r) is a valid solution of LowerOp. Notice that E[S] ≤ E[max(S,B)] = +1 and E[B] ≤ E[max(B,S)] = 1. Therefore, constraints (2), (3) and (4) directly follows from inequality (12) +and (13). What’s more, we could see (6) holds by the definition of r. +Now by property (17), we have +n� +i=1 +n� +j=1 +sib j ·max(pi,p j) ≥ +n� +i=1 +n� +j=1 +E +S∼FS +B∼FB +� +max(S,B)1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] +� += 1, +so constraint (5) is satisfied. +Finally, we are only left to show that the best price in {p1,p2,··· ,pk} must obtain an approximation +ratio that is at least r on instance I, i.e., r ≤ maxp∈{p1,p2,···,pk}W (I,p). Inequality (18) states that +W (I,pt) ≥ +n� +i=1 +si pi + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi). +Taking maximum over t ∈ [n], we then get that +r = max +t∈[n] +n� +i=1 +si · pi + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi) ≤ max +t∈[n] W (I,pt) +which finishes our proof. +B.3 +Proof of Lemma 3.3 +In the following, we complete the proof of Lemma 3.3. +5The price p∗ depends on c, but the approximation ratio to the optimal welfare does not. +19 + +The Optimization Problem UpperOp +min +s1,s2···,sn +b1,b2,···,bn,r +r +s.t. +si,bi ≥ 0 +∀i ∈ [n] +n� +i=1 +si = 1 +and +n� +i=1 +bi = 1 +n� +i=1 +n� +j=1 +sib j max(pi,p j) ≥ 1 +n� +i=1 +si pi + +t� +i=1 +n� +j=t+1 +sib j(p j − pi) ≤ r +∀t ∈ [n] +For any fixed support 0 = p1 < p2 < ··· < pn and a valid solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r), define +an instance I = (FS,FB) satisfying +S ∼ FS,S = + + + + + + + + + + + +p1 +ε +w.p. +s1 +p2 +ε +w.p. +s2 +··· +pn +ε +w.p. +sn +B ∼ FB,B = + + + + + + + + + + + +p1 +w.p. +b1 +p2 +w.p. +b2 +··· +pn +w.p. +bn +where ε > 0 is a constant that small enough. +It is easy to see that both FS and FB are valid distributions since the UpperOp requires the non- +negativity of si,bi and �n +i=1 si = �n +i=1bi = 1. Next, we aim to show that no fixed-price mechanism have +an approximation ratio of r +ε on this instance I = (FS,FB). For any x ∈ R≥0, we could first see that x < ε +would never be a optimal price. Thus let pi be the largest p ∈ {p1,p2,··· ,pn} that is not greater than x −ε. +Notice that both FS is a distribution on support {pi +ε}i∈[n] and FB is a discrete distribution on support +{pi}i∈[n]. This means choosing pi + ε instead of x would never become worse. Therefore, we could see +that the optimal fixed-price mechanism on this instance is simply choosing one pt ∈ {p1,p2,··· ,pk} that +maximizes W (I,pt +ε). Again, by the fact that FS and FB are discrete distributions, W (I,pt +ε) could +be written as: +W (I,pt +ε) = +n� +i=1 +(pi +ε)si + +t� +i=1 +n� +j=t+1 +sib j(p j − pi −ε) +≤ +n� +i=1 +pi si + +t� +i=1 +n� +j=t+1 +sib j(p j − pi)+ε. +Also notice that the constraints of UpperOp guarantee that +OPT-W (I) = +n� +i=1 +n� +j=1 +sib j max(pi +ε,p j) ≥ +n� +i=1 +n� +j=1 +sib j max(pi,p j) ≥ 1. +Therefore, the approximation ratio of the optimal fixed-price mechanism on instance I = (FS,FB) is +upper bounded by +maxt∈[n] W (I,pt +ε) +OPT-W (I) +≤ max +t∈[n] +n� +i=1 +pi si + +t� +i=1 +n� +j=t +sib j(p j − pi)+ε = r +ε. +And this finishes our proof. +20 + +B.4 +Proof of Lemma 3.4 +In this section, we assume that ε > 0 is a small enough constant such that ε2 ≪ ε. +We first show that, for any ε > 0, there exists a set of support {p1,p2,··· ,pn} such that UpperOp has +an optimal value of at most r ∗ +ε. +As we show before, we could assume that the instance I = (FS,FB) has optimal welfare 1. Thus, the +approximation ratio of the optimal fixed-price mechanism is +r ∗ = +min +I =(FS ,FB ) +OPT-W (I )=1 +max +p∈R W (I,p). +Suppose r ∗ is attained at I ∗ = (F ∗ +S ,F ∗ +B). Now define n = 1/ε4, and pi = i ·(1/ε2)+ε/2 for i ∈ [n +1]. +Our idea is to construct a valid solution {si,bi}i∈[n+1] by rounding up I ∗ to pi and show that this solution +has an objective value that is close to r ∗. +Suppose p0 = 0. Now we define +si = Pr +S∼F ∗ +S +� +S ∈ [(i −1)ε2,iε2) +� +and bi = Pr +B∼F ∗ +B +� +B ∈ [(i −1)ε2,iε2) +� +for i ∈ [n]. Especially, let +sn+1 = +E +S∼F ∗ +S +� +S · +1[S ≥ nε2] +� +/ +� +nε2� +and bn+1 = +E +B∼F ∗ +B +� +B · +1[B ≥ nε2] +� +/ +� +nε2� +. +Since E[S] and E[B] are upper bounded by 1, we could see that sn+1,bn+1 ≤ ε2. In the last, let s = +�n+1 +i=1 si and b = �n+1 +i=1 be the normalization factors. It’s also straightforward to see that s ≤ �n +i=1 si +sn+1 ≤ +1+ε2. Following the same argument, it also holds that b ≤ 1+ε2. Now define +r = max +t∈[n] +n� +i=1 +(si/s)pi + +t� +i=1 +n� +j=t+1 +(si/s)(b j/b)(p j − pi). +We aim to verify that (s1/s,s2/s,··· ,sn+1/s,b1/b,b2/b,··· ,bn+1/b,r) is a valid solution of UpperOp. +It is easy to see the non-negativity of si,bi and �n+1 +i=1 si/s = �n+1 +i=1 bi/b = 1. What’s more, from the +definition of r, we could see the last constraint holds. Now we only need to check the third constraint. +For any i, j ∈ [n], it holds that +E +S∼F∗ +S +B∼F∗ +B +� +max(S,B)1[S ∈ [(i −1)ε2,iε2)] 1[B ∈ [(j −1)ε2, jε2)] +� +≤ (max(pi,p j)−ε/4)si b j +When one of i, j equals to n +1(we can assume i = n +1 w.l.o.g.), it is true that +E +S∼F∗ +S +B∼F∗ +B +� +max(S,B)1[S ≥ nε2] 1[B ∈ [(j −1)ε2, jε2)] +� += +E +S∼F ∗ +S +[S · +1[S ≥ nε2]] Pr +B∼F ∗ +B +[B ∈ [(j −1)ε2, jε2)] += +� +nε2� +sn+1b j +≤ (pn+1 −ε/4)sib j +21 + +Finally, for the special case that i = j = n +1, we could see that +E +S∼F∗ +S +B∼F∗ +B +� +max(S,B)1[S ≥ nε2] 1[B ≥ nε2] +� +≤ +E +S∼F∗ +S +B∼F∗ +B +� +BS/ +� +nε2� +· +1[S ≥ nε2] 1[B ≥ nε2] +� += +� +nε2� +sn+1bn+1 +≤ (pn+1 −ε/4)sn+1bn+1 +Summing up all the inequalities above, we then get that +E +S∼F∗ +S +B∼F∗ +B +[max(S,B)] ≤ +n+1 +� +i=1 +n+1 +� +j=1 +� +max(pi,p j)−ε/4 +� +sib j +This implies that +n+1 +� +i=1 +n+1 +� +j=1 +(si/s)(b j/b)max(pi,p j) +≥ +n+1 +� +i=1 +n+1 +� +j=1 +sib j max(pi,p j)· +� +1+ε2�−2 +≥ + + E +S∼F∗ +S +B∼F∗ +B +[max(S,B)]+ +n+1 +� +i=1 +n+1 +� +j=1 +ε/4sib j + +·(1−ε2)2 +≥ 1+ε/8. +which means that (s1/s,s2/s,··· ,sn+1/s,b1/b,b2/b,··· ,bn+1/b,r) is truly a valid solution. +Next, we give an upper bound of r. To start with, notice that +n+1 +� +i=1 +si pi = +n� +i=1 +Pr +� +S ∈ [(i −1)ε2,iε2) +� +· +� +(i −1)ε+ε2 +ε/4 +� ++E[S · +1[S ≥ nε2]]+ sn+1 ·(ε2 +ε/4) +≤ +n� +i=1 +E +� +S · +1 +� +S ∈ [(i −1)ε2,iε2) +�� ++E[S · +1[S ≥ nε2]]+ +n+1 +� +i=1 +si(ε2 +ε/4) +≤ E[S]+ε/2. +(24) +For the term of gain from trade, it holds that +t� +i=1 +n+1 +� +j=t+1 +sib j(p j − pi) = +t� +i=1 +n� +j=t+1 +sib j(jε2 −iε2)+ +t� +i=1 +sibn+1 +� +nε2 −(i −1)ε2� +≤ +t� +i=1 +n� +j=t+1 +sib j((j −1)ε2 −iε2)+ +t� +i=1 +sibn+1 +� +nε2 −iε2� ++ε2(1+ε2)2 +≤ +t� +i=1 +n� +j=t+1 +E[(B −S)· +1[S ∈ [(i −1)ε2,iε2)]] 1[B ∈ [(j −1)ε2, jε2]] ++E[(B −S)· +1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε/2 +≤ E[(B −S)· +1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε/2. +(25) +22 + +Combining (24) and (25), we know that for any t ∈ [n +1], +n+1 +� +i=1 +(si/s)pi + +t� +i=1 +n+1 +� +j=t+1 +(si/s)(b j/b)(p j − pi) +≤ +n+1 +� +i=1 +si pi + +t� +i=1 +n+1 +� +j=t+1 +sib j(p j − pi) += E[S]+E[(B −S)· +1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε +≤ W (I,t ·ε2)+ε. +Taking maximum over [n +1], we then get that +r = max +t∈[n+1] +n+1 +� +i=1 +(si/s)pi + +t� +i=1 +n+1 +� +j=t +(si/s)(b j/b)(p j − pi) ≤ max +t∈[n+1]W (I,t ·ε2)+ε ≤ r ∗ +ε. +This means that the optimal value of UpperOp with respect to {pi}i∈[n+1] is at most r ∗ + ε, and this +finishes our proof. +Next, we aim to show that for any ε > 0, there exists 0 = p0 < p1 < p2 < ···pn such that LowerOp +has an optimal value of at least r ∗ − ε. Now define n = +� +ε−6� ++ 1, and pi = (i − 1) · ε3 for i ∈ [n]. Let +(s1,··· ,sn,b1,··· ,bn,r) be the optimal solution of the optimization problem LowerOp with respect to +{pi}i∈n. It is equivalent to show that there exists an instance I = (FS,FB) such that the optimal approxi- +mation ratio of I, i.e. maxx∈R +W (I ,x) +OPT-W (I ), is at most r +ε. +We construct the instance as follows. Let n′ = n + +� 4 +ε +� +, and si = bi = 0 for n < i ≤ n′. Now define {s′ +i} +where +s′ +j = +j� +i=max +� +1,j− +� 4 +ε +� ++1 +�si/ +�4 +ε +� +. +It follows that +n′ +� +j=1 +s′ +j = +n� +j=1 +j� +i=max +� +1,j− +� 4 +ε +� ++1 +�si/ +�4 +ε +� += +n′ +� +i=1 +si. +Let s = �n′ +i=1 s′ +i and b = �n′ +i=1bi be the normalization factors. LowerOp guarantees that 1 ≤ s,b ≤ 1+ε3. +What’s more, we could also see that +s′ +j = +j� +i=max +� +1,j− +� 4 +ε +� ++1 +�si/ +�4 +ε +� +≤ (1+ε3) +�4 +ε +� +≤ ε/3. +holds for all j ∈ [n′]. +Consider the following instance I = (FS,FB): +S ∼ FS,S = + + + + + + + +p1 +ε4 +w.p. +s′ +1/s +··· +pn′ +ε4 +w.p. +s′ +n′/s +B ∼ FB,B = + + + + + +p1 +w.p. +b1/b +··· +pn′ +w.p. +bn′/b +23 + +First, it is straight forward to verify this is a valid distribution. We first calculate OPT-W (I): +OPT-W (I) = +n′ +� +i=1 +n′ +� +j=1 +max(pi +ε4,p j)·(s′ +i/s)(b j/b) +≥ +n′ +� +i=1 +n′ +� +j=1 +max(pi,p j)·(s′ +i/s)(b j/b)−ε4 +≥ +n′ +� +i=1 +n′ +� +j=1 +max(pi,p j) + + +i� +k=max +� +1,i− +� 4 +ε +� ++1 +� +� +sk/ +�4 +ε +�� +/s + +·(b j/b)−ε4 +≥ +n′ +� +i=1 +n′ +� +j=1 +max(pi,p j)· sib j/(bs)−ε4 +≥ (1+ε3)−2 −ε4 +≥ 1−ε2 +Now consider the optimal fixed-price mechanism for the instance. As we have shown in the proof of +Lemma 3.3, the optimal mechanism only need to choose price from the support of the discrete distribu- +tion. This implies that +max +p∈R W (I,p) = max +t∈[n] +n′ +� +i=1 +(pi +ε4)(s′ +i/s)+ +t� +i=1 +n′ +� +j=t+1 +(s′ +i/s)(b j/b)(p j − pi −ε4) +For any t ∈ [n], one could see that +n′ +� +i=1 +pi s′ +i = +n′ +� +i=1 + + +i� +k=max +� +1,i− +� 4 +ε +� ++1 +� +� +sk/ +�4 +ε +�� +pi +≤ +n′ +� +i=1 + + +i� +k=max +� +1,i− +� 4 +ε +� ++1 +� +� +sk/ +�4 +ε +�� +· +� +pk + +�4 +ε +� +·ε3 +� + +≤ +n′ +� +i=1 +(pi +5ε2)si +(26) +For the term of gain from trade, it follows that +t� +i=1 +n′ +� +j=t+1 +s′ +ib j(p j − pi) = +t−1 +� +i=1 +n′ +� +j=t+1 +s′ +ib j(p j − pi)+ +n′ +� +j=t+1 +s′ +tb j(p j − pi) +≤ +t−1 +� +i=1 +n′ +� +j=t+1 + + +i� +k=max +� +1,i− +� 4 +ε +� ++1 +� +� +sk/ +�4 +ε +�� +b j(p j − pi)+ s′ +t +n′ +� +j=1 +b j p j +≤ +t−1 +� +i=1 +n′ +� +j=t+1 +sib j(p j − pi)+ε/3. +(27) +where we use the fact that skb j(p j − pi) ≤ skb j(p j − pk) for j > i ≥ k, s′ +t ≤ ε/3, and �n′ +i=1b j p j ≤ 1. +24 + +Again by combining the two inequalities above, we know that +n′ +� +i=1 +(pi +ε4)(s′ +i/s)+ +t� +i=1 +n′ +� +j=t+1 +(s′ +i/s)(b j/b)(p j − pi −ε4) +≤ +n′ +� +i=1 +pi s′ +i + +t� +i=1 +n′ +� +j=t+1 +s′ +ib j(p j − pi)+ε4 +≤ +n′ +� +i=1 +(pi +5ε2)si + +t−1 +� +i=1 +n′ +� +j=t+1 +sib j(p j − pi)+ε/3 +≤ +n′ +� +i=1 +pi si + +t−1 +� +i=1 +n′ +� +j=t+1 +sib j(p j − pi)+ε/2 +where we apply (26) and (27) in the second inequality. +We could see that the optimal solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r) of LowerOp must satisfy that +r = maxt∈[n] +�n +i=1 pi si +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi). +Therefore, taking the maximum over t ∈ [n′], we then get that +max +p∈R W (I,p) = max +t∈[n′] +n′ +� +i=1 +(pi +ε4)(s′ +i/s)+ +t� +i=1 +n′ +� +j=t+1 +(s′ +i/s)(b j/b)(p j − pi −ε4) +≤ max +t∈[n′] +n′ +� +i=1 +pi si + +t−1 +� +i=1 +n′ +� +j=t+1 +sib j(p j − pi)+ε/2 += max +t∈[n] +n� +i=1 +pi si + +t−1 +� +i=1 +n� +j=t+1 +sib j(p j − pi)+ε/2 += r + ε +2. +where the second equation follows from si = bi = 0 when i > n. +Therefore, on instance I, it holds that +maxp∈R W (I,p) +OPT-W (I) +≤ r +ε/2 +1−ε2 ≤ r +ε. +And this completes our proof. +C +Proof of Theorem 4.1 +We start with the case when we only know E[S]. +We consider discrete distribution PS. Suppose x ∼ PS equals to pi with probabiltiy wi for i ∈ [n] +where �n +i=1 wi = 1. This means that our mechanism would choose pi with probability wi and use pi · +ES∼FS[S] as the price. Fixing {pi,wi}i∈[n], we claim that the optimal solution of the following program +lower bounds the approximation ratio of this mechanism. +25 + +min +s1,s2···,sn +b1,b2,···,bn +�n +t=1 wt +��n +i=1 si pi +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi) +� +�n +i=1 +�n +j=1 sib j max(pi,p j) +s.t. +si,bi ≥ 0 +∀i ∈ [n] +(28) +n� +i=1 +si ≥ 1 +and +n� +i=1 +si ≤ 1+ 1 +pn +and +n−1 +� +i=1 +si ≤ 1 +(29) +n� +i=1 +bi ≥ 1 +and +n−1 +� +i=1 +bi ≤ 1 +(30) +n� +i=1 +si · pi = 1 +(31) +Similar to the proof of Lemma 3.2, we could assume that E[S] = 1 without loss of generality. The +approximation ratio r ∗ of our mechanism could be written as +r ∗ = +min +I =(FS ,FB ) +E[S]=1 +�n +i=1 wi ·W (I,pi) +OPT-W (I) +. +Suppose r ∗ is attained at I ∗ = +� +F ∗ +S ,F ∗ +B +� +. Applying Lemma B.1 with I ∗, we know that there exists +{s1,s2,··· ,sn,b1,b2,··· ,bn} satisfying all the properties in the statement. Notice that ES∼F ∗ +S [S] = 1. There- +fore, we could directly verify that constraints (28), (29), (30) and (31) are satisfied by all the properties in +Lemma B.1. Again, by Lemma B.1, it holds that �n +i=1 si pi +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi ) ≤ W (I ′,pt) for all +t ∈ [n] and �n +i=1 +�n +j=1 sib j max(pi,p j) ≥ OPT-W (I ′). This implies that +�n +t=1 wt +��n +i=1 si pi +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi) +� +�n +i=1 +�n +j=1 sib j max(pi,p j) +≤ +�n +i=1 wi ·W (I ∗,pi) +OPT-W (I ∗) += r ∗. +Since {si,bi}i∈[n] is a valid solution of the optimization problem above, we then show that it lower +bounds the approximation ratio. +Finally, we solve this optimization problem numerically and show that there exists {pi,wi}i∈[n] such +that the optimal solution is at least by 0.65. The details of the numerical result could be found at our +GitHub repository. +Now we turn to the case when we know E[B]. The proof uses similar ideas and is almost identical. +Consider the following mechanism: it picks pi with probability wi for i ∈ [n] where �n +i=1 wi = 1, and +chooses pi ·E[B] as the price. Again, for fixed {pi,wi}i∈[n], we aim to show that the following optimization +problem give a lower bound of the approximation ratio. +min +s1,s2···,sn +b1,b2,···,bn +�n +t=1 wt +��n +i=1 sipi +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi) +� +�n +i=1 +�n +j=1 sib j max(pi,p j) +s.t. +si,bi ≥ 0 +∀i ∈ [n] +(32) +n� +i=1 +bi ≥ 1 and +n� +i=1 +bi ≤ 1+ 1 +pn +and +n−1 +� +i=1 +bi ≤ 1 +(33) +n� +i=1 +si ≥ 1 and +n−1 +� +i=1 +si ≤ 1 +(34) +n� +i=1 +bi · pi = 1 +(35) +26 + +Without loss of generality, we could assume that E[B] = 1. Therefore, the approximation ratio of the +mechanism is exactly +r ∗ = +min +I =(FS ,FB ) +E[B]=1 +�n +i=1 wi ·W (I,pi) +OPT-W (I) +. +Suppose r ∗ is attained at I ∗ = +� +F ∗ +S ,F ∗ +B +� +. Again by Lemma B.1 , we could discretize the instance +I ∗ to {s1,s2,··· ,sn,b1,b2,··· ,bn} that satisfies all the properties in the lemma. Since EB∼F ∗ +B [B] = 1, it is +easy to verify that constraints (32), (33), (34) and (35) holds due to Lemma B.1. Again, by Lemma B.1, it +holds that �n +i=1 si pi +�t−1 +i=1 +�n +j=t+1 sib j(p j −pi) ≤ W (I ′,pt) for all t ∈ [n] and �n +i=1 +�n +j=1 sib j max(pi,p j) ≥ +OPT-W (I ′). This implies that +�n +t=1 wt +��n +i=1 si pi +�t−1 +i=1 +�n +j=t+1 sib j(p j − pi) +� +�n +i=1 +�n +j=1 sib j max(pi,p j) +≤ +�n +i=1 wi ·W (I ∗,pi) +OPT-W (I ∗) += r ∗. +Since {si,bi}i∈[n] is a valid solution of the optimization problem above, this means that the optimiza- +tion problem is a lower bound of the approximation ratio. +Finally, we solve this optimization problem numerically and find a set of numbers {pi,wi}i∈[n] such +that the optimal solution is at least by 0.65. The details of the numerical result could be found at our +GitHub repository. +D +Missing Proofs in Section 5 +D.1 +Mechanisms over Buyer’s information +In the sample setting, we only consider mechanisms over seller’s information. We do not consider quan- +tile or order statistics mechanisms over buyer’s information since it is impossible to get any constant +approximation with these family of mechanisms. +Theorem D.1. No quantile mechanism over buyer’s distribution or order statistic mechanism over only +buyer’s samples can achieve a constant fraction of the optimal welfare. +Proof. We first show that there is no constant approximation quantile mechanism Q over buyer’s distri- +bution. Remind that FB and FS respectively stand for the distribution of the buyer and the seller, and we +will also use Q to denote the corresponding distribution over the buyer’s quantile. +To start with, we can assume that distribution Q does not have point mass at 1. That’s because if we +set the 1-quantile of the buyer’s distribution, i.e. F −1 +B (1), as the price, we have PrB∼FB[B ≥ F −1 +B (1)] = 0. +This means that the trade will never happen under such price and thus this price will not increase the +welfare. Therefore, if we move this probability mass to other values, the welfare and also the approxima- +tion ratio will not decrease, and we prove that this assumption is with out loss of generality. +Now, for an arbitarily small ε > 0, we will show that there is no ε-approximation quantile mechanism +over buyer’s distribution. For any quantile mechanism Q over buyer’s distribution, we construct the +following set +X = {t ∈ [0,1] | Pr +x∼Q[x ≥ t] ≤ ε/2} +Since there is no point mass at 1, this set will contain some t ∈ X and t ̸= 1. Consider the following +instance I = (FS,FB): +27 + +B ∼ FB,B = +� +0 +w.p. +t +H +w.p. +1− t +S ∼ FS,S = ε/2 +w.p. +1 +where H = +1 +1−t is a large enough number. +In this instance, the intuition is that all the welfare is hide at some very little probability of the buyer, +and we must make sure that the trade is very likely to happen when the buyer has a very high value. +However, since we don’t know the value of the seller, it is hard for us to make sure that p ≥ S which +means that this trade will not happen. +Remind that we define OPT-W (I) as the optimal welfare, i.e., ES∼FS,B∼FB[max(S,B)] against instance +I = (FS,FB) and W (Q,I) as the welfare of mechanism Q against instance I. Formally speaking, we +have that +OPT-W (I) ≥ H ·(1− t) = 1 +and also +W (Q,I) = E[S]+ +E +p∼F −1 +B (Q) +[(B −S)· +1[B ≥ p ≥ S]] +≤ ε/2+ +E +p∼F −1 +B (Q) +[B · +1[p ≥ S]] +≤ ε/2+ H ·(1− t)·ε/2 = ε +where the last inequality holds since p = F −1 +B (x) ≥ S is equivlent to x ≥ t where x is drawn fromQ, and +this happends w.p. at most ε/2 by the definition of t. So for every distribution Q over buyer’s quantile, +we find an instance I so that W (Q,I) ≤ ε·OPT-W (I), which completes the first part of our proof. +Next we aim to show that for any ε > 0,N > 0, there is no ε-approximation mechanisms using only N +samples from the buyer. +First, for any mechanisms M using N samples from the buyer, it can be formallized as a mapping +f : RN +≥0 �→ ∆(R≥0) +where f (x1,x2,··· ,xN) stands for the distribution of price selected by this mechanism after receiving +N samples (x1,x2,··· ,xN). Let D be f (0,0,...,0), which is the distribution of the price if this mechanism +sees N samples all with value 0. Similarly, we consider the following set: +H ′ = {t ∈ R≥0 | Pr +x∼D[x ≥ t] ≤ ε/2} +Again we know this set is non-empty, so let t be any real positive number in the set H ′. Therefore, +we could construct an instance I = (FS,FB) satisfying +B ∼ FB,B = +� +0 +w.p. +(1−ε/4)1/N +H +w.p. +1−(1−ε/4)1/N +S ∼ FS,S = t +1 +w.p. +1 +where H > +t+1 +ε/4·(1−(1−ε/4)1/N is a large enough number. +In this instance, we can see that with just N samples, no mechanism can distinguish this instance +with another instance whose buyer always have a value of 0. Therefore, it can not get the welfare hidden +at the buyer. Formally speaking: +OPT-W (I) ≥ H · +� +1−(1−ε/4)1/N � +> t +1 +ε/4 +28 + +To calculate W (M,I), we consider the case when all the samples are zero and the case when there +is at least one non-zero number in the samples. In the latter case, the probability that at least one +sample is non-zero is at most 1 − +� +(1−ε/4)1/N �N = ε/4 which is negligible. In the former case, since +Prp∼f (0,0,...,0)[p ≥ t] ≤ ε/2, the trade happends w.p. at most ε/2. Therefore, we could expand W (M,I) +into: +W (M,I) ≤ +E +p∼f (0,0,...,0)[S +(B −S)· +1[B ≥ p ≥ S]]·Pr[All N samples are 0] ++OPT-W (I)·Pr[at least 1 sample is not 0] +≤ (t +1+E[B · +1[p ≥ S]])·1+OPT-W (I)·(ε/4) +≤ OPT-W (I)·(ε/4+ε/2+ε/4) += ε·OPT-W (I) +where the second inequality holds since t + 1 ≤ (ε/4) · OPT-W (I), Prp∼f (0,0,...,0)[p ≥ S] ≤ ε/2 and +EB∼FB[B] ≤ ES∼FS,B∼FB[max(S,B)] = OPT-W (I). +And this finishes our proof. +D.2 +Proof of Lemma 5.1 +The proof here is quite straight forward. As we show in Section 5.1.1, each order statistic mechanism +corresponds to a quantile mechanism. Thus C (P (Q)) is exactly the approximation ratio of the order +statistic mechanism Q. What’s more, we could see that the ∆N enumerates all possible order statistic +mechanisms with N samples. Therefore, this directly implies that argmaxQ∈∆N C (P (Q)) is the optimal +order statistic mechanism with N samples. +D.3 +Proof of Lemma 5.2 +Fix an instance I = (F,F), recall that S and B are the random variables respectively indicating the value +of the seller and the buyer. Define ALG to be the random variable which indicates the welfare of our +mechanism in the realization, which is S + (B − S) · +1[B ≥ p ≥ S] where p is the price chosen by our +quantile mechanism Q. Similarly, let OPT be the random variable which indicates the optimal welfare in +the realization, which is max(B,S). +To prove Lemma 5.2, we introduce the following lemma. +Lemma D.1. For any quantile mechanism Q, let ALG and OPT respectively be the random variables indi- +cating the welfare of the mechanism Q and the optimal welfare in the realization. Let r be +min +I =(F,F) inf +x∈[0,1) +Pr[ALG ≥ F −1(x)] +Pr[OPT ≥ F −1(x)] +where F(x) is the cumulative distribution function of distribution F, and F −1(x) is the quantile function. +The quantile mechanism Q is at least r-approximate. +Proof. We have +Pr[ALG ≥ F −1(x)] ≥ r ·Pr[OPT ≥ F −1(x)] +for all x ∈ [0,1] and quantile function F −1(x). +Without loss of generality, we could assume the distribution has a support over [0,a]. Notice that +since we assume the distribution is continuous w.l.o.g. in the sample setting, F −1(x) is a continuous and +29 + +increasing function over [0,1] and F −1(0) = 0,F −1(1) = a, so we have +W (I,Q) = E[ALG] += +�a +0 +Pr[ALG ≥ x]dx += +�1 +0 +Pr[ALG ≥ F −1(z)]dF −1(z) +≥ +�1 +0 +r ·Pr[OPT ≥ F −1(z)]dF −1(z) += r · +�a +0 +Pr[OPT ≥ x]dx = r ·E[OPT] += r ·OPT-W (I) +holds for any instance I = (F,F), which implies that quantile mechanism Q is at least r-approximate. +With Lemma D.1, we are able to give a lower bound of approximation ratio for any quantile function +Q. +Fixing the buyer and seller’s distribution F, we only need to calculate the term Pr[ALG ≥ F −1(x)] and +Pr[OPT ≥ F −1(x)]. The event OPT ≥ F −1(x) happens if and only if either B or S is greater than F −1(x). +Thus, +Pr[OPT ≥ F −1(x)] = 1− x2 +(36) +The event ALG ≥ F −1(x) happens if and only if one of the following conditions is satisfied: +• S ≥ F −1(x) +• p ≤ F −1(x), S ≤ p and B ≥ F −1(x). Here S ≤ p ≤ B, thus the trade takes place, and B ≥ F −1(x). +• p > F −1(x), S ≤ F −1(x) and B ≥ p. Since S ≤ p ≤ B, the seller trades the item to the buyer,and we +have B ≥ p ≥ F −1(x). +Note that these three events are disjoint, so we could calculate the probability for each event to hap- +pen and add them up. +For the first event, Pr[S ≥ F −1(x)] = 1− x. +For the second event, we just enumerate the quantile of p. Suppose the quantile of p is t, which +means that F −1(t) = p. Then, we have Pr[S ≤ p] = t and Pr[B ≥ F −1(x)] = 1 − x. Thus, this event takes +place w.p. +� +[0,x] t(1− x)dQ(t) where Q(t) is the c.d.f. of distribution Q. +For the third event, we use the same idea. Suppose the quantile of p is t, we have Pr[S ≤ F −1(x)] = x +and Pr[B ≥ p] = 1− t. Therefore, this event happens w.p. +� +(x,1](1− t)x dQ(t). +By adding the terms above up, we have: +Pr[ALG ≥ F −1(x)] = +� +[0,x] +t(1− x)dQ(t)+ +� +(x,1] +(1− t)x dQ(t)+(1− x) +(37) +Therefore, combining Lemma D.1 and Equation (36) and (37), we have that for any quantile mecha- +nism Q with c.d.f. Q(x), the minimum of the following optimization problem lower bounds the approxi- +mation ratio of the quantile mechanism Q. +min +I =(F,F) inf +x∈[0,1) +�x +0 q(t)· t(1− x)dt + +�1 +x q(t)·(1− t)x dt +(1− x) +1− x2 += inf +x∈[0,1) +�x +0 q(t)· t(1− x)dt + +�1 +x q(t)·(1− t)x dt +(1− x) +1− x2 +30 + +where the equality holds since we could see that the term is independent from F(x). +We now left to show that the approximation ratio of Q is also upper bounded by r. It suffices to show +that for any ε > 0 there exists some instance I = (F,F) such that +W (Q,I) ≤ (r +ε)·OPT-W (I) +First, Recall Equation (36) and (37). We could see that both the term Pr[ALG ≥ F −1(x)] and the term +Pr[OPT ≥ F −1(x)] are independent of the distribution F. Thus, +r = min +F +inf +x∈[0,1) +Pr[ALG ≥ F −1(x)] +Pr[OPT ≥ F −1(x)] = inf +x∈[0,1) +�x +0 q(t)· t(1− x)dt + +�1 +x q(t)·(1− t)x dt +(1− x) +1− x2 +Suppose the optimum of infx∈[0,1) +�x +0 q(t)·t(1−x)dt + +�1 +x q(t)·(1−t)x dt +(1−x) +1−x2 +is attained at x∗. Consider the +following instance I = (F,F) satisfying +v ∼ F,v = +� +U[0,r ·(1− x∗)·ε/2] +w.p. +x∗ +U[1,1+ε/2] +w.p. +1− x∗ +Notice that in this instance, The event +1[ALG ≥ F −1(x∗)] is equivalent to +1 +� +ALG ∈ [1,1+ε/2] +� +. Such +argument also holds for OPT. Thus, we can see that +W (Q,I) = E[ALG] +≤ Pr[ALG ∈ [1,1+ε/2]]·(1+ε/2)+Pr[ALG ∈ [0,r ·(1− x∗)·ε/2]]·(r ·(1− x∗)·ε/2) +≤ Pr[ALG ≥ F −1(x∗)]·(1+ε/2)+r ·(1− x∗)·ε/2 += r ·Pr[OPT ≥ F −1(x∗)]·(1+ε/2)+r ·(1− x∗)·ε/2 +≤ r ·E[OPT]·(1+ε/2)+r ·(1− x∗)·ε/2 +≤ (r +ε)·E[OPT] = (r +ε)·OPT-W (I) +where the second equation uses the fact that r ·Pr[OPT ≥ F −1(x∗)] = Pr[ALG ≥ F −1(x∗)] since r is attained +at x∗. +Since the above holds for every ε > 0, this completes our proof of Lemma 5.2. +D.4 +Proof of Theorem 5.1 +The proof of Theorem 5.1 is directly a combination of Lemma 5.1 and Lemma 5.2. From Lemma 5.2, we +know that the approximation ratio of a quantile mechanism Q with c.d.f. Q(x) is exactly +inf +x∈[0,1) +� +[0,x] q(t)t(1− x)dt + +� +(x,1] q(t)(1− t)x dt +(1− x) +1− x2 +. +where we use q(x)dx instead of dQ(x) since we have a continuous probability density function. +We could see that the set of all distributions over [N] is actually +� +{Pi}i∈[n] | Pi ≥ 0 and �n +i=1Pi = 1 +� +where distribution P would choose i w.p. Pi. What’s more, the probabiltiy density function of the quan- +tile of such order statistic mechanism is exactly p(x) = �n +i=1Pi f i +N(x) where f i +N(x) = N +�n−1 +i−1 +� +· xi−1 · (1 − +x)N−i . Therefore, combining it with Lemma 5.1, it follows that Theorem 5.1 holds. +31 + +D.5 +Analysis of Empirical Risk Minimization Mechanism +In this section, we give the upper bounds of the Empirical Risk Minimization mechanism. +When N = 1, the empirical distribution is a one-point distribution, so any price x ∈ R≥0 is optimal. +Therefore, we consider the following instance I = (F,F) s.t. +x ∼ F,x = +� +0 +w.p. +1−1/H +H +w.p. +1/H +Since any price is optimal for ERM, we can assume that it will always select H + 1 so that the trade +will never take place. Therefore taking H → ∞ in this instance, the approximation ratio tends to 0.5. +When N = 2, the empirical distribution is a two-point distribution. Suppose the two samples are +X1 ≤ X2, we can see that any price in [X1, X2) is optimal for this empirical distribution. Thus, we can +assume that it will always select X1. So, ERM is equivalent to an order statistic mechanism that always +selects the smallest sample. Therefore, we can solve the following optimization problem: +inf +x∈[0,1) +1 +3x3 − x2 − 1 +3x +1 +1− x2 += 2 +3 +Then, by Lemma 5.2, we know that there exists an instance such that ERM achieves exactly 2 +3 approx- +imation. +Now we are in the case that N = 3. By some calculations, we know that the second smallest sample +will always be an optimal choice. Therefore, ERM is equivalent to an order statistic mechanism that +always selects the second smallest sample when N = 3. Similarly, we can calculate that +inf +x∈[0,1) +1 +2x4 − x3 − 1 +2x +1 +1− x2 += 3 +4 +Applying Lemma 5.2 again, there is an instance such that ERM has an approximation ratio of exactly +3 +4. +Our proof strategy changes when the number of samples is greater than 5. We consider a particular +instance I = (F,F) and calculate the performance of ERM on such instance. +x ∼ F,x = + + + + + + + + + + + + + + + +0 +w.p. +( +� +2−1)·(1− 1 +n ) +1 +n +w.p. +2− +� +2 +1 +w.p. +1 +n ( +� +2−1) +Actually, this is a counterexample appeared in [24]. They show that +OPT-W (I) = 4( +� +2−1) +n +− 4 +� +2−1 +n2 +Since we will let n → ∞, we will ignore the O( 1 +n2 ) terms in the following calculation. Notice that +Pr[x = 1] = O( 1 +n ), the probability that there is at least 1 sample with value 1 is negligible, so we will also +assume that all samples are 0 or 1 +n . Recall that there will be a tie-breaker coordinate drawn uniformly +from [0,1] for each variable, and we will compare the tie-breaker coordinate if they have the same value. +Now suppose there are k1 samples with value 0 and k2 samples with value 1 +n . We know that the largest 0 +32 + +or the smallest 1 +n is an optimal price for the empirical distribution when k1,k2 ̸= 0. Now, if we choose the +largest 0, as the price p, the expected welfare is: +W1(k1) =Pr[S = 0]·Pr[S < p]·E[B]+Pr +� +S = 1 +n +� +· 1 +n +Pr[S = 1]·1 +=( +� +2−1)· +k1 +k1 +1 +� +(2− +� +2+ +� +2−1)· 1 +n +� ++(2− +� +2)· 1 +n + 1 +n ·( +� +2−1) +(38) +if we choose the smallest 1 +n , as the price p, the expected welfare is: +W2(k2) =Pr[S = 0]·E[B & B > p]+Pr +� +S = 1 +n +� +· +� 1 +n +Pr[S < p]·E[B & B = 1] +� ++Pr[S = 1]·1 +=( +� +2−1)· +� +(2− +� +2)· 1 +n · +k2 +k2 +1 + 1 +n ( +� +2−1)) +� ++(2− +� +2)· +� 1 +n + +1 +k2 +1 ·( +� +2−1) 1 +n +� ++ 1 +n ·( +� +2−1) +(39) +When all the samples have the same value, any price is optimal. Similar to the case when N = 1, the +trade may never happen, so the expected welfare is 1 +n in this case. +Now, when there are 5 samples, suppose the ERM will choose the largest 0 when there are 1 ∼ 3 +samples with value 0, and choose the smallest 1 +n when there are 1 samples with value 1 +n . The expected +welfare of ERM when N = 5 is : +3� +i=1 +( +� +2−1)i (2− +� +2)5−i +� +5 +i +� +·W1(i)+ +1� +i=1 +( +� +2−1)5−i(2− +� +2)i +� +5 +i +� +·W2(i)+(p5 +(1− p)5)· 1 +n +Now compare it to the optimal: +OPT-W (I) = 4( +� +2−1) +n +− 4 +� +2−1 +n2 +By numerical calculations, the ratio is ≈ 0.76 as n → ∞. +Similarly, when there are 10 samples, suppose the ERM will choose the largest 0 when there are 1 ∼ 6 +samples with value 0, and choose the smallest 1 +n when there are 1 ∼ 3 samples with value 1 +n . The expected +welfare of ERM when N = 10 is : +6� +i=1 +( +� +2−1)i (2− +� +2)10−i +� +10 +i +� +·W1(i)+ +3� +i=1 +( +� +2−1)10−i(2− +� +2)i +� +10 +i +� +·W2(i)+(p10 +(1− p)10)· 1 +n +By calculations, the approximation ratio is ≈ 0.80 as n → ∞. +D.6 +Proof of Lemma 5.3 +Now we fix the quantile mechanism Q and suppose its c.d.f. is Q(x). Let r be +min +x∈[0,1] +� +[0,x] +t dQ(t)+1− x. +[5] already prove that r is the lower bound the approximation ratio. We are only left to show that it is +also the upper bound. We aim to show that for any ε > 0, there exists an instance I = (FS,FB) such that +W (I,Q) ≤ (r +ε)·OPT-W (I) +Now suppose the optimization problem above achieves its minimum at x∗. Consider the following +instance I = (FS,FB). +33 + +S ∼ FS,S = +� +U[0,ε] +w.p. +x∗ +H +ε +w.p. +1− x∗ +B ∼ FB,B = H +w.p. +1 +where H > 1 is a sufficiently large number. We can see that in this instance, +OPT-W (I) ≥ H +Now we compute the expected welfare for our mechanism. When its price has a quantile t smaller +than or equal to x∗, the trade will happen with probability exactly t. When the quantile of its price is +greater than x∗, the trade will never happen. +W (I) = E[S]+E[(B −S)1[B ≤ p ≤ S]] +≤ x∗ε+(H +ε)(1− x∗)+ +� +[0,x∗] +tH dQ(t) +≤ H · +� +1− x∗ + +� +[0,x∗] +t dQ(t) +� ++ε +≤ H ·r +ε ≤ H ·(r +ǫ) +≤ (r +ε)·OPT-W (I) +where the third inequality follows from r = +� +[0,x∗] t dQ(t)+(1− x∗). +Therefore, we could find an instance I = (FS,FB) such that W (Q,I) ≤ (r + ε) · OPT-W (I) for any +small enough ε > 0, and this concludes our proof. +D.7 +Proof of Theorem 5.2 +The proof of Theorem 5.2 is nearly identical to Theorem 5.1. From Lemma 5.3, we know that the approx- +imation ratio of a quantile mechanism Q with a continuous p.d.f. q(x) is exactly +C (p) = min +x∈[0,1] +�x +0 +q(t)t dt +1− x. +where we use q(x)dx instead of dQ(x) since we have a continuous probability density function. +Again, we know that the set of all distributions over [N] is actually +� +{Pi}i∈[n] | Pi ≥ 0 and �n +i=1Pi = +1 +� +where distribution P would choose i w.p. Pi. What’s more, the probabiltiy density function of the +quantile of such order statistic mechanism is exactly p(x) = �n +i=1Pi f i +N(x) where f i +N(x) = N +�n−1 +i−1 +� +· xi−1 · +(1− x)N−i . Therefore, combining it with Lemma 5.1, it follows that Theorem 5.2 holds. +D.8 +Proof of Lemma 5.4 +Before we give the proof, we first introduce some notations and lemmas about Bernstein that may useful +to our proof. +Definition D.1 (Stochastic Bernstein Polynomials). The stochastic Bernstein polynomial of degree n for a +continous function f on [0,1] is defined as +� +B X +n f +� +(t) = +n� +k=0 +f (Xk)pn,k(t) +34 + +in which X0, X1,··· Xn are the order statistics of (n + 1) independent copies of the random variable +uniformly distributed in [0,1], and, +pn,k(t) = +� +n +k +� +xk(1− x)n−k,0 ≤ k ≤ n,0 ≤ t ≤ 1 +Now fix the continous function q(x) we aim to approximate, define ω(h) as the following function: +ω(h) = sup +0≤x,y≤1 +|x−y|≤h +|q(x)− q(y)| +Now we can introduce the lemma in [30] that help us approximate the function q(x) by order statis- +tics. +Lemma D.2 (Theorem 2.11 In [30]). Let ε > 0 and f ∈C[0,1] be given. Suppose that ω +� +1 +�n +� +< ε/6.2. Then +the following inequality holds true: +Pr +���B X +n f − f +�� +∞ > ε +� +≤ 2(n +1)exp + +− +2ε2 +ω2 +� +1 +�n +� + + +We are now ready to prove Lemma 5.4. +We first present our mechanism P. For some instance I = (FS,FB), suppose there are n samples +X1 ≤ X2 ≤ ··· ≤ Xn drawn from the distribution. We draw another n samples Y1 ≤ Y2 ≤ ··· ≤ Yn uniformly +and independently from [0,1]. Let s = �n +i=1 q(Yi) be the sum. Then our mechanism will choose Xi with +probability q(Yi)/s +Now, let +g(x) = +n� +i=1 +q(Yi)f i +n(x)/s +where f i +n(x) = n +� +n −1 +i −1 +� +· xi−1(1− x)n−i +be the corresponding probability density function of the order statistic mechanism P. It suffices to +prove that with high probability +|g(x)− q(x)| ≤ ε +∀x ∈ [0,1] +To prove this, we introduce an intermediate function h(x): +h(x) = +n� +i=1 +q(Yi)· +� +n −1 +i −1 +� +xi−1(1− x)N−i = +n� +i=1 +q(Yi)pn−1,i−1(x) +As we can see, h is the stochastic Bernstein polynomial of q with degree n −1 and ω +� +1 +� +n−1 +� +≤ ε/100. +Applying lemma D.2, we know that +Pr +���h − q +�� +∞ > ε/4 +� +≤ 2n exp + +− +ε2 +8·ω2 +� +1 +� +n−1 +� + + ≤ ε +(40) +where the last inequality comes from the assumption in the statement of lemma. +35 + +Thus, we only need to show that the difference between h and g is small. First we have +g(x) = +N +� +i=1 +q(Yi) +S +f i +n(x) += +N +� +i=1 +q(Yi) +� +n −1 +i −1 +� +xi−1(1− x)n−i n +s += n +s ·h(x) +So it is equivalent to prove that n and s are close w.h.p. We have the following lemma. +Lemma D.3. +Pr +�� +1− ε +4M +� +·n ≤ s ≤ +� +1+ ε +4M +� +·n +� +≥ 1−ε +We first use the lemma to continue our proof, before proving the lemma itself. We know that |g(x)− +h(x)| ≤ ε/2 ∀x ∈ [0,1] if n +� +1− +ε +4M +� +≤ s ≤ n +� +1+ +ε +4M +� +and h(x) < 2M ∀x ∈ [0,1]. Therefore, +Pr[|g(x)−h(x)| ≥ ε/2 ∃x ∈ [0,1]] ≤ Pr +� +s > n +� +1+ ε +4M +�� ++Pr +� +s < n +� +1− ε +4M +�� ++Pr[∃x∈[0,1]h(x) > 2M] +≤ 2ε +(41) +where the second inequality is from Lemma D.3 and the fact that with probability at least 1−ε, ∥h − +q∥ ≤ ε/4 and q has a maximum of M on [0,1]. +Now combining inequality (40) and (41), we know that with probability at least 1−3ε, we have: +|g(x)− q(x)| ≤ |g(x)−h(x)|+|h(x)− q(x)| ≤ ε +Since such probability is strictly greater than 0, we know that there exists some order statistic mech- +anism P over N samples such that |g(x) − q(x)| ≤ ε ∀x ∈ [0,1]. Also we show that our construction will +find such order statistics with high probability. Finally, as we assumed in the statement, it holds that +C (P) ≥ C (Q)−c ·|g − q|∞ ≥ C (Q)−cε +This completes the proof of Lemma 5.4. +Proof of Lemma D.3. We know that E[s] = n. Notice that s is the sum of n i.i.d. random variables ranging +in [0,M]. Therefore, by Chernoff bound, it holds that +Pr +�� +1− ε +4M +� +·n ≤ s ≤ +� +1+ ε +4M +� +·n +� +≥ 1−exp +� +− ε2n +48M2 +� +−exp +� +− ε2n +32M2 +� +≥ 1−ε +where the last inequality is from the property in the statement. +D.9 +Application of Lemma 5.4 +D.9.1 +Symmetric Instance +We first study the case when the distributions are symmetric. [24] provide a mechanism that chooses the +mean of the distribution F as the price. They show that in the symmetric setting, this is the optimal fixed +36 + +price mechanism and achieves an approximation ratio of 2+ +� +2 +4 +. However, what we want here is a quan- +tile mechanism and we could not convert such mean-based mechanism directly into an order statistic +mechanism. We show that quantile mechanisms can also reach the optimal 2+ +� +2 +4 +-approximation ratio. +After that, we use the technique in Lemma 5.4 to produce an order statistic mechanism that achieves an +approximation ratio of 2+ +� +2 +4 +−ε with poly +� 1 +ε +� +samples. +To start with, we first show our optimal order statistic mechanism. +Theorem D.2. There is a 2+ +� +2 +4 +-approximation quantile mechanism in the symmetric setting. +Proof. Our quantile mechanism Q runs as following: +• Let F be the c.d.f. of the distribution. +• Output F −1� � +2 +2 +� +, i.e. +� +2 +2 -quantile of the distribution, as the price. +The approximation ratio could be directly calculated using Lemma 5.2. One could see that +inf +x∈[0,1) +� +[0,x] t(1− x)dQ(t)+ +� +(x,1](1− t)x dQ(t)+(1− x) +1− x2 += min + + min +x∈[0, +� +2 +2 ] +x ·(1− +� +2 +2 )+1− x +1− x2 +, +inf +x∈[ +� +2 +2 ,1) +� +2 +2 ·(1− x)+(1− x) +1− x2 + + += 2+ +� +2 +4 +. +which completes the proof. +Now we aim to convert it to an order statistic mechanism. +Theorem D.3. There exists an order statistic mechanism P with N = O +� 1 +ε7 +� +samples that achieves 2+ +� +2 +4 +−ε +approximation. +To start with, we may notice that it is impossible to directly apply Lemma 5.4 to the optimal quantile +mechanism a since it is does not have a continiuous probability density function. So our first step is to +provide a quantile mechanism with continuous distribution. +Lemma D.4. For any ε > 0, let Q′ be the quantile mechanism with p.d.f. ˜q(x) satisfying +˜q(x) = + + + + + + + + + + + + + + + + + + + + + + + +0 +x ∈ +� +0, +� +2 +2 − 1 +32ε +� +� +�� +2 +2 + 1 +32ε,1 +� +(32/ε)2 · +� +x − +� +2 +2 + 1 +32ε +� +x ∈ +�� +2 +2 − 1 +32ε, +� +2 +2 +� +−(32/ε)2 · +� +x − +� +2 +2 − 1 +32ε +� +x ∈ +�� +2 +2 , +� +2 +2 + 1 +32ε +� +The quantile mechanism Q′ has an appriximation ratio of +� +2+2 +4 +−ε/2. +Our last step is to make sure that the approximation ratio would not differ to much for two probability +density functions that are close to each other. +37 + +Lemma D.5. Suppose C (p) is the function that maps every quantile mechanismQ for symmetric instances +with a continuous probabilitiy density function q(x) to its approximation ratio where +C (p) = inf +x∈[0,1) +� +[0,x] q(t)t(1− x)dt + +� +(x,1] q(t)(1− t)x dt +(1− x) +1− x2 +. +For any quantile mechanism Q1 with continuous p.d.f q1(x) and Q2 with continuous p.d.f. q2(x), it +holds that +C (p1)−C (p2) ≥ − +��q1 − q2 +�� +∞. +We first use these lemmas to give a proof of Theorem D.3, and leave the proof of Lemma D.4 and +Lemma D.5 to Appendix D.10 and D.11. +Proof Of Theorem D.3. Let n = c · 1 +ε7 +1 where c is a large enough constant. We could see that ω +� +1 +� +n−1 +� +≤ +322c−0.5 ·ε1.5. This means that 2n exp +� +− +ε2 +8ω2 +� +1 +� +n−1 +� +� +≤ ε/2. Besides, define M = maxx∈[0,1] ˜q(x) = 32/ε, it is +also easy to verify that exp +� +− ε2n +48M3 +� +≤ ε/4. Combining the properties above with Lemma D.5, we could +apply Lemma 5.4, and see that there exists an order statistic mechanism with n samples with an approx- +imation of at least C (Q′)−ε/2. Together with Lemma D.4, it follows that this order statistic mechanism +is 2+ +� +2 +4 +−ε-approximate. This concludes our proof. +We would like to comment that if we always choose the +� � +2 +2 ·n +� +-th order statistic as the price, there is +an argument to prove that we could achieve an approximation ratio of 2+ +� +2 +4 +−ε with ˜O +� 1 +ε2 +� +samples. +D.9.2 +General Instance +We now consider the general instance. [5] provides a 1−1/e approximation quantile mechanism that is +also shown to be optimal by [24]. Using the block-box reduction shown in Lemma 5.4, we show that the +optimal quantile mechanism can be approximated by order statistics mechanism as closely as desired +and also obtain an upper bound on the sample complexity. +Theorem D.4. There exists an order statistic mechanism P with N = O +� 1 +ε5 +� +samples that achieves 1− 1 +e −ε +approximation. +In the following proof, we will use order statistic mechanism to approximate the optimal quantile +mechanism Q with p.d.f. q(x) = 1 +x on [1/e,1]. Similarly, q(x) is not a continous function on [0,1]. Thus, +we need to first convert it to a continous function ˜q(x) on [0,1] and then apply Lemma 5.4. +Similarly, we introduce the following two lemmas first. +Lemma D.6. For any ε > 0, let ˜Q be the quantile mechanism with p.d.f. ˜q(x) satisfying +˜q(x) = + + + + + + + + + + + + + + + + + + + + + +0 +x ∈ +� +0, 1 +e − +� +1−e−1� +� +e − 1 +2ε +� ε +� +(e − 1 +2ε)2 +ε(1−e−1) +� +x − +� +1 +e − +� +1−e−1� +� +e − 1 +2ε +� ε +�� +x ∈ +� +1 +e − +� +1−e−1� +� +e − 1 +2ε +� ε, 1 +e +� +1 +x − 1 +2ε +x ∈ +�1 +e ,1 +� +The quantile mechanism Q′ has an appriximation ratio of 1− 1 +e −ε/2. +38 + +Lemma D.7. Suppose C (p) is the function that maps every quantile mechanism Q for general instances +with a continuous probabilitiy density function q(x) to its approximation ratio where +C (p) = min +x∈[0,1] +�x +0 +q(t)t dt +1− x. +For any quantile mechanism Q1 with continuous p.d.f q1(x) and Q2 with continuous p.d.f. q2(x), it +holds that +C (p1)−C (p2) ≥ − +��q1 − q2 +�� +∞. +The proofs of Lemma D.6 and Lemma D.7 are respectively in Appendix D.12 and D.13. +Proof Of Theorem D.4. We follow the same argument to prove Theorem D.4. +Let n = c · 1 +ε5 + 1 where c is a large enough constant. Again it is easy to see that ω +� +1 +�n +� +≤ c−0.5e2(1 − +e−1)−1 ·ε1.5. Thus we have 2n exp +� +− +ε2 +8ω2 +� +1 +� +n−1 +� +� +≤ ε/2. +Besides, define M = maxx∈[0,1] ˜q(x) ≤ 2e, we could also verify that exp +� +− ε2n +48M3 +� +≤ ε/4. Combining +the properties above with Lemma D.7, we could see the existence of an order statistic mechanism with n +samples with an approximation of at least C (Q′)−ε/2 by applying Lemma 5.4. Together with Lemma D.6, +we know that the approximation ratio of this order statistic mechanism is 1 − 1/e − ε. This finishes our +proof. +D.10 +Proof of Lemma D.4 +One could see that the quantile mechanism ˜Q would choose a price with its quantile in +� � +2 +2 − 1 +32ε, +� +2 +2 + 1 +32ε +� +. +Fix a price p and an instance I = (F,F), ALG is the random variable indicating the welfare for the +fixed price p in the realzation, i.e. S + (B − S)1[B ≥ p ≥ S] where B,S are drawn independently from F. +Thus, it suffices to show that for any instance I = (F,F), the following holds when +� +2 +2 − 1 +32ε ≤ t ≤ +� +2 +2 + 1 +32ε. +E +�ALG | p = F −1(t) +� +≥ +� +2+ +� +2 +4 +−ε/2 +� +·OPT-W (I) +To prove the approximation ratio, we need to use Lemma 5.2. Notice that the term E[ALG | p = F −1(t)] +could be understood as the expected welfare of a quantile mechanism that always selects the t-quantile +as the price, so Lemma 5.2 could also be applied to analyze the ratio between E +�ALG | p = F −1(t) +� +and +OPT-W (I). For x ≥ t, we know that +inf +x∈[t,1) +(1− x)+ t(1− x) +1− x2 += 1+ t +2 +When x < t, it holds that +min +x∈[0,t] +(1− x)+ x(1− t) +1− x2 += 1+ +� +1− t2 +2 +As we can see, 1+t +2 ≥ 1+ +� +1−t 2 +2 +when 1 ≥ t ≥ +� +2 +2 and 1+t +2 ≤ 1+ +� +1−t 2 +2 +when +� +2 +2 ≥ t ≥ 0. By Lemma 5.2, we +know that for any instance I = (F,F), +39 + +E +�ALG | p = F −1(t) +� +≥ + + + + + + + + + + + +1+ +� +1− t2 +2 +·OPT-W (I) +t ∈ +�� +2 +2 ,1 +� +1+ t +2 +·OPT-W (I) +t ∈ +� +0, +� +2 +2 +� +Now if +� +2 +2 − 1 +32ε ≤ t ≤ +� +2 +2 , it holds that +E +�ALG | p = F −1(t) +� +≥ 1+ t +2 +·OPT-W (I) +≥ +� +2+ +� +2 +4 +−ε/2 +� +·OPT-W (I) +. +For +� +2 +2 ≤ t ≤ +� +2 +2 + 1 +32ε, we have +E +�ALG | p = F −1(t) +� +≥ 1+ +� +1− t2 +2 +·OPT-W (I) +≥ +� +2+ +� +2 +4 +−ε/2 +� +·OPT-W (I)W W +which concludes the proof. +D.11 +Proof of Lemma D.5 +Suppose |q1(x)− q2(x)| ≤ c for all x ∈ [0,1]. We could see that +�x +0 (q1(t)− q2(t))t(1− x)dt + +�1 +x (q1(t)− q2(t))(1− t)x dt +1− x2 +≥ − +�x +0 ct(1− x)dt + +�1 +x c(1− t)x dt +1− x2 +≥ − +�x +0 ct(1− x)dt + +�1 +x c(1− x)x dt +1− x2 += − +�x +0 ct dt + +�1 +x cx dt +1+ x +≥ −c. +holds for all x ∈ [0,1]. +Taking the infimum over [0,1), this directly implies that C (Q1)−C (Q2) ≥ −c. +D.12 +Proof of Lemma D.6 +By Lemma 5.3, the approximation ratio of ˜Q can be computed as +min +x∈[0,1] +�x +0 +˜q(t)· t dt +1− x ≥ min +x∈[0,1] +�x +1 +e +�1 +t − 1 +2ε +� +· t dt +1− x +≥ min +x∈[0,1] +�x +1 +e +1 +t · t dt +1− x − 1 +2ε += 1− 1 +e − 1 +2ε. +40 + +D.13 +Proof of Lemma D.7 +Suppose |q1(x)− q2(x)| ≤ c for all x ∈ [0,1]. We could see that +��x +0 +q1(t)t dt +(1− x) +� +− +��x +0 +q2(t)t dt +(1− x) +� +≥ − +�x +0 +ct dt ≥ −c. +holds for all x ∈ [0,1]. +Taking the minimum over [0,1), this directly implies that C (Q1)−C (Q2) ≥ −c. +E +Details of Numerical Experiments +E.1 +Symmetric Instance +We now present the details of numerical experiments for the symmetric instances I = (F,F). We first for- +mally write down the optimization problem that indicates the optimal order statistic mechanisms. As we +proved in Section 5.2.1, suppose the following optimization problem PO achieves its maximum OPTPO +at (P∗ +1 ,P∗ +2 ,...,P∗ +N). Let P∗ be the distribution over [N] such that Prx∼P ∗[x = i] = P∗ +i . Then, P∗ is the op- +timal order statistic mechanism in the symmetric setting with N samples and its approximation ratio is +OPTPO. Notice that since the c.d.f. of the order statistic mechanism Q(x) is differentiable, we use q(t)dt +instead of dQ(t) for ease of computation. Here q(x) is the probality density function of mechanism Q. +The Optimization Problem PO +max +P1,...,PN +min +x∈[0,1] +�x +0 q(t)· t(1− x)dt + +�1 +x q(t)·(1− t)x dt +(1− x) +1− x2 +s.t. +q(x) = +N +� +i=1 +Pi · f i +N(x) +f i +N(x) = N · +� +N −1 +i −1 +� +· xi−1(1− x)n−i +∀i ∈ [N] +n� +i=1 +Pi = 1 +(42) +Pi ≥ 0 +∀i ∈ [N] +(43) +Now we aim to solve the optimization problem PO numerally with different numbers of samples N. +For the inner minimization problem, it must be solved accurately so that it precisely reflect the approxi- +mation ratio of the order statistics mechanism. We use binary search to find its optimum. When we need +to check whether +inf +x∈[0,1) +�x +0 q(t)· t(1− x)dt + +�1 +x q(t)·(1− t)x dt +(1− x) +1− x2 +≥ r +It is equivalent to check if +�x +0 +q(t)· t(1− x)dt + +�1 +x +q(t)·(1− t)x dt +(1− x) −r(1− x2) ≥ 0 +∀x ∈ [0,1] +Notice that q(t) is a polynomial of degree N, thus we only need to find the minimum of a single- +variable polynomial over [0,1], and this could be efficiently done by finding the roots of its derivatives. +41 + +We do the binary search for 100 times, so the error caused by binary search is at most 2−100, which is +much smaller than the floating point errors and can be ignored. Then we use some empirical algorithms +to search for parameters in the outer maximization problem. Unlike the inner minimization problem, +we do not need to get an exact optimum of the outer maximization problem since it only reflects the ratio +of the order statistic mechanism we have found. The code can be found at our GitHub repository(https: +//github.com/BilateralTradeAnonymous/On-the-Optimal-Fixed-Price-Mechanism-in-Bilateral-Trade). +E.2 +General Instance +Now again we formally write down the optimization problem that characterizes the optimal order statis- +tic mechanism with N samples. Similarly, as we proved in Section 5.2.2, suppose the following optimiza- +tion problem QO achieves its maximum OPTQO at (P∗ +1 ,P∗ +2 ,...,P∗ +n). Let P∗ be the distribution over [N] +such that Prx∼P ∗[x = i] = P∗ +i . Then, P∗ is the optimal order statistic mechanism in the general setting +with N samples and its approximation ratio is exactly OPTQO. Again notice that since the c.d.f. of the +order statistic mechanism Q(x) is differentiable, we use q(t)dt instead of dQ(t) where q(x) is the p.d.f. +of the order statistic mechanism. +The Optimization Problem QO +max +P1,...,PN +min +x∈[0,1] +�x +0 +q(t)· t dt +1− x +s.t. +q(x) = +N� +i=1 +Pi · f i +N (x) +f i +N(x) = N · +� +N −1 +i −1 +� +· xi−1(1− x)n−i +∀i ∈ [N] +n� +i=1 +Pi = 1 +(44) +Pi ≥ 0 +∀i ∈ [N] +(45) +Now we solve the optimization problem numerically for different fixed number N. Again the inner +minimization problem could be solved efficiently by calculating the zero point of its derivatives and we +search for the parameters in the outer maximization problem to get a good enough solution. The code +could be found at our Github repository. +42 + diff --git a/TtE4T4oBgHgl3EQfmQ2m/content/tmp_files/load_file.txt b/TtE4T4oBgHgl3EQfmQ2m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f67a7f9ade85648ed9d49552b290c0733444024f --- /dev/null +++ b/TtE4T4oBgHgl3EQfmQ2m/content/tmp_files/load_file.txt @@ -0,0 +1,1463 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf,len=1462 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='05167v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='GT] 12 Jan 2023 On the Optimal Fixed-Price Mechanism in Bilateral Trade Yang Cai* Yale University, USA yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='cai@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='edu Jinzhao Wu Yale University, USA jinzhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='wu@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='edu January 13, 2023 Abstract We study the problem of social welfare maximization in bilateral trade, where two agents, a buyer and a seller, trade an indivisible item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The seminal result of Myerson and Satterthwaite [29] shows that no incentive compatible and budget balanced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', the mechanism does not run a deficit) mech- anism can achieve the optimal social welfare in bilateral trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Motivated by this impossibility result, we focus on approximating the optimal social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We consider arguably the simplest form of mechanisms – the fixed-price mechanisms, where the designer offers trade at a fixed price to the seller and buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Besides the simple form, fixed-price mechanisms are also the only dominant strat- egy incentive compatible and budget balanced mechanisms in bilateral trade [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We obtain improved approximation ratios of fixed-price mechanisms in both (i) the full infor- mation setting, where the designer knows the value distributions of both the seller and buyer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' and (ii) the limited information settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the full information setting, we show that the optimal fixed- price mechanism can achieve at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72 of the optimal welfare, and no fixed-price mechanism can achieve more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7381 of the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Prior to our result the state of the art approximation ratio was 1 − 1 e + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0001 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='632 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We further consider two limited information settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the first one, the designer is only given the mean of the buyer’s value (or the mean of the seller’s value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show that with such minimal information, one can already design a fixed-price mechanism that achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65 of the optimal social welfare, which surpasses the previous state of the art ratio in the full information setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the second limited information setting, we assume that the designer has access to finitely many samples from the value distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Recent results show that one can al- ready obtain a constant factor approximation to the optimal welfare using a single sample from the seller’s distribution [3, 16, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our goal is to understand what approximation ratios are possible if the designer has more than one but still finitely many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This is usually a technically more challenging regime and requires tools different from the single-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We propose a new family of sample-based fixed-price mechanisms that we refer to as the order statistic mechanisms and provide a complete characterization of their approximation ratios for any fixed number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using the characterization, we provide the optimal approximation ratios obtainable by order statis- tic mechanism for small sample sizes (no more than 10 samples) and observe that they significantly outperform the single sample mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Yang Cai is supported by a Sloan Foundation Research Fellowship and the National Science Foundation Award CCF-1942583 (CAREER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Part of this work was done while the author was visiting the Simons Institute for the Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 Introduction We study a fundamental problem in mechanism design – maximizing social welfare in bilateral trade, in which two agents, a seller and a buyer, trade an indivisible item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' More specifically, we consider the Bayesian setting where the seller’s private value S for the item that is drawn from distribution FS, and the buyer’s private value B for the item is drawn from distribution FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The social welfare is therefore defined as EB,S [S +(B −S)· x(B,S)], where x(B,S) denotes the probability that the trade happens when the seller’s value is S and the buyer’s value is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Surprisingly, exactly maximizing the social welfare in bilateral trade is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The seminal re- sult by Myerson and Satterthwaite [29] shows that no mechanism can simultaneously be (i) incentive compatible (to the buyer and the seller), (ii) budget balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', the mechanism does not run a deficit, and (iii) maximizes the social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For example, the VCG mechanism is incentive compatible and maximizes the social welfare but is not budget balanced in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Motivated by this impossibility re- sult, our goal is to design incentive compatible and budget balanced mechanisms to approximate the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We focus on the fixed-price mechanisms, in which the designer offers trade at a fixed price to the seller and buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is also known that fixed-price mechanisms are the only dominant strategy incentive compatible and budge balance mechanisms in bilateral trade [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Our Contributions We make progress on this problem on multiple fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first consider the full information setting, where the designer knows both FS and FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show how to use a factor revealing min-max program to improve the approximation ratio of achievable by a fixed-price mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Contribution 1: For any FS,FB, there exists a fixed-price mechanism whose welfare is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72 ·OPT, where OPT = ES,B[max{S,B}] is the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Moreover, there exists a FS and FB such that no fixed-price mechanism can attain welfare more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7381 ·OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The formal statement of our result can be found in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We also have a “constant time” algorithm for computing the fixed-price mechanism that achieves the welfare guarantee above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' More specifically, we construct a collection of numbers p1,··· ,pn in [0,1], so that for any FS, FB, our algorithm chooses the best price in the set � p1 ·OPT,··· ,pn ·OPT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Clearly, the approximation ratio will be better when we increase n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show that when n = 16, our algorithm already computes a fixed-price mechanism that has welfare at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72·OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our result significantly improves on the state-of-the-art approximation 1− 1 e +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0001 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6322 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our new hardness result also strengthens the previous best bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7385 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our upper and lower bounds are obtained by considering two discretized variants of an infinite-dimensional min-max opti- mization problem defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 that, in the limit when the discretization accuracy approaches 0, the upper bound and lower bound obtainable by our method will converge to the optimal approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Of course, the factor-revealing program become more expensive to solve with finer discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our upper and lower bounds are derived using the finest discretization that we can computationally solve, but one could further close the gap with more computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixed-price mechanism based on only E[B] or E[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our first result requires the designer to know both FS and FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 However, information about the underlying distributions of the agents’ values is often scarce in practice, thus it is more desirable to design approximately optimal mechanisms using limited prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our second contribution concerns the case where the designer does not have the full infor- mation of the underlying distributions but only knows the mean of FS or FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1Our first result uses FS and FB in two places: (1) to compute OPT and (2) to identify the best price in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 Contribution 2: Given access to E[B] or E[S], we can design a randomized fixed-price mechanism whose welfare is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65·OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that the ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65 already exceeds the previous state-of-the-art approximation ratio in the full information setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [5, 24] consider the setting where only FS is known to the designer and show that a quantile mechanism (Mechanism 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', a fixed-price mechanism that chooses the trading price according to a distribution of quantiles of the seller’s distribution, can obtain at least 1− 1 e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6321 frac- tion of the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [24] further shows that no quantile mechanism can obtain more than 1− 1 e fraction of the optimal welfare in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This result is sometimes interpreted as saying no mech- anism can obtain an approximation ratio better than 1− 1 e with only information about the seller’s value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our second result shows that there is a strictly better way to use the information about seller’s value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Indeed, with minimal information about FS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', its mean E[S], one can design a fixed-price mechanism that strictly outperforms the optimal quantile mechanism that requires full knowledge of FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Moreover, the quantile mechanism is asymmetric and only defined when we know the seller’s value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 that this is unavoidable, as no quantile mechanism over buyer’s value distribution can guarantee a constant fraction of the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 In contrast, our second result holds when the designer only knows the mean of the buyer’s value distribution FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Mechanism 1: Quantile mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 Input: A distribution Q over the interval [0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 2 Randomly choose a quantile x ∈ [0,1] according to Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3 Output the x-quantile of the seller’s distribution as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let FS be the seller’s distribution and F −1 S (·) be seller’s quantile function mapping any quantile to its corresponding value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The quantile mechanism outputs F −1 S (x) as the price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixed-price mechanism using finitely many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we consider a different limited infor- mation model and initiate the study of approximating the optimal social welfare in using finitely many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Namely, we are given a finite and limited number of samples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', 3 or 5 samples, and the goal is to design the best mechanism possible using these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is important to distinguish this set- ting from the more standard large sample setting, where the goal is to determine the number of samples needed to design a (1−ε)-optimal mechanism (or optimal in a certain mechanism class) grows as a func- tion of 1 ε and other parameters of the mechanism design environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The sample complexity in large sample settings is usually stated using the big-O notation and ignores the accompanying constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As a result, these bounds are often vacuous when apply to the small sample regime, where there are only a small finite number of samples available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Contribution 3: We introduce a new family of mechanisms – order statistic mechanisms (Mecha- nism 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' and provide an exact characterization of the optimal order statistic mechanisms for any fixed number of samples (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using our characterization, we can compute the optimal approximation ratio obtainable for any sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Recent results show that one can already obtain a constant factor approximation to the optimal wel- fare using a single sample from the seller’s distribution [3, 16, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, techniques from these pa- pers are tailored for the single sample setting and are difficult to generalize to even the case when two samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We provide a rich family of mechanisms that is well-defined for any number of samples and characterize their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using the characterization, we manage to optimize within 2This asymmetry is due to the asymmetry of the initial allocation – the item is owned by the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 2 this family of mechanisms for any fixed number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Mechanism 2: Order statistic mechanism with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 Input: A distribution P over [N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 2 Randomly choose a number i ∈ [N] according to the distribution P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3 Given N samples from the seller, select the i-th smallest sample as the price, which is the i-th order statistic of these samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' By numerically computing the optimal approximation ratios of order statistic mechanisms, we ob- serve that the optimal order statistic mechanism with a small number of samples is usually sufficient to significantly boost the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For example, in the symmetric setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', FS = FB, five samples is sufficient to obtain an approximation ratio that is within 1% of the optimal ratio achievable by any fixed-price mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' in the asymmetric setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', FS ̸= FB, the approximation ratio improves from 1/2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='578 when the sample size increases from one to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Another natural mechanism is the empirical risk minimization (ERM) mechanism, where one selects a price to maximize the social welfare w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We compare the performance of the optimal order statistic mechanism with ERM for sample size N = 2,3,5,10 in the symmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In all cases, the order statistic mecha- nism substantially outperform the ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' See Table 1 and 2 for our computed ratios in the symmetric and asymmetric cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our analysis of the order statistic mechanisms builds on an interesting connection between the or- der statistic mechanisms and the quantile mechanisms, that is, any order statistic mechanism is also a quantile mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that the i-th order statistic over N samples drawn uniformly and indepen- dently from [0,1] has density f i N(x) = N �N−1 i−1 � xi−1 ·(1− x)N−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose we use the i-th order statistic as the price, then it is equivalent to the quantile mechanism who selects a quantile corresponding to the density function f i N(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' More generally, if we choose the i-th order statistic with probability Pi, then the order statistic mechanism is equivalent to the quantile mechanism that chooses the quantile according to the density function q(·) = �N i=1Pi f i N(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' With this connection, we can focus on quantile mechanisms, and we characterize the approximation ratio of any quantile mechanism as the solution of a minimiza- tion problem (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' By applying this characterization for quantile mechanisms to order statistic mechanisms, we show that for any fixed sample size N, the ratio of the optimal order statistic mechanism is exactly the solution of a max-min optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Despite that the optimization problem seems intractable in general, we manage to solve it with sufficient numerical accuracy for N ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Although we only study approximating social welfare in bilateral trade in this paper, we believe this perspective of viewing sample-based mechanisms through the lens of quantile mechanisms is novel and has broader applications, especially in the small sample regime where the designer only has access to finitely many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Related Work Gains from Trade Maximization in Two-Sided Markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Another important objective in two-sided mar- kets is the gains from trade (GFT), which measures the increment of the welfare after the trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that [29] also implies that optimal GFT is not achievable in bilateral trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There has been increasing interest from the algorithmic mechanism design community to study the approximability of the optimal GFT [6, 8, 12, 2, 3, 10, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It will be interesting to study the optimal approximation ratio obtainable for GFT maximization in both the full information and the limited information settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Sample-Based Mechanism Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Sample-based mechanism design has become a central topic in algorithmic mechanism design as it provides an alternative model that weakens the classical but some- times unrealistic Bayesian assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The results in this direction can be roughly partition into two groups: (1) Large sample results, where the goal is to determine the number of samples needed to design 3 a 1 − ε-optimal mechanism (or optimal in a certain mechanism class) as a function of 1 ε and other pa- rameters of the mechanism design environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', [17, 11, 26, 21, 31, 28, 27, 9, 7] or (2) Single sample results, where the goal is to determine the optimal approximation ratio obtainable using a single sample, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',[18, 15, 19, 24, 20, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our result does not fit in to either of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In particular, we study the regime where the designer has a small fixed number of samples, as a result, the machinery developed for large number of samples or a single sample does not apply to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' A recent line of works focus on the same regime as ours but for the monopolist pricing problem [4, 13, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Due to the different nature of the studied problems, their techniques also do not apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 2 Preliminaries Bilateral Trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We study the bilateral trade problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In this setting, there are two agents, a buyer and a seller, trade a single indivisible item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The seller owns the item and values it at S while the buyer values the item at B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Both S and B are non-negative and unknown to us but they are respectively drawn from distributions FS and FB independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We assume that FS and FB are continous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Actually, such assumption is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' and we discuss the reduction from distributions with point masses to continous ones in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixed-price Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We consider fixed-price mechanisms, which offer a price p to trade the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The trade happens if and only if both the seller and the buyer accept the price, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', B ≥ p ≥ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As shown by [23], fixed-price mechanism is the only dominant-strategy incentive-compatibility mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In this paper, we consider (possibly randomized) fixed-price mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We abuse notation and use M(FS,FB) or M(I) where I = (FS,FB) to denote the distribution of prices p selected by mechanism M on instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Welfare and Approximation Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We consider the objective of social welfare in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For an instance I = (FS,FB), the optimal welfare is defined as: OPT-W (I) = E S∼FS,B∼FB [max(S,B)] Similarly, for a fixed-price mechanism M, the expected welfare on instance I can be written as: W (M,I) = E S∼FS ,B∼FB p∼M(FS ,FB ) � S + 1[S ≤ p ≤ B]·(B −S) � Our goal is to maximize the approximiation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' That is, find some mechanism M maximize the following ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' min I =(FS,FB) W (M,I) OPT-W (I) Quantile Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose F(·) is the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of a distribution, and we define F −1(·) as the quantile func- tion mapping the quantile to its corresponding value in this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' That is, F −1(x) = inf{y | F(y) = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3 A Near-Optimal Mechanism in the Full Information Setting In this section, we show a near-optimal fixed-price mechanism when given the full information of the buyer and the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There exists a DSIC, individually rational, budget balanced mechanism that achieves at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72 fraction of the optimal welfare for any instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Moreover, no such mechanism has an approximation ratio better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To prove this, we first identify the best fixed-price mechanism when given the instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then, the approximation ratio is determined by the mechanism’s performance on the worst-case in- stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Such a worst-case instance could be characterized by an infinite dimensional quadratically con- strained quadratic program (QCQP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, the infinite dimensional program is hard to solve directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Instead, we use two finite programs that can be solved numerically to upper bound and lower bound the infinite dimensional program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Additionally, we show that the optimal solutions of these two programs converge to the optimal solution of the infinite dimensional program as the number of variables tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Characterizing the Optimal Mechanism We first characterize the optimal fixed-price mechanism via an infinite dimensional QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Given any instance I = (FS,FB), we could assume that OPT-W (I) = 1 without loss of generality since we can al- ways scale the instance so that this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The optimal fixed-price mechanism corresponds to choosing a price p ∈ argmaxp W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The following program captures the worst-case instance for fixed-price mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The Optimization Problem FullOp min µ,ν,r r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' µ,ν are probability measures defined on R≥0 (1) OPT-W (I) def= � R≥0 � R≥0 max(x, y)ν(d y)µ(dx) ≥ 1 W (I,t) def = � R≥0 x µ(dx)+ � R≥0 � R≥0 (y − x)· 1[x ≤ t ≤ y]ν(d y)µ(dx) ≤ r ∀t ≥ 0 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The value of the optimal solution of FullOp is the tight worst-case approximation ratio achievable by a fixed-price mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 is postponed to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since it is difficult to directly solve an infinite dimensional program like FullOp, we approximate FullOp from both above and below by constructing two families of finite programs which provide an upper bound and a lower bound respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Factor Revealing Program for the Approximation Ratio under Full Information We show that the approximation ratio of the optimal fixed price mechanism is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72, which signif- icantly improves the previous state of the art bound of 1−1/e +ε with ε ≈ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our approach is to find a fixed-price mechanism whose performance under the worst distribution is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This is exactly captured by the optimization problem FullOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, it is an infinite-dimensional program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In this section, we consider a discretized version of FullOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' More specifically, we assume that OPT-W (I) = 1, and we restrict the mechanism to only choose price from a finite set P = {p1,p2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',pk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What we manage to show is that the optimal value of the optimization problem LowerOp is indeed a lower bound on the maximum approximation ratio one can obtain using prices from P for instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We establish the fol- lowing two crucial properties: (i) For any I = (FS,FB) satisfying OPT-W (I) = 1, we can carefully round 5 FS and FB to two discrete distributions supported on P, where {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',sn} and {b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',bn} can be viewed as the corresponding “probability mass function” for the discretized distributions of the seller and the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Importantly, {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',sn} and {b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',bn} satisfy inequalities (2) - (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (ii) For any price pt, the wel- fare from the corresponding fixed-price mechanism under I is at least the welfare under the rounded distributions �n i=1 si pi + �t−1 i=1 �n j=t+1 sib j(p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, if we choose r to be maxt∈[n] W (I,pt), {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',sn}, {b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',bn}, and r form a feasible solution of LowerOp, which implies that the optimal value of LowerOp is no greater than the constructed r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As the rounded distribution needs to satisfy a sequence of constraints (especially constraint (5)), the procedure we use to round FS and FB is subtle and does not simply round things up or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The Optimization Problem LowerOp min s1,s2···,sn b1,b2,···,bn,r r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' si,bi ≥ 0 ∀i ∈ [n] (2) n� i=1 si ≥ 1 and n� i=1 bi ≥ 1 (3) n� i=1 si ≤ 1+ 1 pn and n� i=1 bi ≤ 1+ 1 pn (4) n� i=1 n� j=1 sib j max(pi,p j) ≥ 1 (5) n� i=1 si pi + t−1 � i=1 n� j=t+1 sib j(p j − pi) ≤ r ∀t ∈ [n] (6) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any 0 = p1 < p2 < ··· < pn, let r ∗ be the optimal value of LowerOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose M is the mechanism that chooses the best price from the set � p1·E[max(S,B)],p2·E[max(S,B)],··· ,pn·E[max(S,B)] � to maximize the welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The welfare obtained by M is at least r ∗ ·OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We defer the proof of the lemma to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Hardness Result under Full Information In this section, our goal is to find a threshold and an instance such that no fixed-price mechanism has an approximation ratio better than the threshold on this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We focus on discrete distributions and consider an instance I = (FS,FB) where FS is a discrete distribution supported on {p1+ε,p2+ε,··· ,pn + ε}, and FB is a discrete distributions supported on P = {p1,p2,··· ,pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For such instance, the optimal price must also lie in the set {pi +ε}i∈[n], as choosing a price x where pi +ε ≤ x < pi+1+ε is equivalent to choosing a price of pi +ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, any valid solution for the optimization problem below corresponds to a hard instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any valid solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r) of UpperOp (defined in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3) satisfying r = maxt∈[n] �n i=1 si pi +�t i=1 �n j=t+1 sib j(p j −pi) and ε > 0, there exists an instance I = (FS,FB) such that no fixed-price mechanism can achieve more than (r +ε)-fraction of the optimal welfare on this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 3For technical reasons, {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',sn} and {b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',bn} do not exactly correspond to probability mass functions, but viewing them as the probability mass functions gives the right intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 6 The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' With Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, we are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the numerical results, our anonymous GitHub repository(https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='com/BilateralTradeAnonymou s/On-the-Optimal-Fixed-Price-Mechanism-in-Bilateral-Trade) provides all the certificates and codes and also carefully explains all the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the lower bound, we choose n = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using Gurobi [22], we obtain a lower bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72 for the optimization problem LowerOp for a carefully chosen set of price {p1,p2,··· ,pn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, there exists a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72-approximate fixed-price mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Things become much easier for the upper bound since we only need to find a feasible solution in- stead of proving a lower bound of the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We choose n = 100 and numerically solve UpperOp with a specific support {p1,p2,··· ,pn} and find a feasible solution that satisfies the constraints in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 where r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, we then find a hard instance such that no fixed-price mech- anism attains a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7381-approximation of the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Please check our GitHub repository for the detailed specification of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we would like to point out that the optimal value obtained by LowerOp and UpperOp will converge to the optimal value as the discretization accuracy tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let r ∗ be the optimal value of FullOp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the optimal approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any ε > 0, there exists two sets numbers 0 = p1 < p2 < ··· < pn and {p′ 1,p′ 2,··· ,p′ n′} such that the optimal value of LowerOp with respect to {p1,p2,··· ,pn} is at least r ∗−ε and the optimal value of UpperOp w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' {p′ 1,p′ 2,··· ,p′ n′} is at most r ∗ +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 4 Breaking 1−1/e with Limited Information We consider the limited information setting where we only knows the mean of the seller or the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [24] shows that any mechanism that only uses quantile information from the seller can not achieve a better performance of 1−1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, we observe that even with minimal information of FS such as its mean E[S] (or similarly E[B]), we can break the 1−1/e barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We again provide a factor revealing program for this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Although it looks similar to LowerOp, there is some subtle differences in how we discretize a continuous distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' See Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following fixed-price mechanism: Given E[B] (or E[S]), it randomly pick a number x ∼ P according to a distribution P, and selects x · E[B] (or x · E[S]) as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There exists a distribution PS for the seller and a distribution PB for the buyer such that the corresponding fixed-price mechanism achieves at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65·OPT welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The high level idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 shows us how to discretize a continuous distribution so that E[max(S,B)] increases and E[W (I,pt)] decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In other words, the discretization worsens the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we could use a similar technique to derive a lower bound of the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The complete proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 is in Appendix C 4We choose {p1,p2,··· ,p16} to be {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='19,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='27,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='315,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='355,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='395,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='44,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='485,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='535,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='595,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='665,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='74,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='875,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='195,1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0} to derive the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' These numbers are chosen heuristically to provide good coverage between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5, which is the region with concentration of probability mass in some bad instances we encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 7 5 Fixed-Price Mechanism with Different Numbers of Samples In this section, we consider the limited information setting where we only have sample access to the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We focus on order statistic mechanisms which is defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and our results cover different number of samples for both symmetric and general instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the small sample regime, we are able to characterize the optimal order statistic mechanism with any fixed number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When the number of samples goes to infinity, we show that the optimal quantile mechanism can be approximated by order statistics mechanism as closely as desired and also obtain an upper bound on the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, recall that we assume the distributions for the seller and the buyer are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' See Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Order Statistic Mechanisms To start with, we briefly discuss these two families of mechanisms that is used in the sample setting and give high level ideas on how to design the order statistic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Order statistic mechanisms will be used when we only have samples from the distribution and quantile mechanisms will help us analyze the performance of order statistic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Actually, we will point out that quantile mechanisms and order statistic mechanisms are equivalent in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Connection Between Two Mechanisms Next we aim to show the connection between these two mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Such observations give us insights on designing mechanisms with small or large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The order statistic mechanism is a special kind of quantile mechanisms First, we can see that the following two operations are equivalent: Draw a sample from distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Uniformly sample a quantile x from [0,1], and use F −1(x) as the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now suppose f i N(x) = N �N−1 i−1 � xi−1 ·(1− x)N−i be the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the i-th order statistic over N samples drawn uniformly and independently from [0,1] and let Pi to be Prx∼P[x = i] for any distribution P over [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using similar ideas above, it can be proved that any order statistic mechanism P is equivalent to a quantile mechanism Q with probability density function q(x) = N� i=1 Pi f i N(x) Therefore, we can analyze the approximation ratio of quantile mechanism Q instead of order statistic mechanism P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' If we are able to compute the approximation ratio of any quantile mechanism Q, it fol- lows that we can also characterize the optimal order statistic mechanism exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When the number of samples are small, we can have a fine-grained analysis of the order statistic mechanisms and use these limited samples carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 actually follow such intuitions to characterize the best possible order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Quantile mechanisms can be approximated by order statistic mechanisms within any small error Our goal is that for any quantile mechanism Q with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q(x), we need to find some integer N and a distribution P over [N], such that q(x) ≈ N� i=1 Pi f i N(x) 8 Since �N i=1Pi f i N(x) is a polynomial of degree N − 1, this could be done for any continuous q(x) on [0,1] since the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more interesting is that {f i N (x)}N i=1 are Bernstein basis polynomials and there are a series of work show- ing that (stochastic) Bernstein polynomials can efficiently and uniformly approximate to any continous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we can have an asymptotic analysis of the order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, such observation also shows that we have a block-box transformation from any quantile mechanism to mechanisms only using samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 uses such techniques and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Small Sample Regime In this section, we characterize the optimal order statistic mechanisms with any fixed number of sam- ples for both symmetric and general instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first show that, in any setting, if we are able to give a tight analysis of the quantile mechanism, we could directly characterize the optimal order statistic mechanism with any fixed number of samples via an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the next, we show a tight analysis of the quantile mechanism on both symmetric and general instances, and thus we obtain the characterization of the optimal order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Recall that an order statistics mechanism with N samples randomly choose a number i ∈ [N] ac- cording to a previously defined distribution P and select the i-th smallest sample as the price, and a quantile mechanism randomly choose a quantile x ∈ [0,1] from a determined distribution Q and choose the x-quantile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' F −1 S (x), as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since every quantile mechanism and order statistic mechanism is determined by the previously defined distribution, we abuse the notation and use distribution P over [N] denote its corresponding order statistic mechanism and distribution Q over [0,1] denote its corre- sponding quantile mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose C : ∆([0,1]) �→ R maps every quantile mechanism P to its exact approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let P (Q) be the corresponding quantile mechanism of the order statisticmechanismQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixing the number of samples N, the optimal order statistic mechanism with N samples Q∗ N is characterized by the following optimization problem: Q∗ N = arg max Q∈∆N C (P (Q)) where ∆([0,1]) is the set of all distributions over [0,1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the set of all quantile mechanisms, and ∆N is the set of all distributions over [N], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the set of all order statistic mechanisms with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 is quite straightforward and thus is postponed to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Symmetric Instances Now we study the case when the distributions are symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', FS = FB, which means that the seller’s value S and the buyer’s value B are drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For simplification, we will use F to refer to their distributions in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In order to find out the optimal order statistic mechanism, we need to first give a tight analysis of the quantile mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism for symmetric instance with distribution Q over [0,1], the ap- proximation ratio is exactly inf x∈[0,1) � [0,x] t(1− x)dQ(t)+ � (x,1](1− t)x dQ(t)+(1− x) 1− x2 where Q(t) is the cumulative distribution function of distributionQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 9 Therefore, combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we could characterize the optimal order statistic mechanism via an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The optimal order statistic mechanism with N samples for symmetric instances is the solu- tion to the following optimization problem: P∗ N = max P1,P2,···,PN ≥0 �N i=1 Pi =1 inf x∈[0,1) �x 0 p(t)t(1− x)dt + �1 x p(t)(1− t)x dt +(1− x) 1− x2 where p(t) = �N i=1Pi f i N(x) and f i N(x) = N �N−1 i−1 � xi−1 ·(1− x)N−i is the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the i-th order statistic over N samples drawn uniformly and independently from [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 is in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It turns out the optimization above is computationally tractable when N is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We solve the optimization problem and find out the optimal order statistic mechanism numerally with different numbers of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To compare with the order statistic mechanisms, we also consider the most natural sample-based mechanism – the Empirical Risk Minimization mechanism (ERM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first provide the formal definition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 (Empirical Risk Minimization Mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Given N samples X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', XN drawn from F, define ˜F be the empirical distribution of these N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' That is to say, ˜F is the distribution with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' ˜F(x) satisfying: ˜F(x) = 1 N N � i=1 1[x ≥ Xi] The Empirical Risk Minimization mechanism (ERM) is the mechanism that computes the optimal price according the empirical distribution ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In particular, for N samples X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', XN, ERM(X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', XN) = argmax p E S∼ ˜F,B∼ ˜F [S +(B −S)· 1[B ≥ p ≥ S]] If there are multiple prices p that maximize the expected welfare, the ERM mechanism may select any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For N = 1,2,3,5,10, we compute the approximation ratio of order statistic mechanisms and also show the upper bound of ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The results are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To prove the upper bound, we use a counter example in [24] and show that ERM has a bad performance on this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We defer the complete proof of the upper bound of ERM to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 and the details of numerical results to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' #Samples Order Statistics Mechanism ERM 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='821 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='67 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='822 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='75 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='847 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='76 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='852 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='80 ∞ 2+ � 2 4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='8536 / Table 1: Approximation Ratios with Different Number of Samples in the Symmetric Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 General Instances Now we consider the general setting, where the buyer’s distribution may be different from the seller’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Recall that we only consider mechanisms over seller’s information since there is no constant quantile or order statistic mechanism over seller’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using similar ideas, we first show a tight analysis re- garding quantile mechanisms, which would guide us to discover the optimal order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 of [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism Q (over seller’s distribution) with cumula- tive distribution function Q, its approximation ratio is exactly min x∈[0,1] � [0,x] t dQ(t)+1− x Similarly, combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, we are able to charaterize the optimal order statistic mechanism over with N samples from seller’s distribution by an optimization problem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The optimal order statistic mechanism with N samples for symmetric instances is the solu- tion to the following optimization problem: P∗ N = max P1,P2,···,PN ≥0 �N i=1 Pi =1 min x∈[0,1] � [0,x] t · p(t)dt +1− x, where p(x) = �N i=1Pi · f i N (x) and f i N (x) = N �N−1 i−1 � xi−1·(1−x)N−i is the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the i-th order statistic over N samples drawn uniformly and independently from [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 is deferred to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, such optimization problem is easy to solve when the number of samples N is not to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We solve the optimization problem numerically for N = 1,2,3,5,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that we do not compare our mechanism to the Empirical Risk Minimization mechanism in the general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This is because we only have sample access to the seller’s distribution, and the ERM can not be implemented without the buyer’s samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The details of numerical results is defered to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' #Samples Order Statistic Mechanism 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='531 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='578 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='601 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='615 ∞ 1− 1 e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='632 Table 2: Approximation Ratios with Different Number of Samples In the General Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Asymptotic Analysis: From Quantile to Order Statistics In this section, we turn to the case when the number of samples tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As we show in sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, we could approximate any quantile mechanisms by order statistic mechanisms within any small error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using such ideas, we provide a "black-box" reduction that allows us to convert any quantile mechanism with continuous probability density function q(x) to order statistic mechanism with N sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Here N is usually a polynomial of 1 ε, as long as the probability density function is not too crazy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We now formally write it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 11 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let C : ∆([0,1]) �→ R be a function that maps every quantile mechanism Q with continuous probability density function to its approximation ratio such that for any quantile mechanism Q1 with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q1(x) and quantile mechanism Q2 with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q2(x), it holds that C (Q1)−C (Q2) ≥ −c · ��q1 − q2 �� ∞ where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now let Q be any quantile mechanism with continuous probability density function q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Define M as maxx∈[0,1] q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any ε > 0, suppose n is a positive integer satisfying that ω � 1 � n −1 � ≤ ε/100 (7) 2n exp \uf8eb \uf8ed− ε2 8ω2 � 1 � n−1 � \uf8f6 \uf8f8 ≤ ε (8) exp � − ε2n 48M3 � ≤ ε/2 (9) where w(h) = sup x,y∈[0,1] |x−y|≤h ��q(x)− q(y) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then, there exists an order statistic mechanism with n samples that achieves an approximation ratio of C (Q)−c ·ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The high level idea of the proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since we know that probability density functions of order statistics form Bernstein basis polynomials, we could approximate the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the quantile mechanism q(x) within any small error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Inequality (7), (8) and (9) actually help us to get an order statistic mechanism whose corresponding distribution of quantile is close to the desired quantile mechanism Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, by the property of C , we know that their approximation ratio is also close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof is postponed to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we show that we could apply lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 to both the symmetric and general settings and convert the optimal quantile mechanism to order statistic mechanism within a error of at most ε using poly � 1 ε � samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We leave the details of such applications to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' References [1] Amine Allouah, Achraf Bahamou, and Omar Besbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Revenue Maximization from Finite Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In Proceedings of the 22nd ACM Conference on Economics and Computation, EC ’21, page 51, New York, NY, USA, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Journal of Economic Theory, 42(1):94–107, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [24] Zi Yang Kang, Francisco Pernice, and Jan Vondrák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixed-price approximations in bilateral trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2964– 2985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' SIAM, 2022.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In ICML, pages 262–270, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [27] Jamie Morgenstern and Tim Roughgarden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Learning simple auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In Proceedings of the 30th Annual Conference on Learning Theory (COLT), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [28] Jamie H Morgenstern and Tim Roughgarden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' On the pseudo-dimension of nearly optimal auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In Proceedings of the the 29th Annual Conference on Neural Information Processing Systems (NIPS), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [29] Roger B Myerson and Mark A Satterthwaite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Efficient mechanisms for bilateral trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Journal of economic theory, 29(2):265–281, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Publisher: Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [30] Xingping Sun, Zongmin Wu, and Xuan Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' On probabilistic convergence rates of stochastic bern- stein polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', 90(328):813–830, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [31] Vasilis Syrgkanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' A sample complexity measure with applications to learning optimal auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 5352–5359, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' A Tie Breaking For distribution D with point masses, the following reduction will convert it to continuous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We will overload the notation of D and think of it as a bivariate distribution with the first coordinate drawn from the previous single-variate distribution D and the second tie-breaker coordinate drawn independently and uniformly from [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' And (X1,t1) > (X2,t2) if and only if either X1 > X2, or X1 = X2 and t1 > t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since 14 the tie-breaker coordinate is continuous, the probability of having (X1,t1) = (X2,t2) for any two values during a run of any mechanism is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore we could define the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of D as FD(X ,t) = Pr (Y,u)∼(D,U[0,1])[(Y ,u) < (X ,t)] Remind the second coordinate is only used to break ties, and it does not affect the calculation of welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' After including the additional random variable, we can see that D has been converted into a continuous distribution since its second coordinate is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' B Missing Proofs in Section 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 We first show the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The approximation ratio of the optimal fixed-price mechanism could be written as min I =(FS,FB)max p∈R W (I,p) OPT-W (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first show that for any instance I = (FS,FB), there is a valid solution (u,v,r) such that r = maxp∈R W (I ,p) OPT-W (I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could first simply scale the instance by 1 OPT-W (I ) to I ′ = (F ′ S,F ′ B) where OPT-W (I ′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Such scaling means that W (I,OPT-W (I)· p) = OPT-W (I)·W (I ′,p) for all p ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This implies that max p∈R W (I,p) OPT-W (I) = max p∈R W (I ′,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, let u and v be the probability measures of F ′ S and F ′ B and r be maxp∈R W (I ′,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is easy to verify that (u,v,r) is a valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let r ∗ be the optimal value of FullOp, this implies that r ∗ ≤ maxp∈R W (I ,p) OPT-W (I ) for any instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Taking the minimum over all possible I, we then get that r ∗ ≤ min I =(FS,FB)max p∈R W (I,p) OPT-W (I) (10) Next, let (u∗,v∗,r ∗) be the optimal solution of FullOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since u∗,v∗ are both probability measures, let F ∗ S ,F ∗ B be the corresponding distributions of u and v and I ∗ = (F ∗ S ,F ∗ B) be the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now by the constraint of FullOp, we know that OPT-W (I ∗) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Besides, (u∗,v∗,r ∗) is an optimal solution implies that r ∗ = maxp∈R W (I ∗,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, r ∗ = max p∈R W (I ∗,p) ≥ max p∈R W (I ∗,p) OPT-W (I ∗) ≥ min I =(FS,FB)max p∈R W (I,p) OPT-W (I) (11) Now combining inequality (10) and (11), it follows that r ∗ = min I =(FS,FB)max p∈R W (I,p) OPT-W (I) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 15 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Before we give the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we first show prove a lemma that helps us discretize a continuous distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any instance I = (FS,FB), and 0 = p1 < p2 < ··· < pn, there exists a set of numbers {si}i∈[n],{bi}i∈[n] satisfying the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' si,bi ≥ 0 ∀i ∈ [n] (12) 1 ≤ n� i=1 si ≤ 1+ E[S] pn 1 ≤ n� i=1 bi ≤ 1+ E[B] pn (13) n−1 � i=1 si ≤ 1 n−1 � i=1 bi ≤ 1 (14) E S∼FS [S] = n� i=1 sipi (15) E B∼FB [B] = n� j=1 b j p j (16) OPT-W (I) def= E S∼FS B∼FB [max(S,B)] ≤ n� i=1 n� j=1 sib j ·max(pi,p j) (17) W (I,pt) def= E S∼FS [S]+ E S∼FS B∼FB � (B −S)· 1[S ≤ pt ≤ B] � ≥ n� i=1 sipi + t−1 � i=1 n� j=t+1 sib j(p j − pi) ∀t ∈ [n] (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We construct (s1,··· ,sn,b1,··· ,bn) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the seller, define qs,i = Pr S∼FS [pi ≤ S < pi+1] and Es,i = E S∼FS [S · 1[pi ≤ S < pi+1]], ∀i ∈ [n], where we assume that pn+1 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is clear from the definition that qs,i · pi ≤ Es,i ≤ qs,i · pi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, for any i ∈ [n −1], there exists non-negative numbers si,LEFT and si+1,RIGHT such that si,LEFT + si+1,RIGHT = qs,i and si,LEFT · pi + si+1,RIGHT · pi+1 = Es,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (19) We further define sn,LEFT as Es,n/pn and s1,RIGHT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now set si = si,LEFT + si,RIGHT for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the buyer, we define {bi}i∈[n] similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We now verify that ({si}i∈[n],{bi}i∈[n]) satisfies the properties above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The non-negativity of si and bi is immediately derived from si,LEFT,si,RIGHT ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' From our definition, it is clear that �n i=1 qs,i = 1, therefore n� i=1 si = n� i=1 si,LEFT + si,RIGHT ≥ n� i=1 qs,i = 1, and n−1 � i=1 si = n−1 � i=1 si,LEFT + si,RIGHT ≤ n� i=1 qs,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could also see that n� i=1 si = n� i=1 si,LEFT + si,RIGHT ≤ n� i=1 qs,i + sn,LEFT = 1+ Es,n pn ≤ 1+ E[S] pn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 16 For the expectations, it holds that n� i=1 si pi = n� i=1 � si,LEFT + si,RIGHT � pi = n� i=1 Es,i = E[S] By symmetry, similar inequalities also holds for for {bi}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' So far, we have verified that properties (12), (13) , (14), (15) and (16) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It only remains to show that (17) and(18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any i ̸= j ∈ [n], w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' we can assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that E S∼FS B∼FB � max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='B)· 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] 1[B ∈ [p j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j+1)] � = E S∼FS B∼FB � B · 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] 1[B ∈ [p j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j+1)] � = E B∼FB � B · 1[B ∈ [p j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j+1)] � Pr S∼FS � S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1) � = Eb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='j · qs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i = (b j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT · p j +b j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT · p j+1)·(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT + si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT) = si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT ·b j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFTmax(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j)+ si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT·b j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFTmax(pi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j) + si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT ·b j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHTmax(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j+1)+ si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT·b j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHTmax(pi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j+1) (20) The second equality is due to the independence between S and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now consider the case when i = j ≤ n −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any x, y ∈ [pi,pi+1], we have max(x, y) ≤ max(pi, y)· pi+1 − x pi+1 − pi +max(pi+1, y)· x − pi pi+1 − pi , as pi+1−x pi+1−pi + x−pi pi+1−pi = 1 and pi+1−x pi+1−pi · pi + x−pi pi+1−pi · pi+1 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Based on the inequality above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' for any fixed y ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' we have E S∼FS � max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' y)· 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � ≤ E S∼FS �� max(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' y)· pi+1 −S pi+1 − pi +max(pi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' y)· S − pi pi+1 − pi � 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � =max(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' y) E S∼FS � pi+1 −S pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � +max(pi+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' y) E S∼FS � S − pi pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � =y · si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT + pi+1 · si+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT The last equality is because of the following identities: E S∼FS � pi+1 −S pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � + E S∼FS � S − pi pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � = Pr[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)]] pi · E S∼FS � pi+1 −S pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � + pi+1 · E S∼FS � S − pi pi+1 − pi 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)] � = E[S · 1[S ∈ [pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pi+1)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Hence, we can conclude that � ES∼FS � pi+1−S pi+1−pi · 1[S ∈ [pi,pi+1)] � ,ES∼FS � S−pi pi+1−pi · 1[S ∈ [pi,pi+1)] �� is the unique solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, these two numbers respectively equal to si,LEFT and si+1,RIGHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 17 Due to the inequality above, we have E B∼FB � E S∼FS � max(S,B)· 1[S ∈ [pi,pi+1)] � 1[B ∈ [pi,pi+1)] � ≤ E B∼FB �� B · si,LEFT + pi+1 · si+1,RIGHT � 1[B ∈ [pi,pi+1)] � =bi,LEFTsi,LEFT · pi +bi+1,RIGHTsi,LEFT · pi+1 +bi,LEFTsi+1,RIGHT· pi+1 +bi+1,RIGHTsi+1,RIGHT· pi+1 (21) The last special case is when i = j = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' E S∼FS B∼FB � max(S,B)· 1[S ≥ pn] 1[B ≥ pn] � ≤ E S∼FS B∼FB � BS/pn · 1[S ≥ pn] 1[B ≥ pn] � = pn · � E S∼FS � S · 1[S ≥ pn] � /pn � � E S∼FB � B · 1[B ≥ pn] � /pn � = sn,LEFT ·bn,LEFT · pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (22) Combining inequality (20), (21) and (22), we have n� i=1 n� j=1 sib j ·max(pi,p j) ≥ n� i=1 n� j=1 E S∼FS B∼FB � max(S,B)1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] � = E S∼FS B∼FB [max(S,B)], so inequality (17) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we are only left to show that property (18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any t ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' it follows that W (I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='pt) = E S∼FS [S]+ E S∼FS B∼FB � (B −S)· 1[S ≤ pt ≤ B] � ≥ E S∼FS [S]+ E S∼FS B∼FB � (B −S)· 1[S < pt ≤ B] � = n� i=1 Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i + E B∼FB [B · 1[B ≥ pt]]· Pr S∼FS [S < pt]− E S∼FS [S · 1[S < pt]]· Pr B∼FB [B ≥ pt] ≥ n� i=1 si · pi + � n� j=t+1 bj · p j +bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT · pt � � t−1 � i=1 si + st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT � − � t−1 � i=1 si · pi + st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT · pt � � n� j=t+1 bj +bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT � = n� i=1 si · pi + t−1 � i=1 n� j=t+1 si bj (p j − pi)+ n� j=t+1 bj · st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='RIGHT ·(p j − pt)+ t−1 � i=1 si ·bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='LEFT ·(pt − pi) ≥ n� i=1 si · pi + t−1 � i=1 n� j=t+1 si bj (p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (23) where the second inequality follows from the fact that Pr[B ≥ pt] = n−1 � j=t (b j,LEFT +b j+1,RIGHT)+Pr[B ≥ pn]≤ n−1 � j=t (b j,LEFT +b j+1,RIGHT)+bn,LEFT ≤ n� j=t+1 b j +bt,LEFT Therefore, we could see that inequality (18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 18 With the lemma above, we are ready to give the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following fixed-price mechanism: Given any instance I = (FS,FB), we first compute the optimal welfare of the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose OPT-W (I) = c, we choose the fixed price p∗ from {cp1,··· ,cpn} to maximizes the welfare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', p∗ ∈ argmaxp∈{cp1,···,cpn}W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the following, we show that this mech- anism is an r ∗-approximation to the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that the approximation ratio of our mechanism is independent of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 To keep our analysis clean, we first assume that the instance I = (FS,FB) has optimal welfare 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The approximation ratio of our mechanism could be written as min I =(FS,FB ) OPT-W (I )=1 max p∈{p1,p2,···,pn}W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Next, we argue that for any instance I = (FS,FB) satisfying OPT-W (I) = 1, there exists a valid solu- tion (s1,··· ,sn,b1,··· ,bn,r) of LowerOp such that r ≤ maxp∈{p1,p2,···,pn} W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This immediately implies that r ∗ is a lower bound of the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Given an instance I = (FS,FB) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' OPT-W (I) = 1, the solution (s1,··· ,sn,b1,··· ,bn,r) is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let (s1,s2,··· ,sn,b1,b2,··· ,bn) be the set of numbers that satisfies all the properties stated in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let r be max t∈[n] n� i=1 si pi + t−1 � i=1 n� j=t+1 sib j(p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first verify that ({si}i∈[n],{bi}i∈[n],r) is a valid solution of LowerOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that E[S] ≤ E[max(S,B)] = 1 and E[B] ≤ E[max(B,S)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, constraints (2), (3) and (4) directly follows from inequality (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, we could see (6) holds by the definition of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now by property (17), we have n� i=1 n� j=1 sib j ·max(pi,p j) ≥ n� i=1 n� j=1 E S∼FS B∼FB � max(S,B)1[S ∈ [pi,pi+1)] 1[B ∈ [p j,p j+1)] � = 1, so constraint (5) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we are only left to show that the best price in {p1,p2,··· ,pk} must obtain an approximation ratio that is at least r on instance I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', r ≤ maxp∈{p1,p2,···,pk}W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Inequality (18) states that W (I,pt) ≥ n� i=1 si pi + t−1 � i=1 n� j=t+1 sib j(p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Taking maximum over t ∈ [n], we then get that r = max t∈[n] n� i=1 si · pi + t−1 � i=1 n� j=t+1 sib j(p j − pi) ≤ max t∈[n] W (I,pt) which finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 In the following, we complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 5The price p∗ depends on c, but the approximation ratio to the optimal welfare does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 19 The Optimization Problem UpperOp min s1,s2···,sn b1,b2,···,bn,r r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' si,bi ≥ 0 ∀i ∈ [n] n� i=1 si = 1 and n� i=1 bi = 1 n� i=1 n� j=1 sib j max(pi,p j) ≥ 1 n� i=1 si pi + t� i=1 n� j=t+1 sib j(p j − pi) ≤ r ∀t ∈ [n] For any fixed support 0 = p1 < p2 < ··· < pn and a valid solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r), define an instance I = (FS,FB) satisfying S ∼ FS,S = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 p1 +ε w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' s1 p2 +ε w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' s2 ··· pn +ε w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' sn B ∼ FB,B = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 p1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' b1 p2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' b2 ··· pn w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' bn where ε > 0 is a constant that small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is easy to see that both FS and FB are valid distributions since the UpperOp requires the non- negativity of si,bi and �n i=1 si = �n i=1bi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Next, we aim to show that no fixed-price mechanism have an approximation ratio of r +ε on this instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any x ∈ R≥0, we could first see that x < ε would never be a optimal price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus let pi be the largest p ∈ {p1,p2,··· ,pn} that is not greater than x −ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that both FS is a distribution on support {pi +ε}i∈[n] and FB is a discrete distribution on support {pi}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This means choosing pi + ε instead of x would never become worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we could see that the optimal fixed-price mechanism on this instance is simply choosing one pt ∈ {p1,p2,··· ,pk} that maximizes W (I,pt +ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again, by the fact that FS and FB are discrete distributions, W (I,pt +ε) could be written as: W (I,pt +ε) = n� i=1 (pi +ε)si + t� i=1 n� j=t+1 sib j(p j − pi −ε) ≤ n� i=1 pi si + t� i=1 n� j=t+1 sib j(p j − pi)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Also notice that the constraints of UpperOp guarantee that OPT-W (I) = n� i=1 n� j=1 sib j max(pi +ε,p j) ≥ n� i=1 n� j=1 sib j max(pi,p j) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, the approximation ratio of the optimal fixed-price mechanism on instance I = (FS,FB) is upper bounded by maxt∈[n] W (I,pt +ε) OPT-W (I) ≤ max t∈[n] n� i=1 pi si + t� i=1 n� j=t sib j(p j − pi)+ε = r +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' And this finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 20 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 In this section, we assume that ε > 0 is a small enough constant such that ε2 ≪ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first show that, for any ε > 0, there exists a set of support {p1,p2,··· ,pn} such that UpperOp has an optimal value of at most r ∗ +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As we show before, we could assume that the instance I = (FS,FB) has optimal welfare 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, the approximation ratio of the optimal fixed-price mechanism is r ∗ = min I =(FS ,FB ) OPT-W (I )=1 max p∈R W (I,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose r ∗ is attained at I ∗ = (F ∗ S ,F ∗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now define n = 1/ε4, and pi = i ·(1/ε2)+ε/2 for i ∈ [n +1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our idea is to construct a valid solution {si,bi}i∈[n+1] by rounding up I ∗ to pi and show that this solution has an objective value that is close to r ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now we define si = Pr S∼F ∗ S � S ∈ [(i −1)ε2,iε2) � and bi = Pr B∼F ∗ B � B ∈ [(i −1)ε2,iε2) � for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Especially, let sn+1 = E S∼F ∗ S � S · 1[S ≥ nε2] � / � nε2� and bn+1 = E B∼F ∗ B � B · 1[B ≥ nε2] � / � nε2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since E[S] and E[B] are upper bounded by 1, we could see that sn+1,bn+1 ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the last, let s = �n+1 i=1 si and b = �n+1 i=1 be the normalization factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It’s also straightforward to see that s ≤ �n i=1 si +sn+1 ≤ 1+ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Following the same argument, it also holds that b ≤ 1+ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now define r = max t∈[n] n� i=1 (si/s)pi + t� i=1 n� j=t+1 (si/s)(b j/b)(p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We aim to verify that (s1/s,s2/s,··· ,sn+1/s,b1/b,b2/b,··· ,bn+1/b,r) is a valid solution of UpperOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is easy to see the non-negativity of si,bi and �n+1 i=1 si/s = �n+1 i=1 bi/b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, from the definition of r, we could see the last constraint holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now we only need to check the third constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any i, j ∈ [n], it holds that E S∼F∗ S B∼F∗ B � max(S,B)1[S ∈ [(i −1)ε2,iε2)] 1[B ∈ [(j −1)ε2, jε2)] � ≤ (max(pi,p j)−ε/4)si b j When one of i, j equals to n +1(we can assume i = n +1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' it is true that E S∼F∗ S B∼F∗ B � max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='B)1[S ≥ nε2] 1[B ∈ [(j −1)ε2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' jε2)] � = E S∼F ∗ S [S · 1[S ≥ nε2]] Pr B∼F ∗ B [B ∈ [(j −1)ε2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' jε2)] = � nε2� sn+1b j ≤ (pn+1 −ε/4)sib j 21 Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' for the special case that i = j = n +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' we could see that E S∼F∗ S B∼F∗ B � max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='B)1[S ≥ nε2] 1[B ≥ nε2] � ≤ E S∼F∗ S B∼F∗ B � BS/ � nε2� 1[S ≥ nε2] 1[B ≥ nε2] � = � nε2� sn+1bn+1 ≤ (pn+1 −ε/4)sn+1bn+1 Summing up all the inequalities above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' we then get that E S∼F∗ S B∼F∗ B [max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='B)] ≤ n+1 � i=1 n+1 � j=1 � max(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j)−ε/4 � sib j This implies that n+1 � i=1 n+1 � j=1 (si/s)(b j/b)max(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j) ≥ n+1 � i=1 n+1 � j=1 sib j max(pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p j)· � 1+ε2�−2 ≥ \uf8eb \uf8ec\uf8ed E S∼F∗ S B∼F∗ B [max(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='B)]+ n+1 � i=1 n+1 � j=1 ε/4sib j \uf8f6 \uf8f7\uf8f8·(1−ε2)2 ≥ 1+ε/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' which means that (s1/s,s2/s,··· ,sn+1/s,b1/b,b2/b,··· ,bn+1/b,r) is truly a valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Next, we give an upper bound of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To start with, notice that n+1 � i=1 si pi = n� i=1 Pr � S ∈ [(i −1)ε2,iε2) � � (i −1)ε+ε2 +ε/4 � +E[S · 1[S ≥ nε2]]+ sn+1 ·(ε2 +ε/4) ≤ n� i=1 E � S · 1 � S ∈ [(i −1)ε2,iε2) �� +E[S · 1[S ≥ nε2]]+ n+1 � i=1 si(ε2 +ε/4) ≤ E[S]+ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (24) For the term of gain from trade, it holds that t� i=1 n+1 � j=t+1 sib j(p j − pi) = t� i=1 n� j=t+1 sib j(jε2 −iε2)+ t� i=1 sibn+1 � nε2 −(i −1)ε2� ≤ t� i=1 n� j=t+1 sib j((j −1)ε2 −iε2)+ t� i=1 sibn+1 � nε2 −iε2� +ε2(1+ε2)2 ≤ t� i=1 n� j=t+1 E[(B −S)· 1[S ∈ [(i −1)ε2,iε2)]] 1[B ∈ [(j −1)ε2, jε2]] +E[(B −S)· 1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε/2 ≤ E[(B −S)· 1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (25) 22 Combining (24) and (25), we know that for any t ∈ [n +1], n+1 � i=1 (si/s)pi + t� i=1 n+1 � j=t+1 (si/s)(b j/b)(p j − pi) ≤ n+1 � i=1 si pi + t� i=1 n+1 � j=t+1 sib j(p j − pi) = E[S]+E[(B −S)· 1[S ∈ [0,t ·ε2)] 1[B ≥ t ·ε2]]+ε ≤ W (I,t ·ε2)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Taking maximum over [n +1], we then get that r = max t∈[n+1] n+1 � i=1 (si/s)pi + t� i=1 n+1 � j=t (si/s)(b j/b)(p j − pi) ≤ max t∈[n+1]W (I,t ·ε2)+ε ≤ r ∗ +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This means that the optimal value of UpperOp with respect to {pi}i∈[n+1] is at most r ∗ + ε, and this finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Next, we aim to show that for any ε > 0, there exists 0 = p0 < p1 < p2 < ···pn such that LowerOp has an optimal value of at least r ∗ − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now define n = � ε−6� + 1, and pi = (i − 1) · ε3 for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let (s1,··· ,sn,b1,··· ,bn,r) be the optimal solution of the optimization problem LowerOp with respect to {pi}i∈n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It is equivalent to show that there exists an instance I = (FS,FB) such that the optimal approxi- mation ratio of I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' maxx∈R W (I ,x) OPT-W (I ), is at most r +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We construct the instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let n′ = n + � 4 ε � , and si = bi = 0 for n < i ≤ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now define {s′ i} where s′ j = j� i=max � 1,j− � 4 ε � +1 �si/ �4 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It follows that n′ � j=1 s′ j = n� j=1 j� i=max � 1,j− � 4 ε � +1 �si/ �4 ε � = n′ � i=1 si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let s = �n′ i=1 s′ i and b = �n′ i=1bi be the normalization factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' LowerOp guarantees that 1 ≤ s,b ≤ 1+ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, we could also see that s′ j = j� i=max � 1,j− � 4 ε � +1 �si/ �4 ε � ≤ (1+ε3) �4 ε � ≤ ε/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' holds for all j ∈ [n′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following instance I = (FS,FB): S ∼ FS,S = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 p1 +ε4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' s′ 1/s ··· pn′ +ε4 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' s′ n′/s B ∼ FB,B = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 p1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' b1/b ··· pn′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' bn′/b 23 First, it is straight forward to verify this is a valid distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first calculate OPT-W (I): OPT-W (I) = n′ � i=1 n′ � j=1 max(pi +ε4,p j)·(s′ i/s)(b j/b) ≥ n′ � i=1 n′ � j=1 max(pi,p j)·(s′ i/s)(b j/b)−ε4 ≥ n′ � i=1 n′ � j=1 max(pi,p j) \uf8eb \uf8ed i� k=max � 1,i− � 4 ε � +1 � � sk/ �4 ε �� /s \uf8f6 \uf8f8·(b j/b)−ε4 ≥ n′ � i=1 n′ � j=1 max(pi,p j)· sib j/(bs)−ε4 ≥ (1+ε3)−2 −ε4 ≥ 1−ε2 Now consider the optimal fixed-price mechanism for the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As we have shown in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, the optimal mechanism only need to choose price from the support of the discrete distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This implies that max p∈R W (I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p) = max t∈[n] n′ � i=1 (pi +ε4)(s′ i/s)+ t� i=1 n′ � j=t+1 (s′ i/s)(b j/b)(p j − pi −ε4) For any t ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' one could see that n′ � i=1 pi s′ i = n′ � i=1 \uf8eb \uf8ed i� k=max � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i− � 4 ε � +1 � � sk/ �4 ε ��\uf8f6 \uf8f8pi ≤ n′ � i=1 \uf8eb \uf8ed i� k=max � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i− � 4 ε � +1 � � sk/ �4 ε �� � pk + �4 ε � ε3 �\uf8f6 \uf8f8 ≤ n′ � i=1 (pi +5ε2)si (26) For the term of gain from trade,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' it follows that t� i=1 n′ � j=t+1 s′ ib j(p j − pi) = t−1 � i=1 n′ � j=t+1 s′ ib j(p j − pi)+ n′ � j=t+1 s′ tb j(p j − pi) ≤ t−1 � i=1 n′ � j=t+1 \uf8eb \uf8ed i� k=max � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i− � 4 ε � +1 � � sk/ �4 ε ��\uf8f6 \uf8f8b j(p j − pi)+ s′ t n′ � j=1 b j p j ≤ t−1 � i=1 n′ � j=t+1 sib j(p j − pi)+ε/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (27) where we use the fact that skb j(p j − pi) ≤ skb j(p j − pk) for j > i ≥ k, s′ t ≤ ε/3, and �n′ i=1b j p j ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 24 Again by combining the two inequalities above, we know that n′ � i=1 (pi +ε4)(s′ i/s)+ t� i=1 n′ � j=t+1 (s′ i/s)(b j/b)(p j − pi −ε4) ≤ n′ � i=1 pi s′ i + t� i=1 n′ � j=t+1 s′ ib j(p j − pi)+ε4 ≤ n′ � i=1 (pi +5ε2)si + t−1 � i=1 n′ � j=t+1 sib j(p j − pi)+ε/3 ≤ n′ � i=1 pi si + t−1 � i=1 n′ � j=t+1 sib j(p j − pi)+ε/2 where we apply (26) and (27) in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that the optimal solution (s1,s2,··· ,sn,b1,b2,··· ,bn,r) of LowerOp must satisfy that r = maxt∈[n] �n i=1 pi si +�t−1 i=1 �n j=t+1 sib j(p j − pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, taking the maximum over t ∈ [n′], we then get that max p∈R W (I,p) = max t∈[n′] n′ � i=1 (pi +ε4)(s′ i/s)+ t� i=1 n′ � j=t+1 (s′ i/s)(b j/b)(p j − pi −ε4) ≤ max t∈[n′] n′ � i=1 pi si + t−1 � i=1 n′ � j=t+1 sib j(p j − pi)+ε/2 = max t∈[n] n� i=1 pi si + t−1 � i=1 n� j=t+1 sib j(p j − pi)+ε/2 = r + ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' where the second equation follows from si = bi = 0 when i > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, on instance I, it holds that maxp∈R W (I,p) OPT-W (I) ≤ r +ε/2 1−ε2 ≤ r +ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' And this completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' C Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 We start with the case when we only know E[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We consider discrete distribution PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose x ∼ PS equals to pi with probabiltiy wi for i ∈ [n] where �n i=1 wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This means that our mechanism would choose pi with probability wi and use pi · ES∼FS[S] as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixing {pi,wi}i∈[n], we claim that the optimal solution of the following program lower bounds the approximation ratio of this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 25 min s1,s2···,sn b1,b2,···,bn �n t=1 wt ��n i=1 si pi +�t−1 i=1 �n j=t+1 sib j(p j − pi) � �n i=1 �n j=1 sib j max(pi,p j) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' si,bi ≥ 0 ∀i ∈ [n] (28) n� i=1 si ≥ 1 and n� i=1 si ≤ 1+ 1 pn and n−1 � i=1 si ≤ 1 (29) n� i=1 bi ≥ 1 and n−1 � i=1 bi ≤ 1 (30) n� i=1 si · pi = 1 (31) Similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we could assume that E[S] = 1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The approximation ratio r ∗ of our mechanism could be written as r ∗ = min I =(FS ,FB ) E[S]=1 �n i=1 wi ·W (I,pi) OPT-W (I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose r ∗ is attained at I ∗ = � F ∗ S ,F ∗ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 with I ∗, we know that there exists {s1,s2,··· ,sn,b1,b2,··· ,bn} satisfying all the properties in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that ES∼F ∗ S [S] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There- fore, we could directly verify that constraints (28), (29), (30) and (31) are satisfied by all the properties in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, it holds that �n i=1 si pi +�t−1 i=1 �n j=t+1 sib j(p j − pi ) ≤ W (I ′,pt) for all t ∈ [n] and �n i=1 �n j=1 sib j max(pi,p j) ≥ OPT-W (I ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This implies that �n t=1 wt ��n i=1 si pi +�t−1 i=1 �n j=t+1 sib j(p j − pi) � �n i=1 �n j=1 sib j max(pi,p j) ≤ �n i=1 wi ·W (I ∗,pi) OPT-W (I ∗) = r ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since {si,bi}i∈[n] is a valid solution of the optimization problem above, we then show that it lower bounds the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we solve this optimization problem numerically and show that there exists {pi,wi}i∈[n] such that the optimal solution is at least by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The details of the numerical result could be found at our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now we turn to the case when we know E[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proof uses similar ideas and is almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following mechanism: it picks pi with probability wi for i ∈ [n] where �n i=1 wi = 1, and chooses pi ·E[B] as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again, for fixed {pi,wi}i∈[n], we aim to show that the following optimization problem give a lower bound of the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' min s1,s2···,sn b1,b2,···,bn �n t=1 wt ��n i=1 sipi +�t−1 i=1 �n j=t+1 sib j(p j − pi) � �n i=1 �n j=1 sib j max(pi,p j) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' si,bi ≥ 0 ∀i ∈ [n] (32) n� i=1 bi ≥ 1 and n� i=1 bi ≤ 1+ 1 pn and n−1 � i=1 bi ≤ 1 (33) n� i=1 si ≥ 1 and n−1 � i=1 si ≤ 1 (34) n� i=1 bi · pi = 1 (35) 26 Without loss of generality, we could assume that E[B] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, the approximation ratio of the mechanism is exactly r ∗ = min I =(FS ,FB ) E[B]=1 �n i=1 wi ·W (I,pi) OPT-W (I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose r ∗ is attained at I ∗ = � F ∗ S ,F ∗ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 , we could discretize the instance I ∗ to {s1,s2,··· ,sn,b1,b2,··· ,bn} that satisfies all the properties in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since EB∼F ∗ B [B] = 1, it is easy to verify that constraints (32), (33), (34) and (35) holds due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, it holds that �n i=1 si pi +�t−1 i=1 �n j=t+1 sib j(p j −pi) ≤ W (I ′,pt) for all t ∈ [n] and �n i=1 �n j=1 sib j max(pi,p j) ≥ OPT-W (I ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This implies that �n t=1 wt ��n i=1 si pi +�t−1 i=1 �n j=t+1 sib j(p j − pi) � �n i=1 �n j=1 sib j max(pi,p j) ≤ �n i=1 wi ·W (I ∗,pi) OPT-W (I ∗) = r ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since {si,bi}i∈[n] is a valid solution of the optimization problem above, this means that the optimiza- tion problem is a lower bound of the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, we solve this optimization problem numerically and find a set of numbers {pi,wi}i∈[n] such that the optimal solution is at least by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The details of the numerical result could be found at our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D Missing Proofs in Section 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Mechanisms over Buyer’s information In the sample setting, we only consider mechanisms over seller’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We do not consider quan- tile or order statistics mechanisms over buyer’s information since it is impossible to get any constant approximation with these family of mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' No quantile mechanism over buyer’s distribution or order statistic mechanism over only buyer’s samples can achieve a constant fraction of the optimal welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first show that there is no constant approximation quantile mechanism Q over buyer’s distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Remind that FB and FS respectively stand for the distribution of the buyer and the seller, and we will also use Q to denote the corresponding distribution over the buyer’s quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To start with, we can assume that distribution Q does not have point mass at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' That’s because if we set the 1-quantile of the buyer’s distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' F −1 B (1), as the price, we have PrB∼FB[B ≥ F −1 B (1)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This means that the trade will never happen under such price and thus this price will not increase the welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, if we move this probability mass to other values, the welfare and also the approxima- tion ratio will not decrease, and we prove that this assumption is with out loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now, for an arbitarily small ε > 0, we will show that there is no ε-approximation quantile mechanism over buyer’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism Q over buyer’s distribution, we construct the following set X = {t ∈ [0,1] | Pr x∼Q[x ≥ t] ≤ ε/2} Since there is no point mass at 1, this set will contain some t ∈ X and t ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following instance I = (FS,FB): 27 B ∼ FB,B = � 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' t H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1− t S ∼ FS,S = ε/2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 where H = 1 1−t is a large enough number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In this instance, the intuition is that all the welfare is hide at some very little probability of the buyer, and we must make sure that the trade is very likely to happen when the buyer has a very high value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, since we don’t know the value of the seller, it is hard for us to make sure that p ≥ S which means that this trade will not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Remind that we define OPT-W (I) as the optimal welfare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=', ES∼FS,B∼FB[max(S,B)] against instance I = (FS,FB) and W (Q,I) as the welfare of mechanism Q against instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Formally speaking, we have that OPT-W (I) ≥ H ·(1− t) = 1 and also W (Q,I) = E[S]+ E p∼F −1 B (Q) [(B −S)· 1[B ≥ p ≥ S]] ≤ ε/2+ E p∼F −1 B (Q) [B · 1[p ≥ S]] ≤ ε/2+ H ·(1− t)·ε/2 = ε where the last inequality holds since p = F −1 B (x) ≥ S is equivlent to x ≥ t where x is drawn fromQ, and this happends w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' at most ε/2 by the definition of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' So for every distribution Q over buyer’s quantile, we find an instance I so that W (Q,I) ≤ ε·OPT-W (I), which completes the first part of our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Next we aim to show that for any ε > 0,N > 0, there is no ε-approximation mechanisms using only N samples from the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' First, for any mechanisms M using N samples from the buyer, it can be formallized as a mapping f : RN ≥0 �→ ∆(R≥0) where f (x1,x2,··· ,xN) stands for the distribution of price selected by this mechanism after receiving N samples (x1,x2,··· ,xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let D be f (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',0), which is the distribution of the price if this mechanism sees N samples all with value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, we consider the following set: H ′ = {t ∈ R≥0 | Pr x∼D[x ≥ t] ≤ ε/2} Again we know this set is non-empty, so let t be any real positive number in the set H ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we could construct an instance I = (FS,FB) satisfying B ∼ FB,B = � 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' (1−ε/4)1/N H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1−(1−ε/4)1/N S ∼ FS,S = t +1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 where H > t+1 ε/4·(1−(1−ε/4)1/N is a large enough number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In this instance, we can see that with just N samples, no mechanism can distinguish this instance with another instance whose buyer always have a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, it can not get the welfare hidden at the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Formally speaking: OPT-W (I) ≥ H · � 1−(1−ε/4)1/N � > t +1 ε/4 28 To calculate W (M,I), we consider the case when all the samples are zero and the case when there is at least one non-zero number in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the latter case, the probability that at least one sample is non-zero is at most 1 − � (1−ε/4)1/N �N = ε/4 which is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the former case, since Prp∼f (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',0)[p ≥ t] ≤ ε/2, the trade happends w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' at most ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we could expand W (M,I) into: W (M,I) ≤ E p∼f (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',0)[S +(B −S)· 1[B ≥ p ≥ S]]·Pr[All N samples are 0] +OPT-W (I)·Pr[at least 1 sample is not 0] ≤ (t +1+E[B · 1[p ≥ S]])·1+OPT-W (I)·(ε/4) ≤ OPT-W (I)·(ε/4+ε/2+ε/4) = ε·OPT-W (I) where the second inequality holds since t + 1 ≤ (ε/4) · OPT-W (I), Prp∼f (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',0)[p ≥ S] ≤ ε/2 and EB∼FB[B] ≤ ES∼FS,B∼FB[max(S,B)] = OPT-W (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' And this finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 The proof here is quite straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As we show in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, each order statistic mechanism corresponds to a quantile mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus C (P (Q)) is exactly the approximation ratio of the order statistic mechanism Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, we could see that the ∆N enumerates all possible order statistic mechanisms with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, this directly implies that argmaxQ∈∆N C (P (Q)) is the optimal order statistic mechanism with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 Fix an instance I = (F,F), recall that S and B are the random variables respectively indicating the value of the seller and the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Define ALG to be the random variable which indicates the welfare of our mechanism in the realization, which is S + (B − S) · 1[B ≥ p ≥ S] where p is the price chosen by our quantile mechanism Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, let OPT be the random variable which indicates the optimal welfare in the realization, which is max(B,S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we introduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism Q, let ALG and OPT respectively be the random variables indi- cating the welfare of the mechanism Q and the optimal welfare in the realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let r be min I =(F,F) inf x∈[0,1) Pr[ALG ≥ F −1(x)] Pr[OPT ≥ F −1(x)] where F(x) is the cumulative distribution function of distribution F, and F −1(x) is the quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The quantile mechanism Q is at least r-approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We have Pr[ALG ≥ F −1(x)] ≥ r ·Pr[OPT ≥ F −1(x)] for all x ∈ [0,1] and quantile function F −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Without loss of generality, we could assume the distribution has a support over [0,a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that since we assume the distribution is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' in the sample setting, F −1(x) is a continuous and 29 increasing function over [0,1] and F −1(0) = 0,F −1(1) = a, so we have W (I,Q) = E[ALG] = �a 0 Pr[ALG ≥ x]dx = �1 0 Pr[ALG ≥ F −1(z)]dF −1(z) ≥ �1 0 r ·Pr[OPT ≥ F −1(z)]dF −1(z) = r · �a 0 Pr[OPT ≥ x]dx = r ·E[OPT] = r ·OPT-W (I) holds for any instance I = (F,F), which implies that quantile mechanism Q is at least r-approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' With Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, we are able to give a lower bound of approximation ratio for any quantile function Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fixing the buyer and seller’s distribution F, we only need to calculate the term Pr[ALG ≥ F −1(x)] and Pr[OPT ≥ F −1(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The event OPT ≥ F −1(x) happens if and only if either B or S is greater than F −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, Pr[OPT ≥ F −1(x)] = 1− x2 (36) The event ALG ≥ F −1(x) happens if and only if one of the following conditions is satisfied: S ≥ F −1(x) p ≤ F −1(x), S ≤ p and B ≥ F −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Here S ≤ p ≤ B, thus the trade takes place, and B ≥ F −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' p > F −1(x), S ≤ F −1(x) and B ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since S ≤ p ≤ B, the seller trades the item to the buyer,and we have B ≥ p ≥ F −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Note that these three events are disjoint, so we could calculate the probability for each event to hap- pen and add them up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the first event, Pr[S ≥ F −1(x)] = 1− x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the second event, we just enumerate the quantile of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose the quantile of p is t, which means that F −1(t) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then, we have Pr[S ≤ p] = t and Pr[B ≥ F −1(x)] = 1 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, this event takes place w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' � [0,x] t(1− x)dQ(t) where Q(t) is the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of distribution Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the third event, we use the same idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose the quantile of p is t, we have Pr[S ≤ F −1(x)] = x and Pr[B ≥ p] = 1− t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, this event happens w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' � (x,1](1− t)x dQ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' By adding the terms above up, we have: Pr[ALG ≥ F −1(x)] = � [0,x] t(1− x)dQ(t)+ � (x,1] (1− t)x dQ(t)+(1− x) (37) Therefore, combining Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and Equation (36) and (37), we have that for any quantile mecha- nism Q with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Q(x), the minimum of the following optimization problem lower bounds the approxi- mation ratio of the quantile mechanism Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' min I =(F,F) inf x∈[0,1) �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) 1− x2 = inf x∈[0,1) �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) 1− x2 30 where the equality holds since we could see that the term is independent from F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We now left to show that the approximation ratio of Q is also upper bounded by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It suffices to show that for any ε > 0 there exists some instance I = (F,F) such that W (Q,I) ≤ (r +ε)·OPT-W (I) First, Recall Equation (36) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that both the term Pr[ALG ≥ F −1(x)] and the term Pr[OPT ≥ F −1(x)] are independent of the distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, r = min F inf x∈[0,1) Pr[ALG ≥ F −1(x)] Pr[OPT ≥ F −1(x)] = inf x∈[0,1) �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) 1− x2 Suppose the optimum of infx∈[0,1) �x 0 q(t)·t(1−x)dt + �1 x q(t)·(1−t)x dt +(1−x) 1−x2 is attained at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following instance I = (F,F) satisfying v ∼ F,v = � U[0,r ·(1− x∗)·ε/2] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' x∗ U[1,1+ε/2] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1− x∗ Notice that in this instance, The event 1[ALG ≥ F −1(x∗)] is equivalent to 1 � ALG ∈ [1,1+ε/2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Such argument also holds for OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, we can see that W (Q,I) = E[ALG] ≤ Pr[ALG ∈ [1,1+ε/2]]·(1+ε/2)+Pr[ALG ∈ [0,r ·(1− x∗)·ε/2]]·(r ·(1− x∗)·ε/2) ≤ Pr[ALG ≥ F −1(x∗)]·(1+ε/2)+r ·(1− x∗)·ε/2 = r ·Pr[OPT ≥ F −1(x∗)]·(1+ε/2)+r ·(1− x∗)·ε/2 ≤ r ·E[OPT]·(1+ε/2)+r ·(1− x∗)·ε/2 ≤ (r +ε)·E[OPT] = (r +ε)·OPT-W (I) where the second equation uses the fact that r ·Pr[OPT ≥ F −1(x∗)] = Pr[ALG ≥ F −1(x∗)] since r is attained at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Since the above holds for every ε > 0, this completes our proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 is directly a combination of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we know that the approximation ratio of a quantile mechanism Q with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Q(x) is exactly inf x∈[0,1) � [0,x] q(t)t(1− x)dt + � (x,1] q(t)(1− t)x dt +(1− x) 1− x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' where we use q(x)dx instead of dQ(x) since we have a continuous probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that the set of all distributions over [N] is actually � {Pi}i∈[n] | Pi ≥ 0 and �n i=1Pi = 1 � where distribution P would choose i w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, the probabiltiy density function of the quan- tile of such order statistic mechanism is exactly p(x) = �n i=1Pi f i N(x) where f i N(x) = N �n−1 i−1 � xi−1 · (1 − x)N−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, combining it with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, it follows that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 Analysis of Empirical Risk Minimization Mechanism In this section, we give the upper bounds of the Empirical Risk Minimization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When N = 1, the empirical distribution is a one-point distribution, so any price x ∈ R≥0 is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we consider the following instance I = (F,F) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' x ∼ F,x = � 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1−1/H H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1/H Since any price is optimal for ERM, we can assume that it will always select H + 1 so that the trade will never take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore taking H → ∞ in this instance, the approximation ratio tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When N = 2, the empirical distribution is a two-point distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose the two samples are X1 ≤ X2, we can see that any price in [X1, X2) is optimal for this empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, we can assume that it will always select X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' So, ERM is equivalent to an order statistic mechanism that always selects the smallest sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we can solve the following optimization problem: inf x∈[0,1) 1 3x3 − x2 − 1 3x +1 1− x2 = 2 3 Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we know that there exists an instance such that ERM achieves exactly 2 3 approx- imation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now we are in the case that N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' By some calculations, we know that the second smallest sample will always be an optimal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, ERM is equivalent to an order statistic mechanism that always selects the second smallest sample when N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, we can calculate that inf x∈[0,1) 1 2x4 − x3 − 1 2x +1 1− x2 = 3 4 Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 again, there is an instance such that ERM has an approximation ratio of exactly 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our proof strategy changes when the number of samples is greater than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We consider a particular instance I = (F,F) and calculate the performance of ERM on such instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' x ∼ F,x = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' ( � 2−1)·(1− 1 n ) 1 n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 2− � 2 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 n ( � 2−1) Actually, this is a counterexample appeared in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' They show that OPT-W (I) = 4( � 2−1) n − 4 � 2−1 n2 Since we will let n → ∞, we will ignore the O( 1 n2 ) terms in the following calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that Pr[x = 1] = O( 1 n ), the probability that there is at least 1 sample with value 1 is negligible, so we will also assume that all samples are 0 or 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Recall that there will be a tie-breaker coordinate drawn uniformly from [0,1] for each variable, and we will compare the tie-breaker coordinate if they have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now suppose there are k1 samples with value 0 and k2 samples with value 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We know that the largest 0 32 or the smallest 1 n is an optimal price for the empirical distribution when k1,k2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' if we choose the largest 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' as the price p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the expected welfare is: W1(k1) =Pr[S = 0]·Pr[S < p]·E[B]+Pr � S = 1 n � 1 n +Pr[S = 1]·1 =( � 2−1)· k1 k1 +1 � (2− � 2+ � 2−1)· 1 n � +(2− � 2)· 1 n + 1 n ·( � 2−1) (38) if we choose the smallest 1 n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' as the price p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' the expected welfare is: W2(k2) =Pr[S = 0]·E[B & B > p]+Pr � S = 1 n � � 1 n +Pr[S < p]·E[B & B = 1] � +Pr[S = 1]·1 =( � 2−1)· � (2− � 2)· 1 n · k2 k2 +1 + 1 n ( � 2−1)) � +(2− � 2)· � 1 n + 1 k2 +1 ·( � 2−1) 1 n � + 1 n ·( � 2−1) (39) When all the samples have the same value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' any price is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similar to the case when N = 1, the trade may never happen, so the expected welfare is 1 n in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now, when there are 5 samples, suppose the ERM will choose the largest 0 when there are 1 ∼ 3 samples with value 0, and choose the smallest 1 n when there are 1 samples with value 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The expected welfare of ERM when N = 5 is : 3� i=1 ( � 2−1)i (2− � 2)5−i � 5 i � W1(i)+ 1� i=1 ( � 2−1)5−i(2− � 2)i � 5 i � W2(i)+(p5 +(1− p)5)· 1 n Now compare it to the optimal: OPT-W (I) = 4( � 2−1) n − 4 � 2−1 n2 By numerical calculations, the ratio is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='76 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, when there are 10 samples, suppose the ERM will choose the largest 0 when there are 1 ∼ 6 samples with value 0, and choose the smallest 1 n when there are 1 ∼ 3 samples with value 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The expected welfare of ERM when N = 10 is : 6� i=1 ( � 2−1)i (2− � 2)10−i � 10 i � W1(i)+ 3� i=1 ( � 2−1)10−i(2− � 2)i � 10 i � W2(i)+(p10 +(1− p)10)· 1 n By calculations, the approximation ratio is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='80 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 Now we fix the quantile mechanism Q and suppose its c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' is Q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let r be min x∈[0,1] � [0,x] t dQ(t)+1− x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [5] already prove that r is the lower bound the approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We are only left to show that it is also the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We aim to show that for any ε > 0, there exists an instance I = (FS,FB) such that W (I,Q) ≤ (r +ε)·OPT-W (I) Now suppose the optimization problem above achieves its minimum at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Consider the following instance I = (FS,FB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 33 S ∼ FS,S = � U[0,ε] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' x∗ H +ε w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1− x∗ B ∼ FB,B = H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 1 where H > 1 is a sufficiently large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We can see that in this instance, OPT-W (I) ≥ H Now we compute the expected welfare for our mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When its price has a quantile t smaller than or equal to x∗, the trade will happen with probability exactly t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When the quantile of its price is greater than x∗, the trade will never happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' W (I) = E[S]+E[(B −S)1[B ≤ p ≤ S]] ≤ x∗ε+(H +ε)(1− x∗)+ � [0,x∗] tH dQ(t) ≤ H · � 1− x∗ + � [0,x∗] t dQ(t) � +ε ≤ H ·r +ε ≤ H ·(r +ǫ) ≤ (r +ε)·OPT-W (I) where the third inequality follows from r = � [0,x∗] t dQ(t)+(1− x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, we could find an instance I = (FS,FB) such that W (Q,I) ≤ (r + ε) · OPT-W (I) for any small enough ε > 0, and this concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 is nearly identical to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, we know that the approx- imation ratio of a quantile mechanism Q with a continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q(x) is exactly C (p) = min x∈[0,1] �x 0 q(t)t dt +1− x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' where we use q(x)dx instead of dQ(x) since we have a continuous probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again, we know that the set of all distributions over [N] is actually � {Pi}i∈[n] | Pi ≥ 0 and �n i=1Pi = 1 � where distribution P would choose i w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' What’s more, the probabiltiy density function of the quantile of such order statistic mechanism is exactly p(x) = �n i=1Pi f i N(x) where f i N(x) = N �n−1 i−1 � xi−1 · (1− x)N−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, combining it with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, it follows that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='8 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 Before we give the proof, we first introduce some notations and lemmas about Bernstein that may useful to our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Definition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 (Stochastic Bernstein Polynomials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The stochastic Bernstein polynomial of degree n for a continous function f on [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1] is defined as � B X n f � (t) = n� k=0 f (Xk)pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='k(t) 34 in which X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='··· Xn are the order statistics of (n + 1) independent copies of the random variable uniformly distributed in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='k(t) = � n k � xk(1− x)n−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0 ≤ k ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='0 ≤ t ≤ 1 Now fix the continous function q(x) we aim to approximate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' define ω(h) as the following function: ω(h) = sup 0≤x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='y≤1 |x−y|≤h |q(x)− q(y)| Now we can introduce the lemma in [30] that help us approximate the function q(x) by order statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='11 In [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let ε > 0 and f ∈C[0,1] be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose that ω � 1 �n � < ε/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then the following inequality holds true: Pr ���B X n f − f �� ∞ > ε � ≤ 2(n +1)exp \uf8eb \uf8ed− 2ε2 ω2 � 1 �n � \uf8f6 \uf8f8 We are now ready to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first present our mechanism P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For some instance I = (FS,FB), suppose there are n samples X1 ≤ X2 ≤ ··· ≤ Xn drawn from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We draw another n samples Y1 ≤ Y2 ≤ ··· ≤ Yn uniformly and independently from [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let s = �n i=1 q(Yi) be the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then our mechanism will choose Xi with probability q(Yi)/s Now, let g(x) = n� i=1 q(Yi)f i n(x)/s where f i n(x) = n � n −1 i −1 � xi−1(1− x)n−i be the corresponding probability density function of the order statistic mechanism P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' It suffices to prove that with high probability |g(x)− q(x)| ≤ ε ∀x ∈ [0,1] To prove this, we introduce an intermediate function h(x): h(x) = n� i=1 q(Yi)· � n −1 i −1 � xi−1(1− x)N−i = n� i=1 q(Yi)pn−1,i−1(x) As we can see, h is the stochastic Bernstein polynomial of q with degree n −1 and ω � 1 � n−1 � ≤ ε/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Applying lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we know that Pr ���h − q �� ∞ > ε/4 � ≤ 2n exp \uf8eb \uf8ed− ε2 8·ω2 � 1 � n−1 � \uf8f6 \uf8f8 ≤ ε (40) where the last inequality comes from the assumption in the statement of lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 35 Thus, we only need to show that the difference between h and g is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' First we have g(x) = N � i=1 q(Yi) S f i n(x) = N � i=1 q(Yi) � n −1 i −1 � xi−1(1− x)n−i n s = n s ·h(x) So it is equivalent to prove that n and s are close w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Pr �� 1− ε 4M � n ≤ s ≤ � 1+ ε 4M � n � ≥ 1−ε We first use the lemma to continue our proof, before proving the lemma itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We know that |g(x)− h(x)| ≤ ε/2 ∀x ∈ [0,1] if n � 1− ε 4M � ≤ s ≤ n � 1+ ε 4M � and h(x) < 2M ∀x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, Pr[|g(x)−h(x)| ≥ ε/2 ∃x ∈ [0,1]] ≤ Pr � s > n � 1+ ε 4M �� +Pr � s < n � 1− ε 4M �� +Pr[∃x∈[0,1]h(x) > 2M] ≤ 2ε (41) where the second inequality is from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3 and the fact that with probability at least 1−ε, ∥h − q∥ ≤ ε/4 and q has a maximum of M on [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now combining inequality (40) and (41), we know that with probability at least 1−3ε, we have: |g(x)− q(x)| ≤ |g(x)−h(x)|+|h(x)− q(x)| ≤ ε Since such probability is strictly greater than 0, we know that there exists some order statistic mech- anism P over N samples such that |g(x) − q(x)| ≤ ε ∀x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Also we show that our construction will find such order statistics with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Finally, as we assumed in the statement, it holds that C (P) ≥ C (Q)−c ·|g − q|∞ ≥ C (Q)−cε This completes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We know that E[s] = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that s is the sum of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' random variables ranging in [0,M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Therefore, by Chernoff bound, it holds that Pr �� 1− ε 4M � n ≤ s ≤ � 1+ ε 4M � n � ≥ 1−exp � − ε2n 48M2 � −exp � − ε2n 32M2 � ≥ 1−ε where the last inequality is from the property in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='9 Application of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Symmetric Instance We first study the case when the distributions are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [24] provide a mechanism that chooses the mean of the distribution F as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' They show that in the symmetric setting, this is the optimal fixed 36 price mechanism and achieves an approximation ratio of 2+ � 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' However, what we want here is a quan- tile mechanism and we could not convert such mean-based mechanism directly into an order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We show that quantile mechanisms can also reach the optimal 2+ � 2 4 approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' After that, we use the technique in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 to produce an order statistic mechanism that achieves an approximation ratio of 2+ � 2 4 −ε with poly � 1 ε � samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To start with, we first show our optimal order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There is a 2+ � 2 4 approximation quantile mechanism in the symmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our quantile mechanism Q runs as following: Let F be the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Output F −1� � 2 2 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' � 2 2 -quantile of the distribution, as the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The approximation ratio could be directly calculated using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' One could see that inf x∈[0,1) � [0,x] t(1− x)dQ(t)+ � (x,1](1− t)x dQ(t)+(1− x) 1− x2 = min \uf8eb \uf8ed min x∈[0, � 2 2 ] x ·(1− � 2 2 )+1− x 1− x2 , inf x∈[ � 2 2 ,1) � 2 2 ·(1− x)+(1− x) 1− x2 \uf8f6 \uf8f8 = 2+ � 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Now we aim to convert it to an order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There exists an order statistic mechanism P with N = O � 1 ε7 � samples that achieves 2+ � 2 4 −ε approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' To start with, we may notice that it is impossible to directly apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 to the optimal quantile mechanism a since it is does not have a continiuous probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' So our first step is to provide a quantile mechanism with continuous distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any ε > 0, let Q′ be the quantile mechanism with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' ˜q(x) satisfying ˜q(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 x ∈ � 0, � 2 2 − 1 32ε � � �� 2 2 + 1 32ε,1 � (32/ε)2 · � x − � 2 2 + 1 32ε � x ∈ �� 2 2 − 1 32ε, � 2 2 � −(32/ε)2 · � x − � 2 2 − 1 32ε � x ∈ �� 2 2 , � 2 2 + 1 32ε � The quantile mechanism Q′ has an appriximation ratio of � 2+2 4 −ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Our last step is to make sure that the approximation ratio would not differ to much for two probability density functions that are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 37 Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose C (p) is the function that maps every quantile mechanismQ for symmetric instances with a continuous probabilitiy density function q(x) to its approximation ratio where C (p) = inf x∈[0,1) � [0,x] q(t)t(1− x)dt + � (x,1] q(t)(1− t)x dt +(1− x) 1− x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism Q1 with continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f q1(x) and Q2 with continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q2(x), it holds that C (p1)−C (p2) ≥ − ��q1 − q2 �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first use these lemmas to give a proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, and leave the proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 and Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='10 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof Of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let n = c · 1 ε7 +1 where c is a large enough constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that ω � 1 � n−1 � ≤ 322c−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 ·ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This means that 2n exp � − ε2 8ω2 � 1 � n−1 � � ≤ ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Besides, define M = maxx∈[0,1] ˜q(x) = 32/ε, it is also easy to verify that exp � − ε2n 48M3 � ≤ ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Combining the properties above with Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5, we could apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4, and see that there exists an order statistic mechanism with n samples with an approx- imation of at least C (Q′)−ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Together with Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4, it follows that this order statistic mechanism is 2+ � 2 4 −ε-approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We would like to comment that if we always choose the � � 2 2 ·n � th order statistic as the price, there is an argument to prove that we could achieve an approximation ratio of 2+ � 2 4 −ε with ˜O � 1 ε2 � samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 General Instance We now consider the general instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' [5] provides a 1−1/e approximation quantile mechanism that is also shown to be optimal by [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Using the block-box reduction shown in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4, we show that the optimal quantile mechanism can be approximated by order statistics mechanism as closely as desired and also obtain an upper bound on the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' There exists an order statistic mechanism P with N = O � 1 ε5 � samples that achieves 1− 1 e −ε approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' In the following proof, we will use order statistic mechanism to approximate the optimal quantile mechanism Q with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q(x) = 1 x on [1/e,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, q(x) is not a continous function on [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, we need to first convert it to a continous function ˜q(x) on [0,1] and then apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, we introduce the following two lemmas first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any ε > 0, let ˜Q be the quantile mechanism with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' ˜q(x) satisfying ˜q(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0 x ∈ � 0, 1 e − � 1−e−1� � e − 1 2ε � ε � (e − 1 2ε)2 ε(1−e−1) � x − � 1 e − � 1−e−1� � e − 1 2ε � ε �� x ∈ � 1 e − � 1−e−1� � e − 1 2ε � ε, 1 e � 1 x − 1 2ε x ∈ �1 e ,1 � The quantile mechanism Q′ has an appriximation ratio of 1− 1 e −ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 38 Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Suppose C (p) is the function that maps every quantile mechanism Q for general instances with a continuous probabilitiy density function q(x) to its approximation ratio where C (p) = min x∈[0,1] �x 0 q(t)t dt +1− x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For any quantile mechanism Q1 with continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f q1(x) and Q2 with continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q2(x), it holds that C (p1)−C (p2) ≥ − ��q1 − q2 �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The proofs of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6 and Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7 are respectively in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='12 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Proof Of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We follow the same argument to prove Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let n = c · 1 ε5 + 1 where c is a large enough constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again it is easy to see that ω � 1 �n � ≤ c−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5e2(1 − e−1)−1 ·ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus we have 2n exp � − ε2 8ω2 � 1 � n−1 � � ≤ ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Besides, define M = maxx∈[0,1] ˜q(x) ≤ 2e, we could also verify that exp � − ε2n 48M3 � ≤ ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Combining the properties above with Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7, we could see the existence of an order statistic mechanism with n samples with an approximation of at least C (Q′)−ε/2 by applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Together with Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6, we know that the approximation ratio of this order statistic mechanism is 1 − 1/e − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' This finishes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='10 Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='4 One could see that the quantile mechanism ˜Q would choose a price with its quantile in � � 2 2 − 1 32ε, � 2 2 + 1 32ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Fix a price p and an instance I = (F,F), ALG is the random variable indicating the welfare for the fixed price p in the realzation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' S + (B − S)1[B ≥ p ≥ S] where B,S are drawn independently from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Thus, it suffices to show that for any instance I = (F,F), the following holds when � 2 2 − 1 32ε ≤ t ≤ � 2 2 + 1 32ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' E �ALG | p = F −1(t) � ≥ � 2+ � 2 4 −ε/2 � OPT-W (I) To prove the approximation ratio, we need to use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that the term E[ALG | p = F −1(t)] could be understood as the expected welfare of a quantile mechanism that always selects the t-quantile as the price, so Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 could also be applied to analyze the ratio between E �ALG | p = F −1(t) � and OPT-W (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For x ≥ t, we know that inf x∈[t,1) (1− x)+ t(1− x) 1− x2 = 1+ t 2 When x < t, it holds that min x∈[0,t] (1− x)+ x(1− t) 1− x2 = 1+ � 1− t2 2 As we can see, 1+t 2 ≥ 1+ � 1−t 2 2 when 1 ≥ t ≥ � 2 2 and 1+t 2 ≤ 1+ � 1−t 2 2 when � 2 2 ≥ t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, we know that for any instance I = (F,F), 39 E �ALG | p = F −1(t) � ≥ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1+ � 1− t2 2 OPT-W (I) t ∈ �� 2 2 ,1 � 1+ t 2 OPT-W (I) t ∈ � 0, � 2 2 � Now if � 2 2 − 1 32ε ≤ t ≤ � 2 2 , it holds that E �ALG | p = F −1(t) � ≥ 1+ t 2 OPT-W (I) ≥ � 2+ � 2 4 −ε/2 � OPT-W (I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For � 2 2 ≤ t ≤ � 2 2 + 1 32ε, we have E �ALG | p = F −1(t) � ≥ 1+ � 1− t2 2 OPT-W (I) ≥ � 2+ � 2 4 −ε/2 � OPT-W (I)W W which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='11 Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='5 Suppose |q1(x)− q2(x)| ≤ c for all x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that �x 0 (q1(t)− q2(t))t(1− x)dt + �1 x (q1(t)− q2(t))(1− t)x dt 1− x2 ≥ − �x 0 ct(1− x)dt + �1 x c(1− t)x dt 1− x2 ≥ − �x 0 ct(1− x)dt + �1 x c(1− x)x dt 1− x2 = − �x 0 ct dt + �1 x cx dt 1+ x ≥ −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' holds for all x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Taking the infimum over [0,1), this directly implies that C (Q1)−C (Q2) ≥ −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='12 Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='6 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='3, the approximation ratio of ˜Q can be computed as min x∈[0,1] �x 0 ˜q(t)· t dt +1− x ≥ min x∈[0,1] �x 1 e �1 t − 1 2ε � t dt +1− x ≥ min x∈[0,1] �x 1 e 1 t · t dt +1− x − 1 2ε = 1− 1 e − 1 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 40 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='13 Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='7 Suppose |q1(x)− q2(x)| ≤ c for all x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We could see that ��x 0 q1(t)t dt +(1− x) � − ��x 0 q2(t)t dt +(1− x) � ≥ − �x 0 ct dt ≥ −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' holds for all x ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Taking the minimum over [0,1), this directly implies that C (Q1)−C (Q2) ≥ −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' E Details of Numerical Experiments E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1 Symmetric Instance We now present the details of numerical experiments for the symmetric instances I = (F,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We first for- mally write down the optimization problem that indicates the optimal order statistic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' As we proved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='1, suppose the following optimization problem PO achieves its maximum OPTPO at (P∗ 1 ,P∗ 2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',P∗ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let P∗ be the distribution over [N] such that Prx∼P ∗[x = i] = P∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then, P∗ is the op- timal order statistic mechanism in the symmetric setting with N samples and its approximation ratio is OPTPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Notice that since the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the order statistic mechanism Q(x) is differentiable, we use q(t)dt instead of dQ(t) for ease of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Here q(x) is the probality density function of mechanism Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The Optimization Problem PO max P1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',PN min x∈[0,1] �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) 1− x2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q(x) = N � i=1 Pi · f i N(x) f i N(x) = N · � N −1 i −1 � xi−1(1− x)n−i ∀i ∈ [N] n� i=1 Pi = 1 (42) Pi ≥ 0 ∀i ∈ [N] (43) Now we aim to solve the optimization problem PO numerally with different numbers of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' For the inner minimization problem, it must be solved accurately so that it precisely reflect the approxi- mation ratio of the order statistics mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' We use binary search to find its optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' When we need to check whether inf x∈[0,1) �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) 1− x2 ≥ r It is equivalent to check if �x 0 q(t)· t(1− x)dt + �1 x q(t)·(1− t)x dt +(1− x) −r(1− x2) ≥ 0 ∀x ∈ [0,1] Notice that q(t) is a polynomial of degree N, thus we only need to find the minimum of a single- variable polynomial over [0,1], and this could be efficiently done by finding the roots of its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 41 We do the binary search for 100 times, so the error caused by binary search is at most 2−100, which is much smaller than the floating point errors and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then we use some empirical algorithms to search for parameters in the outer maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Unlike the inner minimization problem, we do not need to get an exact optimum of the outer maximization problem since it only reflects the ratio of the order statistic mechanism we have found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The code can be found at our GitHub repository(https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='com/BilateralTradeAnonymous/On-the-Optimal-Fixed-Price-Mechanism-in-Bilateral-Trade).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2 General Instance Now again we formally write down the optimization problem that characterizes the optimal order statis- tic mechanism with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Similarly, as we proved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='2, suppose the following optimiza- tion problem QO achieves its maximum OPTQO at (P∗ 1 ,P∗ 2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',P∗ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Let P∗ be the distribution over [N] such that Prx∼P ∗[x = i] = P∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Then, P∗ is the optimal order statistic mechanism in the general setting with N samples and its approximation ratio is exactly OPTQO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again notice that since the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the order statistic mechanism Q(x) is differentiable, we use q(t)dt instead of dQ(t) where q(x) is the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' of the order statistic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The Optimization Problem QO max P1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=',PN min x∈[0,1] �x 0 q(t)· t dt +1− x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' q(x) = N� i=1 Pi · f i N (x) f i N(x) = N · � N −1 i −1 � xi−1(1− x)n−i ∀i ∈ [N] n� i=1 Pi = 1 (44) Pi ≥ 0 ∀i ∈ [N] (45) Now we solve the optimization problem numerically for different fixed number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' Again the inner minimization problem could be solved efficiently by calculating the zero point of its derivatives and we search for the parameters in the outer maximization problem to get a good enough solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' The code could be found at our Github repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} +page_content=' 42' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE4T4oBgHgl3EQfmQ2m/content/2301.05167v1.pdf'} diff --git a/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/2301.02949v1.pdf.txt b/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/2301.02949v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79f6029824d6a1cae33bb9c68af0cb1f4d1611fe --- /dev/null +++ b/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/2301.02949v1.pdf.txt @@ -0,0 +1,974 @@ +arXiv:2301.02949v1 [cs.CG] 8 Jan 2023 +MAXIMUM OVERLAP AREA OF A CONVEX POLYHEDRON AND A +CONVEX POLYGON UNDER TRANSLATION +HYUK JUN KWEON AND HONGLIN ZHU +Abstract. Let P be a convex polyhedron and Q be a convex polygon with n vertices +in total in three-dimensional space. +We present a deterministic algorithm that finds a +translation vector v ∈ R3 maximizing the overlap area |P ∩ (Q + v)| in O(n log2 n) time. +We then apply our algorithm to solve two related problems. We give an O(n log3 n) time +algorithm that finds the maximum overlap area of three convex polygons with n vertices in +total. We also give an O(n log2 n) time algorithm that minimizes the symmetric difference +of two convex polygons under scaling and translation. +Contents +1. +Introduction +1 +2. +Preliminaries +3 +3. +Generalized two-dimensional prune-and-search +4 +4. +Maximum overlap of convex polyhedron and convex polygon +7 +5. +Maximum overlap of three convex polygons +14 +6. +Minimum symmetric difference of two convex polygons under homothety +17 +References +19 +1. Introduction +1.1. Background. Shape matching is an important topic in computational geometry, with +useful applications in areas such as computer graphics. In a typical problem of shape match- +ing, we are supplied two or more shapes, and we want to determine how much the shapes +resemble each other. More precisely, given a similarity measure and a set of allowed trans- +formations, we want to transform the shapes to maximize their similarity measure. +There are many candidates for the similarity measure, such as the Hausdorff distance +and the Fr´echet distance between the boundaries of the shapes. We can also consider the +area/volume of overlap or of symmetric difference. The advantage to these is that they are +more robust against noise on the boundary of the images [Ber+98]. +The maximum overlap problem of convex polytopes has been studied by many. In dimen- +sion 2, de Berg et al. [Ber+98] give an O(n log n) time algorithm for finding a translation +maximizing the area of intersection of two convex polygons (where n denotes the total number +of vertices of the polygons). In dimension 3, Ahn et al. [ABS08] give an O(n3 log4 n) expected +time algorithm finding the maximum overlap of two convex polyhedra under translation. For +the same problem, Ahn et al. [ACR13] present an algorithm that runs in O(n log3.5 n) time +with probability 1 − n−O(1) and an additive error. For d > 3, given two convex polytopes +1 + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +2 +of dimension d with n facets in total, Ahn et al. [ACR13] give an algorithm that finds the +maximum overlap under translation in O(n⌊d/2⌋+1 logd n) time with probability 1−nO(1) and +an additive error. +When all rigid motions are allowed, Ahn et al. [Ahn+07] give an approximate algorithm +that finds a rigid motion realizing at least 1−ǫ times the maximal overlap in O((1/ǫ) log n+ +(1/ǫ2) log(1/ǫ)) time. In dimension 3, Ahn et al. [Ahn+14] present an approximate algorithm +that finds a rigid motion realizing at least 1−ǫ times the maximal overlap in O(ǫ−3n log3.5 n) +with probability 1 − n−O(1). +When considering the maximum overlap as a similarity measure, we obviously can only +allow area/volume-preserving transformations. However, we may want to allow scaling as a +transformation—two similar triangles are supposed to be very “similar,” though they may +have different sizes. In this case, the area of symmetric difference is a better measure of +similarity. Yon et al. [Yon+16] give an algorithm minimizing the symmetric difference of +two convex polygons under translation and scaling in O(n log3 n) expected time. +1.2. Our Results. While many have studied the matching problem for two convex poly- +topes of the same dimension, few have examined the problem for polytopes of different +dimensions or matching more than two polytopes. +Our main result in this paper is a deterministic algorithm for the problem of matching a +convex polyhedron and a convex polygon under translation in three-dimensional space. +Theorem 1.1. Let P be a convex polyhedron and Q a convex polygon with n vertices in total. +We can find a vector v ∈ R3 that maximizes the overlap area |P ∩ (Q + v)| in O(n log2 n) +time. +We also present two applications of our algorithm to other problems in computational +geometry. +First, we give a deterministic algorithm for maximizing the overlap of three +convex polygons. +Theorem 1.2. Let P, Q, R be three convex polygons with n vertices in total in the plane. +We can find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area |P ∩ (Q + +vQ) ∩ (R + vR)| in O(n log3 n) time. +We also give a deterministic O(n log2 n) time algorithm for minimizing the symmetric +difference of two convex polygons under homothety, which is an improvement to Yon et al.’s +randomized algorithm [Yon+16]. +Theorem 1.3. Let P and Q be convex polygons with n vertices in total. Suppose κ ∈ (0, 1) +is a constant. We can find a homothety ϕ that minimizes +h(ϕ) = 2(1 − κ)|P \ ϕ(Q)| + 2κ|ϕ(Q) \ P| +in O(n log2 n) time. +1.3. Organization of the Paper. In §2, we introduce the problem of matching a convex +polyhedron and a convex polygon under translation in three-dimensional space. In §3, we +present a core technique we use in our algorithm, which is a generalization of Megiddo’s +prune-and-search technique [Meg84]. In §4, we present the algorithm for Theorem 1.1. In +§5, we apply our algorithm to solve the problem of maximizing the intersection of three +polygons under translation. +In §6, we give the algorithm for minimizing the symmetric +difference of two convex polygons under homothety. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +3 +Acknowledgements. This paper is the result of the MIT SPUR 2022, a summer under- +graduate research program organized by the MIT math department, where Kweon mentored +Zhu. The authors would like to thank the faculty advisors Professor David Jerison and Pro- +fessor Ankur Moitra for their support. They would like to thank the math department for +providing this research opportunity. +2. Preliminaries +Let P ⊂ R3 be a convex polyhedron and Q ⊂ R2 be a convex polygon with n vertices +in total. Throughout the paper, we assume that Q is in the xy-plane, and that the lowest +point of P is on the xy-plane. We want to find a translation vector v = (x, y, z) ∈ R3 that +maximizes the overlap area f(v) = |P ∩ (Q + v)|. +It is easy to observe that f(v) is continuous and piecewise quadratic on the interior of its +support. As noted in [Ber+98; ABS08; ACR13], f is smooth on a region R if P ∩ (Q + v) +is combinatorially equivalent for all v ∈ R, that is, if we have the same set of face-edge +incidences between P and Q. Following the convention of [ABS08], we call the polygons +that form the boundaries of these regions the event polygons, and as in [Ber+98], we call the +space of translations of Q the configuration space. The arrangement of the event polygons +partition the configuration space into cells with disjoint interiors. The overlap function f(v) +is quadratic on each cell. Thus, to locate a translation maximizing f, we need to characterize +the event polygons. +For two sets A, B ⊂ Rd, we write the Minkowski sum of A and B as +A + B := {a + b|a ∈ A, b ∈ B}. +We will make no distinction between the translation A + v and the Minkowski sum A + {v} +for a vector v. We also write A − B for the Minkowski sum of A with −B = {−b|b ∈ B}. +We categorize the event polygons into three types and describe them in terms of Minkowski +sums: +(I) When Q+v contains a vertex of P. For each vertex u of P, we have an event polygon +u − Q. There are O(n) event polygons of this type. +(II) When a vertex of Q + v is contained in a face of P. For each face F of P and each +vertex v of Q, we have an event polygon F − v. There are O(n2) event polygons of +this type. +(III) When an edge of Q+v intersects an edge of P. For each edge e of P and each edge e′ +of Q, we have an event polygon e − e′. There are O(n2) event polygons of this type. +The reason that convexity is fundamental to our solution (and to all the shape-matching +algorithms mentioned in the previous section) is due to the following proposition. +Proposition 2.1. Let P be a d′-dimensional convex polytope and let Q be a d-dimensional +convex polytope. Suppose d′ ≥ d. Let f(v) = Vol(P ∩ (Q + v)) be the volume of the overlap +function. Then, f(v)1/d is concave on its support supp(f) = {v|f(v) > 0}. +Proof. The support of f is the interior of the Minkowski sum P − Q, hence it is convex. +Now, to show that f(v)1/d is concave, it suffices to show that it is concave on any line l. +We parameterize l by p + vt for some point p ∈ Rd′ and some vector v ∈ Rd′, and let +fl(t) = f(p + vt). + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +4 +If Q + l is d + 1 dimensional, let I = P ∩ (Q + l). Then I is a convex d + 1-dimensional +polytope. Moreover, the cross-sections of I parallel to the d-flat containing Q are precisely +P ∩ (Q + p + vt) for t ∈ R, and they have area fl(t). By the Brunn-Minkowski theorem, the +d-th root of the cross-sectional volume of a convex d + 1-dimensional polytope is a concave +function. +If Q + l is d dimensional, that is, if l is contained in the d-flat containing Q, we may just +consider the intersection of P with this d-flat, which is also a convex d-dimensional polytope, +say P ′. Now, consider the d + 1-dimensional polytope I = {(u, t)|u ∈ (P ′ ∩ (Q + p + vt))}. +This is the intersection of two convex polytopes, hence convex. Again, the cross-sections of +I have volume fl(t) and convexity of f 1/d follows from the Brunn-Minkowski theorem. +□ +As in [Avi+96], we say a function f : R → R is unimodal if it increases to a maximum +value, possibly stays there for some interval, and then decreases. It is strictly unimodal if +it strictly increases to the maximum and then strictly decreases. Furthermore, we say a +function f : Rd → R is (strictly) unimodal if its restriction to any line is (strictly) unimodal. +The following corollary of Proposition 2.1 allows us to employ a divide-and-conquer strat- +egy in our algorithm. +Corollary 2.2 ([Avi+96]). For any line l parameterized by l = p + vt in Rd′, the function +fl(t) = f(p + vt) is strictly unimodal. +We also introduce two techniques that we apply in our algorithm. +Lemma 2.3 ([FJ84]). Let M be an m × n matrix of real numbers, where m ≤ n. If every +row and every column of M is in increasing order, then we say M is a sorted matrix. For +any positive integer k smaller or equal to mn, the k-th smallest entry of M can be found in +O(m log(2n/m)) time, assuming an entry of M can be accessed in O(1) time. +For our purposes, we will use this result in the weaker form of O(m + n). +Lemma 2.4 ([Cha93b]). Given n hyperplanes in Rd and a region R ⊂ Rd, a (1/r)-cutting +is a collection of simplices with disjoint interiors, which together cover R and such that the +interior of each simplex intersects at most n/r hyperplanes. A (1/r)-cutting of size O(rd) +can be computed deterministically in O(nrd−1) time. +In addition, the set of hyperplanes +intersecting each simplex of the cutting is reported in the same time. +3. Generalized two-dimensional prune-and-search +In this section, we give a generalization of Megiddo’s prune-and-search technique [Meg84] +that we use in our algorithm. This technique is of independent interest and can likely be +applied to other problems. +Theorem 3.1 ([Meg84]). Suppose there exists a points p∗ ∈ R2 not known to us. Suppose +further that we have an oracle that can tell us for any line l ⊂ R2 whether p∗ ∈ l, and if +p∗ /∈ l, which of the two open half-planes separated by l that p∗ belongs to. Let T be the +running time of the oracle. Then given n lines in the plane. We can find the position of p∗ +relative to each of the n lines in O(n + T log n) time. +We are interested in a generalized version of Megiddo’s problem. Suppose, instead of n +lines, we are given n sets of parallel lines S1, S2, . . . , Sn, each of size O(m). In addition, + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +5 +suppose the lines in each Si are indexed from left to right (assuming none of the lines are +parallel to the x-axis). Again, we want to know the position of p∗ relative to every line in +S = �n +i=1 Si. Megiddo’s algorithm solves this problem in O(mn + T log(mn)) time, but we +want a faster algorithm for large m by exploiting the structure of S. +Before we state and prove our result, we give a related definition. For n distinct sorted real +numbers x1, . . . xn each with positive weights w1, . . . wn such that �n +i=1 wi = 1, the weighted +median is the element xk with +k−1 +� +i=1 +wi ≤ 1/2 +and +n +� +i=k+1 +wi ≤ 1/2. +If there are two elements xk and xk+1 satisfying the condition, then the weighted median +is the mean of xk and xk+1. In the case the weights are all 1/n, this is just the ordinary +median. We have the following well-known result: +Lemma 3.2 ([Cor+09]). Suppose we are given n distinct real numbers with positive weights +that sum to 1. Then we can find the weighted median of these numbers in O(n) time. +Theorem 3.3. Let S = �n +i=1 Si be a union of n sets of O(m) parallel lines in the plane, +none of which are parallel to the x-axis. Suppose that the lines in each Si are indexed from +left to right. +Suppose there is an unknown point p∗ ∈ R2 and we are given an oracle that decides in +time T the relative position of p∗ to any line l. We can find the relative position of p∗ to +every line in S in O(n log2 m + (T + n) log(mn)) time. +Proof. Without loss of generality, suppose that there are no lines parallel to the y-axis. For +each i between 1 and n, suppose Si = {lj +i |la +i lies strictly to the left of lb +i iff a < b}. Suppose +that p∗ = (x∗, y∗) ∈ R2. +To report our final answer, we simply need to provide, for each Si, the two consecutive +indices a and a + 1 such that p∗ lies strictly between la +i and la+1 +i +or the single index a such +that p∗ ∈ la +i . +In our algorithm, we keep track of a feasible region R containing P ∗, which is either the +interior of a (possibly unbounded) triangle or an open line segment if we find a line l that +p∗ lies on. Together with R, we keep track of the 2n indices lower(i) and upper(i) such that +SR = �n +i=1 SR +i = {lj +i|j ∈ (lower(i), upper(i)]} is the set of lines intersecting R, which is also +the set of lines we do not yet know the relative position to p∗. In the beginning, R = R2. +Each step, we find O(1) lines to run the oracle on to find a new feasible region R′ ⊂ R such +that |SR| ≤ (1 − ǫ)|SR′| for some constant ǫ > 0 and recurse on R′. +One extra computational effort is updating SR′ by computing lower(i) and upper(i). Since +the feasible region is always a convex set of constant complexity, we can use binary search +on SR +i to find the new bounds for SR′ +i +in O(log |SR +i |) time. Thus, the total time involved in + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +6 +Algorithm 3.1: Pseudocode for Theorem 3.3 +input : A set S = �n +i=1 Si = {lj +i} of O(mn) lines +output: A list of indices that indicate the position of p∗ to each Si +1 R ←− R2 +2 SR ←− S +3 while SR ̸= ∅ do +4 +Find O(1) lines to run the oracle on +5 +Compute the piece R′ ⊂ R containing p∗ +/* We guarantee that R′ intersects at most (1 − ǫ) of the lines that +intersect R +*/ +6 +Triangulate R′ with O(1) lines to run the oracle on +7 +Update SR ←− SR′ +8 end +this process, assuming |SR| decreases by at least ǫ each iteration, is +n +� +i=1 +O(log |Si|) + +n +� +i=1 +O(log |SR1 +i |) + +n +� +i=1 +O(log |SR2 +i |) + · · · +=O(log( +n +� +i=1 +|Si|)) + O(log( +n +� +i=1 +|SR1 +i |)) + · · · +=O(n log( 1 +n|S|)) + O(n log( 1 +n|SR1|)) + · · · +=O(n log(m)) + O(n log(m(1 − ǫ))) + O(n log(m(1 − ǫ)2)) + · · · +=O(n log2 m). +Given SR and R, we describe how to find R′ ⊂ R to recurse on. We first find the weighted +median of the slopes of the lines in SR in O(n) time. +If this slope is equal to the slope of some line in SR +i and |SR +i | ≥ δ|SR| for some constant +δ > 0 to be specified later, then we simply run the oracle on the median line of SR +i +and +then run the oracle O(1) time to triangulate the feasible region to get some R′ with |SR′| ≤ +(1 − δ/2)|SR|. +Otherwise, at least (1 − δ)/2 of the lines have slopes strictly greater than/less than the +median slope. For convenience, we may assume that at least (1 − δ)/2 of the lines have +positive/negative slope. Now, let SR ++ be the set of lines with positive slope and SR +− the set +with negative slope. Suppose SR ++ = �k +i=1 SR +i and SR +− = �n +i=k+1 SR +i . We remove lines from +one of the sets such that |SR ++| = |SR +−| ≥ (1−δ)/2·|SR|, and ignore the lines we have removed +for this step. +Now, we partition SR ++ ∪ SR +− into O(n) subsets Pi each containing the same number of +lines from SR ++ and SR +− in the following way: going in lexicographical order by the indices of +the lines, we put a line from SR +1 and a line from SR +k+1 into P1 until we exhaust one of the +sets (say it is SR +k+1). Then, we move on to put a line from the remaining SR +1 and a line +from SR +k+2 into P2 until we exhaust one of them, and so on. Each Pi is then of the form + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +7 +{lb(i) +a(i), . . . , lb(i)+|Pi|/2−1 +a(i) +, ld(i) +c(i), . . . , ld(i)+|Pi|/2−1 +c(i) +}, and can be represented by the indices (a(i), b(i)) +and (c(i), d(i)). We can compute this partition in O(n) time. For each Pi, we compute the +intersection pi = (xi, yi) of the median line in Pi with positive slope and the median line +with negative slope, and assign pi a weight wi = |Pi|/(2|SR ++|). Then, the weights of the pi +sum to 1. The significance of this is that if we know the relative position of p∗ to the lines +x = xi and y = yi, then we know the relative position of p∗ to at least 1/4 of the lines in Pi, +or 1 +4wi(1 − δ) of all the lines in |SR|. +We find the median point q = (xq, yq) of the pi’s by weight in x-coordinate in O(n) time +by Lemma 3.2. We run the oracle on the line x = xq. Let pk1, pk2, . . . , pkl be the points such +that we now know the relative position of p∗ to xki. Then the weights of these points sum +to at least 1/2. We find the median point q′ = (xq′, yq′) of these by weight in y-coordinate +in O(n) time. We run the oracle on the line y = yq′. Then, for points with weights that +sum to at least 1/4, we now know the relative position of p∗ to the vertical line and the +horizontal line through those points. This means that we know the relative position of p∗ to +at least +1 +16(1 − δ) of all the lines in |SR|. We get a new feasible region according to the two +oracle calls, and we triangulate it with O(1) more oracle calls to get our desired R′. Setting +δ = 1/9, we get that |SR′| ≤ (17/18)|SR|. The O(n) median finding time contributes to +O(n log mn) total time. +After O(log mn) recursive iterations, we arrive at a feasible region intersecting no line in +S, and we are done. Therefore, our algorithm runs in O(n log2 m + (T + n) log(mn)) time. +If m is polynomial in n, then our algorithm runs in O(n log2 n + T log n) time. +□ +Remark 3.4. A simpler and probably more practical algorithm for Theorem 3.3 is simply +choosing a random line from SR ++ and SR +− to intersect and run the oracle on the horizontal and +vertical line through the intersection. This method gives the same run time in expectation. +4. Maximum overlap of convex polyhedron and convex polygon +4.1. Overview. In this subsection, we give an overview of our algorithm that finds a transla- +tion v ∈ R3 maximizing the area of overlap function f. Following the convention in [Ber+98], +we call such a translation a goal placement. In the algorithm, we keep track of a closed target +region R which we know contains a goal placement and decrease its size until for each event +polygon F, either F ∩ interior(R) = ∅ or F ⊃ R. Then, f is quadratic on R and we can +find the maximum of f on R using standard calculus. Thus, the goal of our algorithm is to +efficiently trim R to eliminate event polygons that intersect it. +In the beginning of the algorithm, the target region is the interior of the Minkowski sum +P − Q, where the overlap function is positive. By the unimodality of the overlap function, +the set of goal placements is convex. Thus, for a plane in the configuration space, either +it contains a goal placement, or all goal placements lie on one of the two open half spaces +separated by the plane. If we have a way of knowing which case it is for any plane, we +can decrease the size of our target region by cutting it with planes and finding the piece +to recurse. More precisely, we need an algorithm PlaneDecision that decides the relative +position of the set of goal placements to a plane S. +Whenever PlaneDecision reports that a goal placement is found on a plane, we can let +the algorithm terminate. Thus, we can assume it always reports a half-space containing a +goal placement. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +8 +In the first stage of our algorithm, we sort the vertices of P by z-coordinate in increasing +order and sort the vertices of Q in counterclockwise order. Next, we trim the target region +with horizontal planes (planes parallel to the xy-plane) to get to a slice that does not contain +any vertices of P. +Lemma 4.1. In O(n log2 n) time, we can locate a strip R = {(x, y, z)|z ∈ [z0, z1]} whose +interior contains a goal placement and does not contain any vertices of P. +Figure 1. The intersection of P and the strip R. +Proof. Starting with the median z-coordinate of the vertices of P, we perform a binary search +on the levels containing a vertex of P, running PlaneDecision on those horizontal planes +to obtain the desired strip R. For a horizontal plane S, [Ber+98, Theorem 3.8] allows us to +compute the maximum overlap of P ∩ S and Q under translation in O(n log n)-time. The +two planes S1 and S2 with the largest maximum values will be the bounding planes for the +slice containing a goal placement by the unimodality of f. Thus, by a binary search, we can +locate this slice in O(n log2 n) time. +□ +By Chazelle’s algorithm [Cha92], the convex polyhedron P ′ = P ∩ R can be computed in +O(n) time. From now on, we replace P with P ′. Without loss of generality, assume z0 = 0 +and z1 = 1. +The region in the configuration space where |P ∩(Q+v)| > 0 is the Minkowski sum P −Q. +Since P only has two levels P0 = {(x, y, z) ∈ P|z = 0} and P1 = {(x, y, z) ∈ P|z = 1} that +contain vertices, the Minkowski sum P − Q is simply the convex hull of (P0 − Q) ∪ (P1 − Q), +which has O(n) vertices. We can compute P0 − Q and P1 − Q in O(n) time and compute +their convex hull in O(n log n) time by Chazelle’s algorithm [Cha93a]. +4.2. PlaneDecision. In this subsection, we show that PlaneDecision can be performed +in O(n log n) time. Let S be a fixed plane in the configuration space. We call a translation +v that achieves maxv∈S f(v) a good placement. First, we can compute the intersection of S +with P − Q in O(n). If the intersection is empty, we just report the side of S containing +P − Q. From now on assume this is not the case. +The following lemma shows that PlaneDecision runs in the same time bound as the +algorithm that just finds the maximum of f on a plane. +Lemma 4.2. Suppose we can compute maxv∈S f(v) for any plane S ⊂ R3 in time T, then +we can perform PlaneDecision for any plane in time O(T). + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +9 +Algorithm 4.1: Pseudocode for Theorem 1.1 +input : A convex polyhedron P ∈ R3 and a convex polygon Q ∈ R3 with n vertices +in total +output: A translation v ∈ R3 maximizing the area |P ∩ (Q + v)| +1 Locate a horizontal slice containing a goal placement that does not contain any +vertices of P. +2 Find a “tube” D + ly whose interior contains a goal placement and intersects O(n) +event polygons, where D is a triangle in the xz-plane and ly is the y-axis. +3 Recursively construct a (1/2)-cutting of the target region D + ly to find a simplex +containing a goal placement that does not intersect any event polygon +Proof. The idea is to compute maxv∈S′ f(v) for certain S′ that are perturbed slightly from +S to see in which direction relative to S does f increase. +We compute over an extension of the reals R[ω]/(ω3), where ω > 0 is smaller than any +real number. Let A > 0 be the maximum of f over a plane S. Let S+ and S− be the two +planes parallel to S that have distance ω from S. We compute A+ = maxv∈S+ f(v) and +A− = maxv∈S− f(v) in O(T) time. Since f is piecewise quadratic, A+ and A− as symbolic +expression will only involve quadratic terms in ω. Since f is strictly unimodal on P − Q, +there are three possibilities: +(1) If A+ > A, then halfspace on the side of S+ contains the set of goal placements. +(2) If A− > A, then halfspace on the side of S− contains the set of goal placements. +(3) If A ≥ A+ and A ≥ A−, then A is the global maximum of f. +Thus, in O(T) time, we can finish PlaneDecision. +□ +Finding a good placement on S is similar to finding a goal placement on the whole config- +uration space. S is partitioned into cells by the intersections of event polygons with S. We +need to find a region on S containing a good placement that does not intersect any event +polygons. +We present a subroutine LineDecision that finds, for a line l ⊂ S, the relative position +of the set of good placements on S to l. +Proposition 4.3. For a line l ⊂ S, we can perform LineDecision in O(n) time. +P +Q + l +Figure 2. The convex polyhedron I = P ∩ (Q + l). + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +10 +Proof. First, we compute maxv∈l f(v) and a vector achieving the maximum. We parameterize +the line l by p+vt where t is the parameter and p, v ∈ R3. As in the proof of Proposition 2.1, +the horizontal cross-section of I = P ∩ (Q + l) at height t has area f(p + vt). Since I is +the intersection of two convex polytopes with O(n) vertices, Chazelle’s algorithm [Cha92] +computes I in O(n) time. Then, [Avi+96, Theorem 3.2] computes the maximum cross-section +in O(n) time. +Now, by the same argument and method as in the proof of Lemma 4.2, we can finish +LineDecision in O(n) time. In the case where maxv∈l f(v) = 0, we report the side of l +containing S ∩ (P − Q). +□ +We now present PlaneDecision. If S is horizontal, then we only need to find the maxi- +mum overlap of the convex polygons P ∩S and Q using De Berg et al.’s algorithm [Ber+98], +which takes O(n log n) time. Thus, we assume S is non-horizontal. +We break down PlaneDecision into three steps. Whenever our subroutine LineDecision +reports a good placement is found on a line, we can let the algorithm terminate. Thus, we +can assume it always reports a half-plane of S containing a good placement. +Algorithm 4.2: Pseudocode for PlaneDecision +input : A plane S ⊂ R3 +output: A translation v ∈ S maximizing the area |P ∩ (Q + v)| +1 Compute S ∩ (P − Q) and set it to be our initial target region. +2 Locate a strip on S containing a good placement whose interior intersects O(n) event +polygons. +3 Recursively construct a (1/2)-cutting of the strip to find a triangle containing a good +placement that does not intersect any event polygon +We have already explained step one, so we begin with step two. +4.2.1. Step 2. We want to find a strip on S strictly between z = 0 and z = 1 that intersects +O(n) event polygons. Since there are no vertices of P with z-coordinate in the interval (0, 1), +there are no event polygons of type (I) in this range, and we will only need to consider event +polygons of type (II) and type (III). +We look at the intersection points of S with the edges of the event polygons. These come +from the set {ei − vj|ei ⊂ P non-horizontal edge, vj ∈ Q vertex}. We are interested in the +z-coordinates of these intersections, so we project everything into the xz-plane. Then, S +becomes a line, which we denote by lS, and each edge ei − vj becomes a segment whose +endpoints lie on z = 0 and z = 1. Suppose each edge ei projects to a segment si, and each +vj projects to a point xj on the x-axis. Then, we get O(n2) segments si − xj with endpoints +on z = 0 and z = 1, and the line lS that intersect them in some places. +Lemma 4.4. In O(n log n) time, we can locate a strip R = {(x, y, z) ∈ S|z ∈ [z0, z1]} whose +interior contains a good placement and intersects none of the edges of the event polygons. +Proof. By our setup from above, we want to find a segment on lS whose interior does not +intersect any segment of the form si − xj. +Since si are projections of edges of a convex polyhedron, we can separate them into two +sets such that edges from the same set do not intersect (we take the segments that are + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +11 +projections of the edges of the “front” side and “back” side, respectively), allowing the two +extremal edges to appear in both sets. We will process each set separately. This can be +done by identifying the extremal points points on the top and bottom faces of P in the x +direction, which can be done in O(log n) time. +For a set of non-intersecting segment, since they all have endpoints on the line z = 0 and +z = 1, we can sort them by the sum of the x-coordinates of their two endpoints. This takes +O(n log n) time. We further separate these segments into two sets by slope: those that make +a smaller angle than lS (the projection of S) with the positive x-axis, and those that make +a larger angle. +Suppose we now have a set of non-intersecting segments that all make larger angles than +lS with the positive x-axis, s1, s2, . . . , sm, where m = O(n). We also sort the projections +of the vertices of Q, x1, . . . , xq, in decreasing order by x-coordinate. This can be done in +O(log n) by identifying the extremal vertices of Q in the x-direction. +Let zij be the z-coordinate of the intersection of the line containing si − xj with lS. Let +M be an m × q matrix with (i, j)-th entry given by +Mij = + + + + + +0 +zij ≤ 0 +zij +zij ∈ (0, 1) +1 +zij ≥ 1 +. +We claim that M is a sorted matrix. To see this, consider any fixed row r and indices i < j. +Then the line containing sr − xi lies strictly to the left of the line containing sr − xj since +xi > xj. This means that zri < zrj. Thus, every row of M is in increasing order. Similarly, for +a fixed column c and indices i < j, the segment si − xc lies strictly to the left of the segment +sj − xc. Then, if they both intersect lS, we must have zic < zjc. If si − xc does not intersect +lS and sj − xc does, then si − xc must lie on the left of lS and thus Mic = a < zjc = Mjc. +Similarly, if si − xc intersects lS and sj − xc does not, then sj − xc must lie on the right of lS +and thus Mic = zic < b = Mjc. If they both do not intersect lS, then still Mic ≤ Mjc since it +is impossible to have Mic = b and Mjc = a. This proves our claim. +By Lemma 2.3, we can find the k-th smallest value in M in O(m + q) = O(n) time. +Thus, we can perform a binary search on these z-coordinates of the intersections of the +edges ei − vj with S. Each time we perform a LineDecision on the line with the median +z-coordinate of the remaining entries to eliminate half of the intersections. After O(log n) +iterations or O(n log n) time, we find a strip on S containing a good placement that contains +no intersections with this group of edges. +We repeat the same procedure for the other three groups and compute the intersection +of the four strips to find a strip containing a good placement that contains no intersections +with any edge of the event polygons. +□ +Our current target region, the strip R we obtained from Lemma 4.4, intersects few event +polygons and we can compute them efficiently. +Lemma 4.5. The interior of the region R intersects O(n) event polygons, and we can com- +pute them in O(n log n) time. +Proof. For a vertex v of Q, it contributes the O(n) event polygons of type (II) that are the +faces of P − v. The intersection of the boundary of P − v with S is a convex polygon. Since + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +12 +there are no intersections with edges of event polygons inside the strip R, at most two edges +of the convex polygon can lie inside R, one on the “front side” and the other on the “back +side.” +To compute these two segments on R, we first consider the two sorted matrices that +together describe the edges on the “front side” and look at the column associated to −v. We +find, for each column, the two (or zero) adjacent entries that contain the z-coordinates of R +in between. The two of the at most four that are closest to the strip will be the endpoints +of the segment that intersect the strip on the “front side.” Computing this segment takes +O(log n) time since we can use binary search on the columns to find the desired entries. We +do the same to find the segment on the “back side.” We do this for all vertices of Q to find +the O(n) intersections with the event polygons of type (II) in O(n log n) time. +For an edge e of P, it contributes O(n) event polygons of type (III) that form the sur- +rounding sides of a “cylinder” with base congruent to −Q. Again, each of these “cylinders” +intersect the strip R in at most two faces, so there are O(n) intersections of R with event +polygons of type (III). We can compute these segments by performing the binary search on +the row of one of the sorted matrices associated to the edge e. The two entries immediately +below the strip and the two immediately above the strip define the at most two segments +intersecting R. Similar to the procedure above, this takes O(log n) time for each edge of P, +thus O(n log n) time in total. +□ +4.2.2. Step 3. Now, we have a target region R and the O(n) intersections it makes with the +event polygons. +Lemma 4.6. In O(n log n) time, we can find a region R′ ⊂ R containing a good placement +that does not intersect any of the O(n) event polygons. +Proof. We recursively construct a (1/2)-cutting of the target region. By Lemma 2.4, a (1/2)- +cutting of constant size can be computed in O(n) time. We perform LineDecision on the +lines of the cutting to decide on which riangle to recurse. After O(log n) iterations, we have +a target region R′ that intersects no event polygons. +This procedure runs in O(n log n) +time. +□ +Finally, since the overlap function is quadratic on our final region R′, we can solve for +the maximum using standard calculus. After finding maxv∈S f(v) and a vector achieving +it O(n log n) time, by Lemma 4.2, we can perform PlaneDecision on S in the same time +bound. +Proposition 4.7. For a plane S, we can perform PlaneDecision in O(n log n) time. +4.3. Complete Algorithm. With PlaneDecision as a subroutine, we now present the +complete algorithm. As in Algorithm 4.1, we break down the algorithm into three stages. +4.3.1. Stage 1. This stage is already done in Lemma 4.1. Recall that we now have P with +all its vertices on z = 0 and z = 1, and we have computed P − Q. +4.3.2. Stage 2. The next stage is the main component of our algorithm. We project the +entire configuration space and the event polygons onto the xz-plane in order to find a target +region D whose preimage D + ly intersects few event polygons, where ly is the y-axis. +The non-horizontal edges of the event polygons project to segments on the strip 0 < z < 1 +on the xz-plane. We characterize our desired region D in the following lemma. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +13 +(a) Projection of P +(b) Projection of Q +(c) Projection of the configuration space, and the target region D +Figure 3. Projections onto the xz-plane. +Lemma 4.8. For a region D that does not intersect any of the segments that are the projec- +tions of the non-horizontal edges of the event polygons, the preimage D + ly intersects O(n) +event polygons. +Proof. For any region D on the xz-plane, the set of event polygons that the “tube” D + ly +intersects is precisely the set of projected event polygons that D intersects. Now, let D be +a region that does not intersect any segment from the projections of the event polygons. +Let s1, s2, . . . , sm be the segments that are the projections of the non-horizontal edges +of P, and let x1, . . . , xq be the projections of the vertices of Q on the x-axis and assume +that they are sorted by decreasing x-coordinate. Then, the projections of the non-horizontal +edges of the event polygons are precisely si − xj. +We first split the segments into four groups. Let s1, . . . , sm1 be the projections of the non- +horizontal edges of P on the “front side,” and sm1+1, . . . , sm be those on the “back side.” +The at most two edges visible on both the front and the back may be repeated. Then the +segments from either group are pairwise non-intersecting. Similarly, we split the vertices of +Q into a front side and a backside, including the at most two vertices visible on both the +front and back in both sets. We consider the segments in the configuration space made by +one of the two groups of edges of P and one of the two groups of vertices of Q. The other +three sets of segments are processed similarly. +Suppose that the segments we consider are s1, . . . , sm1, and the projected vertices are +x1, . . . , xq1. Suppose the segments are sorted by increasing sum of the x-coordinates of their +endpoints, and that the vertices are sorted by decreasing x-coordinate. The event polygons of +type (II) are the trapezoids or triangles between segments si−xj and si+1−xj for each of the +four groups of segments. For each fixed projected vertex x, the region D intersects at most +one event polygon of type (II) for each group. Thus, D intersects O(n) event polygons of +type (II). Similarly, the event polygons of type (III) are the parallelograms between segments +si − xj and si − xj+1 for each of the four groups of segments. For each fixed segment si, D + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +14 +intersects at most one event polygon of type (III), thus it intersects O(n) event polygons of +type (III) in total. +□ +Now it remains to efficiently find such a region D with D + ly containing a goal placement +and compute the O(n) event polygons that intersect its interior. +Lemma 4.9. In O(n log2 n) time, we can find a triangle D in the xz-plane such that the +interior of D + ly contains a goal placement and intersects O(n) event polygons. We can +compute these O(n) event polygons in the same time bound. +Proof. The computation of D is a direct application of Theorem 3.3, where m = O(n). +Calling the oracle on a line l in the xz-plane is running the PlaneDecision algorithm on +the plane parallel to the y-axis that projects to l. We compute a triangle for each of the four +groups of segments, take their intersection, and triangulate the intersection using O(1) calls +to PlaneDecision. Thus, we can compute the desired triangle D in O(n log2 n) time. +To compute the event polygons intersecting the interior of D + ly is simple, since we have +shown in the proof of Lemma 4.8 that D intersects at most one projection of an event polygon +of each type in each of the four groups for a fixed vertex xj (for type (II)) or segment si +(for type (III)). Once we have D, we can compute these polygons by binary search on each +of the O(n) groups of O(n) non-intersecting segments to find the two between which R lies. +Also, the event polygons all have constant complexity so computing all of them takes linear +time. We can recover the event polygons from their projections and compute the planes that +contain them in linear time. Thus, this entire process can be done in O(n log n) time. +□ +4.3.3. Stage 3. Now, we have a target region R = D + ly whose interior contains a goal +placement, and we have the O(n) event polygons that intersect it. +Lemma 4.10. In O(n log2 n) time, we can find a region R′ ⊂ R containing a goal placement +that does not intersect any of the O(n) event polygons. +Proof. We recursively construct a (1/2)-cutting of the target region. By Lemma 2.4, a (1/2)- +cutting of constant size can be computed in O(n) time. We perform PlaneDecision on the +planes of the cutting to decide on which simplex to recurse. After O(log n) iterations, we +have a target region R′ that intersects no event polygons. This procedure runs in O(n log2 n) +time. +□ +Finally, since the overlap function is quadratic on our final region R′, we can solve for the +maximum using standard calculus. This concludes the proof of our main theorem. +Theorem 1.1. Let P be a convex polyhedron and Q a convex polygon with n vertices in total. +We can find a vector v ∈ R3 that maximizes the overlap area |P ∩ (Q + v)| in O(n log2 n) +time. +5. Maximum overlap of three convex polygons +Let P, Q, R be three convex polygons with n vertices in total in the plane. We want +to find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area g(vQ, vR) = +|P ∩ (Q + vQ) ∩ (R + vR)|. +In this problem, the configuration space is four-dimensional. An easy extension of Propo- +sition 2.1 and Corollary 2.2 shows that the function of overlap area is again unimodal. This + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +15 +time, we have four-dimensional event polyhedra instead of event polygons that divide the +configuration space into four-dimensional cells on which g(vQ, vR) is quadratic. We call a +hyperplane containing an event polyhedron an event hyperplane, and they are defined by +two types of events: +(I) When one vertex of P, Q + vQ or R + vR lies on an edge of another polygon. There +are O(n) groups of O(n) parallel event hyperplanes of this type. +(II) When an edge from each of the three polygons intersect at one point. There are +O(n3) event hyperplanes of this type. +To overcome the difficulty of dealing with the O(n3) event hyperplanes of type (II), we +first prune the configuration space to a region intersecting no event hyperplanes of type (I). +We then show that the resulting region only intersects O(n) event hyperplanes of type (II). +Similar to Theorem 1.1, we want an algorithm HyperplaneDecision that computes, for +a hyperplane H ⊂ R4, the maximum max(vQ,vR)∈H g(vQ, vR) and the relative location of the +goal placement to H. In fact, we will only need to perform HyperplaneDecision on some +hyperplanes. +Proposition 5.1. Suppose H is a hyperplane that satisfies one of the following three condi- +tions: +(1) H is orthogonal to (x1, y1, 0, 0) for some x1, y1 ∈ R. +(2) H is orthogonal to (0, 0, x2, y2) for some x2, y2 ∈ R. +(3) H is orthogonal to (x1, y1, −x1, −y1) for some x1, y1 ∈ R. +Then, we can perform HyperplaneDecision on H in O(n log2 n) time. +Proof. We provide the algorithm for H orthogonal to (x1, y1, 0, 0) for some x1, y1 ∈ R, and +the other two types follow similarly. +We reinterpret the problem of finding max(vQ,vR)∈H g(vQ, vR) as a polyhedron-polygon +matching problem. In H, we allow R to move freely, and Q moves in a line l perpendicular +to (x1, y1). We parameterize l by l = p + vt, and form the convex polyhedron +IP Q = {(x, y, t)|(x, y) ∈ P} ∩ {(x, y, t)|(x, y) ∈ (Q + p + vt)}. +By [Cha92], I can be computed in O(n) time. In addition, the cross-section of I at t = t0 +is P ∩ (Q + p + vt). Then, we see that finding max(vQ,vR)∈H g(vQ, vR) is the same as finding +a translation maximizing the intersection of I and R. By Theorem 1.1, this can be done in +O(n log2 n) time. +Using the formal perturbation argument in Lemma 4.2, HyperplaneDecision on H can +be completed in the same time bound. +□ +Using Proposition 5.1, we can prune the configuration space to a region that intersects no +event hyperplanes of type (I) and O(n) event hyperplanes of type (II). +Proposition 5.2. We can compute a 4-polytope TP QR of complexity O(1) in O(n log3 n) +time such that +(1) the goal placement lies on TP QR, +(2) no hyperplane of type (I) intersects the interior of TP QR, and +(3) only O(n) hyperplanes of type (II) passes through TP QR. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +16 +P + (z-axis) +Q + l +Figure 4. The convex polyhedron IP Q is the intersection of these two objects. +Furthermore, the set of hyperplanes of type (II) intersecting the interior of TP QR is obtained +in the same time bound. +Proof. If a HyperplaneDecision reports a goal placement, we are done. Thus, we assume +that HyperplaneDecision always reports a halfspace containing a goal placement. +Each event hyperplane containing an event polyhedron of a vertex of P on an edge of Q+vQ +or an event polyhedron of a vertex of Q+vQ on an edge of P is orthogonal to some (x1, y1, 0, 0). +We project all these event hyperplanes into the 2-flat SP Q = {(x1, y1, 0, 0)|x1, y1 ∈ R}. Then, +the images are O(n) groups of O(n) parallel lines. We can therefore apply Theorem 3.3 +to these lines, where an oracle call on a line l is running HyperplaneDecision on the +hyperplane that projects to l on SP Q, which is orthogonal to some (x1, y1, 0, 0). Thus, by +Proposition 5.1, we can find a triangle TP Q ⊂ SP Q whose interior does not intersect any +event hyperplane as described above in O(n log3 n) time. +Similarly, we can find the triangles +TP R ⊂ {(0, 0, x2, y2)|x2, y2 ∈ R} +and +TQR ⊂ {(x1, y1, −x1, −y1)|x1, y1 ∈ R} +corresponding to the other event hyperplanes of type (I) in O(n log3 n) time. Then, the +interior of +TP QR = {(x1, y1, x2, y2)|(x1, y1, 0, 0) ∈ TP Q, (0, 0, x2, y2) ∈ TP R, +�x1 − x2 +2 +, y1 − y2 +2 +, x2 − x1 +2 +, y2 − y1 +2 +� +∈ TQR} +does not intersect any event hyperplane of type (I) and contains a goal placement. +Since the interior of TP QR intersects no event hyperplane of type (I), the pairwise config- +uration of P and Q, P and R, Q and R are fixed (the pairwise edge incidences are fixed). +Since any edge eP of P intersects at most two edges of Q and at most two edges of R inside +TP QR, there are at most four event hyperplanes of type (II) where eP is concurrent with an +edge of Q and an edge of R. Thus, at most 4n event hyperplanes of type (II) intersect the +interior of TP QR. +□ +With Proposition 5.2, we can finish our algorithm. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +17 +Theorem 1.2. Let P, Q, R be three convex polygons with n vertices in total in the plane. +We can find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area |P ∩ (Q + +vQ) ∩ (R + vR)| in O(n log3 n) time. +Proof. Take TP QR as in Proposition 5.2. Let +f(vP, vQ) = +� +|P ∩ (Q + vQ) ∩ (R + vR)| +if (vQ, vR) ∈ TP QR +0 +otherwise. +Then f is unimodal and the maximum of f is the goal placement. Given an m-flat S, we want +to compute the maximum of f on S in O(n logm−1) time by induction on m ∈ {1, 2, 3, 4}. +If m = 1, this can be done in O(n) time by Proposition 4.3. +Assume that m > 1. +Then S ∩ TP QR can be computed in O(1) time. Given an (m − 1)-flat l ⊂ S, we can use +the perturbation method as in Lemma 4.2 to report the relative position of the maximum +over S. There are O(n) event hyperplane intersecting S ∩ TP QR. Thus, by Lemma 2.4, we +can recursively construct (1/2)-cuttings to give an O(n logm−1) time algorithm to find the +maximum of f on S. +□ +6. Minimum symmetric difference of two convex polygons under homothety +A homothety ϕ: R2 → R2 is a composition of a scaling and a translation. Let λ > 0 be +the scaling factor and v be the translation vector of ϕ. Then +ϕ(A) = λA + v = {λp + v | p ∈ A}. +Define the symmetric difference of sets A, B ⊂ R2 to be +A△B :=(A ∪ B) \ (A ∩ B) +=(A \ B) ∪ (B \ A). +Let P and Q be convex polygons with n vertices in total. We want to find a homothety +ϕ of Q that minimizes the area of symmetric difference +h(ϕ) = h(x, y, λ) = |P△ϕ(Q)|, +where ϕ(Q) = λQ + (x, y). +Yon et al. [Yon+16] consider a slightly more general problem, where they minimize the +function +h(ϕ) = (2 − 2κ)|P \ ϕ(Q)| + 2κ|ϕ(Q) \ P|, +where κ ∈ (0, 1) is some constant. When κ = 1/2, this is the area of symmetric difference +function. They give a randomized algorithm that solves this problem in O(n log3 n) expected +time. We present a faster determinisitc algorithm by relating this problem to the polyhedron- +polygon matching problem and then applying a modified version of Theorem 1.1. +As in [Yon+16], we rewrite the objective function h(ϕ): +h(ϕ) = 2(1 − κ)|P| + 2κ|ϕ(Q)| − 2|P ∩ ϕ(Q)| += 2(1 − κ)|P| + 2κ|Q|λ2 − 2|P ∩ ϕ(Q)|. +Thus, minimizing h(ϕ) is the same as maximizing h′(ϕ) = |P ∩ϕ(Q)|−cλ2, where c = κ|Q|. + +MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON +18 +Q +C +Figure 5. Formation of the cone C. +Consider the cone C = {(x, y, λ)|λ ∈ [0, M], (x, y) ∈ λQ}, where M = |P|/(c|Q|). Then h is +not minimized for λ > M. Since +h′(x, y, λ) = |C ∩ (P + (−x, −y, λ))| − cλ2, +the problem reduces to maximizing the overlap area of the cone C and P under translation +subtracted by a quadratic function. To show that we can still use a divide-and-conquer +strategy, we identify a region where h′ is strictly unimodal. +Lemma 6.1 ([Yon+16]). The closure D of the set {ϕ ∈ R3|h′(ϕ) > 0} is convex. Further- +more, h′(x, y, λ) is strictly unimodal on D. +Proof. This follows from [Yon+16, Lemma 2.2] and [Yon+16, Lemma 2.7]. +□ +Although it is difficult to directly compute D, note that −Q ⊂ D. With this observation, +we show that we can still find the relative position of the set of goal placements to certain +planes S in O(n log n) time with some modifications to LineDecision and PlaneDecision. +Lemma 6.2. For any l ⊂ R3, we can compute maxϕ∈l h′(ϕ) or report it is a negative number +in O(n) time. +Proof. If l is horizontal, then we can apply Proposition 4.3 since cλ is constant. Otherwise, +we parameterize l by l = p + vt and construct the convex polyhedron I whose cross-section +I(t0) at t = t0 has area |C ∩(P +(p+vt0))| as in the proof of Proposition 4.3. It comes down +to maximizing |I(t)| − c(λ(t))2, where λ(t) is the λ-coordinate of p + vt. Since +� +|I(t)| is a +concave function, +� +|I(t)| − √cλ(t) is also concave, and has the same complexity as +� +|I(t)|. +Thus, we can apply [Avi+96, Theorem 3.2] to find the maximum of +� +|I(t)| − √cλ(t). +Supposed it is achieved at t′. Although t′ may not be where the maximum of |I(t)|−c(λ(t))2 +is, it tells us whether the maximum is positive. If not, we can simply terminate the process. +If it is, we know that l intersects D, and p+vt′ ∈ D. This allows us to use divide-and-conquer +as in [Avi+96], since we can recurse in the direction of t′ whenever we query a point t and +find h′(t) < 0. +□ +Proposition 6.3. Let S ⊂ R3 be a plane. If S is horizontal or if S intersects the polygon +−Q ⊂ D, then we can perform PlaneDecision on S in O(n log n) time. +Proof. If S is horizontal, then we can apply Algorithm 4.2. If the maximum is negative, then +we simply report the side of S containing −Q, otherwise we proceed as in Lemma 4.2. Now + +REFERENCES +19 +assume S is non-horizontal and intersects −Q. Let s = S∩(−Q). Then we know that s ⊂ D. +Let l ⊂ S be a line we want to run the subroutine LineDecision on. By Lemma 6.2, we +can find maxϕ∈l h′(ϕ) or report it is negative in O(n) time. If it is the latter case, we report +the side of l containing s. Otherwise, l intersects D, and we can proceed as in Lemma 4.2. +Thus, we can still find maxϕ∈S h′(ϕ) in O(n log n) time. Since S intersects D, we can use +Lemma 4.2 to complete PlaneDecision on S. +□ +Finally, we present the complete algorithm, which applies Theorem 1.1 with slight modi- +fications. +Theorem 1.3. Let P and Q be convex polygons with n vertices in total. Suppose κ ∈ (0, 1) +is a constant. We can find a homothety ϕ that minimizes +h(ϕ) = 2(1 − κ)|P \ ϕ(Q)| + 2κ|ϕ(Q) \ P| +in O(n log2 n) time. +Proof. We want to maximize h′(x, y, λ) = |C ∩ (P + (−x, −y, λ))| − cλ2 over R3, where +c = κ|Q|. In order to apply our algorithm for Theorem 1.1, we need to show that we only +run PlaneDecision on horizontal planes and planes that intersect −Q. +In the first stage (as outlined in Algorithm 4.1), we only run PlaneDecision on horizontal +planes. +In the second stage, we apply Theorem 3.3 to the O(n) groups of O(n) lines that are +the projections of the lines containing edges of event polygons on the xz-plane. Observe +that these lines all intersect the projection of −Q on the xz-plane. In each recursive step +of our algorithm, we query a horizontal (parallel to the x-axis) line and a line that goes +“between” two lines in the O(n2) lines. The planes they represent both satisfy the condition +for Proposition 6.3. Then we run PlaneDecision O(1) more times to triangulate our feasible +region. Here, we make a small modification: instead of maintaining a triangular feasible +region, we maintain a trapezoidal one by making O(1) horizontal cuts to make the region a +trapezoid. +In the third stage, we have a “tube” and O(n) event polygons that intersect it. As usual, +we recursively construct a (1/2)-cutting by Lemma 2.4. Chazelle’s algorithm [Cha93b] picks +O(1) planes intersecting the target region as the cutting, along with O(1) extra planes to +triangulate each piece. All the planes containing the event polygons intersect −Q, so we +can run PlaneDecision on them. Instead of triangulating our target region, it suffices to +reduce it to constant complexity. We do this by cutting it with O(1) horizontal planes such +that the remaining region only has vertices on two levels. Then, let e be any non-horizontal +edge. With O(1) planes through e, we can cut the target region into prisms and pyramids +with triangular bases. These planes all intersect −Q since they are between the two faces of +the target region containing e, and the planes containing them intersect −Q. +Therefore, with slight modifications to Theorem 1.1, we obtain a deterministic O(n log2 n) +algorithm for minimizing h(ϕ). +□ +References +[ABS08] +H.-K. Ahn, P. Brass, and C.-S. Shin. “Maximum overlap and minimum convex +hull of two convex polyhedra under translations”. In: Comput. Geom. 40.2 (2008), +pp. 171–177. + +REFERENCES +20 +[Ahn+14] +H.-K. Ahn, S.-W. Cheng, H. J. Kweon, and J. Yon. “Overlap of convex polytopes +under rigid motion”. In: Comput. Geom. 47.1 (2014), pp. 15–24. +[ACR13] +H.-K. Ahn, S.-W. Cheng, and I. Reinbacher. “Maximum overlap of convex poly- +topes under translation”. In: Comput. Geom. 46.5 (2013), pp. 552–565. +[Ahn+07] +H.-K. Ahn, O. Cheong, C.-D. Park, C.-S. Shin, and A. Vigneron. “Maximizing +the overlap of two planar convex sets under rigid motions”. In: Comput. Geom. +37.1 (2007), pp. 3–15. +[Avi+96] +D. Avis, P. Bose, T. C. Shermer, J. Snoeyink, G. Toussaint, and B. Zhu. “On +the sectional area of convex polytopes”. In: Communication at the 12th Annu. +ACM Sympos. Comput. Geom. 1996, p. C. +[Ber+98] +M. de Berg, O. Cheong, O. Devillers, M. van Kreveld, and M. Teillaud. “Com- +puting the maximum overlap of two convex polygons under translations”. In: +vol. 31. 5. Seventh International Symposium on Algorithms and Computation +(Osaka, 1996). 1998, pp. 613–628. +[Cha92] +B. Chazelle. “An optimal algorithm for intersecting three-dimensional convex +polyhedra”. In: SIAM J. Comput. 21.4 (1992), pp. 671–696. +[Cha93a] +B. Chazelle. “An optimal convex hull algorithm in any fixed dimension”. In: +Discrete Comput. Geom. 10.4 (1993), pp. 377–409. +[Cha93b] +B. Chazelle. “Cutting hyperplanes for divide-and-conquer”. In: Discrete Comput. +Geom. 9.2 (1993), pp. 145–158. +[Cor+09] +T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to algo- +rithms. Third. MIT Press, Cambridge, MA, 2009, pp. xx+1292. +[FJ84] +G. N. Frederickson and D. B. Johnson. “Generalized selection and ranking: sorted +matrices”. In: SIAM J. Comput. 13.1 (1984), pp. 14–30. +[Meg84] +N. Megiddo. “Linear programming in linear time when the dimension is fixed”. +In: J. Assoc. Comput. Mach. 31.1 (1984), pp. 114–127. +[Yon+16] +J. Yon, S. W. Bae, S.-W. Cheng, O. Cheong, and B. T. Wilkinson. “Approxi- +mating convex shapes with respect to symmetric difference under homotheties”. +In: 32nd International Symposium on Computational Geometry. Vol. 51. LIPIcs. +Leibniz Int. Proc. Inform. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016, +Art. No. 63, 15. + diff --git a/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/load_file.txt b/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..165495cced7dbca166acd276009ffe0447d56c9b --- /dev/null +++ b/W9E1T4oBgHgl3EQfJQNW/content/tmp_files/load_file.txt @@ -0,0 +1,864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf,len=863 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='02949v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='CG] 8 Jan 2023 MAXIMUM OVERLAP AREA OF A CONVEX POLYHEDRON AND A CONVEX POLYGON UNDER TRANSLATION HYUK JUN KWEON AND HONGLIN ZHU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P be a convex polyhedron and Q be a convex polygon with n vertices in total in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We present a deterministic algorithm that finds a translation vector v ∈ R3 maximizing the overlap area |P ∩ (Q + v)| in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We then apply our algorithm to solve two related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We give an O(n log3 n) time algorithm that finds the maximum overlap area of three convex polygons with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also give an O(n log2 n) time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Preliminaries 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Generalized two-dimensional prune-and-search 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Maximum overlap of convex polyhedron and convex polygon 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Maximum overlap of three convex polygons 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Minimum symmetric difference of two convex polygons under homothety 17 References 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Shape matching is an important topic in computational geometry, with useful applications in areas such as computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In a typical problem of shape match- ing, we are supplied two or more shapes, and we want to determine how much the shapes resemble each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' More precisely, given a similarity measure and a set of allowed trans- formations, we want to transform the shapes to maximize their similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are many candidates for the similarity measure, such as the Hausdorff distance and the Fr´echet distance between the boundaries of the shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can also consider the area/volume of overlap or of symmetric difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The advantage to these is that they are more robust against noise on the boundary of the images [Ber+98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The maximum overlap problem of convex polytopes has been studied by many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In dimen- sion 2, de Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Ber+98] give an O(n log n) time algorithm for finding a translation maximizing the area of intersection of two convex polygons (where n denotes the total number of vertices of the polygons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In dimension 3, Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [ABS08] give an O(n3 log4 n) expected time algorithm finding the maximum overlap of two convex polyhedra under translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For the same problem, Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [ACR13] present an algorithm that runs in O(n log3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='5 n) time with probability 1 − n−O(1) and an additive error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For d > 3, given two convex polytopes 1 MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 2 of dimension d with n facets in total, Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [ACR13] give an algorithm that finds the maximum overlap under translation in O(n⌊d/2⌋+1 logd n) time with probability 1−nO(1) and an additive error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' When all rigid motions are allowed, Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Ahn+07] give an approximate algorithm that finds a rigid motion realizing at least 1−ǫ times the maximal overlap in O((1/ǫ) log n+ (1/ǫ2) log(1/ǫ)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In dimension 3, Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Ahn+14] present an approximate algorithm that finds a rigid motion realizing at least 1−ǫ times the maximal overlap in O(ǫ−3n log3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='5 n) with probability 1 − n−O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' When considering the maximum overlap as a similarity measure, we obviously can only allow area/volume-preserving transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' However, we may want to allow scaling as a transformation—two similar triangles are supposed to be very “similar,” though they may have different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In this case, the area of symmetric difference is a better measure of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Yon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Yon+16] give an algorithm minimizing the symmetric difference of two convex polygons under translation and scaling in O(n log3 n) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Our Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' While many have studied the matching problem for two convex poly- topes of the same dimension, few have examined the problem for polytopes of different dimensions or matching more than two polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Our main result in this paper is a deterministic algorithm for the problem of matching a convex polyhedron and a convex polygon under translation in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P be a convex polyhedron and Q a convex polygon with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a vector v ∈ R3 that maximizes the overlap area |P ∩ (Q + v)| in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also present two applications of our algorithm to other problems in computational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' First, we give a deterministic algorithm for maximizing the overlap of three convex polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P, Q, R be three convex polygons with n vertices in total in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area |P ∩ (Q + vQ) ∩ (R + vR)| in O(n log3 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also give a deterministic O(n log2 n) time algorithm for minimizing the symmetric difference of two convex polygons under homothety, which is an improvement to Yon et al.’s randomized algorithm [Yon+16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P and Q be convex polygons with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose κ ∈ (0, 1) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a homothety ϕ that minimizes h(ϕ) = 2(1 − κ)|P \\ ϕ(Q)| + 2κ|ϕ(Q) \\ P| in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Organization of the Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In §2, we introduce the problem of matching a convex polyhedron and a convex polygon under translation in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In §3, we present a core technique we use in our algorithm, which is a generalization of Megiddo’s prune-and-search technique [Meg84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In §4, we present the algorithm for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In §5, we apply our algorithm to solve the problem of maximizing the intersection of three polygons under translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In §6, we give the algorithm for minimizing the symmetric difference of two convex polygons under homothety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 3 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This paper is the result of the MIT SPUR 2022, a summer under- graduate research program organized by the MIT math department, where Kweon mentored Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The authors would like to thank the faculty advisors Professor David Jerison and Pro- fessor Ankur Moitra for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' They would like to thank the math department for providing this research opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Preliminaries Let P ⊂ R3 be a convex polyhedron and Q ⊂ R2 be a convex polygon with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Throughout the paper, we assume that Q is in the xy-plane, and that the lowest point of P is on the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We want to find a translation vector v = (x, y, z) ∈ R3 that maximizes the overlap area f(v) = |P ∩ (Q + v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' It is easy to observe that f(v) is continuous and piecewise quadratic on the interior of its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' As noted in [Ber+98;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' ABS08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' ACR13], f is smooth on a region R if P ∩ (Q + v) is combinatorially equivalent for all v ∈ R, that is, if we have the same set of face-edge incidences between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Following the convention of [ABS08], we call the polygons that form the boundaries of these regions the event polygons, and as in [Ber+98], we call the space of translations of Q the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The arrangement of the event polygons partition the configuration space into cells with disjoint interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The overlap function f(v) is quadratic on each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, to locate a translation maximizing f, we need to characterize the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For two sets A, B ⊂ Rd, we write the Minkowski sum of A and B as A + B := {a + b|a ∈ A, b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We will make no distinction between the translation A + v and the Minkowski sum A + {v} for a vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also write A − B for the Minkowski sum of A with −B = {−b|b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We categorize the event polygons into three types and describe them in terms of Minkowski sums: (I) When Q+v contains a vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each vertex u of P, we have an event polygon u − Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n) event polygons of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (II) When a vertex of Q + v is contained in a face of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each face F of P and each vertex v of Q, we have an event polygon F − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n2) event polygons of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (III) When an edge of Q+v intersects an edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each edge e of P and each edge e′ of Q, we have an event polygon e − e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n2) event polygons of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The reason that convexity is fundamental to our solution (and to all the shape-matching algorithms mentioned in the previous section) is due to the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P be a d′-dimensional convex polytope and let Q be a d-dimensional convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose d′ ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let f(v) = Vol(P ∩ (Q + v)) be the volume of the overlap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, f(v)1/d is concave on its support supp(f) = {v|f(v) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The support of f is the interior of the Minkowski sum P − Q, hence it is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, to show that f(v)1/d is concave, it suffices to show that it is concave on any line l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We parameterize l by p + vt for some point p ∈ Rd′ and some vector v ∈ Rd′, and let fl(t) = f(p + vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 4 If Q + l is d + 1 dimensional, let I = P ∩ (Q + l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then I is a convex d + 1-dimensional polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Moreover, the cross-sections of I parallel to the d-flat containing Q are precisely P ∩ (Q + p + vt) for t ∈ R, and they have area fl(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By the Brunn-Minkowski theorem, the d-th root of the cross-sectional volume of a convex d + 1-dimensional polytope is a concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If Q + l is d dimensional, that is, if l is contained in the d-flat containing Q, we may just consider the intersection of P with this d-flat, which is also a convex d-dimensional polytope, say P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, consider the d + 1-dimensional polytope I = {(u, t)|u ∈ (P ′ ∩ (Q + p + vt))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This is the intersection of two convex polytopes, hence convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Again, the cross-sections of I have volume fl(t) and convexity of f 1/d follows from the Brunn-Minkowski theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ As in [Avi+96], we say a function f : R → R is unimodal if it increases to a maximum value, possibly stays there for some interval, and then decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' It is strictly unimodal if it strictly increases to the maximum and then strictly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Furthermore, we say a function f : Rd → R is (strictly) unimodal if its restriction to any line is (strictly) unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The following corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 allows us to employ a divide-and-conquer strat- egy in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 ([Avi+96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For any line l parameterized by l = p + vt in Rd′, the function fl(t) = f(p + vt) is strictly unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also introduce two techniques that we apply in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 ([FJ84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let M be an m × n matrix of real numbers, where m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If every row and every column of M is in increasing order, then we say M is a sorted matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For any positive integer k smaller or equal to mn, the k-th smallest entry of M can be found in O(m log(2n/m)) time, assuming an entry of M can be accessed in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For our purposes, we will use this result in the weaker form of O(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4 ([Cha93b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Given n hyperplanes in Rd and a region R ⊂ Rd, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together cover R and such that the interior of each simplex intersects at most n/r hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' A (1/r)-cutting of size O(rd) can be computed deterministically in O(nrd−1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In addition, the set of hyperplanes intersecting each simplex of the cutting is reported in the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Generalized two-dimensional prune-and-search In this section, we give a generalization of Megiddo’s prune-and-search technique [Meg84] that we use in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This technique is of independent interest and can likely be applied to other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 ([Meg84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose there exists a points p∗ ∈ R2 not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose further that we have an oracle that can tell us for any line l ⊂ R2 whether p∗ ∈ l, and if p∗ /∈ l, which of the two open half-planes separated by l that p∗ belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let T be the running time of the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then given n lines in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find the position of p∗ relative to each of the n lines in O(n + T log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We are interested in a generalized version of Megiddo’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose, instead of n lines, we are given n sets of parallel lines S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , Sn, each of size O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In addition, MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 5 suppose the lines in each Si are indexed from left to right (assuming none of the lines are parallel to the x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Again, we want to know the position of p∗ relative to every line in S = �n i=1 Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Megiddo’s algorithm solves this problem in O(mn + T log(mn)) time, but we want a faster algorithm for large m by exploiting the structure of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Before we state and prove our result, we give a related definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For n distinct sorted real numbers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' xn each with positive weights w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' wn such that �n i=1 wi = 1, the weighted median is the element xk with k−1 � i=1 wi ≤ 1/2 and n � i=k+1 wi ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If there are two elements xk and xk+1 satisfying the condition, then the weighted median is the mean of xk and xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the case the weights are all 1/n, this is just the ordinary median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We have the following well-known result: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 ([Cor+09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose we are given n distinct real numbers with positive weights that sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then we can find the weighted median of these numbers in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let S = �n i=1 Si be a union of n sets of O(m) parallel lines in the plane, none of which are parallel to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose that the lines in each Si are indexed from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose there is an unknown point p∗ ∈ R2 and we are given an oracle that decides in time T the relative position of p∗ to any line l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find the relative position of p∗ to every line in S in O(n log2 m + (T + n) log(mn)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Without loss of generality, suppose that there are no lines parallel to the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each i between 1 and n, suppose Si = {lj i |la i lies strictly to the left of lb i iff a < b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose that p∗ = (x∗, y∗) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' To report our final answer, we simply need to provide, for each Si, the two consecutive indices a and a + 1 such that p∗ lies strictly between la i and la+1 i or the single index a such that p∗ ∈ la i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In our algorithm, we keep track of a feasible region R containing P ∗, which is either the interior of a (possibly unbounded) triangle or an open line segment if we find a line l that p∗ lies on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Together with R, we keep track of the 2n indices lower(i) and upper(i) such that SR = �n i=1 SR i = {lj i|j ∈ (lower(i), upper(i)]} is the set of lines intersecting R, which is also the set of lines we do not yet know the relative position to p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the beginning, R = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Each step, we find O(1) lines to run the oracle on to find a new feasible region R′ ⊂ R such that |SR| ≤ (1 − ǫ)|SR′| for some constant ǫ > 0 and recurse on R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' One extra computational effort is updating SR′ by computing lower(i) and upper(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since the feasible region is always a convex set of constant complexity, we can use binary search on SR i to find the new bounds for SR′ i in O(log |SR i |) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, the total time involved in MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 6 Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1: Pseudocode for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 input : A set S = �n i=1 Si = {lj i} of O(mn) lines output: A list of indices that indicate the position of p∗ to each Si 1 R ←− R2 2 SR ←− S 3 while SR ̸= ∅ do 4 Find O(1) lines to run the oracle on 5 Compute the piece R′ ⊂ R containing p∗ /* We guarantee that R′ intersects at most (1 − ǫ) of the lines that intersect R / 6 Triangulate R′ with O(1) lines to run the oracle on 7 Update SR ←− SR′ 8 end this process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' assuming |SR| decreases by at least ǫ each iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' is n � i=1 O(log |Si|) + n � i=1 O(log |SR1 i |) + n � i=1 O(log |SR2 i |) + · · · =O(log( n � i=1 |Si|)) + O(log( n � i=1 |SR1 i |)) + · · · =O(n log( 1 n|S|)) + O(n log( 1 n|SR1|)) + · · · =O(n log(m)) + O(n log(m(1 − ǫ))) + O(n log(m(1 − ǫ)2)) + · · · =O(n log2 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Given SR and R, we describe how to find R′ ⊂ R to recurse on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We first find the weighted median of the slopes of the lines in SR in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If this slope is equal to the slope of some line in SR i and |SR i | ≥ δ|SR| for some constant δ > 0 to be specified later, then we simply run the oracle on the median line of SR i and then run the oracle O(1) time to triangulate the feasible region to get some R′ with |SR′| ≤ (1 − δ/2)|SR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Otherwise, at least (1 − δ)/2 of the lines have slopes strictly greater than/less than the median slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For convenience, we may assume that at least (1 − δ)/2 of the lines have positive/negative slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, let SR + be the set of lines with positive slope and SR − the set with negative slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose SR + = �k i=1 SR i and SR − = �n i=k+1 SR i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We remove lines from one of the sets such that |SR +| = |SR −| ≥ (1−δ)/2·|SR|, and ignore the lines we have removed for this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, we partition SR + ∪ SR − into O(n) subsets Pi each containing the same number of lines from SR + and SR − in the following way: going in lexicographical order by the indices of the lines, we put a line from SR 1 and a line from SR k+1 into P1 until we exhaust one of the sets (say it is SR k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, we move on to put a line from the remaining SR 1 and a line from SR k+2 into P2 until we exhaust one of them, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Each Pi is then of the form MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 7 {lb(i) a(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , lb(i)+|Pi|/2−1 a(i) , ld(i) c(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , ld(i)+|Pi|/2−1 c(i) }, and can be represented by the indices (a(i), b(i)) and (c(i), d(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can compute this partition in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each Pi, we compute the intersection pi = (xi, yi) of the median line in Pi with positive slope and the median line with negative slope, and assign pi a weight wi = |Pi|/(2|SR +|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, the weights of the pi sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The significance of this is that if we know the relative position of p∗ to the lines x = xi and y = yi, then we know the relative position of p∗ to at least 1/4 of the lines in Pi, or 1 4wi(1 − δ) of all the lines in |SR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We find the median point q = (xq, yq) of the pi’s by weight in x-coordinate in O(n) time by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We run the oracle on the line x = xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let pk1, pk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , pkl be the points such that we now know the relative position of p∗ to xki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then the weights of these points sum to at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We find the median point q′ = (xq′, yq′) of these by weight in y-coordinate in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We run the oracle on the line y = yq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, for points with weights that sum to at least 1/4, we now know the relative position of p∗ to the vertical line and the horizontal line through those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This means that we know the relative position of p∗ to at least 1 16(1 − δ) of all the lines in |SR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We get a new feasible region according to the two oracle calls, and we triangulate it with O(1) more oracle calls to get our desired R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Setting δ = 1/9, we get that |SR′| ≤ (17/18)|SR|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The O(n) median finding time contributes to O(n log mn) total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' After O(log mn) recursive iterations, we arrive at a feasible region intersecting no line in S, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Therefore, our algorithm runs in O(n log2 m + (T + n) log(mn)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If m is polynomial in n, then our algorithm runs in O(n log2 n + T log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' A simpler and probably more practical algorithm for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 is simply choosing a random line from SR + and SR − to intersect and run the oracle on the horizontal and vertical line through the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This method gives the same run time in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Maximum overlap of convex polyhedron and convex polygon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In this subsection, we give an overview of our algorithm that finds a transla- tion v ∈ R3 maximizing the area of overlap function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Following the convention in [Ber+98], we call such a translation a goal placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the algorithm, we keep track of a closed target region R which we know contains a goal placement and decrease its size until for each event polygon F, either F ∩ interior(R) = ∅ or F ⊃ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, f is quadratic on R and we can find the maximum of f on R using standard calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, the goal of our algorithm is to efficiently trim R to eliminate event polygons that intersect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the beginning of the algorithm, the target region is the interior of the Minkowski sum P − Q, where the overlap function is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By the unimodality of the overlap function, the set of goal placements is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, for a plane in the configuration space, either it contains a goal placement, or all goal placements lie on one of the two open half spaces separated by the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If we have a way of knowing which case it is for any plane, we can decrease the size of our target region by cutting it with planes and finding the piece to recurse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' More precisely, we need an algorithm PlaneDecision that decides the relative position of the set of goal placements to a plane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Whenever PlaneDecision reports that a goal placement is found on a plane, we can let the algorithm terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can assume it always reports a half-space containing a goal placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 8 In the first stage of our algorithm, we sort the vertices of P by z-coordinate in increasing order and sort the vertices of Q in counterclockwise order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Next, we trim the target region with horizontal planes (planes parallel to the xy-plane) to get to a slice that does not contain any vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In O(n log2 n) time, we can locate a strip R = {(x, y, z)|z ∈ [z0, z1]} whose interior contains a goal placement and does not contain any vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The intersection of P and the strip R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Starting with the median z-coordinate of the vertices of P, we perform a binary search on the levels containing a vertex of P, running PlaneDecision on those horizontal planes to obtain the desired strip R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a horizontal plane S, [Ber+98, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='8] allows us to compute the maximum overlap of P ∩ S and Q under translation in O(n log n)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The two planes S1 and S2 with the largest maximum values will be the bounding planes for the slice containing a goal placement by the unimodality of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, by a binary search, we can locate this slice in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ By Chazelle’s algorithm [Cha92], the convex polyhedron P ′ = P ∩ R can be computed in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' From now on, we replace P with P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Without loss of generality, assume z0 = 0 and z1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The region in the configuration space where |P ∩(Q+v)| > 0 is the Minkowski sum P −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since P only has two levels P0 = {(x, y, z) ∈ P|z = 0} and P1 = {(x, y, z) ∈ P|z = 1} that contain vertices, the Minkowski sum P − Q is simply the convex hull of (P0 − Q) ∪ (P1 − Q), which has O(n) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can compute P0 − Q and P1 − Q in O(n) time and compute their convex hull in O(n log n) time by Chazelle’s algorithm [Cha93a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' PlaneDecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In this subsection, we show that PlaneDecision can be performed in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let S be a fixed plane in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We call a translation v that achieves maxv∈S f(v) a good placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' First, we can compute the intersection of S with P − Q in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If the intersection is empty, we just report the side of S containing P − Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' From now on assume this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The following lemma shows that PlaneDecision runs in the same time bound as the algorithm that just finds the maximum of f on a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose we can compute maxv∈S f(v) for any plane S ⊂ R3 in time T, then we can perform PlaneDecision for any plane in time O(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 9 Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1: Pseudocode for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 input : A convex polyhedron P ∈ R3 and a convex polygon Q ∈ R3 with n vertices in total output: A translation v ∈ R3 maximizing the area |P ∩ (Q + v)| 1 Locate a horizontal slice containing a goal placement that does not contain any vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 2 Find a “tube” D + ly whose interior contains a goal placement and intersects O(n) event polygons, where D is a triangle in the xz-plane and ly is the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 3 Recursively construct a (1/2)-cutting of the target region D + ly to find a simplex containing a goal placement that does not intersect any event polygon Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The idea is to compute maxv∈S′ f(v) for certain S′ that are perturbed slightly from S to see in which direction relative to S does f increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We compute over an extension of the reals R[ω]/(ω3), where ω > 0 is smaller than any real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let A > 0 be the maximum of f over a plane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let S+ and S− be the two planes parallel to S that have distance ω from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We compute A+ = maxv∈S+ f(v) and A− = maxv∈S− f(v) in O(T) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since f is piecewise quadratic, A+ and A− as symbolic expression will only involve quadratic terms in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since f is strictly unimodal on P − Q, there are three possibilities: (1) If A+ > A, then halfspace on the side of S+ contains the set of goal placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (2) If A− > A, then halfspace on the side of S− contains the set of goal placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (3) If A ≥ A+ and A ≥ A−, then A is the global maximum of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, in O(T) time, we can finish PlaneDecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Finding a good placement on S is similar to finding a goal placement on the whole config- uration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' S is partitioned into cells by the intersections of event polygons with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We need to find a region on S containing a good placement that does not intersect any event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We present a subroutine LineDecision that finds, for a line l ⊂ S, the relative position of the set of good placements on S to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a line l ⊂ S, we can perform LineDecision in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' P Q + l Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The convex polyhedron I = P ∩ (Q + l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' First, we compute maxv∈l f(v) and a vector achieving the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We parameterize the line l by p+vt where t is the parameter and p, v ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' As in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, the horizontal cross-section of I = P ∩ (Q + l) at height t has area f(p + vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since I is the intersection of two convex polytopes with O(n) vertices, Chazelle’s algorithm [Cha92] computes I in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, [Avi+96, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2] computes the maximum cross-section in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, by the same argument and method as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2, we can finish LineDecision in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the case where maxv∈l f(v) = 0, we report the side of l containing S ∩ (P − Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ We now present PlaneDecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If S is horizontal, then we only need to find the maxi- mum overlap of the convex polygons P ∩S and Q using De Berg et al.’s algorithm [Ber+98], which takes O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we assume S is non-horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We break down PlaneDecision into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Whenever our subroutine LineDecision reports a good placement is found on a line, we can let the algorithm terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can assume it always reports a half-plane of S containing a good placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2: Pseudocode for PlaneDecision input : A plane S ⊂ R3 output: A translation v ∈ S maximizing the area |P ∩ (Q + v)| 1 Compute S ∩ (P − Q) and set it to be our initial target region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 2 Locate a strip on S containing a good placement whose interior intersects O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 3 Recursively construct a (1/2)-cutting of the strip to find a triangle containing a good placement that does not intersect any event polygon We have already explained step one, so we begin with step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We want to find a strip on S strictly between z = 0 and z = 1 that intersects O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since there are no vertices of P with z-coordinate in the interval (0, 1), there are no event polygons of type (I) in this range, and we will only need to consider event polygons of type (II) and type (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We look at the intersection points of S with the edges of the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' These come from the set {ei − vj|ei ⊂ P non-horizontal edge, vj ∈ Q vertex}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We are interested in the z-coordinates of these intersections, so we project everything into the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, S becomes a line, which we denote by lS, and each edge ei − vj becomes a segment whose endpoints lie on z = 0 and z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose each edge ei projects to a segment si, and each vj projects to a point xj on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, we get O(n2) segments si − xj with endpoints on z = 0 and z = 1, and the line lS that intersect them in some places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In O(n log n) time, we can locate a strip R = {(x, y, z) ∈ S|z ∈ [z0, z1]} whose interior contains a good placement and intersects none of the edges of the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By our setup from above, we want to find a segment on lS whose interior does not intersect any segment of the form si − xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since si are projections of edges of a convex polyhedron, we can separate them into two sets such that edges from the same set do not intersect (we take the segments that are MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 11 projections of the edges of the “front” side and “back” side, respectively), allowing the two extremal edges to appear in both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We will process each set separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This can be done by identifying the extremal points points on the top and bottom faces of P in the x direction, which can be done in O(log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a set of non-intersecting segment, since they all have endpoints on the line z = 0 and z = 1, we can sort them by the sum of the x-coordinates of their two endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This takes O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We further separate these segments into two sets by slope: those that make a smaller angle than lS (the projection of S) with the positive x-axis, and those that make a larger angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose we now have a set of non-intersecting segments that all make larger angles than lS with the positive x-axis, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , sm, where m = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We also sort the projections of the vertices of Q, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , xq, in decreasing order by x-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This can be done in O(log n) by identifying the extremal vertices of Q in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let zij be the z-coordinate of the intersection of the line containing si − xj with lS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let M be an m × q matrix with (i, j)-th entry given by Mij = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 zij ≤ 0 zij zij ∈ (0, 1) 1 zij ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We claim that M is a sorted matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' To see this, consider any fixed row r and indices i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then the line containing sr − xi lies strictly to the left of the line containing sr − xj since xi > xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This means that zri < zrj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, every row of M is in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similarly, for a fixed column c and indices i < j, the segment si − xc lies strictly to the left of the segment sj − xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, if they both intersect lS, we must have zic < zjc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If si − xc does not intersect lS and sj − xc does, then si − xc must lie on the left of lS and thus Mic = a < zjc = Mjc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similarly, if si − xc intersects lS and sj − xc does not, then sj − xc must lie on the right of lS and thus Mic = zic < b = Mjc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If they both do not intersect lS, then still Mic ≤ Mjc since it is impossible to have Mic = b and Mjc = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3, we can find the k-th smallest value in M in O(m + q) = O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can perform a binary search on these z-coordinates of the intersections of the edges ei − vj with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Each time we perform a LineDecision on the line with the median z-coordinate of the remaining entries to eliminate half of the intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' After O(log n) iterations or O(n log n) time, we find a strip on S containing a good placement that contains no intersections with this group of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We repeat the same procedure for the other three groups and compute the intersection of the four strips to find a strip containing a good placement that contains no intersections with any edge of the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Our current target region, the strip R we obtained from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4, intersects few event polygons and we can compute them efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The interior of the region R intersects O(n) event polygons, and we can com- pute them in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a vertex v of Q, it contributes the O(n) event polygons of type (II) that are the faces of P − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The intersection of the boundary of P − v with S is a convex polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 12 there are no intersections with edges of event polygons inside the strip R, at most two edges of the convex polygon can lie inside R, one on the “front side” and the other on the “back side.” To compute these two segments on R, we first consider the two sorted matrices that together describe the edges on the “front side” and look at the column associated to −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We find, for each column, the two (or zero) adjacent entries that contain the z-coordinates of R in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The two of the at most four that are closest to the strip will be the endpoints of the segment that intersect the strip on the “front side.” Computing this segment takes O(log n) time since we can use binary search on the columns to find the desired entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We do the same to find the segment on the “back side.” We do this for all vertices of Q to find the O(n) intersections with the event polygons of type (II) in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For an edge e of P, it contributes O(n) event polygons of type (III) that form the sur- rounding sides of a “cylinder” with base congruent to −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Again, each of these “cylinders” intersect the strip R in at most two faces, so there are O(n) intersections of R with event polygons of type (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can compute these segments by performing the binary search on the row of one of the sorted matrices associated to the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The two entries immediately below the strip and the two immediately above the strip define the at most two segments intersecting R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similar to the procedure above, this takes O(log n) time for each edge of P, thus O(n log n) time in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, we have a target region R and the O(n) intersections it makes with the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In O(n log n) time, we can find a region R′ ⊂ R containing a good placement that does not intersect any of the O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We recursively construct a (1/2)-cutting of the target region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4, a (1/2)- cutting of constant size can be computed in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We perform LineDecision on the lines of the cutting to decide on which riangle to recurse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' After O(log n) iterations, we have a target region R′ that intersects no event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This procedure runs in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Finally, since the overlap function is quadratic on our final region R′, we can solve for the maximum using standard calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' After finding maxv∈S f(v) and a vector achieving it O(n log n) time, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2, we can perform PlaneDecision on S in the same time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a plane S, we can perform PlaneDecision in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Complete Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' With PlaneDecision as a subroutine, we now present the complete algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' As in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we break down the algorithm into three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This stage is already done in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Recall that we now have P with all its vertices on z = 0 and z = 1, and we have computed P − Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The next stage is the main component of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We project the entire configuration space and the event polygons onto the xz-plane in order to find a target region D whose preimage D + ly intersects few event polygons, where ly is the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The non-horizontal edges of the event polygons project to segments on the strip 0 < z < 1 on the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We characterize our desired region D in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 13 (a) Projection of P (b) Projection of Q (c) Projection of the configuration space, and the target region D Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Projections onto the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For a region D that does not intersect any of the segments that are the projec- tions of the non-horizontal edges of the event polygons, the preimage D + ly intersects O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For any region D on the xz-plane, the set of event polygons that the “tube” D + ly intersects is precisely the set of projected event polygons that D intersects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, let D be a region that does not intersect any segment from the projections of the event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , sm be the segments that are the projections of the non-horizontal edges of P, and let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , xq be the projections of the vertices of Q on the x-axis and assume that they are sorted by decreasing x-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, the projections of the non-horizontal edges of the event polygons are precisely si − xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We first split the segments into four groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , sm1 be the projections of the non- horizontal edges of P on the “front side,” and sm1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , sm be those on the “back side.” The at most two edges visible on both the front and the back may be repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then the segments from either group are pairwise non-intersecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similarly, we split the vertices of Q into a front side and a backside, including the at most two vertices visible on both the front and back in both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We consider the segments in the configuration space made by one of the two groups of edges of P and one of the two groups of vertices of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The other three sets of segments are processed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose that the segments we consider are s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , sm1, and the projected vertices are x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' , xq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose the segments are sorted by increasing sum of the x-coordinates of their endpoints, and that the vertices are sorted by decreasing x-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The event polygons of type (II) are the trapezoids or triangles between segments si−xj and si+1−xj for each of the four groups of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each fixed projected vertex x, the region D intersects at most one event polygon of type (II) for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, D intersects O(n) event polygons of type (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similarly, the event polygons of type (III) are the parallelograms between segments si − xj and si − xj+1 for each of the four groups of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For each fixed segment si, D MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 14 intersects at most one event polygon of type (III), thus it intersects O(n) event polygons of type (III) in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Now it remains to efficiently find such a region D with D + ly containing a goal placement and compute the O(n) event polygons that intersect its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In O(n log2 n) time, we can find a triangle D in the xz-plane such that the interior of D + ly contains a goal placement and intersects O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can compute these O(n) event polygons in the same time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The computation of D is a direct application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3, where m = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Calling the oracle on a line l in the xz-plane is running the PlaneDecision algorithm on the plane parallel to the y-axis that projects to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We compute a triangle for each of the four groups of segments, take their intersection, and triangulate the intersection using O(1) calls to PlaneDecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can compute the desired triangle D in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' To compute the event polygons intersecting the interior of D + ly is simple, since we have shown in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='8 that D intersects at most one projection of an event polygon of each type in each of the four groups for a fixed vertex xj (for type (II)) or segment si (for type (III)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Once we have D, we can compute these polygons by binary search on each of the O(n) groups of O(n) non-intersecting segments to find the two between which R lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Also, the event polygons all have constant complexity so computing all of them takes linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can recover the event polygons from their projections and compute the planes that contain them in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, this entire process can be done in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Stage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now, we have a target region R = D + ly whose interior contains a goal placement, and we have the O(n) event polygons that intersect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In O(n log2 n) time, we can find a region R′ ⊂ R containing a goal placement that does not intersect any of the O(n) event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We recursively construct a (1/2)-cutting of the target region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4, a (1/2)- cutting of constant size can be computed in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We perform PlaneDecision on the planes of the cutting to decide on which simplex to recurse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' After O(log n) iterations, we have a target region R′ that intersects no event polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This procedure runs in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Finally, since the overlap function is quadratic on our final region R′, we can solve for the maximum using standard calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This concludes the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P be a convex polyhedron and Q a convex polygon with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a vector v ∈ R3 that maximizes the overlap area |P ∩ (Q + v)| in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Maximum overlap of three convex polygons Let P, Q, R be three convex polygons with n vertices in total in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We want to find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area g(vQ, vR) = |P ∩ (Q + vQ) ∩ (R + vR)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In this problem, the configuration space is four-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' An easy extension of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 shows that the function of overlap area is again unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 15 time, we have four-dimensional event polyhedra instead of event polygons that divide the configuration space into four-dimensional cells on which g(vQ, vR) is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We call a hyperplane containing an event polyhedron an event hyperplane, and they are defined by two types of events: (I) When one vertex of P, Q + vQ or R + vR lies on an edge of another polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n) groups of O(n) parallel event hyperplanes of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (II) When an edge from each of the three polygons intersect at one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n3) event hyperplanes of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' To overcome the difficulty of dealing with the O(n3) event hyperplanes of type (II), we first prune the configuration space to a region intersecting no event hyperplanes of type (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We then show that the resulting region only intersects O(n) event hyperplanes of type (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we want an algorithm HyperplaneDecision that computes, for a hyperplane H ⊂ R4, the maximum max(vQ,vR)∈H g(vQ, vR) and the relative location of the goal placement to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In fact, we will only need to perform HyperplaneDecision on some hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose H is a hyperplane that satisfies one of the following three condi- tions: (1) H is orthogonal to (x1, y1, 0, 0) for some x1, y1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (2) H is orthogonal to (0, 0, x2, y2) for some x2, y2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' (3) H is orthogonal to (x1, y1, −x1, −y1) for some x1, y1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, we can perform HyperplaneDecision on H in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We provide the algorithm for H orthogonal to (x1, y1, 0, 0) for some x1, y1 ∈ R, and the other two types follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We reinterpret the problem of finding max(vQ,vR)∈H g(vQ, vR) as a polyhedron-polygon matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In H, we allow R to move freely, and Q moves in a line l perpendicular to (x1, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We parameterize l by l = p + vt, and form the convex polyhedron IP Q = {(x, y, t)|(x, y) ∈ P} ∩ {(x, y, t)|(x, y) ∈ (Q + p + vt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By [Cha92], I can be computed in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In addition, the cross-section of I at t = t0 is P ∩ (Q + p + vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, we see that finding max(vQ,vR)∈H g(vQ, vR) is the same as finding a translation maximizing the intersection of I and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, this can be done in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Using the formal perturbation argument in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2, HyperplaneDecision on H can be completed in the same time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we can prune the configuration space to a region that intersects no event hyperplanes of type (I) and O(n) event hyperplanes of type (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can compute a 4-polytope TP QR of complexity O(1) in O(n log3 n) time such that (1) the goal placement lies on TP QR, (2) no hyperplane of type (I) intersects the interior of TP QR, and (3) only O(n) hyperplanes of type (II) passes through TP QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 16 P + (z-axis) Q + l Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The convex polyhedron IP Q is the intersection of these two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Furthermore, the set of hyperplanes of type (II) intersecting the interior of TP QR is obtained in the same time bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If a HyperplaneDecision reports a goal placement, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we assume that HyperplaneDecision always reports a halfspace containing a goal placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Each event hyperplane containing an event polyhedron of a vertex of P on an edge of Q+vQ or an event polyhedron of a vertex of Q+vQ on an edge of P is orthogonal to some (x1, y1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We project all these event hyperplanes into the 2-flat SP Q = {(x1, y1, 0, 0)|x1, y1 ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, the images are O(n) groups of O(n) parallel lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can therefore apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 to these lines, where an oracle call on a line l is running HyperplaneDecision on the hyperplane that projects to l on SP Q, which is orthogonal to some (x1, y1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we can find a triangle TP Q ⊂ SP Q whose interior does not intersect any event hyperplane as described above in O(n log3 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Similarly, we can find the triangles TP R ⊂ {(0, 0, x2, y2)|x2, y2 ∈ R} and TQR ⊂ {(x1, y1, −x1, −y1)|x1, y1 ∈ R} corresponding to the other event hyperplanes of type (I) in O(n log3 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, the interior of TP QR = {(x1, y1, x2, y2)|(x1, y1, 0, 0) ∈ TP Q, (0, 0, x2, y2) ∈ TP R, �x1 − x2 2 , y1 − y2 2 , x2 − x1 2 , y2 − y1 2 � ∈ TQR} does not intersect any event hyperplane of type (I) and contains a goal placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since the interior of TP QR intersects no event hyperplane of type (I), the pairwise config- uration of P and Q, P and R, Q and R are fixed (the pairwise edge incidences are fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since any edge eP of P intersects at most two edges of Q and at most two edges of R inside TP QR, there are at most four event hyperplanes of type (II) where eP is concurrent with an edge of Q and an edge of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, at most 4n event hyperplanes of type (II) intersect the interior of TP QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ With Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2, we can finish our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 17 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P, Q, R be three convex polygons with n vertices in total in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a pair of translations (vQ, vR) ∈ R4 that maximizes the overlap area |P ∩ (Q + vQ) ∩ (R + vR)| in O(n log3 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Take TP QR as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let f(vP, vQ) = � |P ∩ (Q + vQ) ∩ (R + vR)| if (vQ, vR) ∈ TP QR 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then f is unimodal and the maximum of f is the goal placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Given an m-flat S, we want to compute the maximum of f on S in O(n logm−1) time by induction on m ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If m = 1, this can be done in O(n) time by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Assume that m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then S ∩ TP QR can be computed in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Given an (m − 1)-flat l ⊂ S, we can use the perturbation method as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 to report the relative position of the maximum over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' There are O(n) event hyperplane intersecting S ∩ TP QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4, we can recursively construct (1/2)-cuttings to give an O(n logm−1) time algorithm to find the maximum of f on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Minimum symmetric difference of two convex polygons under homothety A homothety ϕ: R2 → R2 is a composition of a scaling and a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let λ > 0 be the scaling factor and v be the translation vector of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then ϕ(A) = λA + v = {λp + v | p ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Define the symmetric difference of sets A, B ⊂ R2 to be A△B :=(A ∪ B) \\ (A ∩ B) =(A \\ B) ∪ (B \\ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P and Q be convex polygons with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We want to find a homothety ϕ of Q that minimizes the area of symmetric difference h(ϕ) = h(x, y, λ) = |P△ϕ(Q)|, where ϕ(Q) = λQ + (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Yon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Yon+16] consider a slightly more general problem, where they minimize the function h(ϕ) = (2 − 2κ)|P \\ ϕ(Q)| + 2κ|ϕ(Q) \\ P|, where κ ∈ (0, 1) is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' When κ = 1/2, this is the area of symmetric difference function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' They give a randomized algorithm that solves this problem in O(n log3 n) expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We present a faster determinisitc algorithm by relating this problem to the polyhedron- polygon matching problem and then applying a modified version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' As in [Yon+16], we rewrite the objective function h(ϕ): h(ϕ) = 2(1 − κ)|P| + 2κ|ϕ(Q)| − 2|P ∩ ϕ(Q)| = 2(1 − κ)|P| + 2κ|Q|λ2 − 2|P ∩ ϕ(Q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, minimizing h(ϕ) is the same as maximizing h′(ϕ) = |P ∩ϕ(Q)|−cλ2, where c = κ|Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' MAXIMUM OVERLAP OF POLYHEDRON AND POLYGON 18 Q C Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Formation of the cone C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Consider the cone C = {(x, y, λ)|λ ∈ [0, M], (x, y) ∈ λQ}, where M = |P|/(c|Q|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then h is not minimized for λ > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since h′(x, y, λ) = |C ∩ (P + (−x, −y, λ))| − cλ2, the problem reduces to maximizing the overlap area of the cone C and P under translation subtracted by a quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' To show that we can still use a divide-and-conquer strategy, we identify a region where h′ is strictly unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 ([Yon+16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The closure D of the set {ϕ ∈ R3|h′(ϕ) > 0} is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Further- more, h′(x, y, λ) is strictly unimodal on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This follows from [Yon+16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2] and [Yon+16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Although it is difficult to directly compute D, note that −Q ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' With this observation, we show that we can still find the relative position of the set of goal placements to certain planes S in O(n log n) time with some modifications to LineDecision and PlaneDecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' For any l ⊂ R3, we can compute maxϕ∈l h′(ϕ) or report it is a negative number in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If l is horizontal, then we can apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 since cλ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Otherwise, we parameterize l by l = p + vt and construct the convex polyhedron I whose cross-section I(t0) at t = t0 has area |C ∩(P +(p+vt0))| as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' It comes down to maximizing |I(t)| − c(λ(t))2, where λ(t) is the λ-coordinate of p + vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since � |I(t)| is a concave function, � |I(t)| − √cλ(t) is also concave, and has the same complexity as � |I(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can apply [Avi+96, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2] to find the maximum of � |I(t)| − √cλ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Supposed it is achieved at t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Although t′ may not be where the maximum of |I(t)|−c(λ(t))2 is, it tells us whether the maximum is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If not, we can simply terminate the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If it is, we know that l intersects D, and p+vt′ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' This allows us to use divide-and-conquer as in [Avi+96], since we can recurse in the direction of t′ whenever we query a point t and find h′(t) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let S ⊂ R3 be a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If S is horizontal or if S intersects the polygon −Q ⊂ D, then we can perform PlaneDecision on S in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If S is horizontal, then we can apply Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If the maximum is negative, then we simply report the side of S containing −Q, otherwise we proceed as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Now REFERENCES 19 assume S is non-horizontal and intersects −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let s = S∩(−Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then we know that s ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let l ⊂ S be a line we want to run the subroutine LineDecision on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2, we can find maxϕ∈l h′(ϕ) or report it is negative in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' If it is the latter case, we report the side of l containing s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Otherwise, l intersects D, and we can proceed as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Thus, we can still find maxϕ∈S h′(ϕ) in O(n log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Since S intersects D, we can use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 to complete PlaneDecision on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ Finally, we present the complete algorithm, which applies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 with slight modi- fications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Let P and Q be convex polygons with n vertices in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Suppose κ ∈ (0, 1) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We can find a homothety ϕ that minimizes h(ϕ) = 2(1 − κ)|P \\ ϕ(Q)| + 2κ|ϕ(Q) \\ P| in O(n log2 n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We want to maximize h′(x, y, λ) = |C ∩ (P + (−x, −y, λ))| − cλ2 over R3, where c = κ|Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In order to apply our algorithm for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we need to show that we only run PlaneDecision on horizontal planes and planes that intersect −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the first stage (as outlined in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1), we only run PlaneDecision on horizontal planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the second stage, we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3 to the O(n) groups of O(n) lines that are the projections of the lines containing edges of event polygons on the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Observe that these lines all intersect the projection of −Q on the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In each recursive step of our algorithm, we query a horizontal (parallel to the x-axis) line and a line that goes “between” two lines in the O(n2) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' The planes they represent both satisfy the condition for Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then we run PlaneDecision O(1) more times to triangulate our feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Here, we make a small modification: instead of maintaining a triangular feasible region, we maintain a trapezoidal one by making O(1) horizontal cuts to make the region a trapezoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In the third stage, we have a “tube” and O(n) event polygons that intersect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' As usual, we recursively construct a (1/2)-cutting by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Chazelle’s algorithm [Cha93b] picks O(1) planes intersecting the target region as the cutting, along with O(1) extra planes to triangulate each piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' All the planes containing the event polygons intersect −Q, so we can run PlaneDecision on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Instead of triangulating our target region, it suffices to reduce it to constant complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' We do this by cutting it with O(1) horizontal planes such that the remaining region only has vertices on two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Then, let e be any non-horizontal edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' With O(1) planes through e, we can cut the target region into prisms and pyramids with triangular bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' These planes all intersect −Q since they are between the two faces of the target region containing e, and the planes containing them intersect −Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Therefore, with slight modifications to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1, we obtain a deterministic O(n log2 n) algorithm for minimizing h(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' □ References [ABS08] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Ahn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Brass, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' “Maximum overlap and minimum convex hull of two convex polyhedra under translations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='2 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 171–177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' REFERENCES 20 [Ahn+14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Cheng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Kweon, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Yon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' “Overlap of convex polytopes under rigid motion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='1 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 15–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [ACR13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Cheng, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Reinbacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' “Maximum overlap of convex poly- topes under translation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='5 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 552–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' [Ahn+07] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Ahn, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Cheong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Shin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Vigneron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' “Maximizing the overlap of two planar convex sets under rigid motions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 37.' metadata={'source': 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with respect to symmetric difference under homotheties”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' In: 32nd International Symposium on Computational Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Leibniz Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Schloss Dagstuhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Leibniz-Zent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=', Wadern, 2016, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} +page_content=' 63, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E1T4oBgHgl3EQfJQNW/content/2301.02949v1.pdf'} diff --git a/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf b/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b4c349ea0118637ff6f48bd0feb0e4f67589d44d Binary files /dev/null and b/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf differ diff --git a/WtFKT4oBgHgl3EQfny5D/content/tmp_files/2301.11863v1.pdf.txt b/WtFKT4oBgHgl3EQfny5D/content/tmp_files/2301.11863v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..22afa5104563f608eae37115e7cd520078be3af9 --- /dev/null +++ b/WtFKT4oBgHgl3EQfny5D/content/tmp_files/2301.11863v1.pdf.txt @@ -0,0 +1,237 @@ +arXiv:2301.11863v1 [math.GR] 27 Jan 2023 +Semiring identities of semigroups of reflexive +relations and upper triangular boolean matrices∗ +S. V. Gusev +Institute of Natural Sciences and Mathematics +Ural Federal University +620000 Ekaterinburg, RUSSIA +sergey.gusb@gmail.com +Abstract +We show that the following semirings satisfy the same identities: +the semiring Rn of all reflexive binary relations on a set with n ele- +ments, the semiring Un of all n×n upper triangular matrices over the +boolean semiring, the semiring Cn of all order preserving and extensive +transformations of a chain with n elements. In view of the result of +Kl´ıma and Pol´ak, which states that Cn has a finite basis of identities +for all n, this implies that the identities of Rn and Un admit a finite +basis as well. +An additively idempotent semiring (ai-semiring, for short) is an algebra +S = (S, +, ·) of type (2, 2) such that the additive reduct (S, +) is a semi- +lattice (that is, a commutative idempotent semigroup), the multiplicative +reduct (S, ·) is a semigroup, and multiplication distributes over addition on +the left and on the right, that is, S satisfies the identities x(y+z) ≈ xy+xz +and (y + z)x ≈ yx + zx. +The set of all reflexive binary relations on a set with n elements forms an +ai-semiring under set-theoretical union and multiplication. This ai-semiring +can be conveniently thought of as a subsemiring of the ai-semiring of all +n × n matrices (with the usual matrix multiplication and addition) over the +boolean semiring B = ⟨{0, 1}; +, ·⟩ with the operations defined by the rules +0 · 0 = 0 · 1 = 1 · 0 = 0 + 0 = 0, +1 · 1 = 1 + 0 = 0 + 1 = 1 + 1 = 1. +∗Supported by the Ministry of Science and Higher Education of the Russian Federation +(project FEUZ-2020-0016). +1 + +Namely, it can be identified with the subsemiring consisting of matrices in +which all diagonal entries are 1. Denote this semiring by Rn = ⟨Rn; +, ·⟩. +By Un = ⟨Un; +, ·⟩ we denote the subsemiring of Rn consisting of upper +triangular matrices, that is, matrices +� +αij +� +with αij = 0 for j < i. +A transformation α of a partially ordered set ⟨Q; ≤⟩ is called order +preserving if q ≤ q′ implies q.α ≤ q′.α for all q, q′ ∈ Q, and extensive +if q ≤ q.α for every q ∈ Q. +We say that an ai-semiring ⟨S; +, ·⟩ is a +join-semiring of order preserving and extensive transformations of a join- +semilattice ⟨Q; ≤⟩ if ⟨S; ·⟩ is a semigroup of order preserving and extensive +transformations of ⟨Q; ≤⟩ and q.(α + β) = sup(q.α, q.β) for all α, β ∈ S +and q ∈ Q. +For example, the set of all order preserving and extensive +transformations of a chain with n elements forms a join-semiring. Denote +this semiring by Cn = ⟨Cn; +, ·⟩. +In [2], Volkov shows that the monoids ⟨Rn; ·⟩, ⟨Un; ·⟩ and ⟨Cn; ·⟩ satisfy +the same identities and these identities admit a finite basis if and only if +n ≤ 4. In contrast, by the result of Kl´ıma and Pol´ak [1], the ai-semiring Cn +has a finite basis of identities for each n. In the present note we prove the +following theorem. +Theorem 1. For each n, the three ai-semirings Un, Rn and Cn satisfy the +same identities and these identities admit a finite basis. +To prove Theorem 1, we need some definitions, notation and auxiliary +results. If u, v are words over the same alphabet Σ, we say that u is a +subword of v whenever there exist words u1, . . . , un, v0, v1, . . . , vn−1, vn ∈ Σ∗ +such that +u = u1 · · · un +and +v = v0u1v1 · · · vn−1unvn; +in other terms, this means that one can extract u treated as a sequence +of letters from the sequence v. Let sk(w) denote the set of all subwords +of w of length ≤ k. Recall that a semiring identity over an alphabet Σ, +or simply identity, is merely a pair (u1 + · · · + uℓ, v1 + · · · + vr), where +u1, . . . , uℓ, v0, . . . , vr ∈ Σ+, usually written as +u1 + · · · + uℓ ≈ v1 + · · · + vr. +(1) +We denote by Jk the set of all identities (1) with �ℓ +i=1 sk(ui) = �r +i=1 sk(vi). +For an ai-semiring S , we denote by Id(S) the set of all identities of S . +Proposition 1. Let S = ⟨S; +, ·⟩ be a join-semiring of order preserving +and extensive transformations of a join-semilattice ⟨Q; ≤⟩. If k + 1 is the +length of the longest chain in ⟨Q; ≤⟩, then S satisfies every identity in Jk . +2 + +Proof. Take any identity (1) in Jk and let Σ be the alphabet of the words +u1, . . . , uℓ and v1, . . . , vr . +We have to show that for every substitution +ϕ: Σ → S one gets (u1 + · · · + uℓ)ϕ = (v1 + · · · + vr)ϕ or, equivalently, +q.(u1 + · · · + uℓ)ϕ = q.(v1 + · · · + vr)ϕ for all q ∈ Q. +Thus, fix an arbitrary substitution ϕ: Σ → S and an arbitrary element +q0 ∈ Q. By symmetry, it suffices to verify that +q0.(u1 + · · · + uℓ)ϕ ≤ q0.(v1 + · · · + vr)ϕ. +If q0.uiϕ = q0 for all i = 1, . . . , ℓ, then +q0.(u1 + · · · + uℓ)ϕ = q0 ≤ q0.(v1 + · · · + vr)ϕ +because the transformation (v1 + · · · + vr)ϕ is extensive. Suppose now that +the set {i1, . . . , ip} = {i | 1 ≤ i ≤ ℓ, q0.uiϕ > q0} is not empty. For any +i = i1, . . . , ip, denote by ui1 the longest prefix of the word ui such that +q0.ui1ϕ = q0 and let xi1 ∈ Σ be the letter that follows ui1 in ui so that +ui = ui1xi1wi1 for some wi1 ∈ Σ∗. Then +qi1 = q0.(ui1xi1)ϕ = q0.ui1ϕxi1ϕ = q0.xi1ϕ ≥ q0 +(2) +because the transformation xi1ϕ is extensive, and by the choice of the prefix +ui1 the inequality q1 ≥ q0 is in fact strict. Now denote by ui2 the longest +prefix of the word wi1 such that qi1.ui2ϕ = qi1 and let xi2 ∈ Σ be the letter +that follows ui2 in wi1 so that ui = ui1xi1ui2xi2wi2 for some wi2 ∈ Σ∗. +Then +qi2 = qi1.(ui2xi2)ϕ = qi1.ui2ϕxi2ϕ = qi1.xi2ϕ > qi1 +(3) +and substituting the expressions for qi1 from (2) in the expressions for qi2 +in (3), we also get +qi2 = q0.(ui1xi1ui2xi2)ϕ = q0.(xi1xi2)ϕ. +Continuing this process, we finally arrive at the decomposition +ui = ui1xi1ui2xi2 · · · ximiumi+1 +(4) +such that q0.uiϕ = q0.(xi1 · · · ximi)ϕ and +qimi > qi,mi−1 > · · · > qi1 > q0 +where qij = q0.(xi1 · · · xij)ϕ for j = 1, 2, . . . , mi. Since the longest chain in +⟨Q; ≤⟩ has k + 1 elements, we conclude that mi ≤ k. As the identity (1) +3 + +is taken from Jk , the word xi1 · · · ximi being in view of (4) a subword of +length ≤ k of the word ui must be a subword of a word in {v1, . . . , vr}. +Thus, there is ri ∈ {1, . . . , r} such that +vri = vi1xi1vi2xi2 · · · ximivi,mi+1 +for some words vi1, . . . , vi,mi+1 ∈ Σ∗. Using the fact that the transforma- +tions in S are extensive and order preserving, we readily obtain that +q0.vriϕ ≥ q0.(x1x2 · · · xm)ϕ = q0.uiϕ, +i = i1, . . . , ip. Since S is a join-semiring of order preserving and extensive +transformations of ⟨Q; ≤⟩, it follows that +q0.(u1 + · · · + uℓ)ϕ = q0.(ui1 + · · · + uip)ϕ +≤ q0.(vri1 + · · · + vrip)ϕ +≤ q0.(v1 + · · · + vr)ϕ +as required. +Corollary 1. Jk ⊆ Id(Rk+1). +Proof. Let Q = Bk+1 \ {(0, . . . , 0)} be the set of all non-zero (k + 1)- +vectors over the boolean semiring B = ⟨{0, 1}; +, ·⟩. We equip the set Q +with the component-wise order ≤ induced by the standard order 0 < 1 in +B. Then ⟨Q, ≤⟩ becomes a join-semilattice in which the longest chain has +length k+1. The semigroup ⟨Rk+1; ·⟩ acts on the set Q by the usual matrix +multiplication on the right: if q = (qi) ∈ Q and α = (αij) ∈ Rk+1, then +q.α = +�k+1 +� +i=1 +qiαi1, . . . , +k+1 +� +i=1 +qiαi k+1 +� +. +As it is noted in [2], this is a faithful representation of the semigroup by +order preserving and extensive transformations of ⟨Q, ≤⟩. Further, for any +q = (qi) ∈ Q and α = (αij), β = (βij) ∈ Rk+1 we have: +q.(α + β) = +�k+1 +� +i=1 +qi(αi1 + βi1), . . . , +k+1 +� +i=1 +qi(αi k+1 + βi k+1) +� += +�k+1 +� +i=1 +qiαi1 + +k+1 +� +i=1 +qiβi1, . . . , +k+1 +� +i=1 +qiαi k+1 + +k+1 +� +i=1 +qiβi k+1 +� += +� +max +�k+1 +� +i=1 +qiαi1, +k+1 +� +i=1 +qiβi1 +� +, . . . , max +�k+1 +� +i=1 +qiαi k+1, +k+1 +� +i=1 +qiβi k+1 +�� += sup(q.α, q.β). +4 + +Now Proposition 1 applies. +Proof of Theorem 1. The ai-semirings U1, R1 and C1 are trivial and +thus admit a finite basis of identities. Denote by Sk+1 = ⟨Sk+1; +, ·⟩ the +subsemiring of Uk+1 consisting of all stair triangular matrices, i.e. matrices +� +αij +� +satisfying: if αij = 1, i < j, then +αii = αii+1 = · · · = αij = αi+1j = · · · = αjj = 1. +It is noticed in [1, Section 5], the monoid ⟨Sk+1; ·⟩ is isomorphic to the +monoid ⟨Ck+1; ·⟩. +In fact, it is easy to see that the ai-semiring Sk+1 is +isomorphic to Ck+1. Further, it is shown in [1, Sections 4.1 and 5] that +Id(Sk+1) = Jk and the ai-semiring Sk+1 is finitely based by the identity +x1 · · · xk+1 ≈ +k+1 +� +i=1 +x1 · · · xi−1xi+1 · · · xk+1, +Since Jk = Id(Ck+1) = Id(Sk+1) ⊇ Id(Uk+1) ⊇ Id(Rk+1), these facts and +Corollary 1 imply that the ai-semirings Uk+1, Rk+1 and Ck+1 satisfy the +same identities and these identities admit a finite basis. Theorem 1 is thus +proved. +References +[1] O. Kl´ıma, L. Pol´ak, Hierarchies of piecewise testable languages, Int. J. Found. +Comput. Sci. 21 (2010), 517–533. +[2] M. V. Volkov, Reflexive relations, extensive transformations and piecewise +testable languages of a given height, Int. J. Algebra Comput. 14 (2004) 817– +827. +5 + diff --git a/WtFKT4oBgHgl3EQfny5D/content/tmp_files/load_file.txt b/WtFKT4oBgHgl3EQfny5D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5adc73adcba5393b16b207764d2b943605089763 --- /dev/null +++ b/WtFKT4oBgHgl3EQfny5D/content/tmp_files/load_file.txt @@ -0,0 +1,204 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf,len=203 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='11863v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='GR] 27 Jan 2023 Semiring identities of semigroups of reflexive relations and upper triangular boolean matrices∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Gusev Institute of Natural Sciences and Mathematics Ural Federal University 620000 Ekaterinburg, RUSSIA sergey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='gusb@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='com Abstract We show that the following semirings satisfy the same identities: the semiring Rn of all reflexive binary relations on a set with n ele- ments, the semiring Un of all n×n upper triangular matrices over the boolean semiring, the semiring Cn of all order preserving and extensive transformations of a chain with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' In view of the result of Kl´ıma and Pol´ak, which states that Cn has a finite basis of identities for all n, this implies that the identities of Rn and Un admit a finite basis as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' An additively idempotent semiring (ai-semiring, for short) is an algebra S = (S, +, ·) of type (2, 2) such that the additive reduct (S, +) is a semi- lattice (that is, a commutative idempotent semigroup), the multiplicative reduct (S, ·) is a semigroup, and multiplication distributes over addition on the left and on the right, that is, S satisfies the identities x(y+z) ≈ xy+xz and (y + z)x ≈ yx + zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' The set of all reflexive binary relations on a set with n elements forms an ai-semiring under set-theoretical union and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' This ai-semiring can be conveniently thought of as a subsemiring of the ai-semiring of all n × n matrices (with the usual matrix multiplication and addition) over the boolean semiring B = ⟨{0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩ with the operations defined by the rules 0 · 0 = 0 · 1 = 1 · 0 = 0 + 0 = 0, 1 · 1 = 1 + 0 = 0 + 1 = 1 + 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ∗Supported by the Ministry of Science and Higher Education of the Russian Federation (project FEUZ-2020-0016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 1 Namely, it can be identified with the subsemiring consisting of matrices in which all diagonal entries are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Denote this semiring by Rn = ⟨Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' By Un = ⟨Un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩ we denote the subsemiring of Rn consisting of upper triangular matrices, that is, matrices � αij � with αij = 0 for j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' A transformation α of a partially ordered set ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩ is called order preserving if q ≤ q′ implies q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α ≤ q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α for all q, q′ ∈ Q, and extensive if q ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α for every q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' We say that an ai-semiring ⟨S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩ is a join-semiring of order preserving and extensive transformations of a join- semilattice ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩ if ⟨S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩ is a semigroup of order preserving and extensive transformations of ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩ and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (α + β) = sup(q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='β) for all α, β ∈ S and q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' For example, the set of all order preserving and extensive transformations of a chain with n elements forms a join-semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Denote this semiring by Cn = ⟨Cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' In [2], Volkov shows that the monoids ⟨Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩, ⟨Un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩ and ⟨Cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩ satisfy the same identities and these identities admit a finite basis if and only if n ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' In contrast, by the result of Kl´ıma and Pol´ak [1], the ai-semiring Cn has a finite basis of identities for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' In the present note we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' For each n, the three ai-semirings Un, Rn and Cn satisfy the same identities and these identities admit a finite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' To prove Theorem 1, we need some definitions, notation and auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' If u, v are words over the same alphabet Σ, we say that u is a subword of v whenever there exist words u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , un, v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , vn−1, vn ∈ Σ∗ such that u = u1 · · · un and v = v0u1v1 · · · vn−1unvn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' in other terms, this means that one can extract u treated as a sequence of letters from the sequence v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Let sk(w) denote the set of all subwords of w of length ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Recall that a semiring identity over an alphabet Σ, or simply identity, is merely a pair (u1 + · · · + uℓ, v1 + · · · + vr), where u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , uℓ, v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , vr ∈ Σ+, usually written as u1 + · · · + uℓ ≈ v1 + · · · + vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (1) We denote by Jk the set of all identities (1) with �ℓ i=1 sk(ui) = �r i=1 sk(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' For an ai-semiring S , we denote by Id(S) the set of all identities of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Let S = ⟨S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩ be a join-semiring of order preserving and extensive transformations of a join-semilattice ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' If k + 1 is the length of the longest chain in ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩, then S satisfies every identity in Jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Take any identity (1) in Jk and let Σ be the alphabet of the words u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , uℓ and v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , vr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' We have to show that for every substitution ϕ: Σ → S one gets (u1 + · · · + uℓ)ϕ = (v1 + · · · + vr)ϕ or, equivalently, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (u1 + · · · + uℓ)ϕ = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (v1 + · · · + vr)ϕ for all q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Thus, fix an arbitrary substitution ϕ: Σ → S and an arbitrary element q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' By symmetry, it suffices to verify that q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (u1 + · · · + uℓ)ϕ ≤ q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (v1 + · · · + vr)ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' If q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='uiϕ = q0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , ℓ, then q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (u1 + · · · + uℓ)ϕ = q0 ≤ q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (v1 + · · · + vr)ϕ because the transformation (v1 + · · · + vr)ϕ is extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Suppose now that the set {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , ip} = {i | 1 ≤ i ≤ ℓ, q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='uiϕ > q0} is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' For any i = i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , ip, denote by ui1 the longest prefix of the word ui such that q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='ui1ϕ = q0 and let xi1 ∈ Σ be the letter that follows ui1 in ui so that ui = ui1xi1wi1 for some wi1 ∈ Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Then qi1 = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (ui1xi1)ϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='ui1ϕxi1ϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='xi1ϕ ≥ q0 (2) because the transformation xi1ϕ is extensive, and by the choice of the prefix ui1 the inequality q1 ≥ q0 is in fact strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Now denote by ui2 the longest prefix of the word wi1 such that qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='ui2ϕ = qi1 and let xi2 ∈ Σ be the letter that follows ui2 in wi1 so that ui = ui1xi1ui2xi2wi2 for some wi2 ∈ Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Then qi2 = qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (ui2xi2)ϕ = qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='ui2ϕxi2ϕ = qi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='xi2ϕ > qi1 (3) and substituting the expressions for qi1 from (2) in the expressions for qi2 in (3), we also get qi2 = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (ui1xi1ui2xi2)ϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (xi1xi2)ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Continuing this process, we finally arrive at the decomposition ui = ui1xi1ui2xi2 · · · ximiumi+1 (4) such that q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='uiϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (xi1 · · · ximi)ϕ and qimi > qi,mi−1 > · · · > qi1 > q0 where qij = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (xi1 · · · xij)ϕ for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Since the longest chain in ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩ has k + 1 elements, we conclude that mi ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' As the identity (1) 3 is taken from Jk , the word xi1 · · · ximi being in view of (4) a subword of length ≤ k of the word ui must be a subword of a word in {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , vr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Thus, there is ri ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , r} such that vri = vi1xi1vi2xi2 · · · ximivi,mi+1 for some words vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , vi,mi+1 ∈ Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Using the fact that the transforma- tions in S are extensive and order preserving, we readily obtain that q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='vriϕ ≥ q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (x1x2 · · · xm)ϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='uiϕ, i = i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Since S is a join-semiring of order preserving and extensive transformations of ⟨Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ≤⟩, it follows that q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (u1 + · · · + uℓ)ϕ = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (ui1 + · · · + uip)ϕ ≤ q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (vri1 + · · · + vrip)ϕ ≤ q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (v1 + · · · + vr)ϕ as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Jk ⊆ Id(Rk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Let Q = Bk+1 \\ {(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , 0)} be the set of all non-zero (k + 1)- vectors over the boolean semiring B = ⟨{0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' We equip the set Q with the component-wise order ≤ induced by the standard order 0 < 1 in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Then ⟨Q, ≤⟩ becomes a join-semilattice in which the longest chain has length k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' The semigroup ⟨Rk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩ acts on the set Q by the usual matrix multiplication on the right: if q = (qi) ∈ Q and α = (αij) ∈ Rk+1, then q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α = �k+1 � i=1 qiαi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , k+1 � i=1 qiαi k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' As it is noted in [2], this is a faithful representation of the semigroup by order preserving and extensive transformations of ⟨Q, ≤⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Further, for any q = (qi) ∈ Q and α = (αij), β = (βij) ∈ Rk+1 we have: q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' (α + β) = �k+1 � i=1 qi(αi1 + βi1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , k+1 � i=1 qi(αi k+1 + βi k+1) � = �k+1 � i=1 qiαi1 + k+1 � i=1 qiβi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , k+1 � i=1 qiαi k+1 + k+1 � i=1 qiβi k+1 � = � max �k+1 � i=1 qiαi1, k+1 � i=1 qiβi1 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' , max �k+1 � i=1 qiαi k+1, k+1 � i=1 qiβi k+1 �� = sup(q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='α, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 4 Now Proposition 1 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' The ai-semirings U1, R1 and C1 are trivial and thus admit a finite basis of identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Denote by Sk+1 = ⟨Sk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' +, ·⟩ the subsemiring of Uk+1 consisting of all stair triangular matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' matrices � αij � satisfying: if αij = 1, i < j, then αii = αii+1 = · · · = αij = αi+1j = · · · = αjj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' It is noticed in [1, Section 5], the monoid ⟨Sk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩ is isomorphic to the monoid ⟨Ck+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' In fact, it is easy to see that the ai-semiring Sk+1 is isomorphic to Ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Further, it is shown in [1, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content='1 and 5] that Id(Sk+1) = Jk and the ai-semiring Sk+1 is finitely based by the identity x1 · · · xk+1 ≈ k+1 � i=1 x1 · · · xi−1xi+1 · · · xk+1, Since Jk = Id(Ck+1) = Id(Sk+1) ⊇ Id(Uk+1) ⊇ Id(Rk+1), these facts and Corollary 1 imply that the ai-semirings Uk+1, Rk+1 and Ck+1 satisfy the same identities and these identities admit a finite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Theorem 1 is thus proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' References [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Kl´ıma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Pol´ak, Hierarchies of piecewise testable languages, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 21 (2010), 517–533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Volkov, Reflexive relations, extensive transformations and piecewise testable languages of a given height, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' Algebra Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 14 (2004) 817– 827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFKT4oBgHgl3EQfny5D/content/2301.11863v1.pdf'} diff --git a/X9E1T4oBgHgl3EQfcARq/content/2301.03179v1.pdf b/X9E1T4oBgHgl3EQfcARq/content/2301.03179v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f25f1b404858adfb7551f0bfcfb19a42d1bd3869 --- /dev/null +++ b/X9E1T4oBgHgl3EQfcARq/content/2301.03179v1.pdf @@ -0,0 +1,3 @@ +version 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Smith2, 4, and Chris Holmes3, 4 +1Barcelona School of Economics, Universitat Pompeu Fabra, Barcelona, Spain +2University of Warwick, Coventry, CV4 7AL +3University of Oxford, Oxford, OX1 3LB +4Alan Turing Institute, London, NW1 2DB +jack.jewson@upf.edu, j.q.smith@warwick.ac.uk, chris.holmes@stats.ox.ac.uk +January 2023 +Abstract +We study the stability of posterior predictive inferences to the specification of the likelihood +model and perturbations of the data generating process. In modern big data analyses, the decision- +maker may elicit useful broad structural judgements but a level of interpolation is required to arrive +at a likelihood model. One model, often a computationally convenient canonical form, is chosen, +when many alternatives would have been equally consistent with the elicited judgements. Equally, +observational datasets often contain unforeseen heterogeneities and recording errors. Acknowledg- +ing such imprecisions, a faithful Bayesian analysis should be stable across reasonable equivalence +classes for these inputs. We show that traditional Bayesian updating provides stability across a +very strict class of likelihood models and DGPs, while a generalised Bayesian alternative using the +β-divergence loss function is shown to be stable across practical and interpretable neighbourhoods. +We illustrate this in linear regression, binary classification, and mixture modelling examples, show- +ing that stable updating does not compromise the ability to learn about the DGP. These stability +results provide a compelling justification for using generalised Bayes to facilitate inference under +simplified canonical models. +Keywords: Stability; Generalised Bayes; β-divergence; Total Variation; Generalised linear models +1 +arXiv:2301.13701v1 [stat.ME] 31 Jan 2023 + +1 +Introduction +Bayesian inferences are driven by the posterior distribution +π(θ|y) = +π(θ)f(y; θ) +� +π(θ)f(y; θ)dθ. +(1) +which provides the provision to update parameter prior π(θ) using observed data y = (y1, . . . , yn) ∈ +Yn assumed to have been generated according to likelihood f(·; θ). The quality of such posterior +inference depends on the specification of the prior, likelihood, and collection of the data. In controlled +experimental environments where time is available to carefully consider such specifications, a posterior +calculated in this way might be credible. However, modern applications often involve high-dimensional +observational data and are undertaken by non-experts. In such scenarios, it is natural to question the +quality of the specification of π(θ) and f(·; θ) and the collection of y and therefore wonder to what +extent posterior inference through (1) can be trusted. Much work has previously investigated the +stability of (1) to the specification of π(θ), therefore our focus here will be on f(·; θ) and y. +The likelihood model captures the decision maker’s (DM’s) beliefs regarding the generation of data +y. However, accurately formulating expert judgements as probability densities is difficult. Even for +a well trained expert, so doing requires many more probability specifications to be made at a much +higher precision than is possible within the time constraints of a typical problem (Goldstein, 1990). +This is not to say that an elicited model is useless. Often domain experts can reliably elicit important +broad judgements. However, the resulting “functional” model f(·; θ) generally involves some form of +interpolating approximation of the DM’s “true” beliefs. So doing is not unreasonable. However, a +consequence of such expediency is that not only does the DM not believe all the judgements made by +f(·; θ), its specific form is likely only one member of an equivalence class of models that also capture +the DM’s elicited beliefs and could have used for inference. +A typical example of the above is when applied practitioners deploy computationally convenient +canonical models, for which there are software and illustrative examples available, to their domain +specific problems. While the broad structure of such models may be suitable across domains, it is the +practitioner’s familiarly with its form, its software implementation or the platform on which it was +published that motivates its use for inference, rather than a careful consideration of how it captures +beliefs about the new environment +Similarly, the data were not necessarily collected exactly how the DM imagined when specifying +f(·; θ). +There may be unforeseen heterogeneities, outliers, or recording errors. +Alternatively, the +2 + +DM may be deploying someone else’s carefully elicited model to an analogous but not necessarily +exchangeable scenario. We therefore also consider the data generating process (DGP) that generated +the DM’s data y to belong to an equivalence class of DGPs to which the DM could have deployed their +inference. +Given the inevitable lack of specificity in f and y, a faithful Bayesian analysis should be able +to demonstrate that it is not overly dependent on arbitrary choices across equivalence classes of its +inputs. Such stability would allow DMs to continue using familiar models in the knowledge that their +selection is not driving the critical posterior inferences. This paper shows that the requirement for +such stability necessitates the consideration of an updating rule different from (1). +Consider, for +example, using a Gaussian distribution, N(y; µ, σ2) to approximate beliefs about data y. While the +Gaussian distribution is ubiquitous, the top of Figure 1 shows that a Student’s-t likelihood t5(y; µ, σ2) +with 5 degrees of freedom would also have sufficed for this specification. The two likelihoods appear +almost indistinguishable for all values of their shared µ and σ2. Therefore, it would be unreasonable to +expect that any DM will strongly prefer one or the other of these. However, the bottom left of Figure 1 +shows that when updating according to (1) each model can result in very different posterior inferences. +Equally, (1) is not stable to perturbations of the data either, as a small proportion of outliers moves +the posterior inferences away from the uncontaminated part of the DGP. We demonstrate that this +is a feature of the fact that implicitly (1) learns about the parameter of the model minimising the +Kullback-Leibler Divergence (KLD) between the data generating process (DGP) and the model, and +that stability can only be expected here when the DM is sure of the tail specification of their model +and the data. See Section 6.1 for full details of this example. +Under traditional Bayesian updating, it is therefore left up to the DM to perform some kind of +post hoc sensitivity analysis to examine the impact their chosen model and particular features of the +data had on the inference (see Box, 1980; Berger et al., 1994, and references within). However, such +analyses are usually unsystematic and limited to the investigation of a small number of alternative +models within the equivalence class. +An alternative, motivated by the M -open world assumption that the model is misspecified for +the DGP (Bernardo and Smith, 2001), is to use general Bayes (Bissiri et al., 2016) to update beliefs +about model parameters minimising a divergence different from the KLD (Jewson et al., 2018). A +particularly convenient alternative is the β-divergence (βD) which has previously been motivated as +providing inference that is robust to outliers (Basu et al., 1998; Ghosh and Basu, 2016) and desirable +3 + +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +x +Density +Gaussian +Student’s-t +-4 +-2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +Cumulative Density +-5 +0 +5 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +KLD +y +Density +(1 − ϵ)N(0, 1) +ϵN(5, 32) +Gaussian +Student’s-t +-5 +0 +5 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +βD +y +Density +(1 − ϵ)N(0, 1) +ϵN(5, 32) +Gaussian +Student’s-t +Figure 1: Top: Probability density function (pdf) and cumulative density function (cdf) of a Gaussian +fσ2 +adj(y; θ) = N +� +y; µ, σ2 +adjσ2� +and a Student’s-t hν(y; η) = tν(y; µ, σ2) random variable, with µ = 0, +σ2 = 1, ν = 5 and σ2 +adj = 1.16. +Bottom: The resulting posterior predictive distributions using +traditional and βD-Bayes updating on n = 1000 observations from an ϵ contamination model g(y) = +0.9 × N (y; 0, 1) + 0.1 × N +� +y; 5, 32� +. +from a decision making point of view (Jewson et al., 2018). In this paper, we extend the motivation for +using βD-Bayes further, showing that its posterior predictive inferences are provably stable across an +interpretable equivalence class of likelihood models and DGPs. We treat stability to f and y separately, +first showing that βD-Bayes inference is stable to the choice likelihood model for a given DGP, and +then that inferences for a fixed model are stable to small perturbations to the DGP. +Importantly, the stability afforded to βD-Bayes inference does not compromise its ability to extract +useful inferences about the DGP. βD-Bayes has the appealing property that if the model is correctly +specified for the DGP, then the data generating parameter will be learned, and there exists a growing +literature that advocates using the βD for applied analyses (e.g. Knoblauch et al., 2018, 2022; Girardi +4 + +et al., 2020; Sugasawa, 2020). This is further demonstrated in our experiments. For example, Figure 1 +shows that as well as producing similar inference for the Gaussian and Student’s-t likelihood models, +the βD-Bayes inferences both capture the modal part of the observed data. Further, inferences must +be also stable to the selection of the βD and its hyperparameter. We discuss methods to select β and +demonstrate reasonable insensitivity to its selection. +Results regarding the stability of (1) have largely focused on the parameter prior. Gustafson and +Wasserman (1995) proved that the total variation divergence (TVD) between two posteriors resulting +from priors in linear and geometric ϵ-contamination neighbourhoods divergences as ϵ → 0 at a rate +exponential in the dimension of the parameter space. However, Smith and Rigat (2012) showed that +the TVD between two posteriors converges to 0 provided the two priors under consideration are close +as measured by the local De Robertis distance. Our first results provide analogies to these for the +specification of the likelihood model. Gilboa and Schmeidler (1989); Whittle and Whittle (1990); +Hansen and Sargent (2001a,b); Watson and Holmes (2016) consider the stability of optimal decision +making and consider minimax decision across neighbourhoods of the posterior. However, they do +not consider what perturbations of the inputs of (1) would leave a DM in such a neighbourhood a +posteriori. Most similar to our work is Miller and Dunson (2018), which considers Bayesian updating +conditioning on data arriving within a KLD ball of the observed data and results concerning ‘global +bias-robustness’ to contaminating observations, for example of the kernel-Stein discrepancy posteriors +of Matsubara et al. (2021). We consider stability to an interpretable neighbourhood of the data which +as a special case contains the globally bias-robust contamination. +Bayes linear methods (Goldstein, 1999), which concern only the sub-collection of probabilities and +expectations the DM considers themselves to be able to specify (Goldstein et al., 2006), is an alternative +to (1) designed to be stable to interpolating approximations. +We prefer, however, to adopt the +general Bayesian paradigm in this analysis. Firstly, the general Bayesian paradigm includes traditional +Bayesian updating as a special case and produces familiar posterior and predictive distributions. +Secondly, linear Bayes requires the elicitation of expectations and variances of unbounded quantities +which are themselves unstable to small perturbations (see discussion on Goldstein and Wooff, 1994). +Lastly, rather than demanding stability across an equivalence class of models, the DM could let the +data guide any decision the DM themselves is not able to make using methods such as penalised +likelihood approaches (e.g. Akaike, 1973; Schwarz et al., 1978), Bayes’ factors (Kass and Raftery, 1995) +or Bayesian model averaging (Hoeting et al., 1999). In particular, Williamson and Goldstein (2015) +5 + +propose methods for combining posterior beliefs across an equivalence class of analyses. However, +such methods can be computationally burdensome to compute across even a finite class of models +(e.g. Rossell et al., 2021) and the DM could reasonably only consider a handful of the models that +might fit with their beliefs, a subset of the full equivalence class. +The rest of the paper is organised as follows: Section 2 presents our inference paradigm, introducing +general Bayesian updating (Bissiri et al., 2016), robustified inference with the βD, and defining how +we will investigate posterior predictive stability. +Section 3 presents our theoretical contributions +surrounding the stability of Bayesian analyses to the choice of the likelihood function and Section 4 +presents our results on the stability of inference to perturbations of the DGP. Proofs of all of our results +are deferred to the supplementary material. Section 5 discusses methods to set the β hyperparameter +and Section 6 illustrates the stability of the βD-Bayes inference in continuous and binary regression +examples from biostatistics and a mixture modelling astrophysics example, where stability is shown +not to compromise the model’s ability to learn about the DGP. Code to reproduce all of the examples +in this paper can be found at https://github.com/jejewson/stabilityGBI. +2 +A paradigm for inference and stability +2.1 +General Bayesian Inference +Under the assumption that the model used for inference f(y; θ) does not exactly capture the DM’s +beliefs, we find it appealing to adopt the general Bayesian perspective of inference. Bissiri et al. (2016) +showed that the posterior update +πℓ(θ|y) = +π(θ) exp (−w �n +i=1 ℓ(θ, yi)) +� +π(θ) exp (−w �n +i=1 ℓ(θ, yi)) dθ. +(2) +provides a coherent means to update prior beliefs about parameter θℓ +g := arg minθ∈Θ +� +ℓ(θ, z)g(z)dz +after observing data y ∼ g(·) without requiring that θ index a model for the data generating density +g(·). +The parameter w > 0 in (2) calibrates the loss with the prior to accounts for the fact that +exp(−ℓ(θ, yi)) is no longer constrained to integrate to 1, as was the likelihood in (1). Lyddon et al. +(2018) set w to match the asymptotic information in the general Bayesian posterior to that of a sample +from the ‘loss-likelihood bootstrap’, while Giummol`e et al. (2019), building on the work of Ribatet +et al. (2012), directly calibrate the curvature of the posterior to match that of the frequentist loss +6 + +minimiser. +We focus on a subset of loss functions, known as scoring rules, that depend upon the DM’s likelihood +model, continuing to allow the DM to use this to encode their beliefs about the DGP. Under the log- +score, ℓ(θ, y) = − log f(y; θ) (2) collapses to (1). The parameter θℓ +g associated with the log-score is the +minimiser of the KLD between the distribution of the sample and the model (Berk et al., 1966). We +therefore call updating using (1) KLD-Bayes. However, it is well known that minimising the log-score +puts large importance on correctly capturing the tails of the data (Bernardo and Smith, 2001) and can +have negative consequences for posterior decision making (Jewson et al., 2018). This is demonstrated +in the bottom left of Figure 1. +2.2 +βD-Bayes +An alternative to the log-score is the β-divergence loss (Basu et al., 1998) +ℓ(β)(y, f(·; θ)) = − +1 +β − 1f(y; θ)β−1 + 1 +β +� +f(z; θ)βdz, +(3) +so called as arg minθ Ey∼g +� +ℓ(β)(y, f(·; θ)) +� += arg minθ D(β) +B (g||f(·; θ)) where D(β) +B (g||f) is the β-divergence +defined in Section A.1. We refer to updating using (2) and loss (3) as βD-Bayes. This was first used +by Ghosh and Basu (2016) to produce a robustified Bayesian posterior (βD-Bayes) and has since been +deployed for a variety of examples (e.g. Knoblauch et al., 2018, 2022; Girardi et al., 2020; Sugasawa, +2020). +The implicit robustness to outliers exhibited by the βD-Bayes is illustrated in the bottom right +of Figure 1, where, unlike the KLD-Bayes, the βD-Bayes continues to captures the distribution of the +majority of observations under outlier contamination. Jewson et al. (2018) argued that updating in +a manner that is automatically robust to outliers, removes the burden on the DM to specify their +beliefs in a way that is robust to outliers is removed. The results of the coming sections provide a +formal rationale for adopting this methodology to provide stability to the canonical model choice and +departures from the DGP. +While Bayesian inference has been proposed minimising several alternative divergences including +the Hellinger divergence, α-divergence, and the TVD (e.g. Hooker and Vidyashankar, 2014; Jewson +et al., 2018; Knoblauch and Vomfell, 2020) such methods require a non-parametric density estimate, +prohibiting their use for high-dimensional problems with continuous data. We restrict our attention to +local methods not requiring such an estimate and in particular to the βD and KLD. The γ-divergence +7 + +(Fujisawa and Eguchi, 2008) has also been shown to produce robust inference without requiring a +non-parametric density estimate (Hung et al., 2018; Knoblauch et al., 2022) and in general behaves +very similarly, see Section B.1.3. +2.3 +Posterior Predictive Stability +Our results will investigate the stability of general Bayesian posterior predictive distributions +mD +f (ynew|y) = +� +f(ynew; θ)πD(θ|y)dθ. +(4) +for exchangeable observation ynew ∈ Y to the specification of the model f, and the DGP g. As a result, +we focus on the stability of the posterior distribution for observables y ∈ Y to perturbations of the +prior for observables, f, and generating distributions for these observables g. +From a decision-making perspective, the posterior predictive is often integrated over to calculate +expected utilities, and therefore stable posterior predictive distributions correspond to stable decision +making. We consider two metrics for stability, the first is the divergence between posterior predictives, +which if small, indicates that a DM with either distribution would make similar decisions. The second +measures the difference between the posterior predictives’ divergence to the DGP. Predictives that are +close to the DGP will make close to optimal decisions and therefore, two predictives that are equally +close will make similarly good decisions +Predictive stability is also a more reasonable requirement than say posterior stability. The param- +eter posteriors for two distinct models/DGPs will generally converge in different places (e.g. Smith, +2007). +However, divergent parameter posteriors do not necessarily imply divergent posterior pre- +dictives, as we show. +Further, focusing on observables allows us to consider interesting cases of +neighbouring models with nested parameter spaces (see Section 6.2) +3 +Stability to the specification of the likelihood function +In this section we consider two potential likelihood models for the data. These could correspond to +the DM’s true and functional beliefs, or two, equally preferable candidates for the later. In both cases, +the DM would not wish their posterior inferences to diverge if one candidate was used in place of the +other. +8 + +3.1 +An interpretable neighbourhood of likelihood models +We first consider the stability of inference to the specification of the DM’s likelihood model. Likelihood +models f and h are considered to be in the same equivalence class of likelihood models for y ∈ Y if +they satisfy Definition 1 +Definition 1 (TVD neighbourhood of likelihood models). Likelihood models f(·; θ) and h(·; η) for +observable y ∈ Y are in the neighbourhood N TVD +ϵ +of size ϵ if +∀θ ∈ Θ, ∃η ∈ A s.t. TVD(f(·; θ), h(·; η)) ≤ ϵ +and +∀η ∈ A, ∃θ ∈ Θ +s.t. +TVD(f(·; θ), h(·; η)) ≤ ϵ +Neighbourhood N TVD +ϵ +demands the existence of functions If : Θ �→ A and Ih : A �→ Θ such that for +all θ, TVD(f(·; θ), h(·; If(θ)) is small and for all η, TVD(h(·; η), f(·; Ih(η)) is also small. The symmetry +of Definition 1 allows Θ and A to have different dimensions. For two likelihoods to be close in terms +of TVD requires that the greatest difference in any of the probability statements made by the two +likelihoods be small on the natural scale. +TVD(f(·; θ), h(·; θ)) := sup +Y ∈Y +|f(Y ; θ) − h(Y ; θ)| = 1 +2 +� +|f(y; θ) − h(y; θ)| dy +(5) +Additionally, TVD neighbourhoods contain ϵ-contaminations considered in the context of prior stability +by Gustafson and Wasserman (1995) and often used as outlier models (e.g. Aitkin and Wilson, 1980). +As a result, it is reasonable for a DM to be able to elicit their beliefs within a N TVD +ϵ +neighbourhood of +their chosen model, and such a neighbourhood contains standard perturbations for sensitivity analysis. +The weak conditions required for the results of the following sections are formally stated in Section +A.3. Briefly, Condition A.1 requires the boundedness of the essential supremum of models f and h +and the DGP g, and Condition A.2 requires sufficient concentration of posterior πD +f (θ|y) around θD +f . +For clarity of argument, we proceed under the assumption that prior πD(θ) and πD(η) are fixed. +3.2 +The stability of the βD-Bayes +In the first of our main results, Theorem 1 bounds the a posteriori divergence between the predic- +tive distributions resulting from likelihood models f and h as a function of the size of the a priori +neighbourhood N TVD +ϵ +. +9 + +Theorem 1 (Stability of the posterior predictive distributions of two models under the βD-Bayes +inference). Given 1 < β ≤ 2 and two likelihood models {f(·; θ) : θ ∈ Θ} and {h(·; η) : η ∈ A} such +that f, h ∈ N TVD +ϵ +for ϵ > 0. Then provided there exists M < ∞ such that Condition A.1 holds, and y, +π(β)(θ) and π(β)(η) satisfy Condition A.2 for D = D(β) +B +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y)) ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c1 ++ 2Mβ−1 +β − 1 +� +TVD(g, f(·; θ))π(β) +f (θ|y)dθ +D(β) +B (m(β) +h (·|y)||m(β) +f (·|y)) ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c2 ++ 2Mβ−1 +β − 1 +� +TVD(g, h(·; η))π(β) +h (η|y)dη, +where c1 and c2 are defined in Condition A.2 . +Further, Theorem 2 bounds the absolute distance between the βD of the posterior predictive +distributions produced from two likelihood models within N TVD +ϵ +from the DGP. +Theorem 2 (The stability in the posterior predictive approximation of two models to the DGP of +βD-Bayes inference). Given 1 < β ≤ 2 and two likelihood models {f(·; θ) : θ ∈ Θ} and {h(·; η) : η ∈ A} +such that f, h ∈ N TVD +ϵ +for ϵ > 0. Then provided there exists M < ∞ such that Condition A.1 holds +and y, π(β)(θ) and π(β)(η) satisfy Condition A.2 for D = D(β) +B +|D(β) +B (g||m(β) +f (·|y)) − D(β) +B (g||m(β) +h (·|y))| ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c + C(β)(f, h, y), +where c = min{c1, c2} as defined in Condition A.2 and +C(β)(f, h, y) : = max +�� +D(β) +B (g||f(·; θ))π(β) +f (θ|y)dθ − D(β) +B (g||m(β) +f (·|y)), +� +D(β) +B (g||h(·; η))π(β) +h (η|y)dη − D(β) +B (g||m(β) +h (·|y)) +� +. +The value M present in both Theorems 1 and 2 is often easy to bound, for example by selecting a +minimum value of the scale of Gaussian or Student’s-t likelihood models, and we expect c1, c2 → ∞ as +n → ∞ (see Section A.3). The final term in Theorem 1 involves the TVD between the models under +consideration and the unknown DGP. While it is difficult to say anything formal about this, Lemma A.6 +shows that the βD can be bounded above by the TVD, and therefore any values of parameters θ and η +that are close to g in TVD should have high posterior mass under the βD posterior. On the other hand, +C(β)(f, h, y) in Theorem 2, is is related to the concentration of the posteriors π(β) +f (θ|y) and π(β) +h (η|y) +with Jensen’s inequality and the convexity of the βD guaranteeing that C(β)(f, h, y) ≥ 0. Under +10 + +suitable regularity conditions as n → ∞ and the posterior collapses to a point mass (Chernozhukov +and Hong, 2003; Lyddon et al., 2018), then this term converges to 0. Importantly, Theorem 2 does +not depend on how well specified the two likelihood models are for the DGP. +3.3 +The stability of the KLD-Bayes +Figure 1 demonstrates that the stability afforded by the βD-Bayes is not afforded by the KLD-Bayes. +The KLD is recovered from the βD as β → 1. However, in such a scenario, the bounds proven in +the previous sections tend to infinity. Instead, Lemma 1 provides an analogous stability result for +traditional Bayesian updating. +Lemma 1 (The stability in the posterior predictive approximation of the DGP of KLD-Bayes infer- +ence). For any two two likelihood models {f(·; θ) : θ ∈ Θ} and {h(·; η) : η ∈ A}, and y, πKLD(θ) and +πKLD(η) satisfying Condition A.2 for D = KLD, we have that +|KLD(g||mKLD +f +(·|y)) − KLD(g||mKLD +h +(·|y))| ≤ CKLD(f, h, y) + 1 +c + T(f, h, y), +where c := min{c1, c2} as defined in Condition A.2 and +T(f, h, y) : = max +�� � +g(·) log +f(·; θ) +h(·; If(θ))dµπKLD +f +(θ|y)dθ, +� � +g(·) log +h(·; η) +f(·; Ih(η))dµπKLD +h +(η|y)dη +� +(6) +CKLD(f, h, y) : = max +�� +KLD(g||f(·; θ))πKLD +f +(θ|y)dθ − KLD(g||mKLD +f +(·|y)), +� +KLD(g||h(·; η))πKLD +h +(η|y)dη − KLD(g||mKLD +h +(·|y)) +� +. +We investigate T(f, h, y), the term not analagous to any of those from Theorem 2. Without loss +of generality assume that the second term in (6) is the largest. Then, the reverse Pinsker’s inequality +(Sason and Verdu, 2016; Binette, 2019) provides +� +g(·) log +h(·; η) +f(·; Ih(η))dµ = +� +g(·) +h(·; η)h(·; η) log +h(·; η) +f(·; Ih(η))dµ ≤ M∗ +hKLD(h(·; η)||f(·; Ih(η))) +≤ M∗ +hKh,f TVD(h(·; η), f(·; Ih(η))) +where M∗ +h = ess sup +g +h(·;θh) and Kh,f = +� +log(a) +a−1 + log(b) +1−b +� +with a = ess inf dF +dH and b = ess sup dF +dH . As +a result, a TVD ball around the likelihood model is not sufficient for posterior stability when using +11 + +Bayes’ rule updating. In fact, posterior stability can only be guaranteed according to Lemma 1 if +|log(h(·; η)) − log(f(·; Ih(η)))| +(7) +is small in regions where g has density. +Without knowledge of g, this requires that (7) be small +everywhere, requiring the DM to be confident in the accuracy of their probability statements on the +log-scale rather than on the natural scale as was the case for N TVD +ϵ +. Logarithms act to inflate the +magnitude of small numbers and thus ensuring that |log(h(·; η)) − log(f(·; Ih(η)))| is small requires +that f and h are increasingly similar as their values decrease. This requires the DM to be more and +more confident of the accuracy of their probability specifications as they get further and further into +the tails, something that is known to already be very difficult for low dimensional problems (Winkler +and Murphy, 1968; O’Hagan et al., 2006), and becomes increasingly difficult as the dimension of the +observation space increases. +4 +Stability to the DGP +4.1 +A reasonable neighbourhood of DGP perturbations +Our second series of results concern the stability of inferences from a single model {f(·; θ); θ ∈ Θ} to +perturbations of the DGP for y ∈ Y. We consider updating on datasets y1 := (y1, . . . , yn1) ∼ g1 or +y2 := (y1, . . . , yn2) ∼ g2 with n1, n2 > 0 and g1 and g2 satisfying Definition 2 +Definition 2 (TVD Neighbourhood of data generating processes). Data generating processes g1 and +g2 for observable y ∈ Y are in the neighbourhood GTVD +ϵ +of size ϵ if TVD(g1, g2) ≤ ϵ +The TVD provides a relevant and reasonable way to describe perturbations of the DGP. It contains +ϵ-contamination neighbourhoods as considered by Matsubara et al. (2021) in the context of ‘global +bias-robustness’ and also in Figure 1. It demands that the data sets were generated under mechanisms +that were absolutely close on the natural scale, rather than the log-score considered in the KLD +neighbourhoods on Miller and Dunson (2018). Conceptually, it is convenient to think about datasets +such that n1 = n2 but this is not necessary. The conditions for the results of the next sections are +similar to those required in Section 3 and are stated in full in Section A.3. +12 + +4.2 +The stability of the βD +Theorem 3 bounds the βD between the posterior predictive distributions resulting from model f and +data from two DGPs in the GTVD +ϵ +neighbourhood. +Theorem 3 (The stability of the posterior predictive distribution under two DGPs of the βD-Bayes in- +ference). Given 1 < β ≤ 2 and likelihood model {f(·; θ) : θ ∈ Θ} and two data sets y1 := (y1, . . . , yn1) ∼ +g1 and y2 := (y1, . . . , yn2) ∼ g2 for n1, n2 > 0 with {g1, g2} ∈ GTVD +ϵ +. Then provided there exists M < ∞ +such that Condition A.3 hold, Condition A.4 holds for D = D(β) +B , y1, y2 and π(β)(θ) then, +D(β) +B (m(β) +f (·|y1)||m(β) +f (·|y2))) ≤2Mβ−1 +β − 1 ϵ + +1 +cS(1) + 2Mβ−1 +β − 1 +� +TVD(g1, f(·; θ1))π(β) +f (θ1|y1)dθ1. +D(β) +B (m(β) +f (·|y2)||m(β) +f (·|y1))) ≤2Mβ−1 +β − 1 ϵ + +1 +cS(2) + 2Mβ−1 +β − 1 +� +TVD(g2, f(·; θ2))π(β) +f (θ2|y2)dθ2. +where cS(1) and cS(2) are defined in Condition A.4 +Further, Theorem 4 bounds the difference in the βD from the DGP of the βD-Bayes posterior +predictive distributions resulting from data from the two DGPs. +Theorem 4 (The stability in the posterior predictive approximation of two DGPs under the same +model of βD-Bayes inference). Given 1 < β ≤ 2 and likelihood model {f(·; θ) : θ ∈ Θ} and two data +sets y1 := (y1, . . . , yn1) ∼ g1 and y2 := (y1, . . . , yn2) ∼ g2 for n1, n2 > 0 with {g1, g2} ∈ GTVD +ϵ +. Then +provided there exists M < ∞ such that Condition A.3 holds, and Condition A.4 holds for D = D(β) +B , +y1, y2 and π(β)(θ) then, +|D(β) +B (g1||m(β) +f (·|y1)) − D(β) +B (g2||m(β) +f (·|y2))| ≤ Mβ−1(β + 2) +β(β − 1) +ϵ + 1 +c + C(β)(f, y1, y2), +where c := min{cS(1), cS(2)} defined in Condition A.4 and +C(β)(f, y1, y2) : = max +�� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1)dθ1 − D(β) +B (g1||m(β) +f (·|y1)), +� +D(β) +B (g2||f(·; θ2))π(β)(θ2|y2)dθ2 − D(β) +B (g2||m(β) +f (·|y2)) +� +Theorems 3 and 4 are the analogous result to Theorems 1 and 2 respectively. The value M is still +easy to bound here and the concentration terms +1 +cS(j) are expected to shrink to 0 as n → ∞. For +Theorem 3, we invoke Lemma A.6 and argue that the βD posterior will place density on parameter +13 + +values of model f that are close to g in TVD. The bound of Theorem 4 depends on C(β)(f, y1, y2), +which under mild regularity conditions goes to 0 as n → ∞, demonstrating that the βD-Bayes is +stable to TVD perturbations of the data, independently of how well the model approximates either of +the DGPs. +4.3 +The stability of the KLD-Bayes +Figure 1 showed that updating using (1) is not stable to perturbations of the DGP. The data considered +is within a GTVD +0.1 neighbourhood of data generated from N(0, 1) and unlike the βD-Bayes, the estimated +posterior predictive is vastly different to what would have been estimated under the uncontaminated +DGP. Lemma 2 investigates perturbations of the DGP that traditional Bayesian inference is stable too. +Lemma 2 (The stability in the posterior predictive approximation of two DGPs under the same model +of KLD-Bayes inference). For likelihood model {f(·; θ) : θ ∈ Θ} and data sets y1 := (y1, . . . , yn1) ∼ g1 +and y2 := (y1, . . . , yn2) ∼ g2 for n1, n2 > 0, given Condition A.4 holds for D = KLD, y1, y2 and +πKLD(θ), we have that +|KLD(g||mKLD +f +(·|y)) − KLD(g||mKLD +h +(·|y))| ≤ CKLD(f, y1, y2) + 1 +c + T1(g1, g2) + T2(f, y1, y2), +where c := min{cS(1), cS(2)} as defined in Condition A.4 and +T1(g1, g2) : = max +�� +g2 log g2 − g1 log g1dµ, +� +g1 log g1 − g2 log g2dµ +� +T2(f, y1, y2) : = max +�� � +(g1 − g2) log f(·; θ1)dµπKLD(θ1|y1)dθ1, +� � +(g2 − g1) log f(·; θ2)dµπKLD(θ2|y2)dθ2 +� +CKLD(f, y1, y2) : = max +�� +KLD(g1||f(·; θ1))πKLD(θ1|y1)dθ1 − KLD(g1||mKLD +f +(·|y1)), +� +KLD(g2||f(·; θ2))πKLD(θ2|y2)dθ2 − KLD(g2||mKLD +f +(·|y2)) +� +Lemma 2 shows that stability of the KLD approximation of DGP by model f to perturbations of +the DGP requires that T1(g1, g2) and T2(f, y1, y2) are small. Small T1(g1, g2) requires g1 and g2 to have +similar entropy, which is not necessarily guaranteed by DGPs according to Definition 2. Alternatively, if +| log f(·; θ)| is bounded then T2(f, y1, y2) can be bounded above by TVD(g1, g2). However, boundedness +14 + +of the log-likelihood is unlikely, as f(y; θ) → 0, | log f(y; θ)| → ∞. Therefore, T2(f, y1, y2) being small +requires g1 and g2 to be increasingly close in the tails of the fitted models, prohibiting, for example, +outlier contaminations such as in Figure 1. +5 +Setting β +The only additional specification required from the DM when implementing the βD-Bayes compared +with the KLD-Bayes is that they select the value of β. This hyperparameter regulates the trade-off +between robustness and efficiency (e.g. Basu et al., 1998). Minimising the KLD (β = 1) provides the +most efficient inference but is very sensitive to outliers. Increasing β away from 1 gains robustness +to outliers at a cost to efficiency. The bounds of the previous theorems all depend on β and we can +therefore additionally interpret β as a sort of meta prior for the DM’s confidence in their elicited model +or data collection. The less confident they are, the greater β will need to be to prevent non-negligible +a posteriori divergence. Eliciting β as such requires the DM to reflect on the value of ϵ associated with +their beliefs or the quality of the data. For the neighbourhoods of Definition 1, this can be obtained +by considering for a given set of parameters what the largest possible error in any of the probability +statements could be, or for Definition 2 by considering the minimal proportion of a population that +they believe is consistent with the DGP. Our results are also informative about when the value of β +might be too large. The DM should want their βD-Bayes inferences be stable because ϵ is small, and +not because the terms involving β that multiply ϵ in the theorems in Sections 3 and 4 are small. +Alternatively, there is increasing interest in data-driven methods to learn β. Warwick and Jones +(2005); Ghosh and Basu (2015); Basak et al. (2021) consider procedures to estimate β to minimise the +mean squared error (MSE) of estimated model parameters, Toma and Broniatowski (2011); Kang and +Lee (2014) estimate β to minimise the maximum perturbation of the parameter estimates resulting +from replacing one observation by the population estimated mean, and Jewson and Rossell (2022); +Yonekura and Sugasawa (2021) estimate β to minimise the Fisher’s divergence to the DGP. +Finally, βD-Bayes inference appears not to be overly sensitive to the exact value of β. Figure 2 +demonstrates that for the example introduced in Section 1, inference for the Gaussian and Student’s-t +models is almost identical for values of β ≥ 1.3. Section B.1.2 provides further demonstration of this. +15 + +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.1 +y +Density +(1 − ϵ)N(0, 1) +ϵN(5, 32) +Gaussian +Student’s-t +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.2 +y +Density +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.3 +y +Density +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.5 +y +Density +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.7 +y +Density +-5 +0 +5 +10 +0.0 +0.2 +0.4 +β = 1.9 +y +Density +Figure 2: Posterior predictive distributions using βD-Bayes updating on n = 1000 observations from +an ϵ-contamination model g(y) = 0.9 × N (y; 0, 1) + 0.1 × N +� +y; 5, 32� +for different values of β. +6 +Experiments +6.1 +Gaussian and Student’s-t likelihood +We revisit the Gaussian and Student’s-t example briefly introduced in Section 1. The likelihood models +considered here are +fσ2 +adj(y; θ) := N +� +y; µ, σ2 × σ2 +adj +� +and hν(y; η) := Student’s − tν +� +y; µ, σ2� +. +(8) +Hyperparameters, ν = 5 and σ2 +adj = 1.16 are fixed to match the quartiles of the two distributions +for all µ and σ2. These were inspired by O’Hagan (2012), who argued that for absolutely continuous +probability distributions, it is only reasonable to ask an expert to make a judgement about the median +and the quartiles of a distribution along with maybe a few specially selected features. This is justified +as adequate as any two distributions with similar percentiles will look very similar, see for example +Figure 1. +However, Section 3.3 suggests that greater precision is required to ensure the stability +of Bayes’ rule updating. +On the other hand, the likelihoods in (8) are contained in N TVD +0.043. +We +generated n = 1000 observations from the ϵ-contamination model g(x) = 0.9 × N (y; 0, 1) + 0.1 × +N +� +y; 5, 32� +contained within the GTVD +0.1 +neighbourhood of N (y; 0, 1). We then conducted Bayesian +updating under the Gaussian and Student’s-t likelihood using both Bayes’ rule and the βD-Bayes +(β = 1.5) under shared priors π(µ, σ2) = N +� +µ; µ0, v0σ2� +IG(σ2; a0, b0), with hyperparameters (a0 = +16 + +-10 +-5 +0 +5 +10 +-2 +-1 +0 +1 +2 +y +Influence Function - µ +KLD - Gaussian +KLD - Student’s-t +βD - Gaussian +βD - Student’s-t +-10 +-5 +0 +5 +10 +-1 +0 +1 +2 +3 +4 +y +Influence Function - σ2 +Figure 3: Influence functions for parameter µ and σ2 of the Gaussian and Student’s-t likelihood models +under the KLD-Bayes and βD-Bayes with β = 1.5. +0.01, b0 = 0.01, µ0 = 0, v0 = 10). +Figure 1 and Figure B.1, which plots the parameter posterior +distributions for both models under both updating mechanisms, clearly demonstrate the stability of +the βD-Bayes across these two models and the lack of stability of traditional Bayesian updating. Not +only is the βD inference more stable across N TVD +ϵ +, the βD predictive better captures the majority of +the DGP than either of the predictive do under traditional Bayesian updating. The capturing of the +N (y; 0, 1) mode further illustrates the βD-Bayes’ stability across neighbourhoods of the DGP. +Figure 3 plots influence functions (West, 1984) for the KLD-Bayes and βD-Bayes under the Gaus- +sian and Student’s-t model. Influence functions are the gradient of the loss function evaluated at +parameter estimates as a function of the observations and show the impact that observation had on +the analysis. Under the βD-Bayes, the influence functions of the Gaussian and Student’s-t likelihoods +are closer for almost every y, illustrating the stability to the model, and additionally, the influence +functions for both models under the βD-Bayes vary less with y, illustrating stability to the DGP. +6.1.1 +DLD data +We consider an RNA-sequencing data set from Yuan et al. (2016) measuring gene expression for +n = 192 patients with different types of cancer. Rossell and Rubio (2018) studied the impact of 57 +predictors on the expression of DLD, a gene that can perform several functions such as metabolism +regulation. To illustrate our results, we selected the 15 variables with the 5 highest loadings in the +first 3 principal components, and fitted regression models using the neighbouring models in (8) for the +residuals. Section B.1.6 lists the selected variables. +17 + +Figure 4 demonstrates that βD-Bayes (β = 1.5) produces more stable estimates of the fitted resid- +uals (top-left), the estimated density of the residuals (top-right), parameter estimates (bottom-left), +and posterior predictive density for the observed data (bottom-right) than the traditional Bayesian in- +ference. Rossell and Rubio (2018) found evidence that this data is heavy-tailed, further demonstrated +in Figure B.5, which caused the KLD-Bayes to estimate very different densities under the Gaussian +and Student’s-t model, while the βD-Bayes is stable to this feature of the data. Figure B.4 shows the +fit of the models to the posterior mean estimates of the standardised residuals, showing that as well as +being stable, the βD-Bayes produces good estimation around the mode of the DLD data under both +models. +Section B.1.5 considers a further regression example showing that even when one of the mod- +els under consideration is ‘well-specified’ for the data, the βD-Bayes inference continues to perform +adequately. +6.2 +Mixture Modeling +An advantage of considering the stability of the distributions for observables rather than parameters +is that it allows ‘neighbouring’ models to have different dimensions to their parameter space. For +example, consider initial model f(·; θ) and then ‘neighbouring’ model +h(·; η) = (1 − ω) × f(·; θ) + ω × h +′(·; κ), +for η = {θ, κ, ω}. +Here, h(·; η) is a mixture model combining the likelihood model f(·; θ), which +could itself already be a mixture model, and some other density h +′(·; κ) with additional parameters +κ. For all θ ∈ Θ and any κ ∈ K we have that TVD(f (·; θ) , h (·; {θ, κ, ω})) < ω and therefore a TVD +neighbourhood can be defined by upper bounding ω. +6.2.1 +Shapley Galaxy Dataset +We examine the Shapley galaxy dataset of Drinkwater et al. (2004), recording the velocities of 4215 +galaxies in the Shapley supercluster, a large concentration of gravitationally-interacting galaxies; see +Figure 5. +The clustering tendency of galaxies continues to be a subject of interest in astronomy. +Miller and Dunson (2018) investigate this data using Gaussian mixture models and use their coarsened +posterior to select the number of mixture components, finding considerable instability in the number +18 + +-4 +-2 +0 +2 +4 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +Gaussian - (y − X ˆθ)/ˆσ +Student’s-t - (y − X ˆθ)/ˆσ +KLD +βD +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +(y − X ˆθ) +Density +KLD - Gaussian +KLD - Student’s-t +βD - Gaussian +βD - Student’s-t +0 +5 +10 +15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Parameter index +|ˆθnorm − ˆθt| +KLD +βD +KLD +βD +-0.2 +-0.1 +0.0 +0.1 +Predictive Density Difference +Figure 4: Posterior mean estimates of standardised residuals (top left), posterior mean estimated +residuals distribution (top-right), absolute difference in posterior mean parameter estimates (bottom +left) and difference in posterior predictive densities of the observations (bottom right) under the +Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes (β = 1.5) for the DLD data. +of estimated components K under different specifications of the coarsening parameter. See Cai et al. +(2021) for further issues with estimating the number of components in mixture models. +We estimate Gaussian mixture models of the form +f(y; θ) = +K +� +k=1 +ωjN(y; µj, σj), +under the KLD-Bayes and βD-Bayes, considering number of components K ∈ {2, 3, 4, 5, 6} and using +the normal-inverse Wishart priors of F´uquene et al. (2019) (full details available in Section B.2). +βD-Bayes inference for such one-dimensional mixture models is easy to implement using adaptive +quadrature to approximate the necessary integral term 1 +β +� +h(z; η)βdz. We do not formally place any +constraint on the estimation of ωk, however, any model that estimates a component with small ωk can +be seen as a neighbour of a model with one fewer component. +19 + +Figure 5 demonstrates the posterior mean approximation to the histogram of the data of the +Gaussian mixture models under the KLD-Bayes and βD-Bayes and Table 1 records the TVD between +the posterior mean predictive distribution of recursively adding components to the model. The βD- +Bayes inference for β = 1.25 and 1.5 is more stable to the addition of an extra component. +In +particular, for K ≥ 3 the βD-Bayes inference stably estimates the biggest components of the data +centered approximately at 5, 000 and 15, 000 km/s, while the KLD-Bayes produces very different +inference for these modes depending on the number of clusters selected. +Table 1: Total variation distances between posterior mean predictive distributions for different number +of mixture components K under the KLD-Bayes and βD for β = 1.25 and 1.5. +Method +K = 2 vs K = 3 +K = 3 vs K = 4 +K = 4 vs K = 5 +K = 5 vs K = 6 +KLD +0.27 +0.12 +0.13 +0.03 +βD (β = 1.25) +0.26 +0.06 +0.06 +0.03 +βD (β = 1.5) +0.23 +0.05 +0.08 +0.02 +6.3 +Binary Classification +Binary classification models predict y ∈ {0, 1} from p-dimensional regressors X. The canonical model +in such a setting is logistic regression where +PLR(y = 1|X, θ) = +1 +1 + exp (−Xθ), +PLR(y = 0|X, θ) = 1 − PLR(Y = 1|X, θ), +where θ ∈ Rp are the regression parameters. +Alternative, less ubiquitous models include, probit +regression, which uses an alternative GLM link function depending on the standard Gaussian CDF +Φ(·), ‘heavier tailed’ t-logistic regression (Ding and Vishwanathan, 2010; Ding et al., 2013) and a +mixture type model that explicitly models the chance of mislabelling of the observed classes. +PPR(y = 1|X, η) = Φ(wPR × Xθ), +PtLR(y = 1|X, η) = expt((wtLR × 0.5Xθ − Gt(wtLR × Xθ))) +PML(y = 1|X, η) = (1 − ν1)PLR(y = 1|X, θ) + ν0(1 − PLR(y = 1|X, θ)) +where 0 < t < 2 and 0 < ν0, ν1 < 1. +The so-called t-exponential ‘expt’ and Gt ensures that +PtLR(y = 1|X, η) is normalised, both are defined in Section B.3.1. Setting t > 1 results in heavier- +tailed probabilities than the logistic model. For the probit and t-logistic models parameters θ are +20 + +Shapley Galaxy Velocities - KLD +Velocity (1000kms/s) +Density +0 +10 +20 +30 +40 +50 +0.00 +0.10 +K = 2 +K = 3 +K = 4 +K = 5 +K = 6 +βD - (β = 1.25) +Velocity (1000kms/s) +Density +0 +10 +20 +30 +40 +50 +0.00 +0.10 +βD - (β = 1.5) +Velocity (1000kms/s) +Density +0 +10 +20 +30 +40 +50 +0.00 +0.10 +Figure 5: Shapley Galaxy Data: Histograms of the data, in units of 1,000 km/s, excluding a small +amount of data extending in a tail up to 80,000 km/s, with fitted Gaussian mixture models with +K = 2−6 components under the KLD-Bayes (top), βD-Bayes with β = 1.25 (middle) and βD-Bayes +with β = 1.5 (bottom). +21 + +-4 +-2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Xθ +P(Y = 1|Xθ) +logistic +t-logistic +probit +mislabelled +-4 +-2 +0 +2 +4 +0.00 +0.02 +0.04 +0.06 +Xθ +TVD +t-logistic +probit +mislabelled +Figure 6: Left: P(y = 1|X, θ) for logistic, probit, t-logistic and mislabelled models. Right: TVD +between the logistic regression canonical model and the probit, t-logistic and mislabelled models. The +θ parameters of the probit and t-logistic models are scalar multiplied in a fashion that minimise the +TVD to the logistic regression +scalar multiples wPR, wtLR ∈ R of the logistic regression parameters θ �→ wθ. These are calculated in +order to minimise the a priori TVD between the models and the logistic regression baseline according +to N TVD +ϵ +(see Section B.3.2). We upper bound ν0 and ν1 by 0.05 making ϵ = 0.05 for these models. +Figure 6 plots P(y = 1|X, θ) as a function of Xθ for all four models (left) and the TVD between each +alternative model and the logistic regression (right), demonstrating that all four produce very similar +binary probabilities. +6.3.1 +Colon Cancer Dataset +To investigate the stability of posterior predictive inferences across the logistic, probit, t-logistic, and +mislabelled binary regression models we consider the colon cancer dataset of Alon et al. (1999). The +dataset contains the expression levels of 2000 genes from 40 tumours and 22 normal tissues and there +is purportedly evidence that certain tissue samples may have been cross-contaminated (Tibshirani and +Manning, 2013). Rather than consider the full 2000 genes we first run a frequentist LASSO procedure, +estimating the hyperparameter via cross-validation, and focus our modelling only on the nine genes +selected by this procedure. We understand that such post-model selection biases parameter estimates, +but the stability of the predictive inference is our focus here. +Figure 7 compares the a posteriori TVD distance between the posterior mean estimated distribution +for each observation with the a priori TVD distance between each of the models (top) and the difference +22 + +between the posterior mean regression parameter estimates of the two models (bottom) under the KLD- +Bayes and βD-Bayes with β = 1.5. The stability of the βD-Bayes is once again demonstrated here, for +almost every observation and every pair of models the posterior predictive inference is as stable as it +was a priori, while the KLD-Bayes inference is more often divergent. For the t-logistic and mislabelled +models the predictive stability of the βD-Bayes also provides greater stability in the posterior mean +parameter estimates. +KLD +βD +0.00 +0.01 +0.02 +0.03 +0.04 +logistic vs probit +|pLR − pP R| +KLD +βD +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +logistic vs t-logistic +|pLR − ptLR| +KLD +βD +0.00 +0.02 +0.04 +0.06 +0.08 +logistic vs mislabelled +|pLR − pML| +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +logistic vs probit +θ-index +|ˆθLR − ˆθP R| +KLD +βD +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +logistic vs t-logistic +θ-index +|ˆθLR − ˆθtLR| +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +logistic vs mislabelled +θ-index +|ˆθLR − ˆθML| +Figure 7: Colon Cancer Data. Top: TVD between the posterior mean estimated probabilities for each +observation of the probit (left), t-logistic (centre) and mislabelled (right) models and the canonical +logistic regression under the KLD-Bayes and βD-Bayes (β = 1.5). The dotted line represented the +a priori TVD distance between the models. Bottom: Absolute differences between posterior mean +parameter estimates and those of the logistic regression. +23 + +7 +Discussion +This paper investigated the posterior predictive stability of traditional Bayesian updating and a gen- +eralised Bayesian alternative minimising the βD. In practice, the model used for inference is usually +a convenient and canonical member of a wider class that capture the broad belief statements made +by the DM and the observed data was not necessarily collected in the manner the DM imagined. +We proved that βD-Bayes inference is provably stable across a class of likelihood models and data +generating processes whose probability statements are absolutely close, a TVD neighbourhood, by +establishing bounds on how far their predictive inferences can diverge. On the other hand, our results +require the DM to be sure about the tail properties of their beliefs and the DGP to guarantee stability +for standard Bayesian inference. +The results of this paper simplify the process of belief elicitation for the βD-Bayes, bounding the +a posteriori consequences for a given level of a priori inaccuracy, leaving the DM free to use the best +guess approximation of their beliefs that they are most comfortable with, rather than switch to a less +familiar model with better outlier rejection properties (O’Hagan, 1979). Such stability is achieved +through a minimal amount of extra work compared with traditional Bayes’ rule inference, and it +provides a similarly recognisable output. We hope such results help to justify the increased use of the +βD to make robust inferences in statistics and machine learning applications. +A key issue motivating the departure from standard Bayesian methods here is a lack of concordance +between the likelihood model and the data. Such an issue can be attributed to either a failure of the +modeller to think carefully enough about the DGP, or errors in data collection. However, we treat +these results separately to exemplify two different manifestations of the instability of Bayes’ rule. +Future work could explore the applicability of such results in multivariate settings where belief +specification and data collection are harder, and further investigate our KLD-Bayes results. While +we argued when you could guarantee the stability of such methods, identifying for which statements +KLD-Bayes is not stable would provide important and useful results to facilitate more focused belief +elicitation. +To continue to facilitate the deployment of βD-Bayes methods in practice, more work is required +to study and build upon existing methods to select β, particularly in high dimensions. While it is +clear that considerable gains can be made over standard methods in certain scenarios, an adversarial +analysis of the βD performance compared with its KLD-Bayes analogue would further motivate its +wider applications. +24 + +Acknowledgements +The authors would like to thank Danny Williamson, Christian Robert, and Sebastian Vollmer for their +insightful discussions on the topics in this paper. JJ was partially funded by the Ayudas Fundaci´on +BBVA a Equipos de Investigaci´on Cientifica 2017, the Government of Spain’s Plan Nacional PGC2018- +101643-B-I00, and a Juan de la Cierva Formaci´on fellowship FJC2020-046348-I. CH was supported by +the EPSRC Bayes4Health programme grant and The Alan Turing Institute, UK. +References +Aitkin, M. and Wilson, G. T. (1980), ‘Mixture models, outliers, and the EM algorithm’, Technometrics +22(3), 325–331. +Akaike, H. (1973), Information theory and an extension of the maximum likelihood principle, in +‘Second International Symposium on Information Theory’, p. 267–281. +Alon, U., Barkai, N., Notterman, D. A., Gish, K., Ybarra, S., Mack, D. and Levine, A. J. (1999), +‘Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues +probed by oligonucleotide arrays’, Proceedings of the National Academy of Sciences 96(12), 6745– +6750. +Basak, S., Basu, A. and Jones, M. 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(2016), ‘Plasma extracellular rna profiles in healthy and cancer patients’, Scientific +reports 6(1), 1–11. +30 + +A +Definitions, Conditions and Proofs +Section A contains additional information for our theoretical analysis, including definitions of the +Kullback-Leibler Divergence (KLD) and β-divergence (βD), full definitions of notation and technical +conditions and proofs of the results of Sections 3 and 4 +A.1 +Divergence Definitions +Here we provide definitions of the KLD and βD. +Definition A.1 (The Kullback-Leibler Divergence (KLD) (Kullback and Leibler, 1951)). The KLD +between probability densities g(·) and f(·) is given by +KLD(g||f) = +� +g log g +f dµ. +Definition A.2 (The β-divergence (βD) (Basu et al., 1998; Mihoko and Eguchi, 2002)). The βD is +defined as +D(β) +B (g||f) = +1 +β(β − 1) +� +gβdµ + 1 +β +� +fβdµ − +1 +β − 1 +� +gfβ−1dµ, +where β ∈ R \ {0, 1}. +The βD is Bregman-divergence (Bregman, 1967) with associated function +ψ(t) = +1 +β(β−1)tβ. When β = 1, D(1) +B (g(x)||f(x)) = KLD(g(x)||f(x)). +The βD has often been referred to as the Density-Power Divergence in the statistics literature +(Basu et al., 1998) where it is often parametrised as β = βDPD + 1. +A.2 +Notation +Now we define the paper’s notation in full. The focus of the paper is on different densities for p- +dimensional observations y ∈ Y ⊂ Rp. Let g, g1 and g2 be potential data generating densities for y. +Consider likelihood models for y +{f(y; θ) : y ∈ Y ⊂ Rp, θ ∈ Θ} , +and where appropriate potential alternative likelihood model +{h(y; η) : y ∈ Y ⊂ Rp, η ∈ A} , +31 + +with functions If : Θ �→ A and Ih : A �→ Θ mapping between their parameter spaces. The parameter +of model f(·; θ) minimising divergence D to DGP g is defined as +θD +g = arg min +θ∈Θ +D(g(·), f(·; θ)) = arg min +θ∈Θ +� +ℓD(x, f(·; θ))dG(·). +The general Bayesian posterior learning about θD +g from y ∼ g given prior πD(·) is +πD(θ|y) = +πD(θ) exp(− �n +i=1 ℓD(xi, f(·; θ))) +� +πD(θ) exp(− �n +i=1 ℓD(xi, f(·; θ)))dθ. +The posterior predictive for exchangeable observation ynew ∈ Y is +mD +f (ynew|y) = +� +f(ynew; θ)πD(θ|y)dθ. +Throughout this section, we will use the · notation within divergence functions to indicate the +variable that is being integrated over in the divergence, i.e. the divergence does not depend on a value +for this variable. +A.3 +Technical Conditions +The results Sections 3 and 4 require the following conditions. +A.3.1 +Stability to the likelihood model +Triangle-type inequalities relating the βD to the TVD will require the bounding of the value of the +density functions according to Condition A.1 +Condition A.1 (Boundedness of g, f and h). For data generating process g(·) and likelihood models +{f(·; θ) : θ ∈ Θ} and {h(·; η) : η ∈ A} there exists 0 < M < ∞ such that +max {ess sup f, ess sup h, ess sup g} ≤ M < ∞ +Given base measure µ - assumed to be the Lebesque measure for continuous random variables +and the counting measures for discrete random variables - M is the essential supremum of density +f(x), ess sup f(x) = M if the set defined by f−1(M, ∞) has measure 0, i.e. µ +� +f−1(M, ∞) +� += 0. For +discrete random variables it is always the case that M ≤ 1, M can be bounded for continuous random +32 + +variables such as a Gaussian or Student’s-t by lower bounding the model’s scale parameter by some +reasonable value. +Additionally, we require the following stochastic concentration condition of the general Bayesian +posterior, which we argue below will hold given sufficient regularity of the observations and the prior +specification. This condition was inspired by the Stochastic Lipschitz continuity assumption of Norkin +(1986). +Condition A.2 (Stochastic Concentration of the posterior for f and h around g). For divergence +D(·| ·) and likelihood models {f(·; θ) : θ ∈ Θ} and {h(·; η) : η ∈ A}, define the subsets of parameters +S(1) +d +: = {θ ∈ Θ, η ∈ A s.t. D(g||h(·|η)) − D(g||h(·|If(θ))) ≤ d} +S(2) +d +: = {θ ∈ Θ, η ∈ A s.t. D(g||f(·|θ))) − D(g||f(·|Ih(η))) ≤ d} , +where S(1) +d , S(2) +d +⊂ Θ × A. Then, for dataset y ∼ g(·) with n > 0 and priors πD(θ) and πD(η) there +exists c1, c2 > 0 such that for all d > 0 the product posterior πD(θ, η|y) = πD(θ|y)πD(η|y) satisfies +πD(S(1) +d |y) ≥ 1 − exp(−c1d) +(9) +πD(S(2) +d |y) ≥ 1 − exp(−c2d). +(10) +Condition A.2 ensures that n is large enough and πD(θ) and πD(η) have sufficient prior mass +at θD +g and ηD +g +for the posterior based on the likelihoods f and h to have concentrated sufficiently +around their optimal parameter such that the a posteriori probabilities that h(y|η) is closer to g than +h(y|If(θ)) and f(y|θ) is closer to g than f(y|Ih(η)) according to divergence D are sufficiently big +The asymptotic normality results of Chernozhukov and Hong (2003); Lyddon et al. (2018) for +the general Bayesian posterior concern convergence in distribution and thus one must be slightly +careful when evoking these to suggest that there must exist some n such that Condition A.2 holds. +However, under the assumption that both likelihood models f(·; θ), h(·; η) and DGP g are all absolutely +continuous and provided the weak conditions for asymptotic normality Chernozhukov and Hong (2003); +Lyddon et al. (2018) are satisfied, then π(β)(S(1) +d |y) +P→ 1 and π(β)(S(2) +d |y) +P→ 1 as n → ∞, as by +definition D(g, f(·; θD +g )) ≤ D(g, f(·; Ih(ηD +g ))) and vice versa. +Conditions A.2 (and A.4 below) are the only part of any of these theorems where the observed +data appears. So the following theorems simply require that the Bayesian updating is being done +33 + +conditional on a dataset satisfying Condition A.2 or A.4 where appropriate. Extensions could look +at whether Condition A.2 and the following theorems hold in expectation under the data generating +process (DGP), however, this may require additional assumptions to be made about the DGP that we +wish to avoid. +A.3.2 +Stability to the DGP +Conditions A.3 and A.4 are required for the results of Section 4 and are analogous to Conditions A.1 +and A.2 introduced in the previous section. +Condition A.3 (Boundedness of g1, g2 and f). For data generating processes g1(·) and g2(·) and +likelihood model {f(·; θ) : θ ∈ Θ} there exists 0 < M < ∞ such that +max {ess sup f, ess sup g1, ess sup g2} ≤ M < ∞ +Condition A.4 (Stochastic Concentration of the posterior for f around g1 and g2). For divergence +D(·||·) and likelihood model {f(·; θ) : θ ∈ Θ}, define subsets of Θ × Θ +S(1) +d +: = {θ1, θ2 ∈ Θ s.t. D(g2||f(·; θ2)) − D(g2||f(·; θ1)) ≤ d} +S(2) +d +: = {θ1, θ2 ∈ Θ s.t. D(g1||f(·; θ1)) − D(g1||f(·; θ2)) ≤ d} . +Then, for datasets y1:n1 ∼ g1(·) and y′ +1:n2 ∼ g2(·) with n1, n2 > 0 and prior πD(θ) there exists +cS(1), cS(2) > 0 such that for all d > 0 the product posterior πD(θ1, θ2|y1, y2) = πD(θ1|y1)πD(θ2|y2) +satisfies +πD(S(1) +d |y1, y2) ≥ 1 − exp(−cS(1)d), +(11) +πD(S(2) +d |y1, y2) ≥ 1 − exp(−cS(2)d). +(12) +A.4 +Proofs: Stability to the Model +Before we prove Theorem 1, Theorem 2 and Lemma 1 we first introduce some useful Lemmas that +simplify their proofs. +34 + +A.4.1 +Useful Lemmas for proving Theorems 1 and 2 +Lemma A.1 establishes a convenient representation for the TVD. +Lemma A.1 (A simplification of the TVD). The following relationship holds for the TVD between +two densities f and h. +TVD(f, h) = +� +A+ (h(y) − f(y)) dy = +� +A− (f(y) − h(y)) dy. +where A+ := {y : h(y) > f(y)} and A− := {y : f(y) > h(y)}. +Proof. Firstly, by definition +TVD(f, h) = 1 +2 +� +|h(y) − f(y)| dy += 1 +2 +� +A+ (h(y) − f(y)) dy + 1 +2 +� +A− (f(y) − h(y)) dy. +Next, consider Lf,h : Y → R with Lf,h(y) := min(f(y), h(y)) as the lower of the two probability +densities for every y. Given that both f and h are probability densities and thus integrate to 1 we +have that +� +A+(h(y) − f(y))dy = 1 − +� +Lf,h(y)dy +� +A−(f(y) − h(y))dy = 1 − +� +Lf,h(y)dy. +The two right-hand sides are identical and therefore the two left-hand sides must be equal. As a result, +TVD(f, h) = 1 +2 +� +A+ (h(y) − f(y)) dy + 1 +2 +� +A− (f(y) − h(y)) dy. += +� +A+ (h(y) − f(y)) dy += +� +A− (f(y) − h(y)) dy, +proving the result. +Lemma A.2 establishes a triangle-type inequality relating the βD and the TVD. Triangle-type +inequalities fit naturally with Section 3’s requirements for stability. If two models are close, then they +ought to provide similar approximations to a third distribution, the DGP. The βD does not strictly +satisfy the triangle inequality. However, we can prove the following results connecting the TVD and +the βD in a triangle-type inequality. The result relies on 1 ≤ β ≤ 2, which places the βD in between +35 + +the KLD at β = 1 and the L2-distance D(2) +B (g||f) = 1 +2 +� +(f − g)2. We are yet to come across scenarios +where setting β outside this range is appropriate from a practical viewpoint (see e.g. Jewson et al., +2018; Knoblauch et al., 2018). +Lemma A.2 (A triangle inequality relating the βD and the TVD). For densities f, h and g with the +property that there exists M < ∞ satisfying Condition A.1 and 1 < β ≤ 2 we have that +��D(β) +B (g||h) − D(β) +B (g||f) +�� ≤ Mβ−1(3β − 2) +β(β − 1) +TVD(h, f). +Proof. By the definition of the βD, we can rearrange +D(β) +B (g||h) +=D(β) +B (g||f) + +�� � 1 +β h(y)β − 1 +β f(y)β − +1 +β − 1g(y)h(y)β−1 + +1 +β − 1g(y)f(x)β−1 +� +dy +� +=D(β) +B (g||f) + +� 1 +β +� � +h(y)β − f(y)β� +dy + +1 +β − 1 +� +g(y) +� +f(y)β−1 − h(y)β−1� +dy +� +As in Lemma A.1, define A+ := {y : h(y) > f(y)} and A− := {y : f(y) > h(y)}. Now by the mono- +tonicity of the function yβ and yβ−1 when 1 ≤ β ≤ 2 we have that +� +A− h(y)β − f(y)βdy < 0 +� +A+ g(y) +� +f(y)β−1 − h(y)β−1� +dy < 0 +therefore removing these two terms provides an upper bound +D(β) +B (g||h) +=D(β) +B (g||f) + 1 +β +� � +h(y)β − f(y)β� +dy + +1 +β − 1 +� +g(y) +� +f(y)β−1 − h(y)β−1� +dy +≤D(β) +B (g||f) + 1 +β +� +A+ +� +h(y)β − f(y)β� +dy + +1 +β − 1 +� +A− g(y) +� +f(y)β−1 − h(y)β−1� +dy. +36 + +Next, adding and subtracting 1 +βh(y)f(y)β−1 provides +D(β) +B (g||h) +≤D(β) +B (g||f) + 1 +β +� +A+ +� +h(y)β − f(y)β� +dy + +1 +β − 1 +� +A− g(y) +� +f(y)β−1 − h(y)β−1� +dy. +=D(β) +B (g||f) + 1 +β +� +A+ +� +h(y)β − h(y)f(y)β−1 + h(y)f(y)β−1 − f(y)β� +dy ++ +1 +β − 1 +� +A− g(y) +� +f(y)β−1 − h(y)β−1� +dy. +=D(β) +B (g||f) + 1 +β +� +A+ h(y) +� +h(y)β−1 − f(y)β−1� +dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y) +� +f(y)β−1 − h(y)β−1� +dy. +=D(β) +B (g||f) + 1 +β +� +A+ h(y)β +� +1 − f(y)β−1 +h(y)β−1 +� +dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y)f(y)β−1 +� +1 − h(y)β−1 +f(y)β−1 +� +dy. +Now on A+ h(y) > f(y) and so +� +f(y) +h(y) +�β−1 +> f(y) +h(y) for 1 ≤ β ≤ 2 so +� +1 − f(y)β−1 +h(y)β−1 +� +≤ +� +1 − f(y) +h(y) +� +with the exact same logic holding in reverse on A−. We can use this to show that +D(β) +B (g||h) ≤ D(β) +B (g||f) + 1 +β +� +A+ h(y)β +� +1 − f(y)β−1 +h(y)β−1 +� +dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y)f(y)β−1 +� +1 − h(y)β−1 +f(y)β−1 +� +dy +≤D(β) +B (g||f) + 1 +β +� +A+ h(y)β +� +1 − f(y) +h(y) +� +dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y)f(y)β−1 +� +1 − h(y) +f(y) +� +dy +=D(β) +B (g||f) + 1 +β +� +A+ h(y)β−1 (h(y) − f(y)) dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y)f(y)β−2 (f(y) − h(y)) dy. +We now use the fact that we defined max {ess sup f, ess sup h, ess sup g} ≤ M < ∞ and Lemma A.1 to +37 + +leave +D(β) +B (g||h) = D(β) +B (g||f) + 1 +β +� +A+ h(y)β−1 (h(y) − f(y)) dy + 1 +β +� +A+ f(y)β−1 (h(y) − f(y)) dy ++ +1 +β − 1 +� +A− g(y)f(y)β−2 (f(y) − h(y)) dy +≤D(β) +B (g||f) + Mβ−1 +β +� +A+ (h(y) − f(y)) dy + Mβ−1 +β +� +A+ (h(y) − f(y)) dy ++ Mβ−1 +β − 1 +� +A− (f(y) − h(y)) dy +=D(β) +B (g||f) + 2Mβ−1 +β +TVD(h, f) + Mβ−1 +β − 1 +TVD(h, f) +=D(β) +B (g||f) + Mβ−1(3β − 2) +β(β − 1) +TVD(h, f). +Lemma A.3 proves the convexity of the D(β) +B (g, f) in both g and f +Lemma A.3 (The convexity of the βD). The βD between two densities g(y) and f(y) is convex in +both densities for 1 < β ≤ 2, when fixing the other. That is to say that for λ ∈ [0, 1] +D(β) +B (λg1 + (1 − λ)g2, f) ≤ λD(β) +B (g1, f) + (1 − λ)D(β) +B (g2, f) for all f +D(β) +B (g, λf1 + (1 − λ)f2) ≤ λD(β) +B (g, f1) + (1 − λ)D(β) +B (g, f2) for all g +for 1 < β ≤ 2. +Proof. First, we fix f and look at convexity in the function g. let λ ∈ [0, 1]. The function xp for x ≥ 0 +and p > 1 is convex and thus satisfies +(λx1 + (1 − λ)x2)p ≤ λxp +1 + (1 − λ)xp +2 +therefore we have that provided D(β) +B (g1||f) < ∞ and D(β) +B (g2||f) < ∞ +D(β) +B (λg1 + (1 − λ)g2||f) += +� +1 +β(β − 1) (λg1 + (1 − λ)g2)β + 1 +β fβ − +1 +β − 1 (λg1 + (1 − λ)g2) fβ−1dµ +≤ +� +1 +β(β − 1) +� +λgβ +1 + (1 − λ)gβ +2 +� ++ 1 +β fβ − +1 +β − 1 (λg1 + (1 − λ)g2) fβ−1dµ +=λD(β) +B (g1||f) + (1 − λ)D(β) +B (g2||f). +38 + +Next, we fix g and look at the convexity in f. Similarly to above we know that when x ≥ 0 and 1 ≤ +p ≤ 2 that 1 +pxp and − +1 +p−1xp−1 are both convex in y. We therefore have that provided D(β) +B (g||f1) < ∞ +and D(β) +B (g||f2) < ∞ +D(β) +B (g||λf1 + (1 − λ)f2) += +� +1 +β(β − 1)gβ + 1 +β (λf1 + (1 − λ)f2)β − +1 +β − 1g (λf1 + (1 − λ)f2)β−1 dµ +≤ +� +1 +β(β − 1)gβ + 1 +β +� +λfβ +1 + (1 − λ)fβ +2 +� +− +1 +β − 1g +� +λfβ−1 +1 ++ (1 − λ)fβ−1 +2 +� +dµ +=λD(β) +B (g||f1) + (1 − λ)D(β) +B (g||f2). +Lemma A.4 introduces a useful the “three-point property” (Cichocki and Amari, 2010) associated +with the βD +Lemma A.4 (Three-point property of the βD). The following relationship for the βD holds for +densities g, f and h +D(β) +B (f||h) = D(β) +B (g||h) − D(β) +B (g||f) + R(g||f||h) +where +R(g||f||h) = +1 +β − 1 +� +(g − f) +� +hβ−1 − fβ−1� +dµ +(13) +Proof. Following the definition of the βD (A.2) +D(β) +B (g||f) + D(β) +B (f||h) += +� +1 +β(β − 1)gβ + 1 +β fβ − +1 +β − 1gfβ−1dµ + +� +1 +β(β − 1)fβ + 1 +β hβ − +1 +β − 1fhβ−1dµ += +� +1 +β(β − 1)gβ + 1 +β hβ + +1 +β − 1fβ − +1 +β − 1gfβ−1 − +1 +β − 1fhβ−1dµ += +� +1 +β(β − 1)gβ + 1 +β hβ − +1 +β − 1ghβ−1 + +1 +β − 1ghβ−1 + +1 +β − 1ffβ−1 − +1 +β − 1gfβ−1dµ +− +1 +β − 1fhβ−1dµ +=D(β) +B (g||h) + +1 +β − 1 +� +(g − f) +� +hβ−1 − fβ−1� +dµ +39 + +Lemma A.5 provides a useful bound for the interpretation of the remainder term from Lemma +A.4. +Lemma A.5 (A bound on R(g||f||h) from Lemma A.4). For densities f, h and g with the property +that there exists M < ∞ satisfying Condition A.1 and 1 < β ≤ 2, the remainder term from Lemma +A.4 can be bounded as +R(g||f||h) ≤ 2Mβ−1 +β − 1 +TVD(g, f) +Proof. Define A+ +f := {y : g(y) ≥ f(y))} and A− +f := {y : g(y) ≤ f(y))} as +R(g||f||h) = +1 +β − 1 +� +(g − f) +� +hβ−1 − fβ−1� +dµ += +1 +β − 1 +� +hβ−1 (g − f) dµ + +1 +β − 1 +� +fβ−1 (f − g) dµ +≤ +1 +β − 1 +� +A+ hβ−1 (g − f) dµ + +1 +β − 1 +� +A− fβ−1 (f − g) dµ +≤ Mβ−1 +β − 1 +� +A+ (g − f) dµ + Mβ−1 +β − 1 +� +A− (f − g) dµ +≤ 2Mβ−1 +β − 1 +TVD(g, f) +by Condition A.1 and Lemma A.1. +Lemma A.6 shows that the βD can be bounded above by the TVD, which is useful when interpreting +the bound in Theorem 1. +Lemma A.6. For densities f, h and g with the property that there exists M < ∞ satisfying Condition +A.1 and 1 < β ≤ 2 we have that +D(β) +B (g||f) ≤ +�Mβ−1 +β − 1 +� +TVD(g, f). +Proof. Firstly, define A− = {y : g(y) < f(y)} and A+ = {y : g(y) ≥ f(y)} and note on A+ that (f(y)− +g(y)) < 0 and on A− that g(y) < f(y) ⇒ gβ−1(y) < fβ−1(y) for 1 ≤ β ≤ 2. The βD can, then, be +40 + +rearranged as +D(β) +B (g||f) += 1 +β +� +f(y)β−1 (f(y) − g(y)) dy + +1 +β(β − 1) +� � +g(y)β−1 − f(y)β−1� +g(y)dy +≤ 1 +β +� +A− +f(y)β−1 (f(y) − g(y)) dy + +1 +β(β − 1) +� +A+ +� +g(y)β−1 − f(y)β−1� +g(y)dy. +Since f ≤ M by Condition A.1 and using Lemma A.1 we can write +� +A− +f(y)β−1 (f(y) − g(y)) dy ≤ Mβ−1 +� +A− +(f(y) − g(y)) dy = Mβ−1TVD(f, g). +Further, on A+ we have that g(y) > f(y) which implies that f(y) +g(y) < 1 and that when 1 < β < 2, +f(y) +g(y) +β−1 > f(y) +g(y) so +� +A+ +� +g(y)β−1 − f(y)β−1� +g(y)dy = +� +A+ +g(y)β−1 +� +1 − +�f(y) +g(y) +�β−1� +g(y)dy +≤ +� +A+ +g(y)β−1 +� +1 − f(y) +g(y) +� +g(y)dy += +� +A+ +g(y)β−1 (g(y) − f(y)) dy +≤Mβ−1TVD(f, g), +since g ≤ M by Condition A.1. Combining the two bounds leaves +D(β) +B (g||f) ≤ Mβ−1 +β +TVD(f, g) + +Mβ−1 +β(β − 1) +TVD(f, g), +which proves the theorem. +The implications of Lemma A.6 are as follows. Provided +� +Mβ−1 +β−1 +� +does not get too small, we can +be confident that any value of θ such that f(·; θ) is close to the data generating density g in terms of +TVD, will receive high posterior mass under an update targeting the βD. +Lastly, Lemma A.7 provides a convenient result for the deployment of Conditions A.2 (and A.4 +below). This result was inspired by part of the proof of Theorem 7 of Dimitrakakis et al. (2017) (p27). +Lemma A.7 (Stochastic Concentration (Norkin, 1986; Dimitrakakis et al., 2017)). If random variable +ω ∈ Ω ⊂ R is distributed according to π and there exists c > 0 such that for all t > t0 +Fω(t) = π({ω ≤ t}) ≥ 1 − exp(−c(t − t0)), +(14) +41 + +then +� +ωπ(ω)dω ≤ t0 + 1 +c +Proof. We can write the expectation of ω in terms of its cumulative distribution function (CDF) as +� +ωπ(ω)dω = +� ∞ +0 +(1 − Fω(t))dt − +� 0 +−∞ +Fω(t)dt +≤ +� ∞ +0 +(1 − Fω(t))dt += +� t0 +0 +(1 − Fω(t))dt + +� ∞ +t0 +(1 − Fω(t))dt +≤ +� t0 +0 +1dt + +� ∞ +t0 +(1 − Fω(t))dt += t0 + +� ∞ +t0 +(1 − Fω(t))dt. +Then, invoking (14) leaves +� +ωπ(ω)dω ≤ t0 + +� ∞ +t0 +(1 − Fω(t))dt +≤ t0 + +� ∞ +t0 +exp(−c(t − t0))dt +≤ t0 + +� ∞ +0 +exp(−ct)dt += t0 + 1 +c +as required. +A.4.2 +Proof of Theorem 1 +We are now able to use the convexity of the βD (Lemma A.3), the triangular relationship between the +βD and the TVD (Lemma A.2) and the three-point property the βD (Lemmas A.4 and A.5) to prove +Theorem 1 which establishes the posterior predictive stability to the likelihood model’s specification +provided by inference using the βD. +Proof. By the convexity of the βD for 1 < β ≤ 2 (Lemma A.3) we can apply Jensen’s inequality to +42 + +show that +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) ≤ +� +D(β) +B (m(β) +f (·|y)||h(·; η))π(β)(η|y)dη +≤ +� �� +D(β) +B (f(·; θ)||h(·; η))π(β)(θ|y)dθ +� +π(β)(η|y)dη. +Now the three-point property associated with the βD (Lemma A.4) gives us that +D(β) +B (f||h) = D(β) +B (g||h) − D(β) +B (g||f) + R(g||f||h) +where R(g||f||h) is defined in (13). Using this here provides +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) +≤ +� �� +D(β) +B (f(·; θ)||h(·; η))π(β)(θ|y)dθ +� +π(β)(η|y)dη += +� �� � +D(β) +B (g||h(·; η)) − D(β) +B (g||f(·; θ)) ++R(g||f(·; θ)||h(·; η)] π(β)(θ|y)dθ +� +π(β)(η|y)dη += +� +D(β) +B (g||h(·; η))π(β)(η|y)dη − +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +Now adding and subtracting +� +D(β) +B (g||h(·; If(θ)))π(β)(θ|y)dθ we have +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) ≤ +� +D(β) +B (g||h(·; η))π(β)(η|y)dη − +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ ++ +� +D(β) +B (g||h(·; If(θ)))π(β)(θ|y)dθ − +� +D(β) +B (g||h(·; If(θ)))π(β)(θ|y)dθ ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. += +� � +D(β) +B (g||h(·; If(θ))) − D(β) +B (g||f(·; θ)) +� +π(β)(θ|y)dθ ++ +� � � +D(β) +B (g||h(·; η)) − D(β) +B (g||h(·; If(θ))) +� +π(β)(θ|y)dθπ(β)(η|y)dη ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +Now we can apply Lemma A.7 to random variable +� +D(β) +B (g||h(·; η)) − D(β) +B (g||h(·; If(θ))) +� +∈ R on Θ×A +which by using (9) of Condition A.2 applied to the βD provides +� � � +D(β) +B (g||h(·; η)) − D(β) +B (g||h(·; If(θ))) +� +π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 +c1 +. +43 + +As a result +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) ≤ +� � +D(β) +B (g||h(·; If(θ))) − D(β) +B (g||f(·; θ)) +� +π(β)(θ|y)dθ ++ 1 +c1 ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +We can now apply the triangle type inequality from Lemma A.2, +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) +≤ +� � +D(β) +B (g||h(·; If(θ))) − D(β) +B (g||f(·; θ)) +� +π(β)(θ|y)dθ ++ 1 +c1 ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +≤ +� Mβ−1(3β − 2) +β(β − 1) +TVD(h(·; If(θ)), f(·; θ))π(β)(θ|y)dθ ++ 1 +c1 ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +Given the neighbourhood of likelihood models N TVD +ϵ +we can then write +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) ≤ +� Mβ−1(3β − 2) +β(β − 1) +TVD(h(·; If(θ)), f(·; θ))π(β)(θ|y)dθ ++ 1 +c1 ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +≤Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c1 ++ +� � +R(g||f(·; θ)||h(·; η))π(β)(θ|y)dθπ(β)(η|y)dη. +Now from Lemma A.5 we have that R(g||f(·; θ)||h(·; η)) ≤ 2Mβ−1 +β−1 TVD(g, f(·; θ)) and as a result we +can bound +D(β) +B (m(β) +f (·|y)||m(β) +h (·|y))) ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c1 ++ 2Mβ−1 +β − 1 +� +TVD(g, f(·; θ))π(β)(θ|y)dθ. +This provides the first part of the required result. We note that we could have instead considered +D(β) +B (m(β) +h (·|y)||m(β) +f (·|y))), applied the corresponding version of the three-point property of Bregman +divergences, with remainder R(g||h||f) = +� +(g − h) +� +1 +β−1fβ−1 − +1 +β−1hβ−1� +dµ, used Lemma A.7 with +(10) of A.2 and Lemma A.5, therefore we also have that +D(β) +B (m(β) +h (·|y)||m(β) +f (·|y))) ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c2 ++ 2Mβ−1 +β − 1 +� +TVD(g, h(·; η))π(β)(η|y)dη. +providing the second part of the required result. +44 + +A.4.3 +Proof of Theorem 2 +Theorem 2 uses the convexity of the βD (Lemma A.3) and the triangular relationship between the βD +and the TVD (Lemma A.2) to prove stability in the posterior predictive approximation to the DGP of +inference using the βD. +Proof. Using the convexity of the βD (Lemma A.3) and Jensen’s inequality, +D(β) +B (g||m(β) +f (·|y)) ≤ +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ +Adding and subtracting +� +D(β) +B (g||f(·; Ih(η)))π(β)(η|y)dη we have that +D(β) +B (g||m(β) +f (·|y)) ≤ +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ += +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ + +� +D(β) +B (g||f(·; Ih(η)))π(β)(η|y)dη − +� +D(β) +B (g||f(·; Ih(η)))π(β)(η|y)dη += +� +D(β) +B (g||f(·; Ih(η)))π(β)(η|y)dη + +� � � +D(β) +B (g||f(·; θ)) − D(β) +B (g||f(·; Ih(η))) +� +π(β)(θ|y)dθπ(β)(η|y)dη +Now we can apply Lemma A.7 to random variable +� +D(β) +B (g||f(·; θ)) − D(β) +B (g||f(·; Ih(η))) +� +∈ R on Θ×A +which by using (10) of Condition A.2 applied to the βD provides +� � � +D(β) +B (g||f(·; θ)) − D(β) +B (g||f(·; Ih(η))) +� +π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 +c2 +. +We then use the the triangular-type relationship between the βD and the TVD (Lemma A.2) to show +that +D(β) +B (g||m(β) +f (·|y)) ≤ +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ +≤ +� +D(β) +B (g||f(·; Ih(η)))π(β)(η|y)dη + 1 +c2 +≤ +� �Mβ−1(3β − 2) +β(β − 1) +TVD(f(·; Ih(η)), h(·; η)) + D(β) +B (g||h(·; η)) +� +π(β)(η|y)dη + 1 +c2 += +� Mβ−1(3β − 2) +β(β − 1) +TVD(f(·; Ih(η)), h(·; η))π(β)(η|y)dη + +� +D(β) +B (g||h(·; η))π(β)(η|y)dη + 1 +c2 ++ D(β) +B (g||m(β) +h (·|y)) − D(β) +B (g||m(β)(·|y)). +45 + +The same arguments this time using (9) of Condition A.2 can also be used to show that +D(β) +B (g||m(β) +h (·|y)) ≤ +� +D(β) +B (g||h(·; η))π(β) +h (η|y)dη +≤ +� +D(β) +B (g||h(·; If(θ)))π(β)(θ|y)dθ + 1 +c1 +≤ +� �Mβ−1(3β − 2) +β(β − 1) +TVD(f(·; θ), h(·; If(θ))) + D(β) +B (g||f(·; θ)) +� +π(β)(θ|y)dθ + 1 +c1 += +� Mβ−1(3β − 2) +β(β − 1) +TVD(f(·; θ), h(·; If(θ)))π(β)(θ|y)dθ + +� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ + 1 +c1 ++ D(β) +B (g||m(β) +f (·|y)) − D(β) +B (g||m(β) +f (·|y)) +Combining the above two results provides the following bound, +|D(β) +B (g||m(β) +f (·|y)) − D(β) +B (g||m(β) +h (·|y))| ≤ Mβ−1(3β − 2) +β(β − 1) +ϵ + 1 +c + C(β)(f, h, y), +where c = min{c1, c2} and +C(β)(f, h, y) : = max +�� +D(β) +B (g||f(·; θ))π(β)(θ|y)dθ − D(β) +B (g||m(β)(·|y)), +� +D(β) +B (g||h(·; η))π(β)(η|y)dη − D(β) +B (g||m(β)(·|y)) +� +as required. +A.4.4 +Proof of Lemma 1 +The KLD is recovered as the parameter β → 1. However, the bounds of Theorems 1 and 2 go to infinity +in this case. The proof of Lemma 1 applies a similar method to that of Theorem 2 to investigate the +stability of KLD-Bayes. +Proof. Firstly, the logarithm is a concave function and therefore the negative logarithm is a convex +function which is sufficient to prove the convexity of KLD in its second argument. Further, by the +definition of the KLD, we can see that +KLD(g||f) = KLD(g||h) + +� +g log h +f dµ. +(15) +Now we can use the convexity of the KLD and Jensen’s inequality, to show that +KLD(g||mKLD +f +(·|y)) ≤ +� +KLD(g||f(·; θ))πKLD(θ|y)dθ +46 + +Now adding and subtracting +� +KLD(g||f(·; Ih(η)))πKLD(η|y)dη provides +KLD(g||mKLD +f +(·|y)) ≤ +� +KLD(g||f(·; θ))πKLD(θ|y)dθ += +� +KLD(g||f(·; θ))πKLD(θ|y)dθ + +� +KLD(g||f(·; Ih(η)))πKLD(η|y)dη − +� +KLD(g||f(·; Ih(η)))πKLD(η|y)dη += +� +KLD(g||f(·; Ih(η)))πKLD(η|y)dη + +� � +{KLD(g||f(·; θ)) − KLD(g||f(·; Ih(η)))} πKLD(θ|y)dθπKLD(η|y)dη +Now we can apply Lemma A.7 to random variable {KLD(g||f(·; θ)) − KLD(g||f(·; Ih(η)))} ∈ R on +Θ × A which by using (10) of Condition A.2 applied to the KLD provides +� � +{KLD(g||f(·; θ)) − KLD(g||f(·; Ih(η)))} π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 +c2 +. +We can now use the triangular-type relationship of (15) to show that +KLD(g||mKLD +f +(·|y)) ≤ +� +KLD(g||f(·; Ih(η)))πKLD(η|y)dη + 1 +c2 += +� �� +g(·) log +h(·; η) +f(·; Ih(η))dµ + KLD(g||h(·; η)) +� +πKLD(η|y)dθ + 1 +c2 += +� � +g(·) log +h(·; η) +f(·; Ih(η))dµπKLD(η|y)dη + +� +KLD(g||h(·; η))πKLD(η|y)dη ++ 1 +c2 ++ KLD(g||mKLD +h +(·|y)) − KLD(g||mKLD +h +(·|y)). +The same arguments of Lemma A.7 and (9) of Condition A.2 show +KLD(g||mKLD +h +(·|y)) ≤ +� +KLD(g||h(·; η))πKLD(η|y)dη +≤ +� +KLD(g||h(·; If(θ)))πKLD(θ|y)dθ + 1 +c1 += +� �� +g(·) log +f(·; θ) +h(·; If(θ))dµ + KLD(g||f(·; θ)) +� +πKLD(θ|y)dθ + 1 +c1 += +� � +g(·) log +f(·; θ) +h(·; If(θ))dµπKLD(θ|y)dθ + +� +KLD(g||f(·; θ))πKLD(θ|y)dθ ++ 1 +c1 ++ KLD(g||mKLD(·|y)) − KLD(g||mKLD(·|y)) +Combining the above two results provides the following bound, +|KLD(g||mKLD +f +(·|y)) − KLD(g||mKLD +h +(·|y))| ≤ CKLD(f, h, y) + 1 +c + T(f, h, y) +47 + +where c := min{c1, c2} and +T(f, h, y) : = max +�� � +g(·) log +f(·; θ) +h(·; If(θ))dµπKLD(θ|y)dθ, +� � +g(·) log +h(·; η) +f(·; Ih(η))dµπKLD(η|y)dη +� +CKLD(f, h, y) : = max +�� +KLD(g||f(·; θ))πKLD(θ|y)dθ − KLD(g||mKLD(·|y)), +� +KLD(g||h(·; η))πKLD(η|y)dη − KLD(g||mKLD +h +(·|y)) +� +as required +A.5 +Proofs: Stability to the DGP +A.5.1 +A useful Lemma for proving Theorems 3 and 4 +In order to prove Theorems 3 and 4, Lemma A.8 provides a second triangle-type inequality for the +βD and TVD in the case where one model is estimated under two DGPs. +Lemma A.8 (Another triangle inequality relating the βD and the TVD). For densities f, g1 and g2 +with the property that there exists M < ∞ satisfying Condition A.3 and 1 < β ≤ 2 we have that +��D(β) +B (g1||f) − D(β) +B (g2||f) +�� ≤ Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2). +Proof. By the definition of the βD, we can rearrange +D(β) +B (g1||f) = D(β) +B (g2||f)+ +�� � +1 +β(β − 1)g1(y)β − +1 +β(β − 1)g2(y)β + +1 +β − 1g2(y)f(y)β−1 − +1 +β − 1g1(y)f(y)β−1 +� +dy +� +=D(β) +B (g2||f) + +� +1 +β(β − 1) +� � +g1(y)β − g2(y)β� +dy + +1 +β − 1 +� +f(y)β−1 (g2(y) − g1(y)) dy +� +As in Lemma A.1, define A+ := {y : g2(y) > g1(y)} and A− := {y : g1(y) > g2(y)}. By the mono- +tonicity of the function yβ when 1 ≤ β ≤ 2 we have that +� +A+ g1(y)β − g2(y)βdy < 0 +� +A− f(x)β−1 (g2(y) − g1(y)) dy < 0 +48 + +therefore removing these two terms provides an upper bound +D(β) +B (g1||f) +=D(β) +B (g2||f) + +� +1 +β(β − 1) +� � +g1(y)β − g2(y)β� +dy + +1 +β − 1 +� +f(x)β−1 (g2(y) − g1(y)) dy +� +≤D(β) +B (g2||f) + +1 +β(β − 1) +� +A− +� +g1(y)β − g2(y)β� +dy + +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy. +Now adding and subtracting g1(y)g2(y)β−1 provides +D(β) +B (g1||f) +≤D(β) +B (g2||f) + +1 +β(β − 1) +� +A− +� +g1(y)β − g2(y)β� +dy + +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +=D(β) +B (g2||f) + +1 +β(β − 1) +� +A− +� +g1(y)β − g1(y)g2(y)β−1 + g1(y)g2(y)β−1 − g2(y)β� +dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +=D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y) +� +g1(y)β−1 − g2(y)β−1� +dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +=D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y)β +� +1 − g2(y)β−1 +g1(y)β−1 +� +dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy. +Next, on A− g1(x) > g2(x) which implies that +� +g2(x) +g1(x) +�β−1 +> g2(x) +g1(x) for 1 ≤ β ≤ 2 and +� +1 − g2(x)β−1 +g1(x)β−1 +� +≤ +� +1 − g2(x) +g1(x) +� +. +We can use this to show that +D(β) +B (g1||f) ≤ D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y)β +� +1 − g2(y)β−1 +g1(y)β−1 +� +dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +≤D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y)β +� +1 − g2(y) +g1(y) +� +dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +=D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y)β−1 (g1(y) − g2(y)) dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy. +49 + +We now use the fact that we defined max {ess sup f, ess sup g1, ess sup g2} ≤ M < ∞ and Lemma A.1 +to leave +D(β) +B (g1||f) = D(β) +B (g2||f) + +1 +β(β − 1) +� +A− g1(y)β−1 (g1(y) − g2(y)) dy + +1 +β(β − 1) +� +A− g2(y)β−1 (g1(y) − g2(y)) dy ++ +1 +β − 1 +� +A+ f(y)β−1 (g2(y) − g1(y)) dy +≤D(β) +B (g2||f) + +Mβ−1 +β(β − 1) +� +A− (g1(y) − g2(y)) dy + +Mβ−1 +β(β − 1) +� +A− (g1(y) − g2(y)) dy ++ Mβ−1 +β − 1 +� +A+ (g2(y) − g1(y)) dy +=D(β) +B (g2||f) + 2 Mβ−1 +β(β − 1) +TVD(g1, g2) + Mβ−1 +β − 1 +TVD(g1, g2) +=D(β) +B (g2||f) + Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2), +providing the required result. +A.5.2 +Proof of Theorem 3 +Similarly to Theorem 1, we use the convexity of the βD (Lemma A.3)and the three-point property +the βD (Lemmas A.4 and A.5) to prove Theorem 3 which establishes the posterior predictive stability +to perturbations of the DGP provided by inference using the βD. +Lemma A.5 is the important lemma for this proof rather than the triangle inequality +Proof. By the convexity of the βD for 1 < β ≤ 2 (Lemma A.3) we can apply Jensen’s inequality to +show that +D(β) +B (m(β) +f (·|y1)||m(β) +f (·|y2))) ≤ +� +D(β) +B (m(β) +f (·|y1)||f(·; θ2))π(β)(θ2|y2)dθ2 +≤ +� �� +D(β) +B (f(·; θ1)||f(·; θ2))π(β)(θ1|y1)dθ1 +� +π(β)(θ2|y2)dθ2. +Now the three-point property associated with the βD (Lemma A.4) gives us that +D(β) +B (f1||f2) = D(β) +B (g2||f2) − D(β) +B (g2||f1) + R(g2||f1||f2) +50 + +where R(g||f||h) is defined in (13). Using this here provides +D(β) +B (m(β) +f (·|y1)||m(β) +f (·|y2))) +≤ +� �� +D(β) +B (f(·; θ1)||f(·; θ2))π(β)(θ1|y1)dθ1 +� +π(β)(θ2|y2)dθ2 += +� �� � +D(β) +B (g2||f(·; θ2)) − D(β) +B (g2||f(·; θ1)) ++R(g2||f(·; θ1)||f(·; θ2)) π(β)(θ1|y1)dθ1 +� +π(β)(θ2|y2)dθ2 += +� � � +D(β) +B (g2||f(·; θ2)) − D(β) +B (g2||f(·; θ1)) +� +π(β)(θ1|y1)dθ2π(β)(θ2|y2)dθ1 ++ +� � +R(g2||f(·; θ1)||f(·; θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2. +Now we can apply Lemma A.7 to random variable +� +D(β) +B (g2||f(·; θ2)) − D(β) +B (g2||f(·; θ1)) +� +∈ R on Θ×A +which by using (11) of Condition A.4 applied to the βD provides +� � � +D(β) +B (g2||f(·; θ2)) − D(β) +B (g2||f(·; θ1)) +� +π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 ≤ +1 +cS(1) . +Therefore +D(β) +B (m(β) +f (·|y1)||m(β) +f (·|y2))) +≤ +� � � +D(β) +B (g2||f(·; θ2)) − D(β) +B (g2||f(·; θ1)) +� +π(β)(θ1|y1)dθ2π(β)(θ2|y2)dθ1 ++ +� � +R(g2||f(·; θ1)||f(·; θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 +≤ +1 +cS(1) + +� � +R(g2||f(·; θ1)||f(·; θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2. +Now from Lemma A.5 we have that R(g2||f(·; θ1)||f(·; θ2)) ≤ 2Mβ−1 +β−1 TVD(g2, f(·; θ1)) which is itself +not necessarily small. We can however apply the triangle inequality to the TVD and say that +R(g2||f(·; θ1)||f(·; θ2)) ≤ 2Mβ−1 +β − 1 +TVD(g2, f(·; θ1)) +≤ 2Mβ−1 +β − 1 (TVD(g1, g2) + TVD(g1, f(·; θ1))) +≤ 2Mβ−1 +β − 1 (ϵ + TVD(g1, f(·; θ1))) , +given the neighbourhood of data generating processes defined by GTVD +ϵ +. As a result +D(β) +B (m(β) +f (·|y1)||m(β) +f (·|y2))) ≤2Mβ−1 +β − 1 ϵ + +1 +cS(1) + 2Mβ−1 +β − 1 +� +TVD(g1, f(·; θ1))π(β)(θ1|y1)dθ1. +51 + +We note that we could have instead considered D(β) +B (m(β) +f (·|y2)||m(β) +f (·|y1))), applied the corre- +sponding version of the three-point property of βD, used Lemma A.7, (12) of Condition A.4 applied +to the βD and Lemma A.5 to also show that +D(β) +B (m(β) +f (·|y2)||m(β) +f (·|y1))) ≤ 2Mβ−1 +β − 1 ϵ + +1 +cS(2) + 2Mβ−1 +β − 1 +� +TVD(g2, f(·; θ2))π(β) +f (θ2|y2)dθ2. +Which proves the required result. +A.5.3 +Proof of Theorem 4 +Theorem 4 uses the convexity of the βD (Lemma A.3) and the second triangular relationship between +the βD and the TVD (Lemma A.8) to prove stability in the posterior predictive approximation to the +DGP of inference using the βD. +Proof. Using the convexity of the βD (Lemma A.3) and Jensen’s inequality, +D(β) +B (g1||m(β) +f (·|y1:n1)) ≤ +� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1:n1)dθ1 +Now, adding and subtracting +� +D(β) +B (g1||f(·; θ2))π(β)(θ2|y′ +1:n:2)dθ2 provides +D(β) +B (g1||m(β) +f (·|y1:n1)) ≤ +� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1:n1)dθ1 += +� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1:n1)dθ1 ++ +� +D(β) +B (g1||f(·; θ2))π(β)(θ2|y′ +1:n:2)dθ2 − +� +D(β) +B (g1||f(·; θ2))π(β)(θ2|y′ +1:n:2)dθ2 += +� +D(β) +B (g1||f(·; θ2))π(β)(θ2|y′ +1:n:2)dθ2 ++ +� � � +D(β) +B (g1||f(·; θ1)) − D(β) +B (g1||f(·; θ2)) +� +π(β)(θ1|y1:n1)dθ1π(β)(θ2|y′ +1:n:2)dθ2 +Now we can apply Lemma A.7 to random variable +� +D(β) +B (g1||f(·; θ1)) − D(β) +B (g1||f(·; θ2)) +� +∈ R on Θ×A +which by using (12) of Condition A.4 applied to the βD provides +� � � +D(β) +B (g1||f(·; θ1)) − D(β) +B (g1||f(·; θ2)) +� +π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 ≤ +1 +cS(2) . +We can now use the triangular-type relationship between the βD and the TVD (Lemma A.8) to show +52 + +that +D(β) +B (g1||m(β) +f (·|y1:n1)) ≤ +� +D(β) +B (g1||f(·; θ2))π(β)(θ2|y′ +1:n:2)dθ2 + +1 +cS(2) +≤ +� �Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2) + D(β) +B (g2||f(·; θ2)) +� +π(β)(θ2|y′ +1:n2)dθ2 + +1 +cS(2) += Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2) + +� +D(β) +B (g2||f(·; θ2))π(β)(θ2|y′ +1:n2)dθ2 ++ +1 +cS(2) + D(β) +B (g2||m(β) +f (·|y′ +1:n2)) − D(β) +B (g2||m(β) +f (·|y′ +1:n2)) +and the same arguments applying Lemma A.7 and (11) of Condition A.4 show +D(β) +B (g2||m(β) +f (·|y′ +1:n2)) ≤ +� +D(β) +B (g2||f(·; θ2))π(β)(θ2|y′ +1:n2)dθ2 +≤ +� +D(β) +B (g2||f(·; θ1))π(β)(θ1|y1:n1)dθ1 + +1 +cS(1) +≤ +� �Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2) + D(β) +B (g1||f(·; θ1)) +� +π(β)(θ1|y)dθ1 + +1 +cS(1) += Mβ−1(β + 2) +β(β − 1) +TVD(g1, g2) + +� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1:n1)dθ1 ++ +1 +cS(1) + D(β) +B (g1||m(β) +f (·|y1:n1)) − D(β) +B (g1||m(β) +f (·|y1:n1)) +Combining the above two results provides the following bound, +|D(β) +B (g1||m(β) +f (·|y1:n1)) − D(β) +B (g2||m(β) +f (·|y′ +1:n2))| ≤ Mβ−1(β + 2) +β(β − 1) +ϵ′ + 1 +c + C(β)(f, y1:n1, y′ +1:n2), +where c := min{cS(1), cS(2)} and +C(β)(f, y1:n1, y′ +1:n2) : = max +�� +D(β) +B (g1||f(·; θ1))π(β)(θ1|y1:n1)dθ1 − D(β) +B (g1||m(β) +f (·|y1:n1)), +� +D(β) +B (g2||f(·; θ2))π(β)(θ2|y′ +1:n2)dθ2 − D(β) +B (g2||m(β) +f (·|y′ +1:n2)) +� +as required. +A.5.4 +Proof of Lemma 2 +The KLD is recovered as the parameter β → 1. However, the bounds of Theorems 3 and 4 go to infinity +in this case. The proof of Lemma 2 applies a similar method to that of Theorem 4 to investigate the +stability of the KLD-Bayes estimation of a model to perturbations of the DGP +53 + +Proof. The proof of Lemma 1 established the convexity of the KLD in its second argument. Further, +by the definition of the KLD, we can see that +KLD(g1||f) = KLD(g2||f) + +� +g1 log g1 − g2 log g2dµ + +� +(g2 − g1) log fdµ. +(16) +Now we can use the convexity of the KLD and Jensen’s inequality, to show that +KLD(g1||mKLD +f +(·|y1:n1)) ≤ +� +KLD(g1||f(·; θ1))πKLD(θ|y1:n1)dθ1 +Now adding and subtracting +� +KLD(g1||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 provides +KLD(g1||mKLD +f +(·|y1:n1)) ≤ +� +KLD(g1||f(·; θ1))πKLD(θ1|y1:n1)dθ1 += +� +KLD(g1||f(·; θ1))πKLD(θ1|y1:n1)dθ1 + +� +KLD(g1||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 +− +� +KLD(g1||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 += +� +KLD(g1||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 ++ +� � +{KLD(g1||f(·; θ1)) − KLD(g1||f(·; θ2))} πKLD(θ1|y1:n1)dθ1πKLD(θ2|y′ +1:n2)dθ2 +Now we can apply Lemma A.7 to random variable {KLD(g1||f(·; θ1)) − KLD(g1||f(·; θ2))} ∈ R on Θ×A +which by using (12) of Condition A.4 applied to the KLD provides +� � +{KLD(g1||f(·; θ1)) − KLD(g1||f(·; θ2))} πKLD(θ1|y1)dθ1πKLD(θ2|y2)dθ2 ≤ +1 +cS(2) . +We can now use the triangular-type relationship of (16) to show that +KLD(g1||mKLD +f +(·|y1:n1)) ≤ +� +KLD(g1||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 + +1 +cS(2) += +� � +KLD(g2||f(·; θ2)) + +� +g1 log g1 − g2 log g2dµ + +� +(g2 − g1) log f(·; θ2)dµ +� +πKLD(θ2|y′ +1:n2)dθ2 + +1 +cS(2) += +� � +(g2 − g1) log f(·; θ2)dµπKLD(θ2|y′ +1:n2)dθ2 + +� +KLD(g2||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 ++ +� +g1 log g1 − g2 log g2dµ + +1 +cS(2) + KLD(g2||mKLD +f +(·|y′ +1:n2)) − KLD(g2||mKLD +f +(·|y′ +1:n2)). +54 + +The same arguments of Lemma A.7 and (11) of Condition A.4 show +KLD(g2||mKLD +f +(·|y′ +1:n2)) ≤ +� +KLD(g2||f(·; θ2))πKLD(θ2|y′ +1:n2)dη +≤ +� +KLD(g2||f(·; θ1))πKLD(θ1|y1:n1)dθ1 + +1 +cS(1) += +� � +KLD(g1||f(·; θ1)) + +� +g2 log g2 − g1 log g1dµ + +� +(g1 − g2) log f(·; θ1)dµ +� +πKLD(θ1|y1:n1)dθ1 + +1 +cS(1) += +� � +(g1 − g2) log f(·; θ1)dµπKLD(θ1|y1:n1)dθ1 + +� +KLD(g1||f(·; θ1))πKLD(θ1|y1:n1)dθ1 ++ +� +g2 log g2 − g1 log g1dµ + +1 +cS(1) + KLD(g1||mKLD +f +(·|y1:n1)) − KLD(g1||mKLD +f +(·|y1:n1)). +Combining the above two results provides the following bound, +|KLD(g||mKLD +f +(·|y)) − KLD(g||mKLD +h +(·|y))| ≤ CKLD(f, y1:n1, y′ +1:n2) + 1 +c + T1(g1, g2) + T2(f, y1:n1, y′ +1:n2) +where c := min{cS(1), cS(2)} and +T1(g1, g2) : = max +�� +g2 log g2 − g1 log g1dµ, +� +g1 log g1 − g2 log g2dµ +� +T2(f, y1:n1, y′ +1:n2) : = max +�� � +(g1 − g2) log f(·; θ1)dµπKLD(θ1|y1:n1)dθ1, +� � +(g2 − g1) log f(·; θ2)dµπKLD(θ2|y′ +1:n2)dθ2 +� +CKLD(f, y1:n1, y′ +1:n2) : = max +�� +KLD(g1||f(·; θ1))πKLD(θ1|y1:n1)dθ1 − KLD(g1||mKLD +f +(·|y1:n1)), +� +KLD(g2||f(·; θ2))πKLD(θ2|y′ +1:n2)dθ2 − KLD(g2||mKLD +f +(·|y′ +1:n2)) +� +as required. +B +Extended Experimental Results +Section B contains additional details of the experimental results of Section 6, including full specifica- +tions of the models and data used, as well as additional sensitivity analysis for β and a comparison +with the γ-Divergence (γD). +55 + +B.1 +Gaussian and Student’s-t likelihood +B.1.1 +Posteriors +Figure B.1 plots the posterior distribution of model parameters µ and σ2 of the Gaussian and Student’s- +t models (8) discussed in Section 6.1. +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +2 +4 +6 +8 +10 +KLD +µ +Density +-0.2 +-0.1 +0.0 +0.1 +0.2 +0 +2 +4 +6 +8 +10 +βD +µ +Density +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +KLD +σ2 +Density +0.6 +0.8 +1.0 +1.2 +1.4 +0 +2 +4 +6 +8 +βD +σ2 +Density +Figure B.1: Parameter posterior distributions for µ and σ2 under Bayes’ rule updating (KLD-Bayes) +(left) and βD-Bayes with β = 1.5 (right) under the likelihood functions f(y; θ) = N +� +y; µ, σ2 +adjσ2� +(red) and h(y; η) = tν(y; µ, σ2) (blue) where ν = 5 and σ2 +adj = 1.16. +The left-hand side of Figure B.1 demonstrates what most statistical practitioners expect when +comparing the performance of a Gaussian and a Student’s-t under outlier contamination (O’Hagan, +1979). Under the Student’s-t likelihood, the inference is much less affected by the outlying contami- +nation than under the Gaussian likelihood. The parameter µ is shifted less towards the contaminant +population and the parameter σ2 is inflated much less. In short, very different inferences are pro- +duced using a Student’s-t and a Gaussian under outlier contamination. Updating using the βD-Bayes +presents a striking juxtaposition to this. The βD-Bayes produces almost identical posteriors for both +56 + +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0 +5 +10 +15 +20 +β +Neighbourhood Multiplier +M = 5 +M = 2 +M = 1 +M = 0.5 +M = 0.25 +M = 0.1 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +0.00 +0.04 +0.08 +0.12 +β +Energy Distance +Figure B.2: Left: The multiplier Mβ−1(3β−2) +β(β−1) +from from Theorems 1 and 2 for different values of M +and 1 < β < 2. Right: The energy-distance between the posterior predictives after fitting a Gaussian +and a Student’s-t model as in Figure 1 under the βD-Bayes for different β. +µ and σ2 under both models resulting in almost identical posterior predictive densities in Figure 1. +Estimating the TVD or the βD between the two predictves distributions is hampered by the fact +that they are not available in closed form. However, the energy distance Sz´ekely and Rizzo (2013) +provides a metric that can be easily estimated from samples of the predictive. The energy distance +between the Gaussian and Student’s-t predictive distributions under traditional Bayesian updating +was 0.125, while under βD-Bayes updating the energy distance was 2.13 × 10−3. +B.1.2 +Sensitivity analysis +As was noted in Section 5, it is encouraging to note the stability of the βD-Bayes inference appears +not to be overly sensitive to the exact value of β. +Complimenting Figure 2, the left hand side of Figure B.2 plots Mβ−1(3β−2) +β(β−1) +, the multiplier of the +TVD from Theorems 1 and 2, as a function of β for various M. We see that as β increases away from 1 +this multiplier initially decreases rapidly, indicating a large increase in guaranteed stability by moving +away from the KLD. However, after this point the multiplier plateaus, indicating that a similar amount +of stability results from a range of values of β. The right-hand plot of Figure B.2 has a very similar +shape. Here we plot the energy distance (Sz´ekely and Rizzo, 2013) between the posterior predictives +of fitting a Gaussian likelihood model and Student’s-t likelihood modes, used in Figure 1 for different +values of β. Once again, we see that taking β > 1 results in a large increase in a posteriori stability +but that after a point that stability achieved is fairly constant with β. +57 + +B.1.3 +Comparison with the γ-divergence +Similarly to the βD, the γ-divergence (γD) provides a loss function that does not require an estimator +of the underlying density, and has been shown to have good robustness properties Hung et al. (2018); +Knoblauch et al. (2022). Here, we show its stability performance appears comparable with the βD. +Firstly the γD is defined as follows. +Definition B.1 (The γ-divergence (D(γ) +G ) (Fujisawa and Eguchi, 2008; Hung et al., 2018)). The γD +is defined as +D(γ) +G (g||f) = +1 +(γ − 1)γ +� +� +� +�� +gγdµ +� 1 +γ +− +� +fγ−1 +�� +fγdµ +� γ−1 +γ +gdµ +� +� +� , +where γ ∈ R \ {0, 1}. +The corresponding loss function allowing generalised Bayesian inference for θ(γ) +g +is +ℓ(γ)(y, f(·; θ)) = − +1 +γ − 1f(y; θ)γ−1 · 1 +γ +1 +�� +f(z; θ)γdz +� γ−1 +γ +. +Similarly to the βD-loss in (3), the γD-loss raises the likelihood to the power γ − 1 and ‘adjusts’ by +the integral of the likelihood to the power γ, except for the γD this ‘adjustment’ term is multiplicative +rather than additive as it was in the βD. The integral term is independent of location parameters e.g. +µ from the Gaussian and Student’s-t examples, and therefore inference for these will be very similar +under the βD and γD (Fujisawa and Eguchi, 2008). Figure B.3 shows that in the example introduced +in Section 6.1, inference for σ2 is also very similar under the γD and βD and as a result they estimate +identical posterior predictives under both the Gaussian and Student’s-t models. +B.1.4 +DLD Data +For the DLD data discussed in Section 6.1.1 we provide additional Q-Q normal and histogram plots. +These demonstrate the heavy-tailed nature of the DLD data, and the reasonable fit of the standardised +residuals produced by the βD-Bayes. +B.1.5 +TGF-β data +We consider another regression example to illustrate the stability to the selection between a Gaussian +and Student’s-t example when using βD-Bayes updating. +58 + +-5 +0 +5 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +γD +y +Density +(1 − ϵ)N(0, 1) +ϵN(5, 32) +Gaussian +Student’s-t +-0.2 +-0.1 +0.0 +0.1 +0.2 +0 +2 +4 +6 +8 +10 +γD +µ +Density +0.6 +0.8 +1.0 +1.2 +1.4 +0 +2 +4 +6 +8 +γD +σ2 +Density +Figure B.3: Posterior predictive and parameter posterior distributions for +� +µ, σ2� +under γD-Bayes +with γ = 1.5 and likelihood functions f(y; θ) = N +� +y; µ, σ2 +adjσ2� +(red) and h(y; η) = tν(y; µ, σ2) +(blue) where ν = 5 and σ2 +adj = 1.16. +59 + +KLD - Gaussian +(y − X ˆθ)/ˆσ +Density +-10 +-5 +0 +5 +10 +0.0 +0.2 +0.4 +0.6 +betaD - Gaussian +(y − X ˆθ)/ˆσ +Density +-10 +-5 +0 +5 +10 +0.0 +0.2 +0.4 +0.6 +KLD - Student’s-t +(y − X ˆθ)/ˆσ +Density +-10 +-5 +0 +5 +10 +0.0 +0.2 +0.4 +0.6 +betaD - Student’s-t +(y − X ˆθ)/ˆσ +Density +-10 +-5 +0 +5 +10 +0.0 +0.2 +0.4 +0.6 +Figure B.4: DLD data - Top: Posterior mean estimates of standardised residuals under the Gaussian +and Student’s-t model of KLD-Bayes (left) and βD-Bayes (right). Bottom left: Absolute different +in posterior mean parameter estimates under the Gaussian and Student’s-t model of KLD-Bayes and +βD-Bayes. Bottom right: Difference in posterior predictive densities of the observations under the +Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes +60 + +-3 +-2 +-1 +0 +1 +2 +3 +-4 +-2 +0 +2 +DLD - Normal Q-Q Plot +Theoretical Quantiles +Sample Quantiles +-3 +-2 +-1 +0 +1 +2 +3 +-4 +-3 +-2 +-1 +0 +1 +2 +TGFβ - Normal Q-Q Plot +Theoretical Quantiles +Sample Quantiles +Figure B.5: Q-Q normal plot of the fitted residuals according to the Gaussian model under the +KLD-Bayes for the DLD data (left) and TGF-β data (right) +The dataset from Calon et al. (2012) concerns gene expression data for n = 262 colon cancer +patients. Previous work (Rossell and Telesca, 2017; Rossell and Rubio, 2018) focused on selecting +genes that affect the expression levels of TGF-β, a gene known to play an important role in colon +cancer progression. Instead, we study the relation between TGF-β and the 7 genes (listed in Section +B.1.6) that appear in the ‘TGF-β 1 pathway’ according to the KEGGREST package in R (Tenenbaum, +2016), so that p = 8 after including the intercept. We fitted regression models using the neighbouring +models in (8) for the residuals. +Figure B.6 shows that both inference procedures are stable to the choice of the model here. The +βD-Bayes (β = 1.5) appears to be marginally more stable in estimating the fitted residuals and +predictive density (top-left and bottom-right), while the KLD-Bayes appears marginally more stable +when estimating parameters (bottom-left). Figure B.7 shows the fit of the models to the standardised +residuals under posterior mean estimates. Rossell and Rubio (2018); Jewson and Rossell (2021) find +considerable evidence that a Gaussian model is compatible with this data, further demonstrated in +Figure B.5. +B.1.6 +Variable selection +When investigating the stability of the βD-Bayes inference to the Gaussian and Student’s-t likelihoods +we regressed the DLD and TGF-β gene expressions on a subset of the variables available in the full data +sets. The procedures for which variables were selected as outlined in Sections 6.1.1 and B.1.5. To ensure +that our results are reproducible, below we indicate the selected covariates and the supplementary +61 + +-4 +-3 +-2 +-1 +0 +1 +2 +-4 +-2 +0 +2 +Gaussian - (y − X ˆθ)/ˆσ +Student’s-t - (y − X ˆθ)/ˆσ +KLD +βD +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +(y − X ˆθ) +Density +KLD - Gaussian +KLD - Student’s-t +βD - Gaussian +βD - Student’s-t +0 +1 +2 +3 +4 +5 +6 +7 +0.00 +0.04 +0.08 +Parameter index +|ˆθnorm − ˆθt| +KLD +βD +KLD +βD +-0.04 +0.00 +0.04 +Figure B.6: TGF-β data - Posterior mean estimates of standardised residuals (top left), posterior +mean estimated residuals distribution (top-right), absolute difference in posterior mean parameter +estimates (bottom left) and difference in posterior predictive densities of the observations (bottom +right) under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes (β = 1.5) for the +TGF-β data. +62 + +KLD - Gaussian +(y − X ˆθ)/ˆσ +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +betaD - Gaussian +(y − X ˆθ)/ˆσ +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +KLD - Student’s-t +(y − X ˆθ)/ˆσ +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +betaD - Student’s-t +(y − X ˆθ)/ˆσ +Density +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +Figure B.7: TGF-β data - Top: Posterior mean estimates of standardised residuals under the Gaussian +and Student’s-t model of KLD-Bayes (left) and βD-Bayes (right). Bottom left: Absolute different +in posterior mean parameter estimates under the Gaussian and Student’s-t model of KLD-Bayes and +βD-Bayes. Bottom right: Difference in posterior predictive densities of the observations under the +Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes +63 + +material contains code for these variable pre-screening steps. +DLD: +For the DLD analysis, we selected the 15 genes with the 5 highest loadings in the first 3 principal +components of the original 57 predictors. +This procedure selected the following genes C15orf52, +BRAT1, CYP26C1, SLC35B4, GRLF1, RXRA, RAB3GAP2, NOTCH2NL, SDC4, TTC22, PTCH2, +ECH1, CSF2RA, TP53AIP1, and RRP1B. +TGF-β: +For the TGF-β analysis we focused on 7 of the 10172 genes available in the data set that +appear in the ‘TGF-β 1 pathway’ according to the KEGGREST package in R (Tenenbaum, 2016). +These were the VIT, PDE4B, ATP8B1, MAGEA11, PDE6C, PDE9A, and SEPTIN4 genes. +64 + +B.2 +Mixture Models +Here we provide full details of the models and priors considered in Section 6.2. We estimated Gaussian +mixture models of the form +f(y; ω, µ, σ, K) = +K +� +k=1 +ωjN(y; µj, σj) +using Normal-Inverse-Gamma-Dirichlet priors +(ω1, . . . , ωK) ∼ Dir(α1, . . . , αK) +σ2 +k ∼ IG +�ν0 +2 , S0 +2 +� +, +k = 1, . . . , K +µk|σk ∼ N(0, √κσj), +k = 1, . . . , K +with αk = a k = 1, . . . , K, ν0 = 5, S0 = 0.2 and κ = 5.68 following the recommendations of F´uquene +et al. (2019). We elicited the parameter a to ensure that the marginal prior probability that any of +the component weights was greater than 0.05 was 0.95. +B.3 +Binary Classification +We provide additional details of the binary classification experiments in Section 6.3. +B.3.1 +t-logistic regression +Following Ding and Vishwanathan (2010); Ding et al. (2013), the t-exponential function used in the +t-logistic regression is defined as +expt(x) := +� +� +� +� +� +exp(x) +if t = 1 +max {1 + (1 − t)x, 0}1/(1−t) +otherwise +, +and Gt(Xθ) is defined as the solution of +expt(0.5Xθ − Gt(Xθ)) + expt(−0.5Xθ − Gt(Xθ)) = 1. +(17) +In general there is no closed form for Gt(Xθ) but Algorithm 1 of Ding et al. (2013) computes it +efficiently. +65 + +B.3.2 +Transformations of probit and t-logistic β’s +To minimise the a priori TVD between the probit and t-logistic alternative models and the logistic +canonical model the β’s of the alternative model are scalar multiplied. For the probit model, the +canonical parameters are multiplied by 0.5876364 and for the t-logistic the canonical parameters are +multiplied by 1.331078. +B.3.3 +Variables selected Colon Cancer data +To prepare the Colon Cancer data for our analysis we first took the natural logarithm of the gene +expression levels to remove some of their skewness. We then used the glmnet package in R Friedman +et al. (2010) to conduct LASSO variable selection using cross-validation to choose the hyperparameter. +This process left us with the intercept and genes +genes.249, genes.377, genes.493, genes.625, genes.1325, genes.1473, genes.1582, +genes.1671, genes.1772. +66 + diff --git a/Z9FST4oBgHgl3EQfBDjV/content/tmp_files/load_file.txt b/Z9FST4oBgHgl3EQfBDjV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66d20fc975a820ef6197159df8e4ef025aafad10 --- /dev/null +++ b/Z9FST4oBgHgl3EQfBDjV/content/tmp_files/load_file.txt @@ -0,0 +1,2099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf,len=2098 +page_content='On the Stability of General Bayesian Inference Jack Jewson1, Jim Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Smith2, 4, and Chris Holmes3, 4 1Barcelona School of Economics, Universitat Pompeu Fabra, Barcelona, Spain 2University of Warwick, Coventry, CV4 7AL 3University of Oxford, Oxford, OX1 3LB 4Alan Turing Institute, London, NW1 2DB jack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='jewson@upf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='edu, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='smith@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='uk, chris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='holmes@stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='uk January 2023 Abstract We study the stability of posterior predictive inferences to the specification of the likelihood model and perturbations of the data generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In modern big data analyses, the decision- maker may elicit useful broad structural judgements but a level of interpolation is required to arrive at a likelihood model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' One model, often a computationally convenient canonical form, is chosen, when many alternatives would have been equally consistent with the elicited judgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Equally, observational datasets often contain unforeseen heterogeneities and recording errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Acknowledg- ing such imprecisions, a faithful Bayesian analysis should be stable across reasonable equivalence classes for these inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We show that traditional Bayesian updating provides stability across a very strict class of likelihood models and DGPs, while a generalised Bayesian alternative using the β-divergence loss function is shown to be stable across practical and interpretable neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We illustrate this in linear regression, binary classification, and mixture modelling examples, show- ing that stable updating does not compromise the ability to learn about the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These stability results provide a compelling justification for using generalised Bayes to facilitate inference under simplified canonical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Keywords: Stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Generalised Bayes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' β-divergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Total Variation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Generalised linear models 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='13701v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='ME] 31 Jan 2023 1 Introduction Bayesian inferences are driven by the posterior distribution π(θ|y) = π(θ)f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) � π(θ)f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (1) which provides the provision to update parameter prior π(θ) using observed data y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn) ∈ Yn assumed to have been generated according to likelihood f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The quality of such posterior inference depends on the specification of the prior, likelihood, and collection of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In controlled experimental environments where time is available to carefully consider such specifications, a posterior calculated in this way might be credible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, modern applications often involve high-dimensional observational data and are undertaken by non-experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In such scenarios, it is natural to question the quality of the specification of π(θ) and f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) and the collection of y and therefore wonder to what extent posterior inference through (1) can be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Much work has previously investigated the stability of (1) to the specification of π(θ), therefore our focus here will be on f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The likelihood model captures the decision maker’s (DM’s) beliefs regarding the generation of data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, accurately formulating expert judgements as probability densities is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Even for a well trained expert, so doing requires many more probability specifications to be made at a much higher precision than is possible within the time constraints of a typical problem (Goldstein, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This is not to say that an elicited model is useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Often domain experts can reliably elicit important broad judgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, the resulting “functional” model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) generally involves some form of interpolating approximation of the DM’s “true” beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' So doing is not unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, a consequence of such expediency is that not only does the DM not believe all the judgements made by f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), its specific form is likely only one member of an equivalence class of models that also capture the DM’s elicited beliefs and could have used for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A typical example of the above is when applied practitioners deploy computationally convenient canonical models, for which there are software and illustrative examples available, to their domain specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While the broad structure of such models may be suitable across domains, it is the practitioner’s familiarly with its form, its software implementation or the platform on which it was published that motivates its use for inference, rather than a careful consideration of how it captures beliefs about the new environment Similarly, the data were not necessarily collected exactly how the DM imagined when specifying f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' There may be unforeseen heterogeneities, outliers, or recording errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Alternatively, the 2 DM may be deploying someone else’s carefully elicited model to an analogous but not necessarily exchangeable scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We therefore also consider the data generating process (DGP) that generated the DM’s data y to belong to an equivalence class of DGPs to which the DM could have deployed their inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given the inevitable lack of specificity in f and y, a faithful Bayesian analysis should be able to demonstrate that it is not overly dependent on arbitrary choices across equivalence classes of its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Such stability would allow DMs to continue using familiar models in the knowledge that their selection is not driving the critical posterior inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This paper shows that the requirement for such stability necessitates the consideration of an updating rule different from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Consider, for example, using a Gaussian distribution, N(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2) to approximate beliefs about data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While the Gaussian distribution is ubiquitous, the top of Figure 1 shows that a Student’s-t likelihood t5(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2) with 5 degrees of freedom would also have sufficed for this specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The two likelihoods appear almost indistinguishable for all values of their shared µ and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Therefore, it would be unreasonable to expect that any DM will strongly prefer one or the other of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, the bottom left of Figure 1 shows that when updating according to (1) each model can result in very different posterior inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Equally, (1) is not stable to perturbations of the data either, as a small proportion of outliers moves the posterior inferences away from the uncontaminated part of the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We demonstrate that this is a feature of the fact that implicitly (1) learns about the parameter of the model minimising the Kullback-Leibler Divergence (KLD) between the data generating process (DGP) and the model, and that stability can only be expected here when the DM is sure of the tail specification of their model and the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 for full details of this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Under traditional Bayesian updating, it is therefore left up to the DM to perform some kind of post hoc sensitivity analysis to examine the impact their chosen model and particular features of the data had on the inference (see Box, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1994, and references within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, such analyses are usually unsystematic and limited to the investigation of a small number of alternative models within the equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' An alternative, motivated by the M -open world assumption that the model is misspecified for the DGP (Bernardo and Smith, 2001), is to use general Bayes (Bissiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2016) to update beliefs about model parameters minimising a divergence different from the KLD (Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A particularly convenient alternative is the β-divergence (βD) which has previously been motivated as providing inference that is robust to outliers (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ghosh and Basu, 2016) and desirable 3 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 x Density Gaussian Student’s-t 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 x Cumulative Density 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 KLD y Density (1 − ϵ)N(0, 1) ϵN(5, 32) Gaussian Student’s-t 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 βD y Density (1 − ϵ)N(0, 1) ϵN(5, 32) Gaussian Student’s-t Figure 1: Top: Probability density function (pdf) and cumulative density function (cdf) of a Gaussian fσ2 adj(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) = N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2 adjσ2� and a Student’s-t hν(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) = tν(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2) random variable, with µ = 0, σ2 = 1, ν = 5 and σ2 adj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom: The resulting posterior predictive distributions using traditional and βD-Bayes updating on n = 1000 observations from an ϵ contamination model g(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='9 × N (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0, 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 × N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 5, 32� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' from a decision making point of view (Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In this paper, we extend the motivation for using βD-Bayes further, showing that its posterior predictive inferences are provably stable across an interpretable equivalence class of likelihood models and DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We treat stability to f and y separately, first showing that βD-Bayes inference is stable to the choice likelihood model for a given DGP, and then that inferences for a fixed model are stable to small perturbations to the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Importantly, the stability afforded to βD-Bayes inference does not compromise its ability to extract useful inferences about the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' βD-Bayes has the appealing property that if the model is correctly specified for the DGP, then the data generating parameter will be learned, and there exists a growing literature that advocates using the βD for applied analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Girardi 4 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Sugasawa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This is further demonstrated in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For example, Figure 1 shows that as well as producing similar inference for the Gaussian and Student’s-t likelihood models, the βD-Bayes inferences both capture the modal part of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, inferences must be also stable to the selection of the βD and its hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We discuss methods to select β and demonstrate reasonable insensitivity to its selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Results regarding the stability of (1) have largely focused on the parameter prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Gustafson and Wasserman (1995) proved that the total variation divergence (TVD) between two posteriors resulting from priors in linear and geometric ϵ-contamination neighbourhoods divergences as ϵ → 0 at a rate exponential in the dimension of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, Smith and Rigat (2012) showed that the TVD between two posteriors converges to 0 provided the two priors under consideration are close as measured by the local De Robertis distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Our first results provide analogies to these for the specification of the likelihood model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Gilboa and Schmeidler (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Whittle and Whittle (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Hansen and Sargent (2001a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Watson and Holmes (2016) consider the stability of optimal decision making and consider minimax decision across neighbourhoods of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, they do not consider what perturbations of the inputs of (1) would leave a DM in such a neighbourhood a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Most similar to our work is Miller and Dunson (2018), which considers Bayesian updating conditioning on data arriving within a KLD ball of the observed data and results concerning ‘global bias-robustness’ to contaminating observations, for example of the kernel-Stein discrepancy posteriors of Matsubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We consider stability to an interpretable neighbourhood of the data which as a special case contains the globally bias-robust contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bayes linear methods (Goldstein, 1999), which concern only the sub-collection of probabilities and expectations the DM considers themselves to be able to specify (Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2006), is an alternative to (1) designed to be stable to interpolating approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We prefer, however, to adopt the general Bayesian paradigm in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Firstly, the general Bayesian paradigm includes traditional Bayesian updating as a special case and produces familiar posterior and predictive distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Secondly, linear Bayes requires the elicitation of expectations and variances of unbounded quantities which are themselves unstable to small perturbations (see discussion on Goldstein and Wooff, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lastly, rather than demanding stability across an equivalence class of models, the DM could let the data guide any decision the DM themselves is not able to make using methods such as penalised likelihood approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Akaike, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1978), Bayes’ factors (Kass and Raftery, 1995) or Bayesian model averaging (Hoeting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In particular, Williamson and Goldstein (2015) 5 propose methods for combining posterior beliefs across an equivalence class of analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, such methods can be computationally burdensome to compute across even a finite class of models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rossell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2021) and the DM could reasonably only consider a handful of the models that might fit with their beliefs, a subset of the full equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The rest of the paper is organised as follows: Section 2 presents our inference paradigm, introducing general Bayesian updating (Bissiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2016), robustified inference with the βD, and defining how we will investigate posterior predictive stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Section 3 presents our theoretical contributions surrounding the stability of Bayesian analyses to the choice of the likelihood function and Section 4 presents our results on the stability of inference to perturbations of the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proofs of all of our results are deferred to the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Section 5 discusses methods to set the β hyperparameter and Section 6 illustrates the stability of the βD-Bayes inference in continuous and binary regression examples from biostatistics and a mixture modelling astrophysics example, where stability is shown not to compromise the model’s ability to learn about the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Code to reproduce all of the examples in this paper can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='com/jejewson/stabilityGBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 2 A paradigm for inference and stability 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 General Bayesian Inference Under the assumption that the model used for inference f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) does not exactly capture the DM’s beliefs, we find it appealing to adopt the general Bayesian perspective of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bissiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2016) showed that the posterior update πℓ(θ|y) = π(θ) exp (−w �n i=1 ℓ(θ, yi)) � π(θ) exp (−w �n i=1 ℓ(θ, yi)) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2) provides a coherent means to update prior beliefs about parameter θℓ g := arg minθ∈Θ � ℓ(θ, z)g(z)dz after observing data y ∼ g(·) without requiring that θ index a model for the data generating density g(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The parameter w > 0 in (2) calibrates the loss with the prior to accounts for the fact that exp(−ℓ(θ, yi)) is no longer constrained to integrate to 1, as was the likelihood in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lyddon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2018) set w to match the asymptotic information in the general Bayesian posterior to that of a sample from the ‘loss-likelihood bootstrap’, while Giummol`e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2019), building on the work of Ribatet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2012), directly calibrate the curvature of the posterior to match that of the frequentist loss 6 minimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We focus on a subset of loss functions, known as scoring rules, that depend upon the DM’s likelihood model, continuing to allow the DM to use this to encode their beliefs about the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Under the log- score, ℓ(θ, y) = − log f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) (2) collapses to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The parameter θℓ g associated with the log-score is the minimiser of the KLD between the distribution of the sample and the model (Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We therefore call updating using (1) KLD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, it is well known that minimising the log-score puts large importance on correctly capturing the tails of the data (Bernardo and Smith, 2001) and can have negative consequences for posterior decision making (Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This is demonstrated in the bottom left of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 βD-Bayes An alternative to the log-score is the β-divergence loss (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1998) ℓ(β)(y, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) = − 1 β − 1f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)β−1 + 1 β � f(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)βdz, (3) so called as arg minθ Ey∼g � ℓ(β)(y, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � = arg minθ D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) where D(β) B (g||f) is the β-divergence defined in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We refer to updating using (2) and loss (3) as βD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This was first used by Ghosh and Basu (2016) to produce a robustified Bayesian posterior (βD-Bayes) and has since been deployed for a variety of examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Sugasawa, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The implicit robustness to outliers exhibited by the βD-Bayes is illustrated in the bottom right of Figure 1, where, unlike the KLD-Bayes, the βD-Bayes continues to captures the distribution of the majority of observations under outlier contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2018) argued that updating in a manner that is automatically robust to outliers, removes the burden on the DM to specify their beliefs in a way that is robust to outliers is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The results of the coming sections provide a formal rationale for adopting this methodology to provide stability to the canonical model choice and departures from the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While Bayesian inference has been proposed minimising several alternative divergences including the Hellinger divergence, α-divergence, and the TVD (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Hooker and Vidyashankar, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch and Vomfell, 2020) such methods require a non-parametric density estimate, prohibiting their use for high-dimensional problems with continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We restrict our attention to local methods not requiring such an estimate and in particular to the βD and KLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The γ-divergence 7 (Fujisawa and Eguchi, 2008) has also been shown to produce robust inference without requiring a non-parametric density estimate (Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2022) and in general behaves very similarly, see Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Posterior Predictive Stability Our results will investigate the stability of general Bayesian posterior predictive distributions mD f (ynew|y) = � f(ynew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)πD(θ|y)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (4) for exchangeable observation ynew ∈ Y to the specification of the model f, and the DGP g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' As a result, we focus on the stability of the posterior distribution for observables y ∈ Y to perturbations of the prior for observables, f, and generating distributions for these observables g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' From a decision-making perspective, the posterior predictive is often integrated over to calculate expected utilities, and therefore stable posterior predictive distributions correspond to stable decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We consider two metrics for stability, the first is the divergence between posterior predictives, which if small, indicates that a DM with either distribution would make similar decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The second measures the difference between the posterior predictives’ divergence to the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Predictives that are close to the DGP will make close to optimal decisions and therefore, two predictives that are equally close will make similarly good decisions Predictive stability is also a more reasonable requirement than say posterior stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The param- eter posteriors for two distinct models/DGPs will generally converge in different places (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Smith, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, divergent parameter posteriors do not necessarily imply divergent posterior pre- dictives, as we show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, focusing on observables allows us to consider interesting cases of neighbouring models with nested parameter spaces (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2) 3 Stability to the specification of the likelihood function In this section we consider two potential likelihood models for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These could correspond to the DM’s true and functional beliefs, or two, equally preferable candidates for the later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In both cases, the DM would not wish their posterior inferences to diverge if one candidate was used in place of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 An interpretable neighbourhood of likelihood models We first consider the stability of inference to the specification of the DM’s likelihood model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Likelihood models f and h are considered to be in the same equivalence class of likelihood models for y ∈ Y if they satisfy Definition 1 Definition 1 (TVD neighbourhood of likelihood models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Likelihood models f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) and h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) for observable y ∈ Y are in the neighbourhood N TVD ϵ of size ϵ if ∀θ ∈ Θ, ∃η ∈ A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) ≤ ϵ and ∀η ∈ A, ∃θ ∈ Θ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) ≤ ϵ Neighbourhood N TVD ϵ demands the existence of functions If : Θ �→ A and Ih : A �→ Θ such that for all θ, TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)) is small and for all η, TVD(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)) is also small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The symmetry of Definition 1 allows Θ and A to have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For two likelihoods to be close in terms of TVD requires that the greatest difference in any of the probability statements made by the two likelihoods be small on the natural scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) := sup Y ∈Y |f(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) − h(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)| = 1 2 � |f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) − h(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)| dy (5) Additionally, TVD neighbourhoods contain ϵ-contaminations considered in the context of prior stability by Gustafson and Wasserman (1995) and often used as outlier models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Aitkin and Wilson, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' As a result, it is reasonable for a DM to be able to elicit their beliefs within a N TVD ϵ neighbourhood of their chosen model, and such a neighbourhood contains standard perturbations for sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The weak conditions required for the results of the following sections are formally stated in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Briefly, Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 requires the boundedness of the essential supremum of models f and h and the DGP g, and Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 requires sufficient concentration of posterior πD f (θ|y) around θD f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For clarity of argument, we proceed under the assumption that prior πD(θ) and πD(η) are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 The stability of the βD-Bayes In the first of our main results, Theorem 1 bounds the a posteriori divergence between the predic- tive distributions resulting from likelihood models f and h as a function of the size of the a priori neighbourhood N TVD ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 9 Theorem 1 (Stability of the posterior predictive distributions of two models under the βD-Bayes inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given 1 < β ≤ 2 and two likelihood models {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and {h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : η ∈ A} such that f, h ∈ N TVD ϵ for ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then provided there exists M < ∞ such that Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 holds, and y, π(β)(θ) and π(β)(η) satisfy Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 for D = D(β) B D(β) B (m(β) f (·|y)||m(β) h (·|y)) ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c1 + 2Mβ−1 β − 1 � TVD(g, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β) f (θ|y)dθ D(β) B (m(β) h (·|y)||m(β) f (·|y)) ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c2 + 2Mβ−1 β − 1 � TVD(g, h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β) h (η|y)dη, where c1 and c2 are defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, Theorem 2 bounds the absolute distance between the βD of the posterior predictive distributions produced from two likelihood models within N TVD ϵ from the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Theorem 2 (The stability in the posterior predictive approximation of two models to the DGP of βD-Bayes inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given 1 < β ≤ 2 and two likelihood models {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and {h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : η ∈ A} such that f, h ∈ N TVD ϵ for ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then provided there exists M < ∞ such that Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 holds and y, π(β)(θ) and π(β)(η) satisfy Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 for D = D(β) B |D(β) B (g||m(β) f (·|y)) − D(β) B (g||m(β) h (·|y))| ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c + C(β)(f, h, y), where c = min{c1, c2} as defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 and C(β)(f, h, y) : = max �� D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β) f (θ|y)dθ − D(β) B (g||m(β) f (·|y)), � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β) h (η|y)dη − D(β) B (g||m(β) h (·|y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The value M present in both Theorems 1 and 2 is often easy to bound, for example by selecting a minimum value of the scale of Gaussian or Student’s-t likelihood models, and we expect c1, c2 → ∞ as n → ∞ (see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The final term in Theorem 1 involves the TVD between the models under consideration and the unknown DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While it is difficult to say anything formal about this, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 shows that the βD can be bounded above by the TVD, and therefore any values of parameters θ and η that are close to g in TVD should have high posterior mass under the βD posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' On the other hand, C(β)(f, h, y) in Theorem 2, is is related to the concentration of the posteriors π(β) f (θ|y) and π(β) h (η|y) with Jensen’s inequality and the convexity of the βD guaranteeing that C(β)(f, h, y) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Under 10 suitable regularity conditions as n → ∞ and the posterior collapses to a point mass (Chernozhukov and Hong, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lyddon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018), then this term converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Importantly, Theorem 2 does not depend on how well specified the two likelihood models are for the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 The stability of the KLD-Bayes Figure 1 demonstrates that the stability afforded by the βD-Bayes is not afforded by the KLD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The KLD is recovered from the βD as β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, in such a scenario, the bounds proven in the previous sections tend to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Instead, Lemma 1 provides an analogous stability result for traditional Bayesian updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma 1 (The stability in the posterior predictive approximation of the DGP of KLD-Bayes infer- ence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For any two two likelihood models {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and {h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : η ∈ A}, and y, πKLD(θ) and πKLD(η) satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 for D = KLD, we have that |KLD(g||mKLD f (·|y)) − KLD(g||mKLD h (·|y))| ≤ CKLD(f, h, y) + 1 c + T(f, h, y), where c := min{c1, c2} as defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 and T(f, h, y) : = max �� � g(·) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))dµπKLD f (θ|y)dθ, � � g(·) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµπKLD h (η|y)dη � (6) CKLD(f, h, y) : = max �� KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD f (θ|y)dθ − KLD(g||mKLD f (·|y)), � KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))πKLD h (η|y)dη − KLD(g||mKLD h (·|y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We investigate T(f, h, y), the term not analagous to any of those from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Without loss of generality assume that the second term in (6) is the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then, the reverse Pinsker’s inequality (Sason and Verdu, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Binette, 2019) provides � g(·) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµ = � g(·) h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµ ≤ M∗ hKLD(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))) ≤ M∗ hKh,f TVD(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))) where M∗ h = ess sup g h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='θh) and Kh,f = � log(a) a−1 + log(b) 1−b � with a = ess inf dF dH and b = ess sup dF dH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' As a result, a TVD ball around the likelihood model is not sufficient for posterior stability when using 11 Bayes’ rule updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In fact, posterior stability can only be guaranteed according to Lemma 1 if |log(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − log(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))| (7) is small in regions where g has density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Without knowledge of g, this requires that (7) be small everywhere, requiring the DM to be confident in the accuracy of their probability statements on the log-scale rather than on the natural scale as was the case for N TVD ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Logarithms act to inflate the magnitude of small numbers and thus ensuring that |log(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − log(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))| is small requires that f and h are increasingly similar as their values decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This requires the DM to be more and more confident of the accuracy of their probability specifications as they get further and further into the tails, something that is known to already be very difficult for low dimensional problems (Winkler and Murphy, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' O’Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2006), and becomes increasingly difficult as the dimension of the observation space increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 4 Stability to the DGP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 A reasonable neighbourhood of DGP perturbations Our second series of results concern the stability of inferences from a single model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ ∈ Θ} to perturbations of the DGP for y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We consider updating on datasets y1 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn1) ∼ g1 or y2 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn2) ∼ g2 with n1, n2 > 0 and g1 and g2 satisfying Definition 2 Definition 2 (TVD Neighbourhood of data generating processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Data generating processes g1 and g2 for observable y ∈ Y are in the neighbourhood GTVD ϵ of size ϵ if TVD(g1, g2) ≤ ϵ The TVD provides a relevant and reasonable way to describe perturbations of the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' It contains ϵ-contamination neighbourhoods as considered by Matsubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2021) in the context of ‘global bias-robustness’ and also in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' It demands that the data sets were generated under mechanisms that were absolutely close on the natural scale, rather than the log-score considered in the KLD neighbourhoods on Miller and Dunson (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Conceptually, it is convenient to think about datasets such that n1 = n2 but this is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The conditions for the results of the next sections are similar to those required in Section 3 and are stated in full in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 The stability of the βD Theorem 3 bounds the βD between the posterior predictive distributions resulting from model f and data from two DGPs in the GTVD ϵ neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Theorem 3 (The stability of the posterior predictive distribution under two DGPs of the βD-Bayes in- ference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given 1 < β ≤ 2 and likelihood model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and two data sets y1 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn1) ∼ g1 and y2 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn2) ∼ g2 for n1, n2 > 0 with {g1, g2} ∈ GTVD ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then provided there exists M < ∞ such that Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 hold, Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 holds for D = D(β) B , y1, y2 and π(β)(θ) then, D(β) B (m(β) f (·|y1)||m(β) f (·|y2))) ≤2Mβ−1 β − 1 ϵ + 1 cS(1) + 2Mβ−1 β − 1 � TVD(g1, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β) f (θ1|y1)dθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' D(β) B (m(β) f (·|y2)||m(β) f (·|y1))) ≤2Mβ−1 β − 1 ϵ + 1 cS(2) + 2Mβ−1 β − 1 � TVD(g2, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β) f (θ2|y2)dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' where cS(1) and cS(2) are defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 Further, Theorem 4 bounds the difference in the βD from the DGP of the βD-Bayes posterior predictive distributions resulting from data from the two DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Theorem 4 (The stability in the posterior predictive approximation of two DGPs under the same model of βD-Bayes inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given 1 < β ≤ 2 and likelihood model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and two data sets y1 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn1) ∼ g1 and y2 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn2) ∼ g2 for n1, n2 > 0 with {g1, g2} ∈ GTVD ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then provided there exists M < ∞ such that Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 holds, and Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 holds for D = D(β) B , y1, y2 and π(β)(θ) then, |D(β) B (g1||m(β) f (·|y1)) − D(β) B (g2||m(β) f (·|y2))| ≤ Mβ−1(β + 2) β(β − 1) ϵ + 1 c + C(β)(f, y1, y2), where c := min{cS(1), cS(2)} defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 and C(β)(f, y1, y2) : = max �� D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1)dθ1 − D(β) B (g1||m(β) f (·|y1)), � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y2)dθ2 − D(β) B (g2||m(β) f (·|y2)) � Theorems 3 and 4 are the analogous result to Theorems 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The value M is still easy to bound here and the concentration terms 1 cS(j) are expected to shrink to 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For Theorem 3, we invoke Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 and argue that the βD posterior will place density on parameter 13 values of model f that are close to g in TVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The bound of Theorem 4 depends on C(β)(f, y1, y2), which under mild regularity conditions goes to 0 as n → ∞, demonstrating that the βD-Bayes is stable to TVD perturbations of the data, independently of how well the model approximates either of the DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 The stability of the KLD-Bayes Figure 1 showed that updating using (1) is not stable to perturbations of the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The data considered is within a GTVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 neighbourhood of data generated from N(0, 1) and unlike the βD-Bayes, the estimated posterior predictive is vastly different to what would have been estimated under the uncontaminated DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma 2 investigates perturbations of the DGP that traditional Bayesian inference is stable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma 2 (The stability in the posterior predictive approximation of two DGPs under the same model of KLD-Bayes inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For likelihood model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and data sets y1 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn1) ∼ g1 and y2 := (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , yn2) ∼ g2 for n1, n2 > 0, given Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 holds for D = KLD, y1, y2 and πKLD(θ), we have that |KLD(g||mKLD f (·|y)) − KLD(g||mKLD h (·|y))| ≤ CKLD(f, y1, y2) + 1 c + T1(g1, g2) + T2(f, y1, y2), where c := min{cS(1), cS(2)} as defined in Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 and T1(g1, g2) : = max �� g2 log g2 − g1 log g1dµ, � g1 log g1 − g2 log g2dµ � T2(f, y1, y2) : = max �� � (g1 − g2) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)dµπKLD(θ1|y1)dθ1, � � (g2 − g1) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)dµπKLD(θ2|y2)dθ2 � CKLD(f, y1, y2) : = max �� KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1)dθ1 − KLD(g1||mKLD f (·|y1)), � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y2)dθ2 − KLD(g2||mKLD f (·|y2)) � Lemma 2 shows that stability of the KLD approximation of DGP by model f to perturbations of the DGP requires that T1(g1, g2) and T2(f, y1, y2) are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Small T1(g1, g2) requires g1 and g2 to have similar entropy, which is not necessarily guaranteed by DGPs according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Alternatively, if | log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)| is bounded then T2(f, y1, y2) can be bounded above by TVD(g1, g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, boundedness 14 of the log-likelihood is unlikely, as f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) → 0, | log f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Therefore, T2(f, y1, y2) being small requires g1 and g2 to be increasingly close in the tails of the fitted models, prohibiting, for example, outlier contaminations such as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 5 Setting β The only additional specification required from the DM when implementing the βD-Bayes compared with the KLD-Bayes is that they select the value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This hyperparameter regulates the trade-off between robustness and efficiency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Minimising the KLD (β = 1) provides the most efficient inference but is very sensitive to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Increasing β away from 1 gains robustness to outliers at a cost to efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The bounds of the previous theorems all depend on β and we can therefore additionally interpret β as a sort of meta prior for the DM’s confidence in their elicited model or data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The less confident they are, the greater β will need to be to prevent non-negligible a posteriori divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Eliciting β as such requires the DM to reflect on the value of ϵ associated with their beliefs or the quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For the neighbourhoods of Definition 1, this can be obtained by considering for a given set of parameters what the largest possible error in any of the probability statements could be, or for Definition 2 by considering the minimal proportion of a population that they believe is consistent with the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Our results are also informative about when the value of β might be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The DM should want their βD-Bayes inferences be stable because ϵ is small, and not because the terms involving β that multiply ϵ in the theorems in Sections 3 and 4 are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Alternatively, there is increasing interest in data-driven methods to learn β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Warwick and Jones (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ghosh and Basu (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Basak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2021) consider procedures to estimate β to minimise the mean squared error (MSE) of estimated model parameters, Toma and Broniatowski (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Kang and Lee (2014) estimate β to minimise the maximum perturbation of the parameter estimates resulting from replacing one observation by the population estimated mean, and Jewson and Rossell (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Yonekura and Sugasawa (2021) estimate β to minimise the Fisher’s divergence to the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Finally, βD-Bayes inference appears not to be overly sensitive to the exact value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure 2 demonstrates that for the example introduced in Section 1, inference for the Gaussian and Student’s-t models is almost identical for values of β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 provides further demonstration of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 15 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 y Density (1 − ϵ)N(0, 1) ϵN(5, 32) Gaussian Student’s-t 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 y Density 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 y Density 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 y Density 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 y Density 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='9 y Density Figure 2: Posterior predictive distributions using βD-Bayes updating on n = 1000 observations from an ϵ-contamination model g(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='9 × N (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0, 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 × N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 5, 32� for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 6 Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Gaussian and Student’s-t likelihood We revisit the Gaussian and Student’s-t example briefly introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The likelihood models considered here are fσ2 adj(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) := N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2 × σ2 adj � and hν(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) := Student’s − tν � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (8) Hyperparameters, ν = 5 and σ2 adj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='16 are fixed to match the quartiles of the two distributions for all µ and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These were inspired by O’Hagan (2012), who argued that for absolutely continuous probability distributions, it is only reasonable to ask an expert to make a judgement about the median and the quartiles of a distribution along with maybe a few specially selected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This is justified as adequate as any two distributions with similar percentiles will look very similar, see for example Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 suggests that greater precision is required to ensure the stability of Bayes’ rule updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' On the other hand, the likelihoods in (8) are contained in N TVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We generated n = 1000 observations from the ϵ-contamination model g(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='9 × N (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0, 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 × N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 5, 32� contained within the GTVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 neighbourhood of N (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We then conducted Bayesian updating under the Gaussian and Student’s-t likelihood using both Bayes’ rule and the βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) under shared priors π(µ, σ2) = N � µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ0, v0σ2� IG(σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' a0, b0), with hyperparameters (a0 = 16 10 5 0 5 10 2 1 0 1 2 y Influence Function - µ KLD - Gaussian KLD - Student’s-t βD - Gaussian βD - Student’s-t 10 5 0 5 10 1 0 1 2 3 4 y Influence Function - σ2 Figure 3: Influence functions for parameter µ and σ2 of the Gaussian and Student’s-t likelihood models under the KLD-Bayes and βD-Bayes with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='01, b0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='01, µ0 = 0, v0 = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure 1 and Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1, which plots the parameter posterior distributions for both models under both updating mechanisms, clearly demonstrate the stability of the βD-Bayes across these two models and the lack of stability of traditional Bayesian updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Not only is the βD inference more stable across N TVD ϵ , the βD predictive better captures the majority of the DGP than either of the predictive do under traditional Bayesian updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The capturing of the N (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0, 1) mode further illustrates the βD-Bayes’ stability across neighbourhoods of the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure 3 plots influence functions (West, 1984) for the KLD-Bayes and βD-Bayes under the Gaus- sian and Student’s-t model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Influence functions are the gradient of the loss function evaluated at parameter estimates as a function of the observations and show the impact that observation had on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Under the βD-Bayes, the influence functions of the Gaussian and Student’s-t likelihoods are closer for almost every y, illustrating the stability to the model, and additionally, the influence functions for both models under the βD-Bayes vary less with y, illustrating stability to the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 DLD data We consider an RNA-sequencing data set from Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2016) measuring gene expression for n = 192 patients with different types of cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rossell and Rubio (2018) studied the impact of 57 predictors on the expression of DLD, a gene that can perform several functions such as metabolism regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' To illustrate our results, we selected the 15 variables with the 5 highest loadings in the first 3 principal components, and fitted regression models using the neighbouring models in (8) for the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 lists the selected variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 17 Figure 4 demonstrates that βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) produces more stable estimates of the fitted resid- uals (top-left), the estimated density of the residuals (top-right), parameter estimates (bottom-left), and posterior predictive density for the observed data (bottom-right) than the traditional Bayesian in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rossell and Rubio (2018) found evidence that this data is heavy-tailed, further demonstrated in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5, which caused the KLD-Bayes to estimate very different densities under the Gaussian and Student’s-t model, while the βD-Bayes is stable to this feature of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 shows the fit of the models to the posterior mean estimates of the standardised residuals, showing that as well as being stable, the βD-Bayes produces good estimation around the mode of the DLD data under both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 considers a further regression example showing that even when one of the mod- els under consideration is ‘well-specified’ for the data, the βD-Bayes inference continues to perform adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Mixture Modeling An advantage of considering the stability of the distributions for observables rather than parameters is that it allows ‘neighbouring’ models to have different dimensions to their parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For example, consider initial model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) and then ‘neighbouring’ model h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) = (1 − ω) × f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) + ω × h ′(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' κ), for η = {θ, κ, ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Here, h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) is a mixture model combining the likelihood model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), which could itself already be a mixture model, and some other density h ′(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' κ) with additional parameters κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For all θ ∈ Θ and any κ ∈ K we have that TVD(f (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) , h (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' {θ, κ, ω})) < ω and therefore a TVD neighbourhood can be defined by upper bounding ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Shapley Galaxy Dataset We examine the Shapley galaxy dataset of Drinkwater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2004), recording the velocities of 4215 galaxies in the Shapley supercluster, a large concentration of gravitationally-interacting galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The clustering tendency of galaxies continues to be a subject of interest in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Miller and Dunson (2018) investigate this data using Gaussian mixture models and use their coarsened posterior to select the number of mixture components, finding considerable instability in the number 18 4 2 0 2 4 8 6 4 2 0 2 4 6 Gaussian - (y − X ˆθ)/ˆσ Student’s-t - (y − X ˆθ)/ˆσ KLD βD 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 (y − X ˆθ) Density KLD - Gaussian KLD - Student’s-t βD - Gaussian βD - Student’s-t 0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 Parameter index |ˆθnorm − ˆθt| KLD βD KLD βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Predictive Density Difference Figure 4: Posterior mean estimates of standardised residuals (top left), posterior mean estimated residuals distribution (top-right), absolute difference in posterior mean parameter estimates (bottom left) and difference in posterior predictive densities of the observations (bottom right) under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) for the DLD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' of estimated components K under different specifications of the coarsening parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' See Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2021) for further issues with estimating the number of components in mixture models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We estimate Gaussian mixture models of the form f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) = K � k=1 ωjN(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µj, σj), under the KLD-Bayes and βD-Bayes, considering number of components K ∈ {2, 3, 4, 5, 6} and using the normal-inverse Wishart priors of F´uquene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2019) (full details available in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' βD-Bayes inference for such one-dimensional mixture models is easy to implement using adaptive quadrature to approximate the necessary integral term 1 β � h(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)βdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We do not formally place any constraint on the estimation of ωk, however, any model that estimates a component with small ωk can be seen as a neighbour of a model with one fewer component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 19 Figure 5 demonstrates the posterior mean approximation to the histogram of the data of the Gaussian mixture models under the KLD-Bayes and βD-Bayes and Table 1 records the TVD between the posterior mean predictive distribution of recursively adding components to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD- Bayes inference for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 is more stable to the addition of an extra component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In particular, for K ≥ 3 the βD-Bayes inference stably estimates the biggest components of the data centered approximately at 5, 000 and 15, 000 km/s, while the KLD-Bayes produces very different inference for these modes depending on the number of clusters selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Table 1: Total variation distances between posterior mean predictive distributions for different number of mixture components K under the KLD-Bayes and βD for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Method K = 2 vs K = 3 K = 3 vs K = 4 K = 4 vs K = 5 K = 5 vs K = 6 KLD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='03 βD (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='03 βD (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Binary Classification Binary classification models predict y ∈ {0, 1} from p-dimensional regressors X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The canonical model in such a setting is logistic regression where PLR(y = 1|X, θ) = 1 1 + exp (−Xθ), PLR(y = 0|X, θ) = 1 − PLR(Y = 1|X, θ), where θ ∈ Rp are the regression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Alternative, less ubiquitous models include, probit regression, which uses an alternative GLM link function depending on the standard Gaussian CDF Φ(·), ‘heavier tailed’ t-logistic regression (Ding and Vishwanathan, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2013) and a mixture type model that explicitly models the chance of mislabelling of the observed classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' PPR(y = 1|X, η) = Φ(wPR × Xθ), PtLR(y = 1|X, η) = expt((wtLR × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5Xθ − Gt(wtLR × Xθ))) PML(y = 1|X, η) = (1 − ν1)PLR(y = 1|X, θ) + ν0(1 − PLR(y = 1|X, θ)) where 0 < t < 2 and 0 < ν0, ν1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The so-called t-exponential ‘expt’ and Gt ensures that PtLR(y = 1|X, η) is normalised, both are defined in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Setting t > 1 results in heavier- tailed probabilities than the logistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For the probit and t-logistic models parameters θ are 20 Shapley Galaxy Velocities - KLD Velocity (1000kms/s) Density 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='10 K = 2 K = 3 K = 4 K = 5 K = 6 βD - (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25) Velocity (1000kms/s) Density 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='10 βD - (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) Velocity (1000kms/s) Density 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='10 Figure 5: Shapley Galaxy Data: Histograms of the data, in units of 1,000 km/s, excluding a small amount of data extending in a tail up to 80,000 km/s, with fitted Gaussian mixture models with K = 2−6 components under the KLD-Bayes (top), βD-Bayes with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25 (middle) and βD-Bayes with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 21 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 Xθ P(Y = 1|Xθ) logistic t-logistic probit mislabelled 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='06 Xθ TVD t-logistic probit mislabelled Figure 6: Left: P(y = 1|X, θ) for logistic, probit, t-logistic and mislabelled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Right: TVD between the logistic regression canonical model and the probit, t-logistic and mislabelled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The θ parameters of the probit and t-logistic models are scalar multiplied in a fashion that minimise the TVD to the logistic regression scalar multiples wPR, wtLR ∈ R of the logistic regression parameters θ �→ wθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These are calculated in order to minimise the a priori TVD between the models and the logistic regression baseline according to N TVD ϵ (see Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We upper bound ν0 and ν1 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='05 making ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='05 for these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure 6 plots P(y = 1|X, θ) as a function of Xθ for all four models (left) and the TVD between each alternative model and the logistic regression (right), demonstrating that all four produce very similar binary probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Colon Cancer Dataset To investigate the stability of posterior predictive inferences across the logistic, probit, t-logistic, and mislabelled binary regression models we consider the colon cancer dataset of Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The dataset contains the expression levels of 2000 genes from 40 tumours and 22 normal tissues and there is purportedly evidence that certain tissue samples may have been cross-contaminated (Tibshirani and Manning, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rather than consider the full 2000 genes we first run a frequentist LASSO procedure, estimating the hyperparameter via cross-validation, and focus our modelling only on the nine genes selected by this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We understand that such post-model selection biases parameter estimates, but the stability of the predictive inference is our focus here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure 7 compares the a posteriori TVD distance between the posterior mean estimated distribution for each observation with the a priori TVD distance between each of the models (top) and the difference 22 between the posterior mean regression parameter estimates of the two models (bottom) under the KLD- Bayes and βD-Bayes with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The stability of the βD-Bayes is once again demonstrated here, for almost every observation and every pair of models the posterior predictive inference is as stable as it was a priori, while the KLD-Bayes inference is more often divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For the t-logistic and mislabelled models the predictive stability of the βD-Bayes also provides greater stability in the posterior mean parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' KLD βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 logistic vs probit |pLR − pP R| KLD βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='05 logistic vs t-logistic |pLR − ptLR| KLD βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='08 logistic vs mislabelled |pLR − pML| 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 logistic vs probit θ-index |ˆθLR − ˆθP R| KLD βD 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 logistic vs t-logistic θ-index |ˆθLR − ˆθtLR| 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 logistic vs mislabelled θ-index |ˆθLR − ˆθML| Figure 7: Colon Cancer Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Top: TVD between the posterior mean estimated probabilities for each observation of the probit (left), t-logistic (centre) and mislabelled (right) models and the canonical logistic regression under the KLD-Bayes and βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The dotted line represented the a priori TVD distance between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom: Absolute differences between posterior mean parameter estimates and those of the logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 23 7 Discussion This paper investigated the posterior predictive stability of traditional Bayesian updating and a gen- eralised Bayesian alternative minimising the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In practice, the model used for inference is usually a convenient and canonical member of a wider class that capture the broad belief statements made by the DM and the observed data was not necessarily collected in the manner the DM imagined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We proved that βD-Bayes inference is provably stable across a class of likelihood models and data generating processes whose probability statements are absolutely close, a TVD neighbourhood, by establishing bounds on how far their predictive inferences can diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' On the other hand, our results require the DM to be sure about the tail properties of their beliefs and the DGP to guarantee stability for standard Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The results of this paper simplify the process of belief elicitation for the βD-Bayes, bounding the a posteriori consequences for a given level of a priori inaccuracy, leaving the DM free to use the best guess approximation of their beliefs that they are most comfortable with, rather than switch to a less familiar model with better outlier rejection properties (O’Hagan, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Such stability is achieved through a minimal amount of extra work compared with traditional Bayes’ rule inference, and it provides a similarly recognisable output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We hope such results help to justify the increased use of the βD to make robust inferences in statistics and machine learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A key issue motivating the departure from standard Bayesian methods here is a lack of concordance between the likelihood model and the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Such an issue can be attributed to either a failure of the modeller to think carefully enough about the DGP, or errors in data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, we treat these results separately to exemplify two different manifestations of the instability of Bayes’ rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Future work could explore the applicability of such results in multivariate settings where belief specification and data collection are harder, and further investigate our KLD-Bayes results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While we argued when you could guarantee the stability of such methods, identifying for which statements KLD-Bayes is not stable would provide important and useful results to facilitate more focused belief elicitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' To continue to facilitate the deployment of βD-Bayes methods in practice, more work is required to study and build upon existing methods to select β, particularly in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' While it is clear that considerable gains can be made over standard methods in certain scenarios, an adversarial analysis of the βD performance compared with its KLD-Bayes analogue would further motivate its wider applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 24 Acknowledgements The authors would like to thank Danny Williamson, Christian Robert, and Sebastian Vollmer for their insightful discussions on the topics in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' JJ was partially funded by the Ayudas Fundaci´on BBVA a Equipos de Investigaci´on Cientifica 2017, the Government of Spain’s Plan Nacional PGC2018- 101643-B-I00, and a Juan de la Cierva Formaci´on fellowship FJC2020-046348-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' CH was supported by the EPSRC Bayes4Health programme grant and The Alan Turing Institute, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' References Aitkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' and Wilson, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 20, Wiley New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Williamson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' and Goldstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2015), ‘Posterior belief assessment: Extracting meaningful sub- jective judgements from bayesian analyses with complex statistical models’, Bayesian Analysis 10(4), 877–908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Winkler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' and Murphy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (1968), Evaluation of subjective precipitation probability forecasts, in ‘Proceedings of the first national conference on statistical meteorology’, American Meteorological Society Boston, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 148–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Yonekura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' and Sugasawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2021), ‘Adaptation of the tuning parameter in general bayesian inference with robust divergence’, arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='06902 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Yuan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Woodcock, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Du, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Dittmar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Tsai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Kohli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Boardman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', Patel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2016), ‘Plasma extracellular rna profiles in healthy and cancer patients’, Scientific reports 6(1), 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 30 A Definitions, Conditions and Proofs Section A contains additional information for our theoretical analysis, including definitions of the Kullback-Leibler Divergence (KLD) and β-divergence (βD), full definitions of notation and technical conditions and proofs of the results of Sections 3 and 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Divergence Definitions Here we provide definitions of the KLD and βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 (The Kullback-Leibler Divergence (KLD) (Kullback and Leibler, 1951)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The KLD between probability densities g(·) and f(·) is given by KLD(g||f) = � g log g f dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 (The β-divergence (βD) (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Mihoko and Eguchi, 2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD is defined as D(β) B (g||f) = 1 β(β − 1) � gβdµ + 1 β � fβdµ − 1 β − 1 � gfβ−1dµ, where β ∈ R \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD is Bregman-divergence (Bregman, 1967) with associated function ψ(t) = 1 β(β−1)tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' When β = 1, D(1) B (g(x)||f(x)) = KLD(g(x)||f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD has often been referred to as the Density-Power Divergence in the statistics literature (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 1998) where it is often parametrised as β = βDPD + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Notation Now we define the paper’s notation in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The focus of the paper is on different densities for p- dimensional observations y ∈ Y ⊂ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Let g, g1 and g2 be potential data generating densities for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Consider likelihood models for y {f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : y ∈ Y ⊂ Rp, θ ∈ Θ} , and where appropriate potential alternative likelihood model {h(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : y ∈ Y ⊂ Rp, η ∈ A} , 31 with functions If : Θ �→ A and Ih : A �→ Θ mapping between their parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The parameter of model f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) minimising divergence D to DGP g is defined as θD g = arg min θ∈Θ D(g(·), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) = arg min θ∈Θ � ℓD(x, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))dG(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The general Bayesian posterior learning about θD g from y ∼ g given prior πD(·) is πD(θ|y) = πD(θ) exp(− �n i=1 ℓD(xi, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))) � πD(θ) exp(− �n i=1 ℓD(xi, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)))dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The posterior predictive for exchangeable observation ynew ∈ Y is mD f (ynew|y) = � f(ynew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)πD(θ|y)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Throughout this section, we will use the · notation within divergence functions to indicate the variable that is being integrated over in the divergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' the divergence does not depend on a value for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Technical Conditions The results Sections 3 and 4 require the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Stability to the likelihood model Triangle-type inequalities relating the βD to the TVD will require the bounding of the value of the density functions according to Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 (Boundedness of g, f and h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For data generating process g(·) and likelihood models {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and {h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : η ∈ A} there exists 0 < M < ∞ such that max {ess sup f, ess sup h, ess sup g} ≤ M < ∞ Given base measure µ - assumed to be the Lebesque measure for continuous random variables and the counting measures for discrete random variables - M is the essential supremum of density f(x), ess sup f(x) = M if the set defined by f−1(M, ∞) has measure 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ � f−1(M, ∞) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For discrete random variables it is always the case that M ≤ 1, M can be bounded for continuous random 32 variables such as a Gaussian or Student’s-t by lower bounding the model’s scale parameter by some reasonable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Additionally, we require the following stochastic concentration condition of the general Bayesian posterior, which we argue below will hold given sufficient regularity of the observations and the prior specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This condition was inspired by the Stochastic Lipschitz continuity assumption of Norkin (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 (Stochastic Concentration of the posterior for f and h around g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For divergence D(·| ·) and likelihood models {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} and {h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) : η ∈ A}, define the subsets of parameters S(1) d : = {θ ∈ Θ, η ∈ A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' D(g||h(·|η)) − D(g||h(·|If(θ))) ≤ d} S(2) d : = {θ ∈ Θ, η ∈ A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' D(g||f(·|θ))) − D(g||f(·|Ih(η))) ≤ d} , where S(1) d , S(2) d ⊂ Θ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then, for dataset y ∼ g(·) with n > 0 and priors πD(θ) and πD(η) there exists c1, c2 > 0 such that for all d > 0 the product posterior πD(θ, η|y) = πD(θ|y)πD(η|y) satisfies πD(S(1) d |y) ≥ 1 − exp(−c1d) (9) πD(S(2) d |y) ≥ 1 − exp(−c2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (10) Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 ensures that n is large enough and πD(θ) and πD(η) have sufficient prior mass at θD g and ηD g for the posterior based on the likelihoods f and h to have concentrated sufficiently around their optimal parameter such that the a posteriori probabilities that h(y|η) is closer to g than h(y|If(θ)) and f(y|θ) is closer to g than f(y|Ih(η)) according to divergence D are sufficiently big The asymptotic normality results of Chernozhukov and Hong (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lyddon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2018) for the general Bayesian posterior concern convergence in distribution and thus one must be slightly careful when evoking these to suggest that there must exist some n such that Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, under the assumption that both likelihood models f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) and DGP g are all absolutely continuous and provided the weak conditions for asymptotic normality Chernozhukov and Hong (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lyddon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2018) are satisfied, then π(β)(S(1) d |y) P→ 1 and π(β)(S(2) d |y) P→ 1 as n → ∞, as by definition D(g, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θD g )) ≤ D(g, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(ηD g ))) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Conditions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 (and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 below) are the only part of any of these theorems where the observed data appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' So the following theorems simply require that the Bayesian updating is being done 33 conditional on a dataset satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 or A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Extensions could look at whether Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 and the following theorems hold in expectation under the data generating process (DGP), however, this may require additional assumptions to be made about the DGP that we wish to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Stability to the DGP Conditions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 are required for the results of Section 4 and are analogous to Conditions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 (Boundedness of g1, g2 and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For data generating processes g1(·) and g2(·) and likelihood model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ} there exists 0 < M < ∞ such that max {ess sup f, ess sup g1, ess sup g2} ≤ M < ∞ Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 (Stochastic Concentration of the posterior for f around g1 and g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For divergence D(·||·) and likelihood model {f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) : θ ∈ Θ}, define subsets of Θ × Θ S(1) d : = {θ1, θ2 ∈ Θ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' D(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) ≤ d} S(2) d : = {θ1, θ2 ∈ Θ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' D(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − D(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) ≤ d} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then, for datasets y1:n1 ∼ g1(·) and y′ 1:n2 ∼ g2(·) with n1, n2 > 0 and prior πD(θ) there exists cS(1), cS(2) > 0 such that for all d > 0 the product posterior πD(θ1, θ2|y1, y2) = πD(θ1|y1)πD(θ2|y2) satisfies πD(S(1) d |y1, y2) ≥ 1 − exp(−cS(1)d), (11) πD(S(2) d |y1, y2) ≥ 1 − exp(−cS(2)d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (12) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 Proofs: Stability to the Model Before we prove Theorem 1, Theorem 2 and Lemma 1 we first introduce some useful Lemmas that simplify their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 34 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Useful Lemmas for proving Theorems 1 and 2 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 establishes a convenient representation for the TVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 (A simplification of the TVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The following relationship holds for the TVD between two densities f and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' TVD(f, h) = � A+ (h(y) − f(y)) dy = � A− (f(y) − h(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' where A+ := {y : h(y) > f(y)} and A− := {y : f(y) > h(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Firstly, by definition TVD(f, h) = 1 2 � |h(y) − f(y)| dy = 1 2 � A+ (h(y) − f(y)) dy + 1 2 � A− (f(y) − h(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Next, consider Lf,h : Y → R with Lf,h(y) := min(f(y), h(y)) as the lower of the two probability densities for every y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given that both f and h are probability densities and thus integrate to 1 we have that � A+(h(y) − f(y))dy = 1 − � Lf,h(y)dy � A−(f(y) − h(y))dy = 1 − � Lf,h(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The two right-hand sides are identical and therefore the two left-hand sides must be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' As a result, TVD(f, h) = 1 2 � A+ (h(y) − f(y)) dy + 1 2 � A− (f(y) − h(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' = � A+ (h(y) − f(y)) dy = � A− (f(y) − h(y)) dy, proving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 establishes a triangle-type inequality relating the βD and the TVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Triangle-type inequalities fit naturally with Section 3’s requirements for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If two models are close, then they ought to provide similar approximations to a third distribution, the DGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD does not strictly satisfy the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, we can prove the following results connecting the TVD and the βD in a triangle-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The result relies on 1 ≤ β ≤ 2, which places the βD in between 35 the KLD at β = 1 and the L2-distance D(2) B (g||f) = 1 2 � (f − g)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We are yet to come across scenarios where setting β outside this range is appropriate from a practical viewpoint (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Jewson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 (A triangle inequality relating the βD and the TVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For densities f, h and g with the property that there exists M < ∞ satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and 1 < β ≤ 2 we have that ��D(β) B (g||h) − D(β) B (g||f) �� ≤ Mβ−1(3β − 2) β(β − 1) TVD(h, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' By the definition of the βD, we can rearrange D(β) B (g||h) =D(β) B (g||f) + �� � 1 β h(y)β − 1 β f(y)β − 1 β − 1g(y)h(y)β−1 + 1 β − 1g(y)f(x)β−1 � dy � =D(β) B (g||f) + � 1 β � � h(y)β − f(y)β� dy + 1 β − 1 � g(y) � f(y)β−1 − h(y)β−1� dy � As in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1, define A+ := {y : h(y) > f(y)} and A− := {y : f(y) > h(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now by the mono- tonicity of the function yβ and yβ−1 when 1 ≤ β ≤ 2 we have that � A− h(y)β − f(y)βdy < 0 � A+ g(y) � f(y)β−1 − h(y)β−1� dy < 0 therefore removing these two terms provides an upper bound D(β) B (g||h) =D(β) B (g||f) + 1 β � � h(y)β − f(y)β� dy + 1 β − 1 � g(y) � f(y)β−1 − h(y)β−1� dy ≤D(β) B (g||f) + 1 β � A+ � h(y)β − f(y)β� dy + 1 β − 1 � A− g(y) � f(y)β−1 − h(y)β−1� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 36 Next, adding and subtracting 1 βh(y)f(y)β−1 provides D(β) B (g||h) ≤D(β) B (g||f) + 1 β � A+ � h(y)β − f(y)β� dy + 1 β − 1 � A− g(y) � f(y)β−1 − h(y)β−1� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' =D(β) B (g||f) + 1 β � A+ � h(y)β − h(y)f(y)β−1 + h(y)f(y)β−1 − f(y)β� dy + 1 β − 1 � A− g(y) � f(y)β−1 − h(y)β−1� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' =D(β) B (g||f) + 1 β � A+ h(y) � h(y)β−1 − f(y)β−1� dy + 1 β � A+ f(y)β−1 (h(y) − f(y)) dy + 1 β − 1 � A− g(y) � f(y)β−1 − h(y)β−1� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' =D(β) B (g||f) + 1 β � A+ h(y)β � 1 − f(y)β−1 h(y)β−1 � dy + 1 β � A+ f(y)β−1 (h(y) − f(y)) dy + 1 β − 1 � A− g(y)f(y)β−1 � 1 − h(y)β−1 f(y)β−1 � dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now on A+ h(y) > f(y) and so � f(y) h(y) �β−1 > f(y) h(y) for 1 ≤ β ≤ 2 so � 1 − f(y)β−1 h(y)β−1 � ≤ � 1 − f(y) h(y) � with the exact same logic holding in reverse on A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can use this to show that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g||h) ≤ D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g||f) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ h(y)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − f(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='h(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (h(y) − f(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g(y)f(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − h(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='f(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='≤D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g||f) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ h(y)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − f(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='h(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (h(y) − f(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g(y)f(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − h(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='f(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='=D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g||f) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ h(y)β−1 (h(y) − f(y)) dy + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (h(y) − f(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g(y)f(y)β−2 (f(y) − h(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We now use the fact that we defined max {ess sup f, ess sup h, ess sup g} ≤ M < ∞ and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 to 37 leave D(β) B (g||h) = D(β) B (g||f) + 1 β � A+ h(y)β−1 (h(y) − f(y)) dy + 1 β � A+ f(y)β−1 (h(y) − f(y)) dy + 1 β − 1 � A− g(y)f(y)β−2 (f(y) − h(y)) dy ≤D(β) B (g||f) + Mβ−1 β � A+ (h(y) − f(y)) dy + Mβ−1 β � A+ (h(y) − f(y)) dy + Mβ−1 β − 1 � A− (f(y) − h(y)) dy =D(β) B (g||f) + 2Mβ−1 β TVD(h, f) + Mβ−1 β − 1 TVD(h, f) =D(β) B (g||f) + Mβ−1(3β − 2) β(β − 1) TVD(h, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 proves the convexity of the D(β) B (g, f) in both g and f Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 (The convexity of the βD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD between two densities g(y) and f(y) is convex in both densities for 1 < β ≤ 2, when fixing the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' That is to say that for λ ∈ [0, 1] D(β) B (λg1 + (1 − λ)g2, f) ≤ λD(β) B (g1, f) + (1 − λ)D(β) B (g2, f) for all f D(β) B (g, λf1 + (1 − λ)f2) ≤ λD(β) B (g, f1) + (1 − λ)D(β) B (g, f2) for all g for 1 < β ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' First, we fix f and look at convexity in the function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' let λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The function xp for x ≥ 0 and p > 1 is convex and thus satisfies (λx1 + (1 − λ)x2)p ≤ λxp 1 + (1 − λ)xp 2 therefore we have that provided D(β) B (g1||f) < ∞ and D(β) B (g2||f) < ∞ D(β) B (λg1 + (1 − λ)g2||f) = � 1 β(β − 1) (λg1 + (1 − λ)g2)β + 1 β fβ − 1 β − 1 (λg1 + (1 − λ)g2) fβ−1dµ ≤ � 1 β(β − 1) � λgβ 1 + (1 − λ)gβ 2 � + 1 β fβ − 1 β − 1 (λg1 + (1 − λ)g2) fβ−1dµ =λD(β) B (g1||f) + (1 − λ)D(β) B (g2||f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 38 Next, we fix g and look at the convexity in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Similarly to above we know that when x ≥ 0 and 1 ≤ p ≤ 2 that 1 pxp and − 1 p−1xp−1 are both convex in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We therefore have that provided D(β) B (g||f1) < ∞ and D(β) B (g||f2) < ∞ D(β) B (g||λf1 + (1 − λ)f2) = � 1 β(β − 1)gβ + 1 β (λf1 + (1 − λ)f2)β − 1 β − 1g (λf1 + (1 − λ)f2)β−1 dµ ≤ � 1 β(β − 1)gβ + 1 β � λfβ 1 + (1 − λ)fβ 2 � − 1 β − 1g � λfβ−1 1 + (1 − λ)fβ−1 2 � dµ =λD(β) B (g||f1) + (1 − λ)D(β) B (g||f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 introduces a useful the “three-point property” (Cichocki and Amari, 2010) associated with the βD Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 (Three-point property of the βD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The following relationship for the βD holds for densities g, f and h D(β) B (f||h) = D(β) B (g||h) − D(β) B (g||f) + R(g||f||h) where R(g||f||h) = 1 β − 1 � (g − f) � hβ−1 − fβ−1� dµ (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Following the definition of the βD (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2) D(β) B (g||f) + D(β) B (f||h) = � 1 β(β − 1)gβ + 1 β fβ − 1 β − 1gfβ−1dµ + � 1 β(β − 1)fβ + 1 β hβ − 1 β − 1fhβ−1dµ = � 1 β(β − 1)gβ + 1 β hβ + 1 β − 1fβ − 1 β − 1gfβ−1 − 1 β − 1fhβ−1dµ = � 1 β(β − 1)gβ + 1 β hβ − 1 β − 1ghβ−1 + 1 β − 1ghβ−1 + 1 β − 1ffβ−1 − 1 β − 1gfβ−1dµ − 1 β − 1fhβ−1dµ =D(β) B (g||h) + 1 β − 1 � (g − f) � hβ−1 − fβ−1� dµ 39 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 provides a useful bound for the interpretation of the remainder term from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 (A bound on R(g||f||h) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For densities f, h and g with the property that there exists M < ∞ satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and 1 < β ≤ 2, the remainder term from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 can be bounded as R(g||f||h) ≤ 2Mβ−1 β − 1 TVD(g, f) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Define A+ f := {y : g(y) ≥ f(y))} and A− f := {y : g(y) ≤ f(y))} as R(g||f||h) = 1 β − 1 � (g − f) � hβ−1 − fβ−1� dµ = 1 β − 1 � hβ−1 (g − f) dµ + 1 β − 1 � fβ−1 (f − g) dµ ≤ 1 β − 1 � A+ hβ−1 (g − f) dµ + 1 β − 1 � A− fβ−1 (f − g) dµ ≤ Mβ−1 β − 1 � A+ (g − f) dµ + Mβ−1 β − 1 � A− (f − g) dµ ≤ 2Mβ−1 β − 1 TVD(g, f) by Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 shows that the βD can be bounded above by the TVD, which is useful when interpreting the bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For densities f, h and g with the property that there exists M < ∞ satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and 1 < β ≤ 2 we have that D(β) B (g||f) ≤ �Mβ−1 β − 1 � TVD(g, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Firstly, define A− = {y : g(y) < f(y)} and A+ = {y : g(y) ≥ f(y)} and note on A+ that (f(y)− g(y)) < 0 and on A− that g(y) < f(y) ⇒ gβ−1(y) < fβ−1(y) for 1 ≤ β ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD can, then, be 40 rearranged as D(β) B (g||f) = 1 β � f(y)β−1 (f(y) − g(y)) dy + 1 β(β − 1) � � g(y)β−1 − f(y)β−1� g(y)dy ≤ 1 β � A− f(y)β−1 (f(y) − g(y)) dy + 1 β(β − 1) � A+ � g(y)β−1 − f(y)β−1� g(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Since f ≤ M by Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 we can write � A− f(y)β−1 (f(y) − g(y)) dy ≤ Mβ−1 � A− (f(y) − g(y)) dy = Mβ−1TVD(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, on A+ we have that g(y) > f(y) which implies that f(y) g(y) < 1 and that when 1 < β < 2, f(y) g(y) β−1 > f(y) g(y) so � A+ � g(y)β−1 − f(y)β−1� g(y)dy = � A+ g(y)β−1 � 1 − �f(y) g(y) �β−1� g(y)dy ≤ � A+ g(y)β−1 � 1 − f(y) g(y) � g(y)dy = � A+ g(y)β−1 (g(y) − f(y)) dy ≤Mβ−1TVD(f, g), since g ≤ M by Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Combining the two bounds leaves D(β) B (g||f) ≤ Mβ−1 β TVD(f, g) + Mβ−1 β(β − 1) TVD(f, g), which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The implications of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Provided � Mβ−1 β−1 � does not get too small, we can be confident that any value of θ such that f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) is close to the data generating density g in terms of TVD, will receive high posterior mass under an update targeting the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lastly, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 provides a convenient result for the deployment of Conditions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 (and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This result was inspired by part of the proof of Theorem 7 of Dimitrakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2017) (p27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 (Stochastic Concentration (Norkin, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Dimitrakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If random variable ω ∈ Ω ⊂ R is distributed according to π and there exists c > 0 such that for all t > t0 Fω(t) = π({ω ≤ t}) ≥ 1 − exp(−c(t − t0)), (14) 41 then � ωπ(ω)dω ≤ t0 + 1 c Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can write the expectation of ω in terms of its cumulative distribution function (CDF) as � ωπ(ω)dω = � ∞ 0 (1 − Fω(t))dt − � 0 −∞ Fω(t)dt ≤ � ∞ 0 (1 − Fω(t))dt = � t0 0 (1 − Fω(t))dt + � ∞ t0 (1 − Fω(t))dt ≤ � t0 0 1dt + � ∞ t0 (1 − Fω(t))dt = t0 + � ∞ t0 (1 − Fω(t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Then, invoking (14) leaves � ωπ(ω)dω ≤ t0 + � ∞ t0 (1 − Fω(t))dt ≤ t0 + � ∞ t0 exp(−c(t − t0))dt ≤ t0 + � ∞ 0 exp(−ct)dt = t0 + 1 c as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Proof of Theorem 1 We are now able to use the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3), the triangular relationship between the βD and the TVD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2) and the three-point property the βD (Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) to prove Theorem 1 which establishes the posterior predictive stability to the likelihood model’s specification provided by inference using the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' By the convexity of the βD for 1 < β ≤ 2 (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) we can apply Jensen’s inequality to 42 show that D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � D(β) B (m(β) f (·|y)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη ≤ � �� D(β) B (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθ � π(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now the three-point property associated with the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4) gives us that D(β) B (f||h) = D(β) B (g||h) − D(β) B (g||f) + R(g||f||h) where R(g||f||h) is defined in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Using this here provides D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � �� D(β) B (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθ � π(β)(η|y)dη = � �� � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) +R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)] π(β)(θ|y)dθ � π(β)(η|y)dη = � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη − � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now adding and subtracting � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))π(β)(θ|y)dθ we have D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη − � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))π(β)(θ|y)dθ − � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))π(β)(θ|y)dθ + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' = � � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � π(β)(θ|y)dθ + � � � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) � π(β)(θ|y)dθπ(β)(η|y)dη + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) � ∈ R on Θ×A which by using (9) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 applied to the βD provides � � � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) − D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) � π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 43 As a result D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � π(β)(θ|y)dθ + 1 c1 + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can now apply the triangle type inequality from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2, D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � π(β)(θ|y)dθ + 1 c1 + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' ≤ � Mβ−1(3β − 2) β(β − 1) TVD(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + 1 c1 + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Given the neighbourhood of likelihood models N TVD ϵ we can then write D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ � Mβ−1(3β − 2) β(β − 1) TVD(h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)), f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + 1 c1 + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' ≤Mβ−1(3β − 2) β(β − 1) ϵ + 1 c1 + � � R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(θ|y)dθπ(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 we have that R(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) ≤ 2Mβ−1 β−1 TVD(g, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) and as a result we can bound D(β) B (m(β) f (·|y)||m(β) h (·|y))) ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c1 + 2Mβ−1 β − 1 � TVD(g, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This provides the first part of the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We note that we could have instead considered D(β) B (m(β) h (·|y)||m(β) f (·|y))), applied the corresponding version of the three-point property of Bregman divergences, with remainder R(g||h||f) = � (g − h) � 1 β−1fβ−1 − 1 β−1hβ−1� dµ, used Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 with (10) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5, therefore we also have that D(β) B (m(β) h (·|y)||m(β) f (·|y))) ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c2 + 2Mβ−1 β − 1 � TVD(g, h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' providing the second part of the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 44 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Proof of Theorem 2 Theorem 2 uses the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) and the triangular relationship between the βD and the TVD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2) to prove stability in the posterior predictive approximation to the DGP of inference using the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Using the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) and Jensen’s inequality, D(β) B (g||m(β) f (·|y)) ≤ � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ Adding and subtracting � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))π(β)(η|y)dη we have that D(β) B (g||m(β) f (·|y)) ≤ � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ = � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))π(β)(η|y)dη − � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))π(β)(η|y)dη = � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))π(β)(η|y)dη + � � � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))) � π(β)(θ|y)dθπ(β)(η|y)dη Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))) � ∈ R on Θ×A which by using (10) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 applied to the βD provides � � � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))) � π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We then use the the triangular-type relationship between the βD and the TVD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2) to show that D(β) B (g||m(β) f (·|y)) ≤ � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ ≤ � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))π(β)(η|y)dη + 1 c2 ≤ � �Mβ−1(3β − 2) β(β − 1) TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) + D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) � π(β)(η|y)dη + 1 c2 = � Mβ−1(3β − 2) β(β − 1) TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη + � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη + 1 c2 + D(β) B (g||m(β) h (·|y)) − D(β) B (g||m(β)(·|y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 45 The same arguments this time using (9) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 can also be used to show that D(β) B (g||m(β) h (·|y)) ≤ � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β) h (η|y)dη ≤ � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))π(β)(θ|y)dθ + 1 c1 ≤ � �Mβ−1(3β − 2) β(β − 1) TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))) + D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � π(β)(θ|y)dθ + 1 c1 = � Mβ−1(3β − 2) β(β − 1) TVD(f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ), h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))π(β)(θ|y)dθ + � D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ + 1 c1 + D(β) B (g||m(β) f (·|y)) − D(β) B (g||m(β) f (·|y)) Combining the above two results provides the following bound, |D(β) B (g||m(β) f (·|y)) − D(β) B (g||m(β) h (·|y))| ≤ Mβ−1(3β − 2) β(β − 1) ϵ + 1 c + C(β)(f, h, y), where c = min{c1, c2} and C(β)(f, h, y) : = max �� D(β) B (g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))π(β)(θ|y)dθ − D(β) B (g||m(β)(·|y)), � D(β) B (g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))π(β)(η|y)dη − D(β) B (g||m(β)(·|y)) � as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 Proof of Lemma 1 The KLD is recovered as the parameter β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, the bounds of Theorems 1 and 2 go to infinity in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The proof of Lemma 1 applies a similar method to that of Theorem 2 to investigate the stability of KLD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Firstly, the logarithm is a concave function and therefore the negative logarithm is a convex function which is sufficient to prove the convexity of KLD in its second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, by the definition of the KLD, we can see that KLD(g||f) = KLD(g||h) + � g log h f dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (15) Now we can use the convexity of the KLD and Jensen’s inequality, to show that KLD(g||mKLD f (·|y)) ≤ � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD(θ|y)dθ 46 Now adding and subtracting � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))πKLD(η|y)dη provides KLD(g||mKLD f (·|y)) ≤ � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD(θ|y)dθ = � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD(θ|y)dθ + � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))πKLD(η|y)dη − � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))πKLD(η|y)dη = � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))πKLD(η|y)dη + � � {KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))} πKLD(θ|y)dθπKLD(η|y)dη Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable {KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))} ∈ R on Θ × A which by using (10) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 applied to the KLD provides � � {KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) − KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))} π(β)(θ|y)dθπ(β)(η|y)dη ≤ 1 c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can now use the triangular-type relationship of (15) to show that KLD(g||mKLD f (·|y)) ≤ � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η)))πKLD(η|y)dη + 1 c2 = � �� g(·) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµ + KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η)) � πKLD(η|y)dθ + 1 c2 = � � g(·) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµπKLD(η|y)dη + � KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))πKLD(η|y)dη + 1 c2 + KLD(g||mKLD h (·|y)) − KLD(g||mKLD h (·|y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The same arguments of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 and (9) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 show KLD(g||mKLD h (·|y)) ≤ � KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))πKLD(η|y)dη ≤ � KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ)))πKLD(θ|y)dθ + 1 c1 = � �� g(·) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))dµ + KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) � πKLD(θ|y)dθ + 1 c1 = � � g(·) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))dµπKLD(θ|y)dθ + � KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD(θ|y)dθ + 1 c1 + KLD(g||mKLD(·|y)) − KLD(g||mKLD(·|y)) Combining the above two results provides the following bound, |KLD(g||mKLD f (·|y)) − KLD(g||mKLD h (·|y))| ≤ CKLD(f, h, y) + 1 c + T(f, h, y) 47 where c := min{c1, c2} and T(f, h, y) : = max �� � g(·) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' If(θ))dµπKLD(θ|y)dθ, � � g(·) log h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ih(η))dµπKLD(η|y)dη � CKLD(f, h, y) : = max �� KLD(g||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ))πKLD(θ|y)dθ − KLD(g||mKLD(·|y)), � KLD(g||h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η))πKLD(η|y)dη − KLD(g||mKLD h (·|y)) � as required A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 Proofs: Stability to the DGP A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 A useful Lemma for proving Theorems 3 and 4 In order to prove Theorems 3 and 4, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 provides a second triangle-type inequality for the βD and TVD in the case where one model is estimated under two DGPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 (Another triangle inequality relating the βD and the TVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For densities f, g1 and g2 with the property that there exists M < ∞ satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 and 1 < β ≤ 2 we have that ��D(β) B (g1||f) − D(β) B (g2||f) �� ≤ Mβ−1(β + 2) β(β − 1) TVD(g1, g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' By the definition of the βD, we can rearrange D(β) B (g1||f) = D(β) B (g2||f)+ �� � 1 β(β − 1)g1(y)β − 1 β(β − 1)g2(y)β + 1 β − 1g2(y)f(y)β−1 − 1 β − 1g1(y)f(y)β−1 � dy � =D(β) B (g2||f) + � 1 β(β − 1) � � g1(y)β − g2(y)β� dy + 1 β − 1 � f(y)β−1 (g2(y) − g1(y)) dy � As in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1, define A+ := {y : g2(y) > g1(y)} and A− := {y : g1(y) > g2(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' By the mono- tonicity of the function yβ when 1 ≤ β ≤ 2 we have that � A+ g1(y)β − g2(y)βdy < 0 � A− f(x)β−1 (g2(y) − g1(y)) dy < 0 48 therefore removing these two terms provides an upper bound D(β) B (g1||f) =D(β) B (g2||f) + � 1 β(β − 1) � � g1(y)β − g2(y)β� dy + 1 β − 1 � f(x)β−1 (g2(y) − g1(y)) dy � ≤D(β) B (g2||f) + 1 β(β − 1) � A− � g1(y)β − g2(y)β� dy + 1 β − 1 � A+ f(y)β−1 (g2(y) − g1(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='Now adding and subtracting g1(y)g2(y)β−1 provides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g1||f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='≤D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y)β − g2(y)β� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='=D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y)β − g1(y)g2(y)β−1 + g1(y)g2(y)β−1 − g2(y)β� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='=D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g1(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y)β−1 − g2(y)β−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g2(y)β−1 (g1(y) − g2(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='=D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g1(y)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − g2(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g2(y)β−1 (g1(y) − g2(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Next, on A− g1(x) > g2(x) which implies that � g2(x) g1(x) �β−1 > g2(x) g1(x) for 1 ≤ β ≤ 2 and � 1 − g2(x)β−1 g1(x)β−1 � ≤ � 1 − g2(x) g1(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='We can use this to show that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g1||f) ≤ D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g1(y)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − g2(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y)β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g2(y)β−1 (g1(y) − g2(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='≤D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g1(y)β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 − g2(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g1(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g2(y)β−1 (g1(y) − g2(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='=D(β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='B (g2||f) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g1(y)β−1 (g1(y) − g2(y)) dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β(β − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A− g2(y)β−1 (g1(y) − g2(y)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='A+ f(y)β−1 (g2(y) − g1(y)) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 49 We now use the fact that we defined max {ess sup f, ess sup g1, ess sup g2} ≤ M < ∞ and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 to leave D(β) B (g1||f) = D(β) B (g2||f) + 1 β(β − 1) � A− g1(y)β−1 (g1(y) − g2(y)) dy + 1 β(β − 1) � A− g2(y)β−1 (g1(y) − g2(y)) dy + 1 β − 1 � A+ f(y)β−1 (g2(y) − g1(y)) dy ≤D(β) B (g2||f) + Mβ−1 β(β − 1) � A− (g1(y) − g2(y)) dy + Mβ−1 β(β − 1) � A− (g1(y) − g2(y)) dy + Mβ−1 β − 1 � A+ (g2(y) − g1(y)) dy =D(β) B (g2||f) + 2 Mβ−1 β(β − 1) TVD(g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' g2) + Mβ−1 β − 1 TVD(g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' g2) =D(β) B (g2||f) + Mβ−1(β + 2) β(β − 1) TVD(g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' g2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' providing the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Proof of Theorem 3 Similarly to Theorem 1, we use the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3)and the three-point property the βD (Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) to prove Theorem 3 which establishes the posterior predictive stability to perturbations of the DGP provided by inference using the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 is the important lemma for this proof rather than the triangle inequality Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' By the convexity of the βD for 1 < β ≤ 2 (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) we can apply Jensen’s inequality to show that D(β) B (m(β) f (·|y1)||m(β) f (·|y2))) ≤ � D(β) B (m(β) f (·|y1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y2)dθ2 ≤ � �� D(β) B (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ1|y1)dθ1 � π(β)(θ2|y2)dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now the three-point property associated with the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4) gives us that D(β) B (f1||f2) = D(β) B (g2||f2) − D(β) B (g2||f1) + R(g2||f1||f2) 50 where R(g||f||h) is defined in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Using this here provides D(β) B (m(β) f (·|y1)||m(β) f (·|y2))) ≤ � �� D(β) B (f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ1|y1)dθ1 � π(β)(θ2|y2)dθ2 = � �� � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) +R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) π(β)(θ1|y1)dθ1 � π(β)(θ2|y2)dθ2 = � � � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) � π(β)(θ1|y1)dθ2π(β)(θ2|y2)dθ1 + � � R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) � ∈ R on Θ×A which by using (11) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 applied to the βD provides � � � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) � π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 ≤ 1 cS(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Therefore D(β) B (m(β) f (·|y1)||m(β) f (·|y2))) ≤ � � � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) − D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) � π(β)(θ1|y1)dθ2π(β)(θ2|y2)dθ1 + � � R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 ≤ 1 cS(1) + � � R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Now from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 we have that R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) ≤ 2Mβ−1 β−1 TVD(g2, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) which is itself not necessarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can however apply the triangle inequality to the TVD and say that R(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) ≤ 2Mβ−1 β − 1 TVD(g2, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) ≤ 2Mβ−1 β − 1 (TVD(g1, g2) + TVD(g1, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))) ≤ 2Mβ−1 β − 1 (ϵ + TVD(g1, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))) , given the neighbourhood of data generating processes defined by GTVD ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' As a result D(β) B (m(β) f (·|y1)||m(β) f (·|y2))) ≤2Mβ−1 β − 1 ϵ + 1 cS(1) + 2Mβ−1 β − 1 � TVD(g1, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1)dθ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 51 We note that we could have instead considered D(β) B (m(β) f (·|y2)||m(β) f (·|y1))), applied the corre- sponding version of the three-point property of βD, used Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7, (12) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 applied to the βD and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 to also show that D(β) B (m(β) f (·|y2)||m(β) f (·|y1))) ≤ 2Mβ−1 β − 1 ϵ + 1 cS(2) + 2Mβ−1 β − 1 � TVD(g2, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β) f (θ2|y2)dθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Which proves the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Proof of Theorem 4 Theorem 4 uses the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) and the second triangular relationship between the βD and the TVD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8) to prove stability in the posterior predictive approximation to the DGP of inference using the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Using the convexity of the βD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3) and Jensen’s inequality, D(β) B (g1||m(β) f (·|y1:n1)) ≤ � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 Now, adding and subtracting � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n:2)dθ2 provides D(β) B (g1||m(β) f (·|y1:n1)) ≤ � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 = � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 + � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n:2)dθ2 − � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n:2)dθ2 = � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n:2)dθ2 + � � � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) � π(β)(θ1|y1:n1)dθ1π(β)(θ2|y′ 1:n:2)dθ2 Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) � ∈ R on Θ×A which by using (12) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 applied to the βD provides � � � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) � π(β)(θ1|y1)dθ1π(β)(θ2|y2)dθ2 ≤ 1 cS(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can now use the triangular-type relationship between the βD and the TVD (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8) to show 52 that D(β) B (g1||m(β) f (·|y1:n1)) ≤ � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n:2)dθ2 + 1 cS(2) ≤ � �Mβ−1(β + 2) β(β − 1) TVD(g1, g2) + D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) � π(β)(θ2|y′ 1:n2)dθ2 + 1 cS(2) = Mβ−1(β + 2) β(β − 1) TVD(g1, g2) + � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n2)dθ2 + 1 cS(2) + D(β) B (g2||m(β) f (·|y′ 1:n2)) − D(β) B (g2||m(β) f (·|y′ 1:n2)) and the same arguments applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 and (11) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 show D(β) B (g2||m(β) f (·|y′ 1:n2)) ≤ � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n2)dθ2 ≤ � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 + 1 cS(1) ≤ � �Mβ−1(β + 2) β(β − 1) TVD(g1, g2) + D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) � π(β)(θ1|y)dθ1 + 1 cS(1) = Mβ−1(β + 2) β(β − 1) TVD(g1, g2) + � D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 + 1 cS(1) + D(β) B (g1||m(β) f (·|y1:n1)) − D(β) B (g1||m(β) f (·|y1:n1)) Combining the above two results provides the following bound, |D(β) B (g1||m(β) f (·|y1:n1)) − D(β) B (g2||m(β) f (·|y′ 1:n2))| ≤ Mβ−1(β + 2) β(β − 1) ϵ′ + 1 c + C(β)(f, y1:n1, y′ 1:n2), where c := min{cS(1), cS(2)} and C(β)(f, y1:n1, y′ 1:n2) : = max �� D(β) B (g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))π(β)(θ1|y1:n1)dθ1 − D(β) B (g1||m(β) f (·|y1:n1)), � D(β) B (g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))π(β)(θ2|y′ 1:n2)dθ2 − D(β) B (g2||m(β) f (·|y′ 1:n2)) � as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 Proof of Lemma 2 The KLD is recovered as the parameter β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, the bounds of Theorems 3 and 4 go to infinity in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The proof of Lemma 2 applies a similar method to that of Theorem 4 to investigate the stability of the KLD-Bayes estimation of a model to perturbations of the DGP 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The proof of Lemma 1 established the convexity of the KLD in its second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Further, by the definition of the KLD, we can see that KLD(g1||f) = KLD(g2||f) + � g1 log g1 − g2 log g2dµ + � (g2 − g1) log fdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (16) Now we can use the convexity of the KLD and Jensen’s inequality, to show that KLD(g1||mKLD f (·|y1:n1)) ≤ � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ|y1:n1)dθ1 Now adding and subtracting � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 provides KLD(g1||mKLD f (·|y1:n1)) ≤ � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1:n1)dθ1 = � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1:n1)dθ1 + � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 − � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 = � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 + � � {KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))} πKLD(θ1|y1:n1)dθ1πKLD(θ2|y′ 1:n2)dθ2 Now we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 to random variable {KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))} ∈ R on Θ×A which by using (12) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 applied to the KLD provides � � {KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) − KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))} πKLD(θ1|y1)dθ1πKLD(θ2|y2)dθ2 ≤ 1 cS(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We can now use the triangular-type relationship of (16) to show that KLD(g1||mKLD f (·|y1:n1)) ≤ � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 + 1 cS(2) = � � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)) + � g1 log g1 − g2 log g2dµ + � (g2 − g1) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)dµ � πKLD(θ2|y′ 1:n2)dθ2 + 1 cS(2) = � � (g2 − g1) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)dµπKLD(θ2|y′ 1:n2)dθ2 + � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 + � g1 log g1 − g2 log g2dµ + 1 cS(2) + KLD(g2||mKLD f (·|y′ 1:n2)) − KLD(g2||mKLD f (·|y′ 1:n2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 54 The same arguments of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 and (11) of Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 show KLD(g2||mKLD f (·|y′ 1:n2)) ≤ � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dη ≤ � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1:n1)dθ1 + 1 cS(1) = � � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)) + � g2 log g2 − g1 log g1dµ + � (g1 − g2) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)dµ � πKLD(θ1|y1:n1)dθ1 + 1 cS(1) = � � (g1 − g2) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)dµπKLD(θ1|y1:n1)dθ1 + � KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1:n1)dθ1 + � g2 log g2 − g1 log g1dµ + 1 cS(1) + KLD(g1||mKLD f (·|y1:n1)) − KLD(g1||mKLD f (·|y1:n1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Combining the above two results provides the following bound, |KLD(g||mKLD f (·|y)) − KLD(g||mKLD h (·|y))| ≤ CKLD(f, y1:n1, y′ 1:n2) + 1 c + T1(g1, g2) + T2(f, y1:n1, y′ 1:n2) where c := min{cS(1), cS(2)} and T1(g1, g2) : = max �� g2 log g2 − g1 log g1dµ, � g1 log g1 − g2 log g2dµ � T2(f, y1:n1, y′ 1:n2) : = max �� � (g1 − g2) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1)dµπKLD(θ1|y1:n1)dθ1, � � (g2 − g1) log f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2)dµπKLD(θ2|y′ 1:n2)dθ2 � CKLD(f, y1:n1, y′ 1:n2) : = max �� KLD(g1||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ1))πKLD(θ1|y1:n1)dθ1 − KLD(g1||mKLD f (·|y1:n1)), � KLD(g2||f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ2))πKLD(θ2|y′ 1:n2)dθ2 − KLD(g2||mKLD f (·|y′ 1:n2)) � as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B Extended Experimental Results Section B contains additional details of the experimental results of Section 6, including full specifica- tions of the models and data used, as well as additional sensitivity analysis for β and a comparison with the γ-Divergence (γD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 55 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Gaussian and Student’s-t likelihood B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 Posteriors Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 plots the posterior distribution of model parameters µ and σ2 of the Gaussian and Student’s- t models (8) discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0 2 4 6 8 10 KLD µ Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0 2 4 6 8 10 βD µ Density 1 2 3 4 0 1 2 3 4 5 KLD σ2 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0 2 4 6 8 βD σ2 Density Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1: Parameter posterior distributions for µ and σ2 under Bayes’ rule updating (KLD-Bayes) (left) and βD-Bayes with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 (right) under the likelihood functions f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) = N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2 adjσ2� (red) and h(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) = tν(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2) (blue) where ν = 5 and σ2 adj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The left-hand side of Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 demonstrates what most statistical practitioners expect when comparing the performance of a Gaussian and a Student’s-t under outlier contamination (O’Hagan, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Under the Student’s-t likelihood, the inference is much less affected by the outlying contami- nation than under the Gaussian likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The parameter µ is shifted less towards the contaminant population and the parameter σ2 is inflated much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' In short, very different inferences are pro- duced using a Student’s-t and a Gaussian under outlier contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Updating using the βD-Bayes presents a striking juxtaposition to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD-Bayes produces almost identical posteriors for both 56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0 5 10 15 20 β Neighbourhood Multiplier M = 5 M = 2 M = 1 M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='25 M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='12 β Energy Distance Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2: Left: The multiplier Mβ−1(3β−2) β(β−1) from from Theorems 1 and 2 for different values of M and 1 < β < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Right: The energy-distance between the posterior predictives after fitting a Gaussian and a Student’s-t model as in Figure 1 under the βD-Bayes for different β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ and σ2 under both models resulting in almost identical posterior predictive densities in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Estimating the TVD or the βD between the two predictves distributions is hampered by the fact that they are not available in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, the energy distance Sz´ekely and Rizzo (2013) provides a metric that can be easily estimated from samples of the predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The energy distance between the Gaussian and Student’s-t predictive distributions under traditional Bayesian updating was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='125, while under βD-Bayes updating the energy distance was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='13 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Sensitivity analysis As was noted in Section 5, it is encouraging to note the stability of the βD-Bayes inference appears not to be overly sensitive to the exact value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Complimenting Figure 2, the left hand side of Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 plots Mβ−1(3β−2) β(β−1) , the multiplier of the TVD from Theorems 1 and 2, as a function of β for various M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We see that as β increases away from 1 this multiplier initially decreases rapidly, indicating a large increase in guaranteed stability by moving away from the KLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' However, after this point the multiplier plateaus, indicating that a similar amount of stability results from a range of values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The right-hand plot of Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 has a very similar shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Here we plot the energy distance (Sz´ekely and Rizzo, 2013) between the posterior predictives of fitting a Gaussian likelihood model and Student’s-t likelihood modes, used in Figure 1 for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Once again, we see that taking β > 1 results in a large increase in a posteriori stability but that after a point that stability achieved is fairly constant with β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 57 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Comparison with the γ-divergence Similarly to the βD, the γ-divergence (γD) provides a loss function that does not require an estimator of the underlying density, and has been shown to have good robustness properties Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Knoblauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Here, we show its stability performance appears comparable with the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Firstly the γD is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 (The γ-divergence (D(γ) G ) (Fujisawa and Eguchi, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=', 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The γD is defined as D(γ) G (g||f) = 1 (γ − 1)γ � � � �� gγdµ � 1 γ − � fγ−1 �� fγdµ � γ−1 γ gdµ � � � , where γ ∈ R \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The corresponding loss function allowing generalised Bayesian inference for θ(γ) g is ℓ(γ)(y, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)) = − 1 γ − 1f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)γ−1 · 1 γ 1 �� f(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ)γdz � γ−1 γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Similarly to the βD-loss in (3), the γD-loss raises the likelihood to the power γ − 1 and ‘adjusts’ by the integral of the likelihood to the power γ, except for the γD this ‘adjustment’ term is multiplicative rather than additive as it was in the βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The integral term is independent of location parameters e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ from the Gaussian and Student’s-t examples, and therefore inference for these will be very similar under the βD and γD (Fujisawa and Eguchi, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 shows that in the example introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1, inference for σ2 is also very similar under the γD and βD and as a result they estimate identical posterior predictives under both the Gaussian and Student’s-t models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 DLD Data For the DLD data discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 we provide additional Q-Q normal and histogram plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These demonstrate the heavy-tailed nature of the DLD data, and the reasonable fit of the standardised residuals produced by the βD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 TGF-β data We consider another regression example to illustrate the stability to the selection between a Gaussian and Student’s-t example when using βD-Bayes updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 58 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 γD y Density (1 − ϵ)N(0, 1) ϵN(5, 32) Gaussian Student’s-t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0 2 4 6 8 10 γD µ Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0 2 4 6 8 γD σ2 Density Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3: Posterior predictive and parameter posterior distributions for � µ, σ2� under γD-Bayes with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 and likelihood functions f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' θ) = N � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2 adjσ2� (red) and h(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' η) = tν(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µ, σ2) (blue) where ν = 5 and σ2 adj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 59 KLD - Gaussian (y − X ˆθ)/ˆσ Density 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 betaD - Gaussian (y − X ˆθ)/ˆσ Density 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 KLD - Student’s-t (y − X ˆθ)/ˆσ Density 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 betaD - Student’s-t (y − X ˆθ)/ˆσ Density 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4: DLD data - Top: Posterior mean estimates of standardised residuals under the Gaussian and Student’s-t model of KLD-Bayes (left) and βD-Bayes (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom left: Absolute different in posterior mean parameter estimates under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom right: Difference in posterior predictive densities of the observations under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes 60 3 2 1 0 1 2 3 4 2 0 2 DLD - Normal Q-Q Plot Theoretical Quantiles Sample Quantiles 3 2 1 0 1 2 3 4 3 2 1 0 1 2 TGFβ - Normal Q-Q Plot Theoretical Quantiles Sample Quantiles Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5: Q-Q normal plot of the fitted residuals according to the Gaussian model under the KLD-Bayes for the DLD data (left) and TGF-β data (right) The dataset from Calon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2012) concerns gene expression data for n = 262 colon cancer patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Previous work (Rossell and Telesca, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rossell and Rubio, 2018) focused on selecting genes that affect the expression levels of TGF-β, a gene known to play an important role in colon cancer progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Instead, we study the relation between TGF-β and the 7 genes (listed in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6) that appear in the ‘TGF-β 1 pathway’ according to the KEGGREST package in R (Tenenbaum, 2016), so that p = 8 after including the intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We fitted regression models using the neighbouring models in (8) for the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 shows that both inference procedures are stable to the choice of the model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) appears to be marginally more stable in estimating the fitted residuals and predictive density (top-left and bottom-right), while the KLD-Bayes appears marginally more stable when estimating parameters (bottom-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7 shows the fit of the models to the standardised residuals under posterior mean estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Rossell and Rubio (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Jewson and Rossell (2021) find considerable evidence that a Gaussian model is compatible with this data, further demonstrated in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6 Variable selection When investigating the stability of the βD-Bayes inference to the Gaussian and Student’s-t likelihoods we regressed the DLD and TGF-β gene expressions on a subset of the variables available in the full data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' The procedures for which variables were selected as outlined in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' To ensure that our results are reproducible, below we indicate the selected covariates and the supplementary 61 4 3 2 1 0 1 2 4 2 0 2 Gaussian - (y − X ˆθ)/ˆσ Student’s-t - (y − X ˆθ)/ˆσ KLD βD 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5 (y − X ˆθ) Density KLD - Gaussian KLD - Student’s-t βD - Gaussian βD - Student’s-t 0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='08 Parameter index |ˆθnorm − ˆθt| KLD βD KLD βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='04 Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='6: TGF-β data - Posterior mean estimates of standardised residuals (top left), posterior mean estimated residuals distribution (top-right), absolute difference in posterior mean parameter estimates (bottom left) and difference in posterior predictive densities of the observations (bottom right) under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes (β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5) for the TGF-β data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 62 KLD - Gaussian (y − X ˆθ)/ˆσ Density 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 betaD - Gaussian (y − X ˆθ)/ˆσ Density 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 KLD - Student’s-t (y − X ˆθ)/ˆσ Density 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 betaD - Student’s-t (y − X ˆθ)/ˆσ Density 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='4 Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='7: TGF-β data - Top: Posterior mean estimates of standardised residuals under the Gaussian and Student’s-t model of KLD-Bayes (left) and βD-Bayes (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom left: Absolute different in posterior mean parameter estimates under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Bottom right: Difference in posterior predictive densities of the observations under the Gaussian and Student’s-t model of KLD-Bayes and βD-Bayes 63 material contains code for these variable pre-screening steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' DLD: For the DLD analysis, we selected the 15 genes with the 5 highest loadings in the first 3 principal components of the original 57 predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This procedure selected the following genes C15orf52, BRAT1, CYP26C1, SLC35B4, GRLF1, RXRA, RAB3GAP2, NOTCH2NL, SDC4, TTC22, PTCH2, ECH1, CSF2RA, TP53AIP1, and RRP1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' TGF-β: For the TGF-β analysis we focused on 7 of the 10172 genes available in the data set that appear in the ‘TGF-β 1 pathway’ according to the KEGGREST package in R (Tenenbaum, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' These were the VIT, PDE4B, ATP8B1, MAGEA11, PDE6C, PDE9A, and SEPTIN4 genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 64 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Mixture Models Here we provide full details of the models and priors considered in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We estimated Gaussian mixture models of the form f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' ω, µ, σ, K) = K � k=1 ωjN(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' µj, σj) using Normal-Inverse-Gamma-Dirichlet priors (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , ωK) ∼ Dir(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , αK) σ2 k ∼ IG �ν0 2 , S0 2 � , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , K µk|σk ∼ N(0, √κσj), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , K with αk = a k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' , K, ν0 = 5, S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 and κ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='68 following the recommendations of F´uquene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We elicited the parameter a to ensure that the marginal prior probability that any of the component weights was greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='05 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Binary Classification We provide additional details of the binary classification experiments in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1 t-logistic regression Following Ding and Vishwanathan (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2013), the t-exponential function used in the t-logistic regression is defined as expt(x) := � � � � � exp(x) if t = 1 max {1 + (1 − t)x, 0}1/(1−t) otherwise , and Gt(Xθ) is defined as the solution of expt(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5Xθ − Gt(Xθ)) + expt(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5Xθ − Gt(Xθ)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (17) In general there is no closed form for Gt(Xθ) but Algorithm 1 of Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2013) computes it efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 65 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='2 Transformations of probit and t-logistic β’s To minimise the a priori TVD between the probit and t-logistic alternative models and the logistic canonical model the β’s of the alternative model are scalar multiplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' For the probit model, the canonical parameters are multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='5876364 and for the t-logistic the canonical parameters are multiplied by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='331078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='3 Variables selected Colon Cancer data To prepare the Colon Cancer data for our analysis we first took the natural logarithm of the gene expression levels to remove some of their skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' We then used the glmnet package in R Friedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' (2010) to conduct LASSO variable selection using cross-validation to choose the hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' This process left us with the intercept and genes genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='249, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='377, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='493, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='625, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1325, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1473, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1582, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1671, genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content='1772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} +page_content=' 66' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FST4oBgHgl3EQfBDjV/content/2301.13701v1.pdf'} diff --git a/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/2301.02732v1.pdf.txt b/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/2301.02732v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e14ba3cde9b8e22616953bc3c5c812529d03fd7 --- /dev/null +++ b/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/2301.02732v1.pdf.txt @@ -0,0 +1,1232 @@ + +XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE +Multimodal Lyrics-Rhythm Matching +Callie C. Liao +McLean High School +McLean, USA +cliao2025@gmail.com + + +Duoduo Liao +School of Computing +George Mason University +Fairfax, USA +dliao2@gmu.edu + +Jesse Guessford +School of Music +George Mason University +Fairfax, USA +jguessfo@gmu.edu + +Abstract—Despite the recent increase in research on artificial +intelligence for music, prominent correlations between key +components of lyrics and rhythm such as keywords, stressed +syllables, and strong beats are not frequently studied. This is likely +due to challenges such as audio misalignment, inaccuracies in +syllabic identification, and most importantly, the need for cross- +disciplinary knowledge. To address this lack of research, we +propose a novel multimodal lyrics-rhythm matching approach in +this paper that specifically matches key components of lyrics and +music with each other without any language limitations. We use +audio instead of sheet music with readily available metadata, +which creates more challenges yet increases the application +flexibility of our method. Furthermore, our approach creatively +generates several patterns involving various multimodalities, +including music strong beats, lyrical syllables, auditory changes in +a singer’s pronunciation, and especially lyrical keywords, which +are utilized for matching key lyrical elements with key rhythmic +elements. This advantageous approach not only provides a unique +way to study auditory lyrics-rhythm correlations including +efficient rhythm-based audio alignment algorithms, but also +bridges computational linguistics with music as well as music +cognition. Our experimental results reveal an 0.81 probability of +matching on average, and around 30% of the songs have a +probability of 0.9 or higher of keywords landing on strong beats, +including 12% of the songs with a perfect landing. Also, the +similarity metrics are used to evaluate the correlation between +lyrics and rhythm. It shows that nearly 50% of the songs have 0.70 +similarity or higher. In conclusion, our approach contributes +significantly to the lyrics-rhythm relationship by computationally +unveiling insightful correlations. +Keywords—multimodal analysis, audio alignment, keyword +extraction, music information retrieval, natural language processing +I. +INTRODUCTION + +In recent years, research interest has increased in utilizing +Artificial Intelligence (AI) technologies for music. Even though +there is heightened attention, songwriters and composers’ +music cognition and intuition are often overlooked in AI music +research. In music, there is a variety of musical elements— +rhythm, melody, harmony, lyrics, dynamics, instrumentation, +timbre, etc.—that are combined to create compelling music. In +particular, lyrics, through the art of literature and the diction of +a vocal performance, can express emotion to the listener. When +examining lyrics, keywords (important words within the text) +and musically stressed syllables (syllables that the composer +has emphasized) help establish musical understanding. +Musically stressed syllables may differ from dictionary-defined +stressed syllables. Additionally, some languages such as + + +Fig. 1. The multimodal lyrics-rhythm matching. The first plot is a song’s +waveform. The second plot shows the first music strong beats (i.e., downbeats) +in red and all other beats in gray over the piano roll where both include the rest +period. The third plot indicates the matched keyword-strong-beat pairs in red +over the waveform where the strong beats in gray color include the rest period. +Chinese, Japanese, and Korean lack stress [1]. Thus, in our +view, keywords are preferred over stressed syllables because +they encompass more forms of literary expression, which aids +musical creativity and analytics. + +When the lyrics are mapped onto a rhythm, the significance +of keywords is reflected. Rhythm is defined as “the +organization of musical events in time” [2] and creates +musically metrical stress. In Fig. 2, the beats within the music +measure are split into a group receiving metrical stress (strong +beats) and a group not receiving metrical stress (weak beats). +Strong beats usually indicate the beginning of a measure or +phrase and convey the forward energy of musical emotion, +although some time signatures, such as 4/4, have 2 or more +strong beats within a measure. Since lyrics and rhythm work +together to create musical meaning, studying the correlation +between the two can serve as groundwork for further +multimodal AI music research. So, we investigate the potential +correlation between lyrical keywords and musical strong beats +as well as between non-keywords and weak beats through audio +with lyrics, which has not been studied in quantitative analytics. + +There are challenges along the way, however. Misalignment +between the music beats and lyrical syllables becomes +unavoidable because music is often produced by humans. A +singer sometimes does not perform the rhythm mechanically, +which creates challenges when aligning the audio to Musical +Digital Instrument Interface (MIDI1) files or other digital sheet +music files. Moreover, some current dictionaries for splitting +words into syllables are occasionally unable to produce correct +1https://www.midi.org/specifications + + + +syllables, and some do not produce any corresponding syllables +for words unknown to the dictionaries (both new and existing). +In particular, new words are a challenge since they are being +invented at a fast-growing rate as technology escalates. +Furthermore, lyrics are a type of free-form figurative language +that contain hidden connotations and potential keywords that +are difficult to determine in the text. Therefore, in general, it is +challenging for current Natural Language Processing (NLP) +techniques to process lyrics successfully. + +Fig. 2. Demonstrations of strong beats in music. (a) The downbeats (i.e., first +strong beats) in “A Little Bird” [3] in 2/4 time signature. The downbeats are +noted as 1 in red, and the weak beats are noted as 2. (b) The first and second +strong beats in “Birds are Flying” [4] in 4/4 time signature. The downbeats +are noted as 1 in red, and the second strong beats are noted as 3 in blue. The +weak beats are noted as 2 and 4. + +Above all, to conduct successful AI music research, +computing technologies as well as deep knowledge and +familiarity with music, literature, linguistics, and cognitive +science are required. This is perhaps the most critical challenge +for researchers. Therefore, making breakthroughs in any of +these challenges will contribute to AI music fields. +We propose an innovative multimodal lyrics-rhythm +matching approach. Our main contributions are as follows: +• +Our +approach +computationally +unveils +strong, +insightful correlations between lyrics and rhythm +through data analytics. +• +Our approach is a generalized method that determines +lyrics-rhythm matching based on keywords without any +language limitations. It indicates that keywords have +stronger associations with strong beats. This can be +employed in any language including some that lack +stressed syllables. +• +Creatively patterning lyrics-music multimodalities and +using various metrics to measure their relationships in +our approach can expand horizons in music cognition +and computational analysis. +• +Our approach enhances the syllabication and syllabic +stress pattern identification for unknown words, which +helps computational linguistics. +• +Audio alignment in music continues to pose many +challenges. Our research contributes to the area through +the development of several novel, efficient rhythm- +based audio alignment algorithms. +• +Our research uses challenging audio instead of sheet +music, which makes our solutions more flexible and +more applicable to AI music research. +• +Our research can serve as a basic building block for AI +music technologies, including song identification, +singing voice analysis, music structure analysis, etc. +• +Our research can inspire connections between +languages of music and linguistic representations in +neuroscientific research, which are perhaps some of the +most important interactions between language and +music. +II. +RELATED WORK +Research on the computational analysis of music cognition +and musicology are not addressed unlike other papers in Music +Information Retrieval (MIR) [5]. As the Linguistic Stress Rule +states, composers prefer to align strong beats with stressed +syllables of text [1][6]. However, surprisingly few quantitative +analyses can be found to support it. A few researchers +conducted some basic statistical analysis and concluded that +stressed syllables tend to fall on relatively strong beats of the +meter in vocal music for languages with lexical stress such as +English and German [7]. Another study on corpora of vocal +French music has shown a strong correlation between the stress +levels of syllables and their metrical strength [8]. The paper [9] +presents an observational study for English popular music that +performs minimal preprocessing on MusicXML2 sheets, +including monosyllabic stopwords [10] removal. It draws +conclusions based on various correlations between categorical +variables such as syllable stress, metric position (i.e., types of +beats), and stopwords. It also helps provide fundamental +statistical support to musicology and music-cognition research +as well as a quantitative correlation between lyrics and +melodies. However, this paper has some limitations. Firstly, it +utilizes MusicXML files, so beat, lyrical syllable, rhythm, and +pitch information have already been stored inside of the files; +thus, extra processing is not needed, which causes additional +limitations on the scope of its research. Secondly, the paper +seeks for a general association between musically accented +notes and stressed syllables. Thirdly, just monosyllabic +stopwords removal is performed on its data to find the +correlation between stopwords and non-salient notes. And +fourthly, it lacks consideration for a variety of time signatures +in their methods other than 4/4. +Additionally, audio alignments are among the most +important steps in audio processing for MIR. Audio alignment +has several distinct types, including audio-to-audio, lyrics-to- +audio, audio-to-score, audio-to-visual, etc. Many researchers +have conducted relevant studies in this field [11][12], including +deep learning [13] based audio alignment in [14][15]. Yet, there +are very few people involved in rhythm-based audio-to-audio +alignment and lyrics-to-audio alignment, which are all under +rhythm alignment. The paper [16] proposes a multi-task +learning approach for lyrics alignment that improves the +alignment accuracy through the incorporation of pitch and the +integration of boundary detection in the forced-alignment +algorithm. However, their performance was improved at the +cost of efficiency. Another paper [17] utilizes a multi-scale +neural network based on end-to-end audio-to-character +architecture. It improves the alignment accuracy by predicting +the character probabilities end-to-end from raw audio but only +if such large datasets are available. In fact, most deep learning +technologies (including the mentioned above) need large +datasets for sufficient training. +1 2 3 4 +1 2 +2https://www.musicxml.com/ +1 2 3 4 +1 2 3 4 +1 2 3 4 +1 2 +1 2 +1 2 + +can +see +the +birds fly - ing, +my eye-balls are +go-ingupand downAhli - ly. +Ahli -ly. +Ali-ttlebirdflies +stothetree. + +Our approach resolves the above limitations and expands +the scope by using multimodal data such as music +accompaniment audio, singing vocal audio, and separate lyrics +text data. We perform simple, efficient, and accurate audio +processing based on multimodal data, including rhythm-based +audio-to-MIDI alignment and lyrics-to-audio alignment, +without the use of deep learning techniques and the need for big +training datasets. Most importantly, we investigate a correlation +between the keywords and music strong beats due to the +keywords’ heightened significance when compared to other +words. This correlation is applicable to other languages as well, +including any language that lacks stressed syllables. Although +some researchers have mentioned the lack of stressed syllables +in particular languages [1], the importance of keywords in +songs was not highlighted. Moreover, we use customized word +syllabication and keywords extraction methods as well and +consider all time signatures, including simple, compound, and +asymmetric. Thus, our research places emphasis on the impact +of lyrical keywords in music, making ours different from +current related work. +III. +THE METHODS +The proposed approach aims to investigate positive +connections between lyrics and rhythm. In this study, we focus +on studying correlations among lyrical keywords and music +strong beats as well as non-keywords and weak beats, and +comparing the keywords approach to the stressed syllables +approach to provide further insights into composers’ thought +processes. The architectural framework of the lyrics-rhythm +matching approach is illustrated in Fig. 3. The highlighted +framework is demonstrated in Fig. 1. The framework consists +of three major components: music pronunable patterning, +lyrical syllabic patterning, and lyrics-rhythm matching. Since +each vocal “syllable” is recorded as a change in pronunciation, +we denote each vocal “syllable” as a pronunable in this study. +In the framework, the first component consists of data input, +beat tracking and patterning, rhythm alignment, pronunable +(i.e., singing vocal “syllable”) locating, and patterning with +music strong beats. The second component contains syllable +splitting, syllabic stress pattern identification, pronunable and +lyrical syllable matching, keyword extraction, and keyword +patterning. The third component targets to match keywords and +strong beats and seek insights through data analytics. +The system receives lyrical texts and audio containing the +singing voice and music accompaniment as input. Then, all +vocal and music beats are tracked and aligned with each other. +Pronunables are retrieved from the singing vocal audio and then +patterned with the strong-beats based on rhythm. The keywords +and syllabic stress patterns are extracted from the lyrics +afterwards. The lyrical syllables need to match with the +retrieved pronunables to bridge the lyrics and the beat +information. Finally, the keyword strong-beat matching and +overall lyrics-rhythm matching are checked using the keyword +pattern and the strong-beat pattern of each pronunable +associated with a word. +Syllable Splitting +Syllabic Stress Pattern +Identification +Pronunable-Syllable Matching +Keyword Extraction +Keyword Patterning +Audio & Lyrics Data Input +Beat Tracking & Patterning +Rhythm Alignment +Pronunable Locating +Pronunable Strong-beat +Patterning +Data Analytics +Non-Keyword Weak Matching +Music Pronunable Patterning +Lyrics Syllabic Patterning +Lyrics-Rhythm Matching +Keyword Strong-Beat Matching +Word-based Pronunable Strong-beat Pattern (WPSP) +Keyword Pattern (KP) +Stressed +Syllables and +Strong Beat +Matching +Pronunable Strong-beat Pattern (PSP) +Pronunable- +Word +Conversion +Fig. 3. The framework of Keyword Strong-Beat Matching + +little beaH + +In the following subsections, we explain the components of +the architecture in greater detail. +A. Music Pronunable Patterning +1) Beat Tracking and Strong Beat Patterning +In this framework, the input audio pair has two separate +forms: a solo-singing vocal audio and an accompaniment MIDI. +Singing vocal beats and music beats are tracked from these two +data files, respectively. That is, two time series record onset +locations (i.e., timestamps) of all tracked beats for vocal and +music audios, respectively. Beat tracking is the task of +identifying and synchronizing with the basic rhythmic pulse of +a piece of music [18]. It has been extensively studied in MIR. +Since our research focuses on lyrics-rhythm correlation +findings, some existing music tools, such as Librosa3, are used +for beat tracking in this study. +After beat tracking, a Strong-beat Pattern (SP) list with its +length as the total number of music beats is created to store the +flags of the corresponding strong music beat locations. If a beat +is a strong beat, the flag is set to 1. Otherwise, the flag is set to +0. The number of strong beats and their positions in one measure +can be deduced from the time signature. Our methods can +calculate the numbers for all distinct types of beats and their +positions based on time signatures. +2) Rhythm Alignment +In this study, rhythm alignment includes rhythm-based audio +to-MIDI alignment and lyrics-to-MIDI alignment (i.e., lyrics-to- +accompaniment-audio alignment). +Due to the unavoidable imperfections of a human singer, +which causes delays or early starts, the vocal rhythm of the +singing audio needs to be aligned with the music rhythm of the +accompaniment MIDI, which is the audio-to-MIDI alignment. +More specifically, the audio-to-MIDI alignment is based on two +sequences, the vocal audio and music audio sequences, and uses +a timing threshold or margin of error, which is set based on the +tempo and time signature from the vocal audio. Additionally, the +length of tracked vocal beats may be different from the length of +tracked music beats. This could be due to the possibility of +unmatched or missing beats. These types of beats are handled +using linear interpolation. After alignments, an index list of +vocal beats aligned with music beats is generated. +As for the rhythm-based lyrics-to-MIDI alignment approach, +it is more complicated and consists of several steps: pronunable +retrieval and locating, pronunable-beat patterning, syllable +pattern identification, syllable pronunable matching, and lyrics- +to-rhythm matching. These steps will be discussed further in the +following sections. This approach, the rhythm-based lyrics-to- +audio alignment, can be applied to any audio other than MIDI. + +3) Pronunable Retrieval and Locating +In the vocal audio, pronunables represent every change in +pronunciation regardless of whether it is an actual syllable of a +word. This includes singing words such as “woah”, “nei”, etc., +as well as words with multiple syllables pronounced as one such +as “o’er”, which represents “over”. Such cases present +challenges for lyrical analytics, as lyrics differ from speech and +other forms of literature. For this study, we only focus on +retrieving the location (i.e., start time and end time) of each +pronunable instead of recognizing syllables or words. +To retrieve pronunables on music beats, the pronunables +associated with vocal beats are first retrieved from the vocal +audio. Once all pronunables’ locations are retrieved, the vocal +beats need to align with the corresponding music beats in the +accompaniment audio through the use of pronunables. The +algorithm of pronunable locating is shown in Algorithm 1. +Since the music and voice audio do not match precisely, a +time buffer or margin for error is needed for alignment. If the +music beats are within the time buffer, then the beats and the +beat indices are stored accordingly. Moreover, to optimize the +algorithms, vocal-audio-to-MIDI alignment is processed at the +same time with pronunable locating on music beats. +Specifically, for each time t1 in the pronunable time series T1, the +times within the dynamic buffer are sought in the music beat +time series T2 and then stored into a list Ts. Their corresponding +indices are collected into another list Ls. In the time list Ts, for +each time t, a certain distance function (e.g., Euclidean distance) +is employed to calculate the distance between t and t1. Once +distance calculations for all the times in Ts are complete, the +minimum distance t* is selected. Then, the corresponding index +i* of t* is used to point to the corresponding music beat index ip +for the pronunable. ip is inserted into the output index list L of +music beats aligned with pronunables. If no beat is found, the +corresponding beat is likely unmatched or missing, so the beat +is linearly interpolated. + +Algorithm 1: Pronunable Locating + +Input: a vocal pronunable time series T1, + + a vocal beat time series T2, + + a vocal-music alignment index list La, + + a time buffer dt. +Output: an index list of music beats aligned with pronunables. +1: Initialize L as an empty list. +2: for each t1 in T1 do +3: (Ts, Ls ) = Search(t1, dt, T2) +4: +if Ts is not empty Then +5: + for i =0, 1, …, length(Ts) do +6: D[i] = Distance(t1, Ts[i]) +7: + end for +8: t* = min(D[i]) where i =0, 1, …, length(Ts) +9: + Get corresponding index i* of t* +10: if La is not empty and i*∈ [0, length( La)-1] Then +11: + ip = La [i*] +13: else // unmatched or missing +14: + ip = LinearInterpolation(i) +15: + end if +16: + Insert ip into L +17: end if +18: end for +19: return L + + +To be more mathematically specific, the target is to find the +music beat that minimizes the distance from the pronunable +location to match the pronunable with the closest music beat: + argmin +𝑡𝑠∈𝑇𝑠 +𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑡1, 𝑡𝑠) (1) +3https://librosa.org/ + + + +Several different distance functions can be used to calculate +D[i], where i = 0, 1, …, length(Ts). In this study, the Manhattan +distance is selected for the implementation of Algorithm 1. +Given two points A and B in n-dimensional space such that +A = (a1, a2, …, an) and B = (b1, b2, …, bn), the Manhattan +distance between A and B is defined as: + 𝐷 (𝐴, 𝐵) = ||𝐴 − 𝐵|| = ∑ +|𝑎𝑖 − +𝑛 +𝑖=1 +𝑏𝑖| (2) +Since the beat time series is a sorted time sequence, many +search algorithms such as binary search [19] can be applied in +Algorithm 1. Finally, a time series representing locations of +aligned pronunables is generated accordingly. +4) Pronunable Strong-Beat Patterning +Upon the completion of rhythm alignment and pronunable +locating on music beats, the output list contains all the beat +indices where the pronunables are located. Based on this index +list and the SP list created through the music strong beats, a new +patterned list is created to store the Pronunable Strong-beat +Pattern (PSP). The PSP contains the noted locations of all the +pronunables on a strong beat. If a pronunable has the same +location as a strong beat, it is set to 1 in the PSP; otherwise, it +is set to 0. +B. Lyrical Syllable Patterning +In this component, we utilize customized word syllabication +and keywords extraction methods to find more lyrical keywords. +1) Syllable Splitting and Stress Pattern Identification +Firstly, special preprocessing for lyrics is conducted due to +the potential occurrence of cases similar to pronunables. +Moreover, some additional critical cases also impact the +splitting of syllables. For example, despite having no syllabic +contributions, the parts after the apostrophe in contractions such +as “re” in “they’re” and “t” in “don’t” are still assigned a +syllable by pronouncing dictionaries (e.g., the Carnegie Mellon +University (CMU) pronouncing dictionary4). As a result, a +customized list of words and/or letters for lyrics preprocessing +is created for these types of situations. Then, utilizing a +pronouncing dictionary produces one or more variants of a +word’s stress pattern consisting of flags. For example, in the +CMU dictionary, the stress pattern is expressed through 0s and +1s. The 1s represent the stressed syllables, whereas the 0s +represent the unstressed. To simplify the process in this study, +only the first most common variant is appended to a list +containing all the words’ stress patterns. Occasionally, the +dictionary may not recognize certain vocabulary and newly +created words. It also may create an incorrect syllable split or +stress pattern. Thus, after performing a basic stemming [20] for +words such as ones that end in “ed”, words are split into +syllables through vowel consideration instead. As a result, +stress patterns are created for such words. +The syllabic stress patterns are identified. The algorithm of +syllabic stress pattern identification is shown in Algorithm 2. +The output of the algorithm is a string list of stress patterns with +each stress pattern being composed of 0s and/or 1s. +2) Pronunable-Syllable Matching +The syllable-pronunable matching is then conducted. The +total number of lyrical syllables is counted by adding up each +stress pattern’s length. Each syllable cannot be individually +compared to the pronunables, because each pronunable +expresses every change in the singer’s pronunciation in the +most pronounceable way. As a result, the spelling of +pronunables may vary drastically for each syllable. For +example, the syllables of “very” are “ve” and “ry”. However, +“ry” can be pronounced as “ree” and “rye”, which both are +pronunables. In this case, the pronunable for “ry” in “very” is +“ree”, yet if “ree” and “ry” are compared to each other, the two +will not match. Thus, comparing the total number of syllables +in the lyrics to the actual number of pronunables in a song is +more accurate for the matching between pronunables and +lyrical syllables. This is the key step in rhythm-based lyrics-to- +audio alignment as mentioned in Section A Rhythm Alignment. + + +Algorithm 2: Syllabic Stress Pattern Identification + +Input: Lyrics text S. +Output: A string list of stress patterns. +1: Initialize L as an empty list. +2: S’ = Preprocessed S with the special lyrics cases. +3: A list T = Tokenize(RemovePunctuation(S’)) +4: for each token in T do +5: Get token’s stress pattern from the dictionary D +6: + if stress pattern exists Then +7: + Append 1st stress variant as a string to L + +8: + else +9: + if token’s end has no syllabic meaning Then +10: + +Stem the token + + +11: + end if +12: A list Ls = WordSyllabication(token) + +13: + Set the first syllable pattern s to 1 + +14: + Append s to L +15: + end if +16: end for +17: return L + + +3) Keyword Extraction and Patterning +The lyrical keywords in the text are extracted through NLP +techniques. Additionally, the patterns of the keywords are +identified and used to build the Keyword Pattern (KP) flag list. +Since KP is now word-based yet PSP is still pronunable-based, +PSP needs to be converted to a word-based pattern, i.e., Word- +based Pronunable Strong-beat Pattern (WPSP). WPSP is then +used to compare with KP. This is one of the key components +for finding the correlation between the keywords and strong +beats, not the stressed syllables. +C. Lyrics-Rhythm Matching +Finally, two patterns, WPSP and KP as shown in Fig. 3, are +checked for whether the lyrics match the rhythm, including +whether keywords land on strong beats, etc. There are multiple +ways to find whether the lyrics match the rhythm, including the +utilization of time series similarity and conditional probability. +1) Conditional probability +Clearly, the conditional probability [21] can be used for +checking the matching probability of lyrics and rhythm. For +example, to check if keywords land on strong beats, let A +represent the keywords, let B represent the strong beats, +let 𝑃(𝐴 ∩ 𝐵) represent the probability of overall keywords +4http://www.speech.cs.cmu.edu/cgi-bin/cmudict + + + +landing on strong beats, let 𝑃(𝐵) represent the probability of +strong beats occurring out of all beats, and let 𝑃(𝐴 | 𝐵) +represent the likelihood of keywords occurring on the +occurrence of strong beats. According to the conditional +probability formula, 𝑃(𝐴 | 𝐵) can be calculated as + 𝑃(𝐴 | 𝐵) = +𝑃(𝐴 ∩ 𝐵) +𝑃(𝐵) . (3) +2) Cosine Similarity +Cosine similarity [22] is one of the similarity measures for +a vector space model. It measures the cosine of an angle formed +by the projection of two vectors in a multi-dimensional space. +To be more specific, given two vectors, A and B, representing +two binary patterns, such as KP and WPSP, respectively, the +cosine similarity is utilized to measure the similarity of two +patterns as shown in the following formula: + 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 = 𝑐𝑜𝑠(𝜃) = +𝐴⋅𝐵 + ||𝐴|| ||𝐵|| (4) +In Equation (4), ||𝐴|| is the Euclidean norm of vector A = +(a1, a2, …, an), defined as √𝑎12 + 𝑎22 + ⋯ + 𝑎𝑛2 . +Conceptually, it is the length of the vector. Similarly, ||𝐵|| is +the Euclidean norm of vector B. +The cosine similarity is a real number between 0.0 and 1.0. +The higher the similarity, the smaller the angle is between two +vectors, A and B, in the vector space. That is, the closer the two +vectors are, the better the match. Otherwise, a lower similarity +represents a lower chance of being matched with each other. +Unlike conditional probability, cosine similarity is an +overall matching check. It can check both whether keywords +land on strong beats as well as whether non-keywords and weak +beats match at the same time. In this paper, we use both +matching methods for the experiments. +IV. +EXPERIMENTAL RESULTS AND ANALYSIS +This study examines lyrics-rhythm matching in using +quantitative data gathered from a systematically selected +dataset of songs. Our system is scalable and can be used for +large datasets. +In this experiment, we use the Children’s Song Dataset +(CSD) [23] to pilot our study. This dataset consists of 50 +English songs and 50 Korean songs. Each song is sung in two +different keys by a professional female singer stored in .wav +file format, and each also includes a MIDI piano file, an English +lyrics text file, and text annotations in .csv file format. For +English songs, there are 100 MIDI and 100 .wav files in total. +The text annotations record every change in the singer’s +pronunciation (i.e., pronunables) using the Korean grapheme +system, which differs from both the English grapheme system +and the English syllabic system. This is the reason behind our +introduction of pronunables. Additionally, each song contains a +silence interval. +The tempo and time signature are retrieved from the +metadata of vocal files beforehand since it is already noted. +From the MIDI files, only the beats are utilized. We use Pretty +MIDI5 to process the MIDI data. + + +Fig. 4. The plot of beat locations over the waveform after beat tracking +Fig. 4 shows the tracked beats that are distributed over the +waveform. In this example of the song file, the graph’s +amplitude range is (-1.0, 1.0), and the time spans over one +minute. The gray lines represent every identified beat in the +singing vocal audio. Based on the gray lines, some of the beats +at the end are not tracked, which can be due to the duration of +the last note. This does not affect our study. + + +Fig. 5. Beat tracking for the music MIDI and visualization with the piano roll +of notes and the song’s strong beats + +Likewise, beat tracking is applied to the accompaniment +music MIDI file separately as shown in Fig. 5. The note range +span an octave (from C4 to C5) [2]. The red lines represent the +first strong beats (i.e., downbeats) from each measure and the +gray lines represent all other beats. Both lines include the rest +period. All the beats are perfectly aligned, which is because +computer-generated MIDI files are error-free. + +Fig. 6. The piano roll of notes and all unaligned vocal beats + +In Fig. 6, both vocal beats and music beats are drawn on the +same piano roll graph, which also includes the rest period. It +clearly indicates that the aqua-colored beat lines appear to be +misaligned with the music beats in some areas, particularly +during the silent interval in this example. This shows that each +slight delay in the singer’s voice accumulates over time, +resulting in larger misalignments until the singer is onbeat again. + + +(a) + +(b) +Fig. 7. Rhythm alignment. The music beats are in gray, and the unaligned vocal +beats are in aqua in (a). Aligned vocal beats are in orange in (b). + +In Fig. 7, (a) displays the unaligned vocal beats with music +beats in the waveform and (b) shows the aligned vocal beats +with music beats after rhythm alignment (i.e., rhythm-based +audio-to-MIDI alignment). From the figures, our alignment +algorithm has demonstrated a high accuracy. +5https://github.com/craffel/pretty-midi + + + +In Fig. 8 (a), it indicates the original vocal beats and original +pronunable locations on the singing vocal wave. The +misalignment and early cutoff are still shown through the gray +lines as they are from the singing vocal file. Upon completion +of pronunable locating in Fig. 8 (b), the pronunable beats are +matched with the music beats since every vocal beat is perfectly +aligned with the music strong beats. Fig. 8 (c) further visualizes +the aligned and matched pronunable beats on the piano roll with +notes. The red lines indicate the first and second strong beats +from the MIDI file and the gray lines indicate all the strong +beats from the MIDI file, including places where no note is +plotted, such as the rest period. On the other hand, the red lines +only show the beats which pronunables are aligned with and +therefore do not include the rest period. + +(a) + +(b) + + +(c) + +Fig. 8. Pronunable locating. +(a) The original unaligned vocal beats (gray) and pronunable beats (orange). +(b) Aligned and matched pronunable beats with music strong beats (orange). +(c) Aligned pronunable beats and all matched corresponding music strong beats +(red) on the piano roll of notes. + +Fig. 9 and Fig. 10 show various conditional probabilities +given strong beats in several cases. Only the first key is marked, +but there are 100 songs displayed in the graph. Fig. 9 describes +the likelihood of the keywords occurring on the occurrence of +strong beats. The bar chart compares the probabilities of each +song with each other. The scatter plot presents a distribution of +the probabilities over all of the song indices. The histogram +displays the count of each probability range. To compare with +the paper [9], we also performed a general association between +musically accented notes and stressed syllables using the same +dataset. Fig. 10 indicates such likelihood of the stressed +syllables landing on strong beats. +More specifically, in Fig. 9 (a), based on the bar chart (a) +and scatter chart (b), most of the probabilities concentrated in +two clusters: Cluster 1, which is between 0.7 and 0.8, and +Cluster 2, which is between 0.9 and 1.0. Additionally, Fig. 9 (c) +indicates that around 12% of songs have 1.0 matching +probability (i.e., all keywords landing on strong beats) and +around 30% of songs have 0.9 or higher matching probabilities. +Thus, it can be concluded that keywords are frequently +associated with strong beats and that strong beats help +emphasize keywords. +By comparing with keyword-strong-beat matching as +shown in Fig. 9, Fig. 10 indicates there are two probability +concentrations for stressed syllables landing on strong beats: +Cluster 1, which is between 0.6 and 0.7, and Cluster 2, which is +between 0.75 and 0.85. Additionally, around 20% of songs have +0.9 or higher matching probabilities. Likewise, it shows that +stressed syllables are also frequently associated with strong +beats and that strong beats help emphasize stressed syllables. +However, it is evident that keywords have stronger +associations with Strong Beats (SB) than stressed syllables with +strong beats as shown in Fig. 9 and Fig. 10 as well as +summarized in Table I. This is because in many pieces of music, +there are instances where the keyword lands on a strong beat, +yet certain stressed syllables do not. +Our experimental results reveal an average of an 0.81 +matching probability for our proposed approach, and around 30% +of the songs have a match between keywords and strong beats +that is at least 0.9, including 12% songs with all keywords +landing on strong beats (probability = 1.0). Regarding the +strong beat matching probability of 0.65 or lower, only 8% of +the songs are under this probability for keywords, whereas for +stressed syllables, it is 18%. We appreciate the CSD dataset, but +it still has some limitations. There are occasional typos and +mispronunciations in the lyrics, so our methods are sometimes +unable to recognize these differences, which affects the +matching accuracy. +To compare both cases for overall matching, cosine +similarity in Formula 4 is utilized in this study. Fig. 11 (a) +shows the overall similarity histogram for two patterns, WPSP +and KP, related to keywords. Since each pattern only includes +1 or 0, it can be considered as a binary vector. 1 in the WPSP +means that at least one of the pronunables of a word land on a +strong beat while 0 means that all of the pronunables of a word +land on at least one weak beat. But as for KP, 1 represents a +keyword while 0 represents a non-keyword. In Table II and Fig. +11 (a), it clearly indicates that 46% of songs have 0.70 or higher +similarity between WPSP and KP. That is, keywords tend to +land on strong beats and non-keywords tend to land on weak +beats for overall matching. Additionally, only around 7% of the +songs have 0.55 or lower similarity between WPSP and KP, +which reinforces the strong overall matching tendency. +Fig. 11 (b) shows the similarity histogram for two patterns, +PSP and the Stressed Syllable Strong-beat Pattern (SSSP). Both +are binary vectors. 1 in the stressed syllable pattern means a +stressed syllable landing on a strong beat while 0 means +unstressed syllable landing on a certain beat (a strong or weak +beat). In the figure, it shows that 21% of the songs have 0.70 or +higher similarity between two patterns. Furthermore, from +Table II, about 28% of songs have 0.55 or lower similarity +between PSP and SSSP. In comparison with Fig. 10, it can be +further concluded that keywords have a stronger association +with strong beats than stressed syllables. +V. CONCLUSIONS AND FUTURE WORK +This paper proposes a novel multimodal lyrics-rhythm +matching approach that uses creative and efficient rhythm +alignment, pronunable locating, syllabic stress identification, +lyrical keyword extraction, and various patterning methods to +computationally unveil strong correlations between lyrics and +rhythm. Our approach can be applied to any language without +limitations because we place emphasis on keywords, not + +121 + +TABLE II. SIMILARITY COMPARISONS BETWEEN KEYWORDS AND STRESSED +SYLLABLES LANDING ON STRONG BEATS +Similarity +Keywords w/ SBs +Stressed Syllables w/ SBs +Average +0.68 +0.62 +<= 0.55 +7% +28% +>= 0.70 +46% +21% +>= 0.80 +7% +4% + + + +(a) + +(b) + +(c) + +Fig. 10. The probabilities of stressed syllables landing on strong beats +(a) The probabilities of stressed syllables landing on strong beats +(b) The probability distribution of stressed syllables landing on SBs +(c) The probability histogram of stressed syllables landing on SBs + + +TABLE I. PROBABILITY COMPARISONS BETWEEN KEYWORDS AND STRESSED +SYLLABLES LANDING ON STRONG BEATS +Probability +Keywords w/ SBs +Stressed Syllables w/ SBs +Average +0.81 +0.77 +Cluster 1 +0.70 - 0.80 +0.60 - 0.70 +Cluster 2 +0.90 - 1.0 +0.75 - 0.85 +>= 0.90 +27% of songs +20% of songs +< 0.65 +8% of songs +18% of songs + + +(a) + +(b) + + (c) + +Fig. 9. The probabilities of keywords landing on strong beats (100 songs) + (a) The probabilities of keywords landing on strong beats + (b) The probability distribution of keywords landing on strong beats + (c) The probability histogram of keywords landing on strong beats + + + + + (a) (b) +Fig. 11. The similarity histograms of KP and SSSP for all songs +(a) The similarity histogram of keywords landing on strong beats + (b) The similarity histogram of stressed syllables landing on strong beats + + +1 +Probability +0.8 + 0.6 +0.4 +0.2 +en001a +en002b +en004a +en005b +en007a +en008b +en010a +en011b +en013a +en014b +en016a +en017b +en019a +en020b +en022a +en023b +en025a +en026b +en028a +en029b +en031a +en032b +en034a +en035b +en037a +en038b +en040a +en041b +en043a +en044b +en046a +en047b +en049a +en050b +Song Name100 +Probability +0.95 +80 +0.9 +Index +0.85 +60 +0.8 +Song +0.75 +40 +0.7 +0.65 +20 +0.6 +0.55 +0 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Probability30 +Probability +■0.85 +25 +■0.55 +■0.95 +20 +■ 1.0 +■ 0.7 +unt +0.65 +15 +■ 0.6 +■ 0.8 +■ 0.5 +10 +0.75 +5 +% +0.2 +0.4 +0.6 +0.8 +1 +Probability1 +Probability +0.8 + 0.4 +0.2 +en001a +en002b +en004a +en005b +en007a +en008b +en010a +en011b +en013a +en014b +en016a +en017b +en019a +en020b +en022a +en023b +en025a +en026b +en028a +en029b +en031a +en032b +en034a +en035b +en037a +en038b +en040a +en041b +en043a +en044b +en046a +en047b +en049a +en050b +SongName100 +Probability +80 +0.9 +Index +60 +0.8 +Song +40 +0.7 +20 +0.6 +0 +0.5 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Probability30 +Probability +■0.85 +25 +■0.75 +■ 0.8 +20 +0.7 +■ 1.0 +0.95 +15 +0.9 +COI +0.6 +0.55 +10 +0.65 +0.5 +5 +00 +0.2 +0.4 +0.6 +0.8 +1 +Probability30 +Similarity +■ 0.7 +25 +■0.85 +0.8 +20 +0.6 +0.65 +unt +0.9 +COU +15 +0.75 +■ 0.5 +0.55 +10 +0.4 +■0.45 +5 +00 +0.2 +0.4 +0.6 +0.8 +1 +Similarity30 +Similarity +■ 0.7 +25 +■ 0.6 +■0.8 +20 +■ 0.65 +■0.55 +unt +■0.85 +15 +■ 0.5 += 0.75 +■0.9 +10 +0.45 +■ 0.4 +5 +00 +0.2 +0.4 +0.6 +0.8 +1 +Similarity + +stressed syllables. Moreover, our experimental results show +that both stressed syllables and keywords tend to land on strong +beats, with the matching between keywords and strong beats +having a higher probability. But most importantly, our method +uses audio instead of sheet music (e.g. MusicXML) with +metadata, which is more challenging because information is not +readily available. Using audio broadens our impact and extends +our method to many more horizons and opportunities that are +yet to be explored. +There are some limitations on our study, however. One +limitation is our reliability on other libraries and dictionaries for +some aspects of our study such as beat tracking and +syllabication. In addition, there is a lack of similar audio +datasets with pronunable annotations, so the size of the data +used for this study is small even though our system is designed +to fit the large datasets. Thus, future work includes adding +singing +audio +recognition +and +automatic +pronunable +annotations to expand the method’s use to larger and more +different datasets, which reduces the overall human error as +well. Furthermore, we will work on lessening and eventually +completely eliminating the reliance on other libraries or +dictionaries. This approach may also be used to generate more +accurate music scores from singing audio because in our study, +the results have shown that our establishment of a precise +connection between the lyrical keywords and strong beats is +accurate. Moreover, our methods can be improved upon in the +future to determine the more appropriate syllabic stress pattern +variant for various situations based on part-of-speech tagging +using NLP techniques, as there may be multiple stress patterns +for one word. + +REFERENCES +[1] D. Temperley, “Music and Language”, Annual Review of Linguistics. +8:153–70. 2022. +[2] A. Latham (Ed.), The Oxford Companion to Music. Oxford University +Press, 2011, DOI: 10.1093/acref/9780199579037.001.0001, eISBN: +9780199579037. +[3] C. C. Liao, “A Little Bird”, Journal of Children’s Music, vol. 345, pp. 40- +41, March 2015. +[4] E. L. Zhang, “Birds are Flying”, Journal of Children’s Music, vol. 365, +pp. 9, November 2016. +[5] J. S. Downie, “Music information retrieval”, Annual review of +information science and technology, 37(1):295–340, 2003. +[6] D. Temperley, The Cognition of Basic Musical Structures. Cambridge, +MA: The MIT Press, 2001. +[7] C. Palmer, M. Kelly, “Linguistic prosody and musical meter in song”. +Journal of Memory and Language. 31:525–42. 1992. +[8] N. Temperley, D. Temperley, Stress-meter alignment in French vocal +music. The Journal of the Acoustical Society of America. 134:520– 27. +2013. +[9] E. Nichols, D. Morris, S. Basu, and C. Raphael, "Relationships Between +Lyrics and Melody in Popular Music," in International Society for Music +Information Retrieval Conference (ISMIR), pp. 471–476, October 2009, +Kobe, Japan. +[10] C. Fox, "A stop list for general text", ACM SIGIR Forum. 24 (1–2): 19– +21, ISSN 0163-5840, 1989. +[11] H. Kirchhoff and A. Lerch, "Evaluation of Features for Audio-to-Audio +Alignment", Journal of New Music Research, 2011, Vol. 40, No. 1, pp. +27–41. +[12] C. Raffel and D. P. W. Ellis, “Optimizing DTW-Based Audio-to-MIDI +Alignment and Matching”, 41st IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP), 2016. +[13] Y. LeCun, Y. Bengio, G. Hinton, “Deep learning”, Nature 521, 436–444, +2015, https://doi.org/10.1038/nature14539. +[14] R. Agrawal, D. Wolff and S. Dixon, "Structure-Aware Audio-to-Score +Alignment +Using +Progressively +Dilated +Convolutional +Neural +Networks," ICASSP 2021 - 2021 IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP), 2021, pp. 571-575, +doi: 10.1109/ICASSP39728.2021.9414049. +[15] J. Zhu, C. Zhang, and D. Jurgens, "Phone-to-Audio Alignment without +Text: A Semi-Supervised Approach," ICASSP 2022 - 2022 IEEE +International Conference on Acoustics, Speech and Signal Processing +(ICASSP), 2022, pp. 8167-8171. +[16] J. Huang, E. Benetos, and S. Ewert, “Improving Lyrics Alignment +Through Joint Pitch Detection”, Proceedings of IEEE International +Conference on Acoustics, Speech and Signal Processing (ICASSP), 2022. +[17] D. Stoller, S. Durand, and S. Ewert, “End-to-End Lyrics Alignment for +Polyphonic Music Using an Audio-to-Character Recognition Model”, +Proceedings of IEEE International Conference on Acoustics, Speech and +Signal Processing (ICASSP), 2019. +[18] S. Dixon, Evaluation of audio beat tracking system Beatroot. Journal of +New Music Research, 36(1):39–51, 2007. +[19] T. H. Cormen, C. E. Leiserson, Introduction to Algorithms, 3rd Edition. +The MIT Press. 2009. ISBN-13: 978-0262033848. +[20] M. F. Porter MF, “An algorithm for suffix stripping”, Program, 14: 130- +137, 1980. +[21] R. Durrett, Probability: Theory and Examples. 4th ed. Cambridge +University Press, 2010. ISBN: 9780521765398. +[22] J. Han, J. Pei, and H. Tong, Data Mining: Concepts and Techniques, 3rd +edition, Morgan Kaufmann, 2012. +[23] S. Choi, W. Kim, S. Park, S. Yong, and J. Nam, “Children’s Song Dataset +for Singing Voice Research”, Proceedings of the 21st International +Society for Music Information Retrieval Conference (ISMIR), 2020. + + diff --git a/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/load_file.txt b/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e06d1930587f22d97beb8d321f2bd0fba82f3df7 --- /dev/null +++ b/ZNE0T4oBgHgl3EQf4AJQ/content/tmp_files/load_file.txt @@ -0,0 +1,719 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf,len=718 +page_content='XXX-X-XXXX-XXXX-X/XX/$XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='00 ©20XX IEEE Multimodal Lyrics-Rhythm Matching Callie C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Liao McLean High School McLean, USA cliao2025@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='com Duoduo Liao School of Computing George Mason University Fairfax, USA dliao2@gmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='edu Jesse Guessford School of Music George Mason University Fairfax, USA jguessfo@gmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='edu Abstract—Despite the recent increase in research on artificial intelligence for music, prominent correlations between key components of lyrics and rhythm such as keywords, stressed syllables, and strong beats are not frequently studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is likely due to challenges such as audio misalignment, inaccuracies in syllabic identification, and most importantly, the need for cross- disciplinary knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To address this lack of research, we propose a novel multimodal lyrics-rhythm matching approach in this paper that specifically matches key components of lyrics and music with each other without any language limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' We use audio instead of sheet music with readily available metadata, which creates more challenges yet increases the application flexibility of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Furthermore, our approach creatively generates several patterns involving various multimodalities, including music strong beats, lyrical syllables, auditory changes in a singer’s pronunciation, and especially lyrical keywords, which are utilized for matching key lyrical elements with key rhythmic elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This advantageous approach not only provides a unique way to study auditory lyrics-rhythm correlations including efficient rhythm-based audio alignment algorithms, but also bridges computational linguistics with music as well as music cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our experimental results reveal an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='81 probability of matching on average, and around 30% of the songs have a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 or higher of keywords landing on strong beats, including 12% of the songs with a perfect landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Also, the similarity metrics are used to evaluate the correlation between lyrics and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It shows that nearly 50% of the songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 similarity or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In conclusion, our approach contributes significantly to the lyrics-rhythm relationship by computationally unveiling insightful correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Keywords—multimodal analysis, audio alignment, keyword extraction, music information retrieval, natural language processing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' INTRODUCTION In recent years, research interest has increased in utilizing Artificial Intelligence (AI) technologies for music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Even though there is heightened attention, songwriters and composers’ music cognition and intuition are often overlooked in AI music research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In music, there is a variety of musical elements— rhythm, melody, harmony, lyrics, dynamics, instrumentation, timbre, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='—that are combined to create compelling music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In particular, lyrics, through the art of literature and the diction of a vocal performance, can express emotion to the listener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' When examining lyrics, keywords (important words within the text) and musically stressed syllables (syllables that the composer has emphasized) help establish musical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Musically stressed syllables may differ from dictionary-defined stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, some languages such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The multimodal lyrics-rhythm matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The first plot is a song’s waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The second plot shows the first music strong beats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', downbeats) in red and all other beats in gray over the piano roll where both include the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The third plot indicates the matched keyword-strong-beat pairs in red over the waveform where the strong beats in gray color include the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Chinese, Japanese, and Korean lack stress [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, in our view, keywords are preferred over stressed syllables because they encompass more forms of literary expression, which aids musical creativity and analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' When the lyrics are mapped onto a rhythm, the significance of keywords is reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Rhythm is defined as “the organization of musical events in time” [2] and creates musically metrical stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2, the beats within the music measure are split into a group receiving metrical stress (strong beats) and a group not receiving metrical stress (weak beats).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Strong beats usually indicate the beginning of a measure or phrase and convey the forward energy of musical emotion, although some time signatures, such as 4/4, have 2 or more strong beats within a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since lyrics and rhythm work together to create musical meaning, studying the correlation between the two can serve as groundwork for further multimodal AI music research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' So, we investigate the potential correlation between lyrical keywords and musical strong beats as well as between non-keywords and weak beats through audio with lyrics, which has not been studied in quantitative analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' There are challenges along the way, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Misalignment between the music beats and lyrical syllables becomes unavoidable because music is often produced by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' A singer sometimes does not perform the rhythm mechanically, which creates challenges when aligning the audio to Musical Digital Instrument Interface (MIDI1) files or other digital sheet music files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, some current dictionaries for splitting words into syllables are occasionally unable to produce correct 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='midi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='org/specifications syllables, and some do not produce any corresponding syllables for words unknown to the dictionaries (both new and existing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In particular, new words are a challenge since they are being invented at a fast-growing rate as technology escalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Furthermore, lyrics are a type of free-form figurative language that contain hidden connotations and potential keywords that are difficult to determine in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Therefore, in general, it is challenging for current Natural Language Processing (NLP) techniques to process lyrics successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Demonstrations of strong beats in music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (a) The downbeats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', first strong beats) in “A Little Bird” [3] in 2/4 time signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The downbeats are noted as 1 in red, and the weak beats are noted as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (b) The first and second strong beats in “Birds are Flying” [4] in 4/4 time signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The downbeats are noted as 1 in red, and the second strong beats are noted as 3 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The weak beats are noted as 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Above all, to conduct successful AI music research, computing technologies as well as deep knowledge and familiarity with music, literature, linguistics, and cognitive science are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is perhaps the most critical challenge for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Therefore, making breakthroughs in any of these challenges will contribute to AI music fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' We propose an innovative multimodal lyrics-rhythm matching approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our main contributions are as follows: Our approach computationally unveils strong, insightful correlations between lyrics and rhythm through data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our approach is a generalized method that determines lyrics-rhythm matching based on keywords without any language limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It indicates that keywords have stronger associations with strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This can be employed in any language including some that lack stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Creatively patterning lyrics-music multimodalities and using various metrics to measure their relationships in our approach can expand horizons in music cognition and computational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our approach enhances the syllabication and syllabic stress pattern identification for unknown words, which helps computational linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Audio alignment in music continues to pose many challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our research contributes to the area through the development of several novel, efficient rhythm- based audio alignment algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our research uses challenging audio instead of sheet music, which makes our solutions more flexible and more applicable to AI music research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our research can serve as a basic building block for AI music technologies, including song identification, singing voice analysis, music structure analysis, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our research can inspire connections between languages of music and linguistic representations in neuroscientific research, which are perhaps some of the most important interactions between language and music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' RELATED WORK Research on the computational analysis of music cognition and musicology are not addressed unlike other papers in Music Information Retrieval (MIR) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' As the Linguistic Stress Rule states, composers prefer to align strong beats with stressed syllables of text [1][6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' However, surprisingly few quantitative analyses can be found to support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' A few researchers conducted some basic statistical analysis and concluded that stressed syllables tend to fall on relatively strong beats of the meter in vocal music for languages with lexical stress such as English and German [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Another study on corpora of vocal French music has shown a strong correlation between the stress levels of syllables and their metrical strength [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The paper [9] presents an observational study for English popular music that performs minimal preprocessing on MusicXML2 sheets, including monosyllabic stopwords [10] removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It draws conclusions based on various correlations between categorical variables such as syllable stress, metric position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', types of beats), and stopwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It also helps provide fundamental statistical support to musicology and music-cognition research as well as a quantitative correlation between lyrics and melodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' However, this paper has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Firstly, it utilizes MusicXML files, so beat, lyrical syllable, rhythm, and pitch information have already been stored inside of the files;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' thus, extra processing is not needed, which causes additional limitations on the scope of its research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Secondly, the paper seeks for a general association between musically accented notes and stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thirdly, just monosyllabic stopwords removal is performed on its data to find the correlation between stopwords and non-salient notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' And fourthly, it lacks consideration for a variety of time signatures in their methods other than 4/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, audio alignments are among the most important steps in audio processing for MIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Audio alignment has several distinct types, including audio-to-audio, lyrics-to- audio, audio-to-score, audio-to-visual, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Many researchers have conducted relevant studies in this field [11][12], including deep learning [13] based audio alignment in [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Yet, there are very few people involved in rhythm-based audio-to-audio alignment and lyrics-to-audio alignment, which are all under rhythm alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The paper [16] proposes a multi-task learning approach for lyrics alignment that improves the alignment accuracy through the incorporation of pitch and the integration of boundary detection in the forced-alignment algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' However, their performance was improved at the cost of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Another paper [17] utilizes a multi-scale neural network based on end-to-end audio-to-character architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It improves the alignment accuracy by predicting the character probabilities end-to-end from raw audio but only if such large datasets are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In fact, most deep learning technologies (including the mentioned above) need large datasets for sufficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1 2 3 4 1 2 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='musicxml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='com/ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 1 2 can see the birds fly - ing, my eye-balls are go-ingupand downAhli - ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Ahli -ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Ali-ttlebirdflies stothetree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our approach resolves the above limitations and expands the scope by using multimodal data such as music accompaniment audio, singing vocal audio, and separate lyrics text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' We perform simple, efficient, and accurate audio processing based on multimodal data, including rhythm-based audio-to-MIDI alignment and lyrics-to-audio alignment, without the use of deep learning techniques and the need for big training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Most importantly, we investigate a correlation between the keywords and music strong beats due to the keywords’ heightened significance when compared to other words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This correlation is applicable to other languages as well, including any language that lacks stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Although some researchers have mentioned the lack of stressed syllables in particular languages [1], the importance of keywords in songs was not highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, we use customized word syllabication and keywords extraction methods as well and consider all time signatures, including simple, compound, and asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, our research places emphasis on the impact of lyrical keywords in music, making ours different from current related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' THE METHODS The proposed approach aims to investigate positive connections between lyrics and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this study, we focus on studying correlations among lyrical keywords and music strong beats as well as non-keywords and weak beats, and comparing the keywords approach to the stressed syllables approach to provide further insights into composers’ thought processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The architectural framework of the lyrics-rhythm matching approach is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The highlighted framework is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The framework consists of three major components: music pronunable patterning, lyrical syllabic patterning, and lyrics-rhythm matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since each vocal “syllable” is recorded as a change in pronunciation, we denote each vocal “syllable” as a pronunable in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In the framework, the first component consists of data input, beat tracking and patterning, rhythm alignment, pronunable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', singing vocal “syllable”) locating, and patterning with music strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The second component contains syllable splitting, syllabic stress pattern identification, pronunable and lyrical syllable matching, keyword extraction, and keyword patterning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The third component targets to match keywords and strong beats and seek insights through data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The system receives lyrical texts and audio containing the singing voice and music accompaniment as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Then, all vocal and music beats are tracked and aligned with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Pronunables are retrieved from the singing vocal audio and then patterned with the strong-beats based on rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The keywords and syllabic stress patterns are extracted from the lyrics afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The lyrical syllables need to match with the retrieved pronunables to bridge the lyrics and the beat information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Finally, the keyword strong-beat matching and overall lyrics-rhythm matching are checked using the keyword pattern and the strong-beat pattern of each pronunable associated with a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Syllable Splitting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Syllabic Stress Pattern ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Identification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Pronunable-Syllable Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Keyword Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Keyword Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Audio & Lyrics Data Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Beat Tracking & Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Rhythm Alignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Pronunable Locating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Pronunable Strong-beat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Data Analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Non-Keyword Weak Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Music Pronunable Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Lyrics Syllabic Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Lyrics-Rhythm Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Keyword Strong-Beat Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Word-based Pronunable Strong-beat Pattern (WPSP) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Keyword Pattern (KP) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Stressed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Syllables and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Strong Beat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Pronunable Strong-beat Pattern (PSP) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Pronunable- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Conversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The framework of Keyword Strong-Beat Matching little beaH In the following subsections, we explain the components of the architecture in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Music Pronunable Patterning 1) Beat Tracking and Strong Beat Patterning In this framework, the input audio pair has two separate forms: a solo-singing vocal audio and an accompaniment MIDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Singing vocal beats and music beats are tracked from these two data files, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' That is, two time series record onset locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', timestamps) of all tracked beats for vocal and music audios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Beat tracking is the task of identifying and synchronizing with the basic rhythmic pulse of a piece of music [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It has been extensively studied in MIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since our research focuses on lyrics-rhythm correlation findings, some existing music tools, such as Librosa3, are used for beat tracking in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' After beat tracking, a Strong-beat Pattern (SP) list with its length as the total number of music beats is created to store the flags of the corresponding strong music beat locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' If a beat is a strong beat, the flag is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Otherwise, the flag is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The number of strong beats and their positions in one measure can be deduced from the time signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our methods can calculate the numbers for all distinct types of beats and their positions based on time signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2) Rhythm Alignment In this study, rhythm alignment includes rhythm-based audio to-MIDI alignment and lyrics-to-MIDI alignment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', lyrics-to- accompaniment-audio alignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Due to the unavoidable imperfections of a human singer, which causes delays or early starts, the vocal rhythm of the singing audio needs to be aligned with the music rhythm of the accompaniment MIDI, which is the audio-to-MIDI alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' More specifically, the audio-to-MIDI alignment is based on two sequences, the vocal audio and music audio sequences, and uses a timing threshold or margin of error, which is set based on the tempo and time signature from the vocal audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, the length of tracked vocal beats may be different from the length of tracked music beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This could be due to the possibility of unmatched or missing beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' These types of beats are handled using linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' After alignments, an index list of vocal beats aligned with music beats is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' As for the rhythm-based lyrics-to-MIDI alignment approach, it is more complicated and consists of several steps: pronunable retrieval and locating, pronunable-beat patterning, syllable pattern identification, syllable pronunable matching, and lyrics- to-rhythm matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' These steps will be discussed further in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This approach, the rhythm-based lyrics-to- audio alignment, can be applied to any audio other than MIDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 3) Pronunable Retrieval and Locating In the vocal audio, pronunables represent every change in pronunciation regardless of whether it is an actual syllable of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This includes singing words such as “woah”, “nei”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', as well as words with multiple syllables pronounced as one such as “o’er”, which represents “over”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Such cases present challenges for lyrical analytics, as lyrics differ from speech and other forms of literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For this study, we only focus on retrieving the location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', start time and end time) of each pronunable instead of recognizing syllables or words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To retrieve pronunables on music beats, the pronunables associated with vocal beats are first retrieved from the vocal audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Once all pronunables’ locations are retrieved, the vocal beats need to align with the corresponding music beats in the accompaniment audio through the use of pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The algorithm of pronunable locating is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since the music and voice audio do not match precisely, a time buffer or margin for error is needed for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' If the music beats are within the time buffer, then the beats and the beat indices are stored accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, to optimize the algorithms, vocal-audio-to-MIDI alignment is processed at the same time with pronunable locating on music beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Specifically, for each time t1 in the pronunable time series T1, the times within the dynamic buffer are sought in the music beat time series T2 and then stored into a list Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Their corresponding indices are collected into another list Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In the time list Ts, for each time t, a certain distance function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', Euclidean distance) is employed to calculate the distance between t and t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Once distance calculations for all the times in Ts are complete, the minimum distance t* is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Then, the corresponding index i* of t* is used to point to the corresponding music beat index ip for the pronunable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' ip is inserted into the output index list L of music beats aligned with pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' If no beat is found, the corresponding beat is likely unmatched or missing, so the beat is linearly interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Algorithm 1: Pronunable Locating Input: a vocal pronunable time series T1, a vocal beat time series T2, a vocal-music alignment index list La, a time buffer dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Output: an index list of music beats aligned with pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1: Initialize L as an empty list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2: for each t1 in T1 do 3: (Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Ls ) = Search(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' T2) 4: if Ts is not empty Then 5: for i =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' length(Ts) do 6: D[i] = Distance(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Ts[i]) 7: end for 8: t* = min(D[i]) where i =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' …,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' length(Ts) 9: Get corresponding index i* of t* 10: if La is not empty and i*∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' length( La)-1] Then 11: ip = La [i*] 13: else // unmatched or missing 14: ip = LinearInterpolation(i) 15: end if 16: Insert ip into L 17: end if 18: end for 19: return L To be more mathematically specific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' the target is to find the music beat that minimizes the distance from the pronunable location to match the pronunable with the closest music beat: argmin 𝑡𝑠∈𝑇𝑠 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑡1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 𝑡𝑠) (1) 3https://librosa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='org/ Several different distance functions can be used to calculate D[i], where i = 0, 1, …, length(Ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this study, the Manhattan distance is selected for the implementation of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Given two points A and B in n-dimensional space such that A = (a1, a2, …, an) and B = (b1, b2, …, bn), the Manhattan distance between A and B is defined as: 𝐷 (𝐴, 𝐵) = ||𝐴 − 𝐵|| = ∑ |𝑎𝑖 − 𝑛 𝑖=1 𝑏𝑖| (2) Since the beat time series is a sorted time sequence, many search algorithms such as binary search [19] can be applied in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Finally, a time series representing locations of aligned pronunables is generated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 4) Pronunable Strong-Beat Patterning Upon the completion of rhythm alignment and pronunable locating on music beats, the output list contains all the beat indices where the pronunables are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Based on this index list and the SP list created through the music strong beats, a new patterned list is created to store the Pronunable Strong-beat Pattern (PSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The PSP contains the noted locations of all the pronunables on a strong beat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' If a pronunable has the same location as a strong beat, it is set to 1 in the PSP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' otherwise, it is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Lyrical Syllable Patterning In this component, we utilize customized word syllabication and keywords extraction methods to find more lyrical keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1) Syllable Splitting and Stress Pattern Identification Firstly, special preprocessing for lyrics is conducted due to the potential occurrence of cases similar to pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, some additional critical cases also impact the splitting of syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For example, despite having no syllabic contributions, the parts after the apostrophe in contractions such as “re” in “they’re” and “t” in “don’t” are still assigned a syllable by pronouncing dictionaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', the Carnegie Mellon University (CMU) pronouncing dictionary4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' As a result, a customized list of words and/or letters for lyrics preprocessing is created for these types of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Then, utilizing a pronouncing dictionary produces one or more variants of a word’s stress pattern consisting of flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For example, in the CMU dictionary, the stress pattern is expressed through 0s and 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The 1s represent the stressed syllables, whereas the 0s represent the unstressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To simplify the process in this study, only the first most common variant is appended to a list containing all the words’ stress patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Occasionally, the dictionary may not recognize certain vocabulary and newly created words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It also may create an incorrect syllable split or stress pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, after performing a basic stemming [20] for words such as ones that end in “ed”, words are split into syllables through vowel consideration instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' As a result, stress patterns are created for such words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The syllabic stress patterns are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The algorithm of syllabic stress pattern identification is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The output of the algorithm is a string list of stress patterns with each stress pattern being composed of 0s and/or 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2) Pronunable-Syllable Matching The syllable-pronunable matching is then conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The total number of lyrical syllables is counted by adding up each stress pattern’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Each syllable cannot be individually compared to the pronunables, because each pronunable expresses every change in the singer’s pronunciation in the most pronounceable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' As a result, the spelling of pronunables may vary drastically for each syllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For example, the syllables of “very” are “ve” and “ry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' However, “ry” can be pronounced as “ree” and “rye”, which both are pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this case, the pronunable for “ry” in “very” is “ree”, yet if “ree” and “ry” are compared to each other, the two will not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, comparing the total number of syllables in the lyrics to the actual number of pronunables in a song is more accurate for the matching between pronunables and lyrical syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is the key step in rhythm-based lyrics-to- audio alignment as mentioned in Section A Rhythm Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Algorithm 2: Syllabic Stress Pattern Identification Input: Lyrics text S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Output: A string list of stress patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1: Initialize L as an empty list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 2: S’ = Preprocessed S with the special lyrics cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='3: A list T = Tokenize(RemovePunctuation(S’)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4: for each token in T do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Get token’s stress pattern from the dictionary D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='if stress pattern exists Then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Append 1st stress variant as a string to L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='if token’s end has no syllabic meaning Then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Stem the token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='A list Ls = WordSyllabication(token) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Set the first syllable pattern s to 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='Append s to L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='16: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='17: return L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='3) Keyword Extraction and Patterning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='The lyrical keywords in the text are extracted through NLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, the patterns of the keywords are identified and used to build the Keyword Pattern (KP) flag list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since KP is now word-based yet PSP is still pronunable-based, PSP needs to be converted to a word-based pattern, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', Word- based Pronunable Strong-beat Pattern (WPSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' WPSP is then used to compare with KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is one of the key components for finding the correlation between the keywords and strong beats, not the stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Lyrics-Rhythm Matching Finally, two patterns, WPSP and KP as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 3, are checked for whether the lyrics match the rhythm, including whether keywords land on strong beats, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' There are multiple ways to find whether the lyrics match the rhythm, including the utilization of time series similarity and conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1) Conditional probability Clearly, the conditional probability [21] can be used for checking the matching probability of lyrics and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For example, to check if keywords land on strong beats, let A represent the keywords, let B represent the strong beats, let 𝑃(𝐴 ∩ 𝐵) represent the probability of overall keywords 4http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='edu/cgi-bin/cmudict landing on strong beats, let 𝑃(𝐵) represent the probability of strong beats occurring out of all beats, and let 𝑃(𝐴 | 𝐵) represent the likelihood of keywords occurring on the occurrence of strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' According to the conditional probability formula, 𝑃(𝐴 | 𝐵) can be calculated as 𝑃(𝐴 | 𝐵) = 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐵) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (3) 2) Cosine Similarity Cosine similarity [22] is one of the similarity measures for a vector space model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It measures the cosine of an angle formed by the projection of two vectors in a multi-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To be more specific, given two vectors, A and B, representing two binary patterns, such as KP and WPSP, respectively, the cosine similarity is utilized to measure the similarity of two patterns as shown in the following formula: 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 = 𝑐𝑜𝑠(𝜃) = 𝐴⋅𝐵 ||𝐴|| ||𝐵|| (4) In Equation (4), ||𝐴|| is the Euclidean norm of vector A = (a1, a2, …, an), defined as √𝑎12 + 𝑎22 + ⋯ + 𝑎𝑛2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Conceptually, it is the length of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Similarly, ||𝐵|| is the Euclidean norm of vector B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The cosine similarity is a real number between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The higher the similarity, the smaller the angle is between two vectors, A and B, in the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' That is, the closer the two vectors are, the better the match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Otherwise, a lower similarity represents a lower chance of being matched with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Unlike conditional probability, cosine similarity is an overall matching check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It can check both whether keywords land on strong beats as well as whether non-keywords and weak beats match at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this paper, we use both matching methods for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' EXPERIMENTAL RESULTS AND ANALYSIS This study examines lyrics-rhythm matching in using quantitative data gathered from a systematically selected dataset of songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our system is scalable and can be used for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this experiment, we use the Children’s Song Dataset (CSD) [23] to pilot our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This dataset consists of 50 English songs and 50 Korean songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Each song is sung in two different keys by a professional female singer stored in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='wav file format, and each also includes a MIDI piano file, an English lyrics text file, and text annotations in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='csv file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' For English songs, there are 100 MIDI and 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='wav files in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The text annotations record every change in the singer’s pronunciation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', pronunables) using the Korean grapheme system, which differs from both the English grapheme system and the English syllabic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is the reason behind our introduction of pronunables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, each song contains a silence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The tempo and time signature are retrieved from the metadata of vocal files beforehand since it is already noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' From the MIDI files, only the beats are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' We use Pretty MIDI5 to process the MIDI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The plot of beat locations over the waveform after beat tracking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 4 shows the tracked beats that are distributed over the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In this example of the song file, the graph’s amplitude range is (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0), and the time spans over one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The gray lines represent every identified beat in the singing vocal audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Based on the gray lines, some of the beats at the end are not tracked, which can be due to the duration of the last note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This does not affect our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Beat tracking for the music MIDI and visualization with the piano roll of notes and the song’s strong beats Likewise, beat tracking is applied to the accompaniment music MIDI file separately as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The note range span an octave (from C4 to C5) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The red lines represent the first strong beats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', downbeats) from each measure and the gray lines represent all other beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Both lines include the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' All the beats are perfectly aligned, which is because computer-generated MIDI files are error-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The piano roll of notes and all unaligned vocal beats In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 6, both vocal beats and music beats are drawn on the same piano roll graph, which also includes the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' It clearly indicates that the aqua-colored beat lines appear to be misaligned with the music beats in some areas, particularly during the silent interval in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This shows that each slight delay in the singer’s voice accumulates over time, resulting in larger misalignments until the singer is onbeat again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Rhythm alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The music beats are in gray, and the unaligned vocal beats are in aqua in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Aligned vocal beats are in orange in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 7, (a) displays the unaligned vocal beats with music beats in the waveform and (b) shows the aligned vocal beats with music beats after rhythm alignment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', rhythm-based audio-to-MIDI alignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' From the figures, our alignment algorithm has demonstrated a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='com/craffel/pretty-midi In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 8 (a), it indicates the original vocal beats and original pronunable locations on the singing vocal wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The misalignment and early cutoff are still shown through the gray lines as they are from the singing vocal file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Upon completion of pronunable locating in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 8 (b), the pronunable beats are matched with the music beats since every vocal beat is perfectly aligned with the music strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 8 (c) further visualizes the aligned and matched pronunable beats on the piano roll with notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The red lines indicate the first and second strong beats from the MIDI file and the gray lines indicate all the strong beats from the MIDI file, including places where no note is plotted, such as the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' On the other hand, the red lines only show the beats which pronunables are aligned with and therefore do not include the rest period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Pronunable locating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (a) The original unaligned vocal beats (gray) and pronunable beats (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (b) Aligned and matched pronunable beats with music strong beats (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' (c) Aligned pronunable beats and all matched corresponding music strong beats (red) on the piano roll of notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10 show various conditional probabilities given strong beats in several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Only the first key is marked, but there are 100 songs displayed in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9 describes the likelihood of the keywords occurring on the occurrence of strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The bar chart compares the probabilities of each song with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The scatter plot presents a distribution of the probabilities over all of the song indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The histogram displays the count of each probability range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To compare with the paper [9], we also performed a general association between musically accented notes and stressed syllables using the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10 indicates such likelihood of the stressed syllables landing on strong beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' More specifically, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9 (a), based on the bar chart (a) and scatter chart (b), most of the probabilities concentrated in two clusters: Cluster 1, which is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8, and Cluster 2, which is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9 (c) indicates that around 12% of songs have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0 matching probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=', all keywords landing on strong beats) and around 30% of songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 or higher matching probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, it can be concluded that keywords are frequently associated with strong beats and that strong beats help emphasize keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' By comparing with keyword-strong-beat matching as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10 indicates there are two probability concentrations for stressed syllables landing on strong beats: Cluster 1, which is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7, and Cluster 2, which is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, around 20% of songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 or higher matching probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Likewise, it shows that stressed syllables are also frequently associated with strong beats and that strong beats help emphasize stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' However, it is evident that keywords have stronger associations with Strong Beats (SB) than stressed syllables with strong beats as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10 as well as summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This is because in many pieces of music, there are instances where the keyword lands on a strong beat, yet certain stressed syllables do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our experimental results reveal an average of an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='81 matching probability for our proposed approach, and around 30% of the songs have a match between keywords and strong beats that is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9, including 12% songs with all keywords landing on strong beats (probability = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Regarding the strong beat matching probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 or lower, only 8% of the songs are under this probability for keywords, whereas for stressed syllables, it is 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' We appreciate the CSD dataset, but it still has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' There are occasional typos and mispronunciations in the lyrics, so our methods are sometimes unable to recognize these differences, which affects the matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' To compare both cases for overall matching, cosine similarity in Formula 4 is utilized in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 11 (a) shows the overall similarity histogram for two patterns, WPSP and KP, related to keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Since each pattern only includes 1 or 0, it can be considered as a binary vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1 in the WPSP means that at least one of the pronunables of a word land on a strong beat while 0 means that all of the pronunables of a word land on at least one weak beat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' But as for KP, 1 represents a keyword while 0 represents a non-keyword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In Table II and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 11 (a), it clearly indicates that 46% of songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 or higher similarity between WPSP and KP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' That is, keywords tend to land on strong beats and non-keywords tend to land on weak beats for overall matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Additionally, only around 7% of the songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 or lower similarity between WPSP and KP, which reinforces the strong overall matching tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 11 (b) shows the similarity histogram for two patterns, PSP and the Stressed Syllable Strong-beat Pattern (SSSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Both are binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 1 in the stressed syllable pattern means a stressed syllable landing on a strong beat while 0 means unstressed syllable landing on a certain beat (a strong or weak beat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In the figure, it shows that 21% of the songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 or higher similarity between two patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Furthermore, from Table II, about 28% of songs have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 or lower similarity between PSP and SSSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10, it can be further concluded that keywords have a stronger association with strong beats than stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK This paper proposes a novel multimodal lyrics-rhythm matching approach that uses creative and efficient rhythm alignment, pronunable locating, syllabic stress identification, lyrical keyword extraction, and various patterning methods to computationally unveil strong correlations between lyrics and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Our approach can be applied to any language without limitations because we place emphasis on keywords, not 121 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' SIMILARITY COMPARISONS BETWEEN KEYWORDS AND STRESSED SYLLABLES LANDING ON STRONG BEATS Similarity Keywords w/ SBs Stressed Syllables w/ SBs Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='62 <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 7% 28% >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 46% 21% >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='80 7% 4% (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The probabilities of stressed syllables landing on strong beats (a) The probabilities of stressed syllables landing on strong beats (b) The probability distribution of stressed syllables landing on SBs (c) The probability histogram of stressed syllables landing on SBs TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' PROBABILITY COMPARISONS BETWEEN KEYWORDS AND STRESSED SYLLABLES LANDING ON STRONG BEATS Probability Keywords w/ SBs Stressed Syllables w/ SBs Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='77 Cluster 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='60 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='70 Cluster 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='90 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='75 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85 >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='90 27% of songs 20% of songs < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 8% of songs 18% of songs (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The probabilities of keywords landing on strong beats (100 songs) (a) The probabilities of keywords landing on strong beats (b) The probability distribution of keywords landing on strong beats (c) The probability histogram of keywords landing on strong beats (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' The similarity histograms of KP and SSSP for all songs (a) The similarity histogram of keywords landing on strong beats (b) The similarity histogram of stressed syllables landing on strong beats 1 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 en001a en002b en004a en005b en007a en008b en010a en011b en013a en014b en016a en017b en019a en020b en022a en023b en025a en026b en028a en029b en031a en032b en034a en035b en037a en038b en040a en041b en043a en044b en046a en047b en049a en050b Song Name100 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='95 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 Song 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='75 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 1 Probability30 Probability ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85 25 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='95 20 ■ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='0 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 unt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 15 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='75 5 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 1 Probability1 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 en001a en002b en004a en005b en007a en008b en010a en011b en013a en014b en016a en017b en019a en020b en022a en023b en025a en026b en028a en029b en031a en032b en034a en035b en037a en038b en040a en041b en043a en044b en046a en047b en049a en050b SongName100 Probability 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 Index 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 Song 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 1 Probability30 Probability ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85 25 ■0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 5 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 1 Probability30 Similarity ■ 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='45 5 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 1 Similarity30 Similarity ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='7 25 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 20 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='65 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='55 unt ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='85 15 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='75 ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='45 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 5 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='8 1 Similarity stressed syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, our experimental results show that both stressed syllables and keywords tend to land on strong beats, with the matching between keywords and strong beats having a higher probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' But most importantly, our method uses audio instead of sheet music (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' MusicXML) with metadata, which is more challenging because information is not readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Using audio broadens our impact and extends our method to many more horizons and opportunities that are yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' There are some limitations on our study, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' One limitation is our reliability on other libraries and dictionaries for some aspects of our study such as beat tracking and syllabication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' In addition, there is a lack of similar audio datasets with pronunable annotations, so the size of the data used for this study is small even though our system is designed to fit the large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Thus, future work includes adding singing audio recognition and automatic pronunable annotations to expand the method’s use to larger and more different datasets, which reduces the overall human error as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Furthermore, we will work on lessening and eventually completely eliminating the reliance on other libraries or dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' This approach may also be used to generate more accurate music scores from singing audio because in our study, the results have shown that our establishment of a precise connection between the lyrical keywords and strong beats is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' Moreover, our methods can be improved upon in the future to determine the more appropriate syllabic stress pattern variant for various situations based on part-of-speech tagging using NLP techniques, as there may be multiple stress patterns for one word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQf4AJQ/content/2301.02732v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': 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b/a9AzT4oBgHgl3EQfZfxk/content/tmp_files/2301.01352v1.pdf.txt @@ -0,0 +1,1128 @@ +WLD-Reg: A Data-dependent Within-layer Diversity Regularizer +Firas Laakom1* Jenni Raitoharju,2 Alexandros Iosifidis,3 Moncef Gabbouj 1 +1 Faculty of Information Technology and Communication Sciences, Tampere University, Finland +2 Faculty of Information Technology, University of Jyv¨askyl¨a, Finland +3 DIGIT, Department of Electrical and Computer Engineering, Aarhus University, Denmark +Abstract +Neural networks are composed of multiple layers arranged in +a hierarchical structure jointly trained with a gradient-based +optimization, where the errors are back-propagated from the +last layer back to the first one. At each optimization step, neu- +rons at a given layer receive feedback from neurons belong- +ing to higher layers of the hierarchy. In this paper, we pro- +pose to complement this traditional ’between-layer’ feedback +with additional ’within-layer’ feedback to encourage the di- +versity of the activations within the same layer. To this end, +we measure the pairwise similarity between the outputs of the +neurons and use it to model the layer’s overall diversity. We +present an extensive empirical study confirming that the pro- +posed approach enhances the performance of several state- +of-the-art neural network models in multiple tasks. The code +is publically available at https://github.com/firasl/AAAI-23- +WLD-Reg +Introduction +Deep learning has been extensively used in the last decade +to solve several tasks (Krizhevsky, Sutskever, and Hinton +2012; Golan and El-Yaniv 2018; Hinton et al. 2012a). A +deep learning model, i.e., a neural network, is formed of +a sequence of layers with parameters optimized during the +training process using training data. Formally, an m-layer +neural network model can be defined as follows: +f(x; W) = φm(W m(φm−1(· · · φ2(W 2φ1(W 1x)))), (1) +where φi(.) is the non-linear activation function of the ith +layer and W = {W 1, . . . , W m} are the model’s weights. +Given a training data {xi, yi}N +i=1, the parameters of f(x; W) +are obtained by minimizing a loss ˆL(·): +ˆL(f) = 1 +N +N +� +i=1 +l +� +f(xi; W), yi +� +. +(2) +However, neural networks are often over-parameterized, +i.e., have more parameters than data. As a result, they tend +to overfit to the training samples and not generalize well +on unseen examples (Goodfellow et al. 2016). While re- +search on double descent (Advani, Saxe, and Sompolinsky +*This work was supported by NSF-Business Finland Center for +Visual and Decision Informatics (CVDI) project AMALIA. +2020; Belkin et al. 2019; Nakkiran et al. 2020) shows that +over-parameterization does not necessarily lead to overfit- +ting, avoiding overfitting has been extensively studied (Dzi- +ugaite and Roy 2017; Foret et al. 2020; Nagarajan and Kolter +2019; Neyshabur et al. 2018; Poggio et al. 2017; Grari et al. +2021) and various approaches and strategies, such as data +augmentation (Goodfellow et al. 2016; Zhang et al. 2018), +regularization (Arora et al. 2019; Bietti et al. 2019; Kukaˇcka, +Golkov, and Cremers 2017; Ouali, Hudelot, and Tami 2021; +Han and Guo 2021), and Dropout (Hinton et al. 2012b; Lee +et al. 2019; Li, Gong, and Yang 2016; Wang et al. 2019), +have been proposed to close the gap between the empirical +loss and the expected loss. +Diversity of learners is widely known to be important +in ensemble learning (Li, Yu, and Zhou 2012; Yu, Li, and +Zhou 2011) and, particularly in deep learning context, di- +versity of information extracted by the network neurons has +been recognized as a viable way to improve generalization +(Xie, Liang, and Song 2017; Xie, Deng, and Xing 2015b). +In most cases, these efforts have focused on making the set +of weights more diverse (Yang, Gkatzelis, and Stoyanovich; +Malkin and Bilmes 2009). However, diversity of the acti- +vations has not received much attention. Here, we argue that +due to the presence of non-linear activations, diverse weights +do not guarantee diverse feature representation. Thus, we +propose focusing on the diversity on top of feature mapping +instead of the weights. +To the best of our knowledge, only (Cogswell et al. 2016; +Laakom et al. 2021a) have considered diversity of the ac- +tivations directly in the neural network context. The work +in (Laakom et al. 2021a) studied theoretically how diver- +sity affects generalization showing that it can reduce over- +fitting. The work in (Cogswell et al. 2016) proposed an ad- +ditional loss term using cross-covariance of hidden activa- +tions, which encourages the neurons to learn diverse or non- +redundant representations. The proposed approach, known +as DeCov, was empirically proven to alleviate overfitting +and to improve the generalization ability of neural network. +However, modeling diversity as the sum of the pairwise +cross-covariance, it is not scale-invariant and can lead to +trivial solutions. Moreover, it can capture only the pairwise +diversity between components and is unable to capture the +”higher-order diversity”. +In this work, we propose a novel approach to encour- +arXiv:2301.01352v1 [cs.LG] 3 Jan 2023 + +age activation diversity within the same layer. We pro- +pose complementing the ’between-layer’ feedback with ad- +ditional ’within-layer’ feedback to penalize similarities be- +tween neurons on the same layer. Thus, we encourage each +neuron to learn a distinctive representation and to enrich the +data representation learned within each layer. We propose +three variants for our approach that are based on different +global diversity definitions. +Our contributions in this paper are as follows: +• We propose a new approach to encourage the ’diversifi- +cation’ of the layers’ output feature maps in neural net- +works. The proposed approach has three variants. The +main intuition is that, by promoting the within-layer ac- +tivation diversity, neurons within a layer learn distinct +patterns and, thus, increase the overall capacity of the +model. +• We show empirically that the proposed within-layer ac- +tivation diversification boosts the performance of neural +networks. Experimental results on several tasks show that +the proposed approach outperforms competing methods. +Within-layer Diversity Regularizer +In this section, we propose a novel diversification strategy, +where we encourage neurons within a layer to activate in a +mutually different manner, i.e., to capture different patterns. +In this paper, we define as “feature layer” the last interme- +diate layer in a neural network. In the rest of the paper, we +focus on this layer and propose a data-dependent regularizer +which forces each unit within this layer to learn a distinct +pattern and penalizes the similarities between the units. In- +tuitively, the proposed approach reduces the reliance of the +model on a single pattern and, thus, can improve generaliza- +tion. +We start by modeling the global similarity between two +units. Let φn(xj) and φm(xj) be the outputs of the nth and +mth unit in the feature layer for the same input sample xj. +The similarity snm between the the nth and mth neurons +can be obtained as the average similarity measure of their +outputs for N input samples. We use the radial basis function +to express the similarity: +snm := 1 +N +N +� +j=1 +exp +� +− γ||φn(xj) − φm(xj)|| +� +, +(3) +where γ is a hyper-parameter. The similarity snm can be +computed over the whole dataset or batch-wise. Intuitively, +if two neurons n and m have similar outputs for many sam- +ples, their corresponding similarity snm will be high. Oth- +erwise, their similarity smn is small and they are considered +“diverse”. +Next, based on these pairwise similarities, we propose +three variants for obtaining the overall similarity J of all the +units within the feature layer: +• Direct: J := � +n̸=m snm. In this variant, we model the +global layer similarity directly as the sum of the pairwise +similarities between the neurons. By minimizing their +sum, we encourage the neurons to learn different repre- +sentations. +• Det: J := −det(S), where S is a similarity matrix de- +fined as Snm = snm. This variant is inspired by the +Determinantal Point Process (DPP) (Kulesza and Taskar +2010, 2012), as the determinant of S measures the global +diversity of the set. Geometrically, det(S) is the volume +of the parallelepiped formed by vectors in the feature +space associated with s. Vectors that result in a larger +volume are considered to be more “diverse”. Thus, max- +imizing det(·) (minimizing −det(·)) encourages the di- +versity of the learned features. +• Logdet: J := −logdet(S)1. This variant has the same +motivation as the second one. We use Logdet instead of +Det as Logdet is a convex function over the positive def- +inite matrix space. +It should be noted here that the first proposed variant, +i.e., direct, similar to DeCov (Cogswell et al. 2016), cap- +tures only the pairwise similarity between components and +is unable to capture the higher-order “diversity”, whereas the +other two variants consider the global similarity and are able +to measure diversity in a more global manner. Promoting di- +versity of activations within a layer can lead to tighter gen- +eralization bound and can theoretically decrease the gap be- +tween the empirical and the true risks (Laakom et al. 2021a). +The proposed global similarity measures J can be min- +imized by using them as an additional loss term. However, +we note that the pair-wise similarity measure snm, expressed +in equation 3, is not scale-invariant. In fact, it can be triv- +ially minimized by making all activations of the feature layer +high, i.e., by multiplying by a high scaling factor, which has +no effect on the performance, since the model can rescale +high activations to normal values simply by learning small +weights on the next layer. To alleviate this problem, we pro- +pose an additional term, which penalizes high activation val- +ues. The total proposed additional loss is defined as follows: +ˆLW LD−Reg := λ1J + λ2 +N +� +i=1 +||Φ(xi)||2 +2, +(4) +where Φ(x) = [φ1(x), · · · , φC(x)] is the feature vector, +C is the number of units within the feature layer, and λ1 +and λ2 are two hyper-parameters controlling the contribu- +tion of each term to the diversity loss. Intuitively, the first +term of equation 4 penalizes the similarity between the units +and promotes diversity, whereas the second term ensures the +scale-invariance of the proposed regularizer. +The total loss function ˆL(f) defined in equation 2 is aug- +mented as follows: +ˆLaug(f) := ˆL(f) + ˆLW LD−Reg +(5) += ˆL(f) + λ1J + λ2 +N +� +i=1 +||Φ(xi)||2 +2. +The proposed approach is summarized in Algorithm 1. We +note that our approach can be incorporated in a plug-and- +1This is defined only if S is positive definite. It can be shown +that in our case S is positive semi-definite. Thus, in practice, we use +a regularized version (S + ϵI) to ensure the positive definiteness. + +Algorithm 1: One epoch of training with WLD-Reg +Model: Given a neural network f(·) with a feature repre- +sentation φ(·), i.e., last intermediate layer. +Input: Training Data: {xi, yi}N +i=1 +Parameters: λ1 and λ2 in equation 4 +1: for every mini-batch: {xi, yi}m +i=1 ∈ {xi, yi}N +i=1 do +2: +Forward pass the inputs {xi}m +i=1 into the model to +obtain the outputs {f(xi)}m +i=1 and the feature repre- +sentations {Φ(xi)}m +i=1 +3: +Compute the standard loss ˆL(f) (equation 2). +4: +Compute the extra loss ˆLW LD−Reg (equation 4). +5: +Compute the total loss ˆLaug(f) (equation 5) +6: +Compute the gradient of the total loss and use it to +update the weights of f. +7: end for +8: return Return f. +play manner into any neural network-based approach to aug- +ment the original loss and to ensure learning diverse fea- +tures. We also note that although in this paper, we focus only +on applying diversity regularizer to a single layer, i.e., the +feature layer, our proposed diversity loss, as in (Cogswell +et al. 2016), can be applied to multiple layers within the +model. +Our newly proposed loss function defined in equation 5 +has two terms. The first term is the classic loss function. It +computes the loss with respect to the ground-truth. In the +back-propagation, this feedback is back-propagated from the +last layer to the first layer of the network. Thus, it can be +considered as a between-layer feedback, whereas the second +term is computed within a layer. From equation 5, we can +see that our proposed approach can be interpreted as a regu- +larization scheme. However, regularization in deep learning +is usually applied directly on the parameters, i.e., weights +(Goodfellow et al. 2016; Kukaˇcka, Golkov, and Cremers +2017), while in our approach a data-dependent additional +term is defined over the output maps of the layers. For a fea- +ture layer with C units and a batch size of m, the additional +computational cost is O(C2(m + 1)) for Direct variant and +O(C3 + C2m)) for both Det and Logdet variants. +Related work +Diversity promoting strategies have been widely used in +ensemble learning (Li, Yu, and Zhou 2012; Yu, Li, and Zhou +2011), sampling (Bıyık et al. 2019; Derezinski, Calandriello, +and Valko 2019; Gartrell et al. 2019), energy-based mod- +els (Laakom et al. 2021b; Zhao, Mathieu, and LeCun 2017), +ranking (Gan et al. 2020; Yang, Gkatzelis, and Stoyanovich), +pruning by reducing redundancy (He et al. 2019; Kondo and +Yamauchi 2014; Lee et al. 2020; Singh et al. 2020), and +semi-supervised learning (Zbontar et al. 2021). In the deep +learning context, various approaches have used diversity as +a direct regularizer on top of the weight parameters. Here, +we present a brief overview of these regularizers. Based on +the way diversity is defined, we can group these approaches +into two categories. The first group considers the regulariz- +ers that are based on the pairwise dissimilarity of the compo- +nents, i.e., the overall set of weights is diverse if every pair of +weights is dissimilar. Given the weight vectors {wm}M +m=1, +(Yu, Li, and Zhou 2011) defines the regularizer as � +mn(1− +θmn), where θmn represents the cosine similarity between +wm and wn. In (Bao et al. 2013), an incoherence score de- +fined as − log +� +1 +M(M−1) +� +mn β|θmn| +1 +β +� +, where β is a pos- +itive hyperparameter, is proposed. In (Xie, Deng, and Xing +2015a; Xie, Zhu, and Xing 2016), mean(θmn) − var(θmn) +is used to regularize Boltzmann machines. The authors the- +oretically analyzed its effect on the generalization error +bounds in (Xie, Deng, and Xing 2015b) and extend it to +kernel space in (Xie, Liang, and Song 2017). The second +group of regularizers considers a more global view of diver- +sity. For example, in (Malkin and Bilmes 2008, 2009; Xie, +Singh, and Xing 2017), a weight regularization based on the +determinant of the weights’ covariance is proposed based +on determinantal point process (Kulesza and Taskar 2012; +Kwok and Adams 2012). +Unlike the aforementioned methods which promote diver- +sity on the weight level and similar to our method, (Cogswell +et al. 2016; Laakom et al. 2022) proposed to enforce dissim- +ilarity on the feature map outputs, i.e., on the activations. +To this end, they proposed an additional loss based on the +pairwise covariance of the activation outputs. Their addi- +tional loss, LDecov, is defined as the squared sum of the non- +diagonal elements of the global covariance matrix C of the +activations: +LDecov = 1 +2(||C||2 +F − ||diag(C)||2 +2), +(6) +where ||.||F is the Frobenius norm. Their approach, De- +cov, yielded superior empirical performance. However, cor- +relation is highly sensitive to noise (Kim, Kim, and Erg¨un +2015), as opposite to the RBF-based distance used in our +approach (Savas and Dovis 2019; Haykin 2010). Moreover, +the Decov approach only captures the pairwise diversity be- +tween the components, whereas we propose variants of our +approach which consider a global view of diversity. More- +over, based on the cross-covariance, their approach i-s not +scale-invariant. In fact, it can be trivially minimized by mak- +ing all activations in the latent representation small, which +has no effect on the generalization since the model can +rescale tiny activations to normal values simply by learning +large weights on the next layer. +Experimental results +CIFAR10 & CIFAR100 +We start by evaluating our proposed diversity approach on +two image datasets: CIFAR10 and CIFAR100 (Krizhevsky, +Hinton +et +al. +2009). +They +contain +60,000 +(50,000 +train/10,000 test) 32 × 32 images grouped into 10 and 100 +distinct categories, respectively. We split the original train- +ing set (50,000) into two sets: we use the first 40,000 images +as the main training set and the last 10,000 as a validation +set for hyperparameters optimization. We use our approach +on two state-of-the-art CNNs: + +• ResNext-29-08-16: we consider the standard ResNext +Model (Xie et al. 2017) with a 29-layer architecture, a +cardinality of 8, and a width of 16. +• ResNet50: we consider the standard ResNet model (He +et al. 2016) with 50 layers. +We compare against the standard networks2 as well as net- +works trained with the DeCov diversity strategy (Cogswell +et al. 2016). All the models are trained using stochastic gra- +dient descent (SGD) with a momentum of 0.9, weight decay +of 0.0001, and a batch size of 128 for 200 epochs. The initial +learning rate is set to 0.1 and is then decreased by a factor of +5 after 60, 120, and 160 epochs, respectively. We also adopt +a standard data augmentation scheme that is widely used for +these two datasets (He et al. 2016; Huang et al. 2017). For +all models, the additional diversity term is applied on top the +last intermediate layer. The penalty coefficients λ1 and λ2, +in equation 4, for our approach and the penalty coefficient +of Decov are chosen from {0.0001, 0.001, 0.01, 0.1}, and γ +in the radial basis function is chosen from {1, 10}. For each +approach, the model with the best validation performance is +used in the test phase. We report the average performance +over three random seeds. +Table 1 reports the average top-1 errors of the different +approaches with the two basis networks. We note that, com- +pared to the standard approach, employing a diversity strat- +egy consistently boosts the results for all the two models and +that our approach consistency outperforms both competing +methods (standard and DeCov) in all the experiments. With +ResNet50, the three variants of our proposed approach sig- +nificantly reduce the test errors compared to standard ap- +proach over both datasets: 0.51% − 0.63% improvement on +CIFAR10 and 1.25% − 1.44% on CIFAR100. +For CIFAR10, the best performance is achieved by the di- +rect variant and the Logdet variant for ResNext and ResNet +models, respectively. For example, with ResNext, our direct +variant yields 0.65 boost compared to the standard approach +and 0.54 boost compared to DeCov. For CIFAR100, the best +performance is acheived by our Logdet variant for both mod- +els. This variant leads to 1.4% and 0.85% boost for ResNet +and ResNext, respectively. Overall, our three variants con- +sistently outperform DeCov and standard approach in all +testing configurations. +ImageNet +To further demonstrate the effectiveness of our approach +and its ability to boost the performance of state-of-the- +art neural networks, we conduct additional image classi- +fication experiments on the ImageNet-2012 classification +dataset (Russakovsky et al. 2015) using four different mod- +els: ResNet50 (He et al. 2016), Wide-ResNet50 (Zagoruyko +and Komodakis 2016), ResNeXt50 (Xie et al. 2017), and +ResNet101 (He et al. 2016). The diversity term is applied on +the last intermediate layer, i.e., the global average pooling +layer for both DeCov and our method. +2For the standard approach, the only difference is not using +an additional diversity loss. The remaining regularizers, data aug- +mentation, weight decay etc., are all applied as specified per- +experiment. +For the hyperparameters, we fix λ1 = λ2 = 0.001 and +γ = 10 for all the different approaches. The Scope of this pa- +per is feature diversity. However, in this experiment, we also +report results with weight diversity approaches. In particular, +we compare with the methods in (Yu, Li, and Zhou 2011), +(Xie, Deng, and Xing 2015b), (Rodr´ıguez et al. 2016), and +(Ayinde, Inanc, and Zurada 2019). +We use the standard augmentation practice for this dataset +as in (Zhang et al. 2018; Huang et al. 2017; Cogswell et al. +2016). All the models are trained with a batch size of 256 for +100 epoch using SGD with Nesterov Momentum of 0.9. The +learning rate is initially set to 0.1 and decreases at epochs 30, +60, 90 by a factor of 10. +Table 2 reports the test errors of the different approaches +on ImageNet dataset. As it can be seen, feature diversity +(our approach and DeCov) reduces the test error of the +model and yields a better performance compared to the stan- +dard approach. We note that, as opposite to feature diver- +sity, weight diversity does not always yield performance im- +provement and it can sometimes hurt generalization. Com- +pared to decov, our three variants consistently reach better +performance. +For ResNet50 and ResNeXt50, the best performance is +achieved by our direct variant, yielding more than 0.5% im- +provement compared to standard approach for both mod- +els. For Wide-ResNet50 and ResNet101, our Det variant +yields the top performance with over 0.6% boost for Wide- +ResNet50. We note that our approach has a small additional +time cost. For example for ResNet50, our direct, Det and +Logdet variants take only 0.29%, 0.39%, and 0.49% extra +training time, respectively. +Sensitivity analysis +To further investigate the effect of the proposed diversity +strategy, we conduct a sensitivity analysis using ImageNet +on the hyperparameters of our methods: λ1 and λ2 which +controls the contribution of the global diversity term to the +global loss. We analyse the effect of the two parameters on +the final performance of ResNet50 on ImageNet dataset. The +analysis is presented in Figure 1. +As shown in Figure 1, using a diversity strategy, i.e., three +variants of our method, consistently outperform the stan- +dard approach and are robust to the hyperparameters. For +the Direct variant, the best performance is reached with +λ1 = 0.005 and λ2 = 0.001. With this configuration, the +model achieve 0.71% improvement compared to the stan- +dard approach. For the Det and the Logdet variants, using +λ1 = 0.001 and λ2 = 0.0005, the model reaches the lowest +error rate (23.09%) corresponding to 0.75% accuracy boost. +Emphasizing diversity and using a high weights (λ1 and λ2) +still lead to better results compared to standard approach but +can make the total loss dominated by the diversity term. In +general, we recommend using λ1 = λ2 = 0.001. However, +this depends on the problem at hand. +Feature diversity reduces overfitting +In (Laakom et al. 2021a; Cogswell et al. 2016), it has been +observed that feature diversity can reduce overfitting. To +study the effect of feature diversity on the generalization + +Table 1: Classification errors of the different approaches on CIFAR10 and CIFAR100 with three different models. Results are +averaged over three random seeds. +ResNext-29-08-16 +ResNet50 +method +CIFAR10 +CIFAR100 +CIFAR10 +CIFAR100 +Standard +6.93 ± 0.10 +26.73 ± 0.10 +8.28 ± 0.41 +33.39 ± 0.42 +DeCov +6.82 ± 0.15 +26.70 ± 0.10 +8.03 ± 0.11 +32.26 ± 0.22 +Ours(Direct) +6.28 ± 0.11 +26.20 ± 0.18 +7.77 ± 0.09 +32.09 ± 0.11 +Ours(Det) +6.51 ± 0.16 +26.35 ± 0.23 +7.75 ± 0.12 +32.14 ± 0.28 +Ours(Logdet) +6.38 ± 0.08 +25.88 ± 0.21 +7.65 ± 0.10 +31.99 ± 0.05 +Table 2: Performance of different models with different diversity strategies on ImageNet dataset +ResNet50 +Wide-ResNet50 +ResNeXt50 +ResNet101 +Standard +23.84 +22.42 +22.70 +22.33 +(Yu, Li, and Zhou 2011) +23.87 +22.48 +22.57 +22.23 +(Ayinde, Inanc, and Zurada 2019) +23.95 +22.41 +22.67 +22.36 +(Rodr´ıguez et al. 2016) +24.23 +22.70 +22.80 +23.10 +(Xie, Deng, and Xing 2015b) +23.79 +22.66 +22.64 +22.71 +DeCov +23.62 +22.68 +22.57 +22.31 +Ours(Direct) +23.24 +21.95 +22.25 +22.14 +Ours(Det) +23.34 +21.75 +22.44 +21.87 +Ours(Logdet) +23.32 +21.96 +22.40 +22.04 +gap, in Table 3, we report the final training errors and the +generalization gap, i.e., training accuracy - test accuracy +for the different feature diversity approaches on ImageNet +dataset. +Table 3: Generalization Gap, i.e., training error - test error, +of different models with different diversity strategies on Im- +ageNet dataset. * denotes our approach. +ERM +DeCov +direct* +det* +logdet* +ResNet50 +2.87 +2.70 +1.15 +1.23 +1.21 +Wide-ResNet50 +6.33 +6.34 +4.44 +4.34 +4.58 +ResNeXt50 +5.99 +5.85 +4.41 +4.59 +4.48 +ResNet101 +4.64 +4.61 +3.68 +3.38 +3.71 +As shown in Table 3, we note that using diversity indeed +can reduces overfitting and decreases the empirical general- +isation gap of neural networks. The three variants of our ap- +proach significantly reduces overfitting for all the four mod- +els by more than 1% compared standard and DeCov for all +the models. For example, our Det variant reduces the empir- +ical generalization gap, compared to the standard approach +and DeCov, by 2% for Wide-ResNet model and over 1.2% +for the ResNet101 model. +MLP-based models +Beyond CNN models, we also evaluate the performance +of our diversity strategy on modern attention-free, multi- +layer perceptron (MLP) based models for image classifica- +tion (Tolstikhin et al. 2021; Liu et al. 2021; Lee-Thorp et al. +2021). Such models are known to exhibit high overfitting +and require regularization. We evaluate how diversity affects +the accuracy of such models on CIFAR10. In particular, we +conduct a simple experiment using two models: MLP-Mixer +(Tolstikhin et al. 2021), gMLP (Liu et al. 2021) with four +blocks each. +For the diversity strategies, i.e., ours and Decov, similar +to our other experiments, the additional loss has been added +on top of the last intermediate layer. The input images are +resized to 72 × 72. We use a patch size of 8 × 8 and an em- +bedding dimension of 256. All models are trained for 100 +epochs using Adam with learning rate of 0.002, weight de- +cay with rate 0.0001, batch size 256. Standard data augmen- +tation, i.e., random horizontal flip and random zoom with +a factor of 20%, is used. We use 10% of the training data +for validation. We also reduce the learning rate by a factor +of 2 if the validation loss does not improve for 5 epochs and +use early stopping when the validation loss does not improve +for 10 epochs. All experiments are repeated over 10 random +seeds and the average results are reported. +The results in Table 4 show that employing a diversity +strategy can indeed improve the performance of these mod- +els, thanks to its ability to help learn rich and robust repre- +sentation of the input. Our proposed approach consistently +outperforms the competing methods for both the MLP- +Mixer and gMLP. For example, our direct variant leads to +1.15% and 0.3% boost for MLP-Mixer and gMLP, respec- +tively. +For the MLP-mixer, the top performance is achieved by +the Det variant of our approach reducing the error rates by + +Figure 1: Sensitivity analysis of λ1 and λ2 on the test error using ResNet50 trained on ImageNet. First row contains experiments +with fixed λ1 and second row contains experiments with fixed λ2. From left to right: our Direct variant, our Det variant and our +Logdet variant. γ is fixed to 10 in all experiments. +Table 4: Classification errors of modern MLP-based ap- +proaches on CIFAR10. Results are averaged over ten ran- +dom seeds. +MLP-Mixer +gMLP +Standard +23.93 +22.26 +DeCov +24.10 +22.00 +Ours(Direct) +22.78 +21.95 +Ours(Det) +22.66 +21.62 +Ours(Logdet) +22.84 +21.56 +1.27% and 1.44% compared to the standard approach and +DeCov, respectively. For the gMLP model, the top perfor- +mance is achieved by the Logdet variant of our approach +boosting the results by 0.7% and 0.44% compared to the +standard approach and DeCov, respectively. +Learning in the presence of label noise +To further demonstrate the usefulness of promoting diver- +sity, we test the robustness of our approach in the presence of +label noise. In such situations, standard neural network tend +to overfit to the noisy samples and not generalize well to the +test set. Enforcing diversity can lead to better and richer rep- +resentations attenuating the effect of noise. To show this, we +performed additional experiments with label noise (20% and +40%) on CIFAR10 and CIFAR100 using ResNet50. We use +the same training protocol used for the original CIFAR10 +and CIFAR100: all models are trained using SGD with a +momentum of 0.9, weight decay of 0.0001, and a batch size +of 128 for 200 epochs. The initial learning rate is set to 0.1 +and is then decreased by a factor of 5 after 60, 120, and 160 +epochs, respectively. We also adopt a standard data augmen- +tation scheme that is widely used for these two datasets (He +et al. 2016; Huang et al. 2017). For all models, the addi- +tional diversity term is applied on top the last intermediate +layer. For the hyperparameters: The loss weights is chosen +from {0.0001, 0.001, 0.01, 0.1} for both our approach (λ1 +and λ2) and Decov and γ in the radial basis function is cho- +sen from {1, 10}. For each approach, the model with the best +validation performance is used in the test phase. The average +errors over three random seed are reported. +The results are reported in Table 5. As it can be seen, in +the presence of noise, the gap between the standard approach +and diversity (Decov and ours) increases. For example, our +Logdet variant boosts the results by 1.91% and 2.29% on + +23.8 +23.7 +23.6 +rror +E +23.5 +Test +23.4 +23.3 +23.2 +Standard +Direct: 入1=0.0001 ++ +Direct: 入1=0.0005 ++ +Direct: 入1=0.001 +Direct: 入1=0.005 +23.1 +0.0001 +0.0005 +0.001 +0.005 +入223.8 +23.7 +23.6 +Error +23.5 +Test +23.4 +23.3 +23.2 +Standard +Det: ^1=0.0001 +Det: 入1=0.0005 +23.1 +Det: A1=0.001 +Det: 入1=0.005 +0.0001 +0.0005 +0.001 +0.005 +入223.8 +23.7 +23.6 +Test Error +23.5 +23.4 +23.3 +23.2 +Standard +Logdet: 入1=0.0001 +Logdet: 入1=0.0005 +23.1 +Logdet: A1=0.001 +Logdet: ^1=0.005 +0.0001 +0.0005 +0.001 +0.005 +入223.8 +23.7 +23.6 + Error +23.5 +Test +23.4 +23.3 +23.2 +Standard +Direct: 入2=0.0001 +Direct: 入2=0.0005 +Direct: 入2=0.001 +Direct: 入2=0.005 +23.1 +0.0001 +0.0005 +0.001 +0.005 +入123.8 +23.7 +23.6 + Error +23.5 +Test +23.4 +23.3 +23.2 +Standard +Det: 入2=0.0001 +Det: 入2=0.0005 +23.1 +Det: 入2=0.001 +Det: 入2=0.005 +0.0001 +0.0005 +0.001 +0.005 +入1- +- +Standard +Logdet: X2=0.0001 +23.8 +Logdet: 入2=0.0005 +Logdet: 入2=0.001 +Logdet: 入2=0.005 +23.7 +23.6 + Error +23.5 +Test I +23.4 +23.3 +23.2 +23.1 +0.0001 +0.0005 +0.001 +0.005 +入1Table 5: Classification errors of ResNet50 using different diversity strategies on CIFAR10 and CIFAR100 datasets with different +label noise ratios. Results are averaged over three random seeds. +20% label noise +40% label noise +Method +CIFAR10 +CIFAR100 +CIFAR10 +CIFAR100 +Standard +14.38 ± 0.29 +45.11 ± 0.52 +19.40 ± 0.80 +48.81 ± 0.57 +DeCov +13.75 ± 0.19 +41.93 ± 0.40 +17.60 ± 0.66 +48.23 ± 0.48 +Ours(Direct) +13.31 ± 0.40 +40.10 ± 0.31 +16.96 ± 0.32 +46.73 ± 0.23 +Ours(Det) +13.21 ± 0.21 +40.35 ± 0.31 +17.49 ± 0.04 +46.93 ± 0.62 +Ours(Logdet) +13.01 ± 0.40 +39.97 ± 0.19 +17.24 ± 0.31 +46.52 ± 0.22 +CIFAR10 and CIFAR100 with 40% noise, respectively. +Conclusions +In this paper, we proposed a new approach to encourage +‘diversification’ of the layer-wise feature map outputs in +neural networks. The main motivation is that by promot- +ing within-layer activation diversity, units within the same +layer learn to capture mutually distinct patterns. We pro- +posed an additional loss term that can be added on top of +any fully-connected layer. 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International Conference on +Learning Representations. + diff --git a/a9AzT4oBgHgl3EQfZfxk/content/tmp_files/load_file.txt b/a9AzT4oBgHgl3EQfZfxk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73c4ff5d730ebb07a1bb9782d0fd3a93a38e273b --- /dev/null +++ b/a9AzT4oBgHgl3EQfZfxk/content/tmp_files/load_file.txt @@ -0,0 +1,1351 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf,len=1350 +page_content='WLD-Reg: A Data-dependent Within-layer Diversity Regularizer Firas Laakom1* Jenni Raitoharju,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Alexandros Iosifidis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 Moncef Gabbouj 1 1 Faculty of Information Technology and Communication Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Tampere University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Finland 2 Faculty of Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' University of Jyv¨askyl¨a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Finland 3 DIGIT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Aarhus University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Denmark Abstract Neural networks are composed of multiple layers arranged in a hierarchical structure jointly trained with a gradient-based optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' where the errors are back-propagated from the last layer back to the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' At each optimization step, neu- rons at a given layer receive feedback from neurons belong- ing to higher layers of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In this paper, we pro- pose to complement this traditional ’between-layer’ feedback with additional ’within-layer’ feedback to encourage the di- versity of the activations within the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To this end, we measure the pairwise similarity between the outputs of the neurons and use it to model the layer’s overall diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We present an extensive empirical study confirming that the pro- posed approach enhances the performance of several state- of-the-art neural network models in multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The code is publically available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='com/firasl/AAAI-23- WLD-Reg Introduction Deep learning has been extensively used in the last decade to solve several tasks (Krizhevsky, Sutskever, and Hinton 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Golan and El-Yaniv 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' A deep learning model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', a neural network, is formed of a sequence of layers with parameters optimized during the training process using training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Formally, an m-layer neural network model can be defined as follows: f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' W) = φm(W m(φm−1(· · · φ2(W 2φ1(W 1x)))), (1) where φi(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=') is the non-linear activation function of the ith layer and W = {W 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' , W m} are the model’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Given a training data {xi, yi}N i=1, the parameters of f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' W) are obtained by minimizing a loss ˆL(·): ˆL(f) = 1 N N � i=1 l � f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' W), yi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' (2) However, neural networks are often over-parameterized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', have more parameters than data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' As a result, they tend to overfit to the training samples and not generalize well on unseen examples (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' While re- search on double descent (Advani, Saxe, and Sompolinsky This work was supported by NSF-Business Finland Center for Visual and Decision Informatics (CVDI) project AMALIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Nakkiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020) shows that over-parameterization does not necessarily lead to overfit- ting, avoiding overfitting has been extensively studied (Dzi- ugaite and Roy 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Foret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Nagarajan and Kolter 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Neyshabur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Poggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Grari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021) and various approaches and strategies, such as data augmentation (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2018), regularization (Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Bietti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Kukaˇcka, Golkov, and Cremers 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Ouali, Hudelot, and Tami 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Han and Guo 2021), and Dropout (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Li, Gong, and Yang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019), have been proposed to close the gap between the empirical loss and the expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Diversity of learners is widely known to be important in ensemble learning (Li, Yu, and Zhou 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Yu, Li, and Zhou 2011) and, particularly in deep learning context, di- versity of information extracted by the network neurons has been recognized as a viable way to improve generalization (Xie, Liang, and Song 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Xie, Deng, and Xing 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In most cases, these efforts have focused on making the set of weights more diverse (Yang, Gkatzelis, and Stoyanovich;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Malkin and Bilmes 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, diversity of the acti- vations has not received much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Here, we argue that due to the presence of non-linear activations, diverse weights do not guarantee diverse feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Thus, we propose focusing on the diversity on top of feature mapping instead of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To the best of our knowledge, only (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021a) have considered diversity of the ac- tivations directly in the neural network context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The work in (Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021a) studied theoretically how diver- sity affects generalization showing that it can reduce over- fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The work in (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016) proposed an ad- ditional loss term using cross-covariance of hidden activa- tions, which encourages the neurons to learn diverse or non- redundant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The proposed approach, known as DeCov, was empirically proven to alleviate overfitting and to improve the generalization ability of neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, modeling diversity as the sum of the pairwise cross-covariance, it is not scale-invariant and can lead to trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Moreover, it can capture only the pairwise diversity between components and is unable to capture the ”higher-order diversity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In this work, we propose a novel approach to encour- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='01352v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='LG] 3 Jan 2023 age activation diversity within the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We pro- pose complementing the ’between-layer’ feedback with ad- ditional ’within-layer’ feedback to penalize similarities be- tween neurons on the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Thus, we encourage each neuron to learn a distinctive representation and to enrich the data representation learned within each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We propose three variants for our approach that are based on different global diversity definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Our contributions in this paper are as follows: We propose a new approach to encourage the ’diversifi- cation’ of the layers’ output feature maps in neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The proposed approach has three variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The main intuition is that, by promoting the within-layer ac- tivation diversity, neurons within a layer learn distinct patterns and, thus, increase the overall capacity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We show empirically that the proposed within-layer ac- tivation diversification boosts the performance of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Experimental results on several tasks show that the proposed approach outperforms competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Within-layer Diversity Regularizer In this section, we propose a novel diversification strategy, where we encourage neurons within a layer to activate in a mutually different manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', to capture different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In this paper, we define as “feature layer” the last interme- diate layer in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In the rest of the paper, we focus on this layer and propose a data-dependent regularizer which forces each unit within this layer to learn a distinct pattern and penalizes the similarities between the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In- tuitively, the proposed approach reduces the reliance of the model on a single pattern and, thus, can improve generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We start by modeling the global similarity between two units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Let φn(xj) and φm(xj) be the outputs of the nth and mth unit in the feature layer for the same input sample xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The similarity snm between the the nth and mth neurons can be obtained as the average similarity measure of their outputs for N input samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use the radial basis function to express the similarity: snm := 1 N N � j=1 exp � − γ||φn(xj) − φm(xj)|| � , (3) where γ is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The similarity snm can be computed over the whole dataset or batch-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Intuitively, if two neurons n and m have similar outputs for many sam- ples, their corresponding similarity snm will be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Oth- erwise, their similarity smn is small and they are considered “diverse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Next, based on these pairwise similarities, we propose three variants for obtaining the overall similarity J of all the units within the feature layer: Direct: J := � n̸=m snm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In this variant, we model the global layer similarity directly as the sum of the pairwise similarities between the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' By minimizing their sum, we encourage the neurons to learn different repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Det: J := −det(S), where S is a similarity matrix de- fined as Snm = snm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' This variant is inspired by the Determinantal Point Process (DPP) (Kulesza and Taskar 2010, 2012), as the determinant of S measures the global diversity of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Geometrically, det(S) is the volume of the parallelepiped formed by vectors in the feature space associated with s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Vectors that result in a larger volume are considered to be more “diverse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Thus, max- imizing det(·) (minimizing −det(·)) encourages the di- versity of the learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Logdet: J := −logdet(S)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' This variant has the same motivation as the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use Logdet instead of Det as Logdet is a convex function over the positive def- inite matrix space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' It should be noted here that the first proposed variant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', direct, similar to DeCov (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016), cap- tures only the pairwise similarity between components and is unable to capture the higher-order “diversity”, whereas the other two variants consider the global similarity and are able to measure diversity in a more global manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Promoting di- versity of activations within a layer can lead to tighter gen- eralization bound and can theoretically decrease the gap be- tween the empirical and the true risks (Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The proposed global similarity measures J can be min- imized by using them as an additional loss term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, we note that the pair-wise similarity measure snm, expressed in equation 3, is not scale-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In fact, it can be triv- ially minimized by making all activations of the feature layer high, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', by multiplying by a high scaling factor, which has no effect on the performance, since the model can rescale high activations to normal values simply by learning small weights on the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To alleviate this problem, we pro- pose an additional term, which penalizes high activation val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The total proposed additional loss is defined as follows: ˆLW LD−Reg := λ1J + λ2 N � i=1 ||Φ(xi)||2 2, (4) where Φ(x) = [φ1(x), · · · , φC(x)] is the feature vector, C is the number of units within the feature layer, and λ1 and λ2 are two hyper-parameters controlling the contribu- tion of each term to the diversity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Intuitively, the first term of equation 4 penalizes the similarity between the units and promotes diversity, whereas the second term ensures the scale-invariance of the proposed regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The total loss function ˆL(f) defined in equation 2 is aug- mented as follows: ˆLaug(f) := ˆL(f) + ˆLW LD−Reg (5) = ˆL(f) + λ1J + λ2 N � i=1 ||Φ(xi)||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The proposed approach is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We note that our approach can be incorporated in a plug-and- 1This is defined only if S is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' It can be shown that in our case S is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Thus, in practice, we use a regularized version (S + ϵI) to ensure the positive definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Algorithm 1: One epoch of training with WLD-Reg Model: Given a neural network f(·) with a feature repre- sentation φ(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', last intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Input: Training Data: {xi, yi}N i=1 Parameters: λ1 and λ2 in equation 4 1: for every mini-batch: {xi, yi}m i=1 ∈ {xi, yi}N i=1 do 2: Forward pass the inputs {xi}m i=1 into the model to obtain the outputs {f(xi)}m i=1 and the feature repre- sentations {Φ(xi)}m i=1 3: Compute the standard loss ˆL(f) (equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 4: Compute the extra loss ˆLW LD−Reg (equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 5: Compute the total loss ˆLaug(f) (equation 5) 6: Compute the gradient of the total loss and use it to update the weights of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 7: end for 8: return Return f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' play manner into any neural network-based approach to aug- ment the original loss and to ensure learning diverse fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We also note that although in this paper, we focus only on applying diversity regularizer to a single layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', the feature layer, our proposed diversity loss, as in (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016), can be applied to multiple layers within the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Our newly proposed loss function defined in equation 5 has two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The first term is the classic loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' It computes the loss with respect to the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In the back-propagation, this feedback is back-propagated from the last layer to the first layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Thus, it can be considered as a between-layer feedback, whereas the second term is computed within a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' From equation 5, we can see that our proposed approach can be interpreted as a regu- larization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, regularization in deep learning is usually applied directly on the parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', weights (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Kukaˇcka, Golkov, and Cremers 2017), while in our approach a data-dependent additional term is defined over the output maps of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For a fea- ture layer with C units and a batch size of m, the additional computational cost is O(C2(m + 1)) for Direct variant and O(C3 + C2m)) for both Det and Logdet variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Related work Diversity promoting strategies have been widely used in ensemble learning (Li, Yu, and Zhou 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Yu, Li, and Zhou 2011), sampling (Bıyık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Derezinski, Calandriello, and Valko 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Gartrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019), energy-based mod- els (Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Zhao, Mathieu, and LeCun 2017), ranking (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Yang, Gkatzelis, and Stoyanovich), pruning by reducing redundancy (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Kondo and Yamauchi 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2020), and semi-supervised learning (Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In the deep learning context, various approaches have used diversity as a direct regularizer on top of the weight parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Here, we present a brief overview of these regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Based on the way diversity is defined, we can group these approaches into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The first group considers the regulariz- ers that are based on the pairwise dissimilarity of the compo- nents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', the overall set of weights is diverse if every pair of weights is dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Given the weight vectors {wm}M m=1, (Yu, Li, and Zhou 2011) defines the regularizer as � mn(1− θmn), where θmn represents the cosine similarity between wm and wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2013), an incoherence score de- fined as − log � 1 M(M−1) � mn β|θmn| 1 β � , where β is a pos- itive hyperparameter, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In (Xie, Deng, and Xing 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Xie, Zhu, and Xing 2016), mean(θmn) − var(θmn) is used to regularize Boltzmann machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The authors the- oretically analyzed its effect on the generalization error bounds in (Xie, Deng, and Xing 2015b) and extend it to kernel space in (Xie, Liang, and Song 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The second group of regularizers considers a more global view of diver- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example, in (Malkin and Bilmes 2008, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Xie, Singh, and Xing 2017), a weight regularization based on the determinant of the weights’ covariance is proposed based on determinantal point process (Kulesza and Taskar 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Kwok and Adams 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Unlike the aforementioned methods which promote diver- sity on the weight level and similar to our method, (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2022) proposed to enforce dissim- ilarity on the feature map outputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', on the activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To this end, they proposed an additional loss based on the pairwise covariance of the activation outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Their addi- tional loss, LDecov, is defined as the squared sum of the non- diagonal elements of the global covariance matrix C of the activations: LDecov = 1 2(||C||2 F − ||diag(C)||2 2), (6) where ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='||F is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Their approach, De- cov, yielded superior empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, cor- relation is highly sensitive to noise (Kim, Kim, and Erg¨un 2015), as opposite to the RBF-based distance used in our approach (Savas and Dovis 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Haykin 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Moreover, the Decov approach only captures the pairwise diversity be- tween the components, whereas we propose variants of our approach which consider a global view of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' More- over, based on the cross-covariance, their approach i-s not scale-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In fact, it can be trivially minimized by mak- ing all activations in the latent representation small, which has no effect on the generalization since the model can rescale tiny activations to normal values simply by learning large weights on the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Experimental results CIFAR10 & CIFAR100 We start by evaluating our proposed diversity approach on two image datasets: CIFAR10 and CIFAR100 (Krizhevsky, Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' They contain 60,000 (50,000 train/10,000 test) 32 × 32 images grouped into 10 and 100 distinct categories, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We split the original train- ing set (50,000) into two sets: we use the first 40,000 images as the main training set and the last 10,000 as a validation set for hyperparameters optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use our approach on two state-of-the-art CNNs: ResNext-29-08-16: we consider the standard ResNext Model (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017) with a 29-layer architecture, a cardinality of 8, and a width of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' ResNet50: we consider the standard ResNet model (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016) with 50 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We compare against the standard networks2 as well as net- works trained with the DeCov diversity strategy (Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' All the models are trained using stochastic gra- dient descent (SGD) with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='9, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001, and a batch size of 128 for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 and is then decreased by a factor of 5 after 60, 120, and 160 epochs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We also adopt a standard data augmentation scheme that is widely used for these two datasets (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For all models, the additional diversity term is applied on top the last intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The penalty coefficients λ1 and λ2, in equation 4, for our approach and the penalty coefficient of Decov are chosen from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1}, and γ in the radial basis function is chosen from {1, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For each approach, the model with the best validation performance is used in the test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We report the average performance over three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Table 1 reports the average top-1 errors of the different approaches with the two basis networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We note that, com- pared to the standard approach, employing a diversity strat- egy consistently boosts the results for all the two models and that our approach consistency outperforms both competing methods (standard and DeCov) in all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' With ResNet50, the three variants of our proposed approach sig- nificantly reduce the test errors compared to standard ap- proach over both datasets: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='51% − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='63% improvement on CIFAR10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='25% − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='44% on CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For CIFAR10, the best performance is achieved by the di- rect variant and the Logdet variant for ResNext and ResNet models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example, with ResNext, our direct variant yields 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='65 boost compared to the standard approach and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='54 boost compared to DeCov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For CIFAR100, the best performance is acheived by our Logdet variant for both mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' This variant leads to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='85% boost for ResNet and ResNext, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Overall, our three variants con- sistently outperform DeCov and standard approach in all testing configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' ImageNet To further demonstrate the effectiveness of our approach and its ability to boost the performance of state-of-the- art neural networks, we conduct additional image classi- fication experiments on the ImageNet-2012 classification dataset (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2015) using four different mod- els: ResNet50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016), Wide-ResNet50 (Zagoruyko and Komodakis 2016), ResNeXt50 (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017), and ResNet101 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The diversity term is applied on the last intermediate layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', the global average pooling layer for both DeCov and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2For the standard approach, the only difference is not using an additional diversity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The remaining regularizers, data aug- mentation, weight decay etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', are all applied as specified per- experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the hyperparameters, we fix λ1 = λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 and γ = 10 for all the different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The Scope of this pa- per is feature diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, in this experiment, we also report results with weight diversity approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In particular, we compare with the methods in (Yu, Li, and Zhou 2011), (Xie, Deng, and Xing 2015b), (Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016), and (Ayinde, Inanc, and Zurada 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use the standard augmentation practice for this dataset as in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' All the models are trained with a batch size of 256 for 100 epoch using SGD with Nesterov Momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The learning rate is initially set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 and decreases at epochs 30, 60, 90 by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Table 2 reports the test errors of the different approaches on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' As it can be seen, feature diversity (our approach and DeCov) reduces the test error of the model and yields a better performance compared to the stan- dard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We note that, as opposite to feature diver- sity, weight diversity does not always yield performance im- provement and it can sometimes hurt generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Com- pared to decov, our three variants consistently reach better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For ResNet50 and ResNeXt50, the best performance is achieved by our direct variant, yielding more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5% im- provement compared to standard approach for both mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For Wide-ResNet50 and ResNet101, our Det variant yields the top performance with over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6% boost for Wide- ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We note that our approach has a small additional time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example for ResNet50, our direct, Det and Logdet variants take only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='29%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='39%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='49% extra training time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Sensitivity analysis To further investigate the effect of the proposed diversity strategy, we conduct a sensitivity analysis using ImageNet on the hyperparameters of our methods: λ1 and λ2 which controls the contribution of the global diversity term to the global loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We analyse the effect of the two parameters on the final performance of ResNet50 on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The analysis is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' As shown in Figure 1, using a diversity strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', three variants of our method, consistently outperform the stan- dard approach and are robust to the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the Direct variant, the best performance is reached with λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' With this configuration, the model achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='71% improvement compared to the stan- dard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the Det and the Logdet variants, using λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005, the model reaches the lowest error rate (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='09%) corresponding to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='75% accuracy boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Emphasizing diversity and using a high weights (λ1 and λ2) still lead to better results compared to standard approach but can make the total loss dominated by the diversity term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In general, we recommend using λ1 = λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' However, this depends on the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Feature diversity reduces overfitting In (Laakom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Cogswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016), it has been observed that feature diversity can reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To study the effect of feature diversity on the generalization Table 1: Classification errors of the different approaches on CIFAR10 and CIFAR100 with three different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Results are averaged over three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' ResNext-29-08-16 ResNet50 method CIFAR10 CIFAR100 CIFAR10 CIFAR100 Standard 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='41 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='42 DeCov 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='15 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='11 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='22 Ours(Direct) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='11 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='09 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='11 Ours(Det) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='16 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='12 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='28 Ours(Logdet) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='08 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='05 Table 2: Performance of different models with different diversity strategies on ImageNet dataset ResNet50 Wide-ResNet50 ResNeXt50 ResNet101 Standard 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='84 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='42 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='70 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='33 (Yu, Li, and Zhou 2011) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='87 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='48 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='57 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 (Ayinde, Inanc, and Zurada 2019) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='95 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='41 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='67 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='36 (Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='70 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='80 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 (Xie, Deng, and Xing 2015b) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='79 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='66 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='64 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='71 DeCov 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='62 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='68 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='57 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='31 Ours(Direct) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='24 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='95 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='25 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='14 Ours(Det) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='34 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='75 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='44 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='87 Ours(Logdet) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='32 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='96 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='40 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='04 gap, in Table 3, we report the final training errors and the generalization gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', training accuracy - test accuracy for the different feature diversity approaches on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Table 3: Generalization Gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', training error - test error, of different models with different diversity strategies on Im- ageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' * denotes our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' ERM DeCov direct* det* logdet* ResNet50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='21 Wide-ResNet50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='58 ResNeXt50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='41 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='48 ResNet101 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='71 As shown in Table 3, we note that using diversity indeed can reduces overfitting and decreases the empirical general- isation gap of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The three variants of our ap- proach significantly reduces overfitting for all the four mod- els by more than 1% compared standard and DeCov for all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example, our Det variant reduces the empir- ical generalization gap, compared to the standard approach and DeCov, by 2% for Wide-ResNet model and over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2% for the ResNet101 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' MLP-based models Beyond CNN models, we also evaluate the performance of our diversity strategy on modern attention-free, multi- layer perceptron (MLP) based models for image classifica- tion (Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Lee-Thorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Such models are known to exhibit high overfitting and require regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We evaluate how diversity affects the accuracy of such models on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In particular, we conduct a simple experiment using two models: MLP-Mixer (Tolstikhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021), gMLP (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2021) with four blocks each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the diversity strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', ours and Decov, similar to our other experiments, the additional loss has been added on top of the last intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The input images are resized to 72 × 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use a patch size of 8 × 8 and an em- bedding dimension of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' All models are trained for 100 epochs using Adam with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='002, weight de- cay with rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001, batch size 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Standard data augmen- tation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', random horizontal flip and random zoom with a factor of 20%, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use 10% of the training data for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We also reduce the learning rate by a factor of 2 if the validation loss does not improve for 5 epochs and use early stopping when the validation loss does not improve for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' All experiments are repeated over 10 random seeds and the average results are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The results in Table 4 show that employing a diversity strategy can indeed improve the performance of these mod- els, thanks to its ability to help learn rich and robust repre- sentation of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Our proposed approach consistently outperforms the competing methods for both the MLP- Mixer and gMLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example, our direct variant leads to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='15% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3% boost for MLP-Mixer and gMLP, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the MLP-mixer, the top performance is achieved by the Det variant of our approach reducing the error rates by Figure 1: Sensitivity analysis of λ1 and λ2 on the test error using ResNet50 trained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' First row contains experiments with fixed λ1 and second row contains experiments with fixed λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' From left to right: our Direct variant, our Det variant and our Logdet variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' γ is fixed to 10 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Table 4: Classification errors of modern MLP-based ap- proaches on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Results are averaged over ten ran- dom seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' MLP-Mixer gMLP Standard 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='93 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='26 DeCov 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='00 Ours(Direct) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='78 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='95 Ours(Det) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='66 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='62 Ours(Logdet) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='84 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='27% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='44% compared to the standard approach and DeCov, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the gMLP model, the top perfor- mance is achieved by the Logdet variant of our approach boosting the results by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='44% compared to the standard approach and DeCov, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Learning in the presence of label noise To further demonstrate the usefulness of promoting diver- sity, we test the robustness of our approach in the presence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' In such situations, standard neural network tend to overfit to the noisy samples and not generalize well to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Enforcing diversity can lead to better and richer rep- resentations attenuating the effect of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' To show this, we performed additional experiments with label noise (20% and 40%) on CIFAR10 and CIFAR100 using ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We use the same training protocol used for the original CIFAR10 and CIFAR100: all models are trained using SGD with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='9, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001, and a batch size of 128 for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 and is then decreased by a factor of 5 after 60, 120, and 160 epochs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We also adopt a standard data augmen- tation scheme that is widely used for these two datasets (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For all models, the addi- tional diversity term is applied on top the last intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For the hyperparameters: The loss weights is chosen from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1} for both our approach (λ1 and λ2) and Decov and γ in the radial basis function is cho- sen from {1, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For each approach, the model with the best validation performance is used in the test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The average errors over three random seed are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The results are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' As it can be seen, in the presence of noise, the gap between the standard approach and diversity (Decov and ours) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' For example, our Logdet variant boosts the results by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='91% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='29% on 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 rror E 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 Test 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Standard Direct: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 + Direct: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 + Direct: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Direct: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 Error 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 Test 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Standard Det: ^1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 Det: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 Det: A1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Det: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 Test Error 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Standard Logdet: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 Logdet: 入1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 Logdet: A1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Logdet: ^1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 Error 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 Test 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Standard Direct: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 Direct: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 Direct: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Direct: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 Error 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 Test 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 Standard Det: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 Det: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 Det: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Det: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入1- Standard Logdet: X2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='8 Logdet: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 Logdet: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 Logdet: 入2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='6 Error 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='5 Test I 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='005 入1Table 5: Classification errors of ResNet50 using different diversity strategies on CIFAR10 and CIFAR100 datasets with different label noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Results are averaged over three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 20% label noise 40% label noise Method CIFAR10 CIFAR100 CIFAR10 CIFAR100 Standard 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='29 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='52 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='80 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='57 DeCov 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='19 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='40 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='66 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='48 Ours(Direct) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='40 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='31 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='32 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='23 Ours(Det) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='21 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='31 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='04 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='62 Ours(Logdet) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='40 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='19 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='31 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='22 CIFAR10 and CIFAR100 with 40% noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Conclusions In this paper, we proposed a new approach to encourage ‘diversification’ of the layer-wise feature map outputs in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' The main motivation is that by promot- ing within-layer activation diversity, units within the same layer learn to capture mutually distinct patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We pro- posed an additional loss term that can be added on top of any fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' This term complements the tradi- tional ‘between-layer’ feedback with an additional ‘within- layer’ feedback encouraging diversity of the activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Ex- tensive experimental results showing that such a strategy can indeed improve the performance of different state-of-the-art networks across different datasets and different tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=', image classification, and label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' We are confident that these results will spark further research in diversity-based approaches to improve the performance of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' References Advani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Saxe, A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' Energy-based generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} +page_content=' International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfZfxk/content/2301.01352v1.pdf'} diff --git a/aNE2T4oBgHgl3EQfZgfV/content/2301.03866v1.pdf b/aNE2T4oBgHgl3EQfZgfV/content/2301.03866v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cfc0d74e4906d15405dd77625cebc914608ee5f6 --- /dev/null +++ b/aNE2T4oBgHgl3EQfZgfV/content/2301.03866v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Introduced by Gr¨atzer and E. Knapp in 2007, a slim semimodular +lattice is a planar semimodular lattice without M3 as a sublattice. We prove +that if K is a slim semimodular lattice and n denotes the number of its join- +irreducible congruence relations, then there exists a slim semimodular lattice +L such that Con L ∼ += Con K, the length of L is at most 2n2, and the number +of elements of L is at most 4n4. (In fact, we prove slightly more.) Also, we +present a new construction under which the class of (isomorphism classes of) +posets of join-irreducible congruences of slim semimodular lattices is closed. +1. Goal +In 2016, Cz´edli, D´ek´any, Gyenizse, and Kulin [9] proved that for a natural num- +ber n, there are asymptotically (n − 2)! · e2/2 many slim rectangular lattices1 of +length n, where e is the famous mathematical constant �∞ +k=0 1/n! ≈ 2.71828. There +are several ways, including the one occurring in [9], to construct and list all of these +lattices of length n. At initial level, this is one of the motivations of our aim to prove +Theorem 5.1, which, roughly saying, asserts that if a finite distributive lattice is +representable by the congruence lattice of a slim semimodular lattice (equivalently, +a slim rectangular lattice) L, then we can find an L with small length for this +purpose. The secondary aim is to prove a related result, Proposition 6.1. +2. Basic concepts +For the history and the importance of (slim) planar semimodular lattices, see +many papers that belong to the more than four dozen papers listed at2 +http://www.math.u-szeged.hu/~czedli/m/listak/publ-psml.pdf . +As usual, semimodularity is upper semimodularity, that is, the satisfaction of +(∀x)(∀y)(x ∧ y ≺ x ⇒ y ≺ x ∨ y). The poset (that is, partially ordered set) of +join-irreducible elements of a lattice L is denoted by J(L) = (J(L); ≤). Note that +0 = 0L /∈ J(L) by definition. The set M(L) of meet-irreducible elements is denoted +by M(L); note that 1 /∈ M(L). The elements of J(L) ∩ M(L) are called double +2020 Mathematics Subject Classification. 06C10 +January 1, 2023. +Key words and phrases. Slim rectangular lattice, slim semimodular lattice, planar semimodular +lattice, congruence lattice, lattice congruence, lamp, C1-diagram. +This research was supported by the National Research, Development and Innovation Fund of +Hungary under funding scheme K 138892. +1To be defined later. +2The list the Appendix of https://arxiv.org/abs/2107.10202 is no longer up-to-date. +1 +arXiv:2301.00401v1 [math.RA] 1 Jan 2023 + +2 +G. CZ´EDLI +irreducible. The following definition combines the original definitions of Gr¨atzer +and Knapp [15] and [16] with Cz´edli and Schmidt [11]. +Definition 2.1. (A) A lattice is planar if it is finite and it has a planar (Hasse) +diagram. (All lattices in this paper are assumed to be finite.) +(B) By [11], a finite lattice L is slim if J(L) is the union of two chains. If, in +addition, L is semimodular, then we say that it is a slim semimodular lattice; see +[11]. +(C) By [15], a planar semimodular lattice L is slim if M3, the five-element +modular lattice of length 2, is not a cover-preserving sublattice of L. A slim planar +semimodular lattice in this sense is, shortly, an SPS lattice. +(D) If a lattice L is slim in the sense of (B), then it is necessarily planar. There- +fore, slim semimodular lattices are the same as SPS lattices. +This paper gives +preference to “slim semimodular lattices” to “SPS lattices”. +(E) Following Gr¨atzer and Knapp [16], a slim rectangular lattice is a slim semi- +modular lattice that has exactly two doubly irreducible elements and these two +elements are complements of each other. +3. Concepts, terminology, and tools from earlier papers +To denote the bottom and the top of an interval I, we prefer Foot(I) and Peak(I) +to 0I and 1I, respectively. (In this way, we can avoid subscripts of subscripts.) In +a planar (Hasse) diagram of a planar lattice L, the edges p = [Foot(p), Peak(p)] are +straight line segments that stand for prime intervals p (that is, for intervals p with +Foot(p) ≺ Peak(p)); we do not allow curves as edges. We always assume that a +usual coordinate system of the plane is fixed. Edges parallel to (1, 1) or (1, −1) are +of normal slopes. Edges parallel to (1, t) for some t ∈ R with |t| > 1 and vertical +edges are said to be precipitous. As in Cz´edli [4, Definition 2.1], where a more +general concept from Cz´edli [3] is tailored to the peculiarities of slim rectangular +lattices, we present the following definition, Part (C) of which is crucial here and in +several earlier paper; Parts (A) and (B) go after Gr¨atzer and Knapp [15] and [16]. +Definition 3.1. (A) Let L♯ be a planar diagram of a slim rectangular lattice L. The +left boundary chain and the right boundary chain of L♯ are denoted by LBnd(L) +and RBnd(L), respectively. (Actually, LBnd(L♯) and RBnd(L♯) would be more +precise but we always fix L♯ in a way that we are going to discuss. This comment +applies for several other concepts we are going to define.) The boundary of L is +Bnd(L) = LBnd(L) ∪ RBnd(L). The elements of Bnd(L) and those of L \ Bnd(L) +are called boundary elements and internal elements. +(B) The left corner lc(L) and the right corner rc(L) are the two doubly irre- +ducible elements3 of L such that lc(L) ∈ RBnd(L) and rc(L) ∈ RBnd(L). The upper +left boundary and the upper right boundary of L are the principal filters ↑Llc(L) +and ↑Lrc(L); note that ↑Llc(L) ⊆ LBnd(L) and ↑Lrc(L) ⊆ RBnd(L). +(C) For x ∈ L, we let lsupp(x) := x ∧ lc(L) and rsupp(x) := x ∧ rc(L); note +that lsupp(x) is on the lower left boundary ↓Llc(L), rsupp(x) is on the lower right +boundary ↓Llc(L), and x = lsupp(x) ∨ rsupp(x).4 +3We know from Gr¨atzer and Knapp [16] that they are boundary elements +4This follows from Gr¨atzer and Knapp [16, Lemmas 3 and 4] and the equivalence of the two +definitions of slimness, see Cz´edli and Schmidt [11, Lemma 2.3]. + +REDUCING THE LENGTHS OF SPS LATTICES +3 +(D) The diagram L♯ of L is a C1-diagram if for every edge p = [Foot(p), Peak(p)] +of the diagram, p is either precipitous or it is of a normal slope and, furthermore, +p is precipitous ⇐⇒ Foot(p) is an internal meet-irreducible element of L. +Convention 3.2. Together with each slim rectangular lattice occurring in the pa- +per, a C1-diagram of the lattice is fixed . Moreover, even if we do not say it all +the times, whenever we construct a lattice (like a sublattice or a larger lattice), +then we always construct its fixed C1-diagram as well . In notation, we rarely dis- +tinguish a slim rectangular lattice from its C1-diagram. +Complying with Convention 3.2, all lattice diagrams in this paper are C1-diagrams. +Let L denote a slim rectangular lattice. Note in advance that +quite often, we do not make a distinction be- +tween lattice theoretic and geometric objects. +� +(3.1) +If a < b ∈ L and C1, C2 are maximal chains of the interval [a, b] such that C1∩C2 = +{a, b} and all elements x of C1 are on the left of C2 (including the possibility of +x ∈ C2), then the elements [a, b] that are simultaneously on the right of C1 and +on the left of C2 form a so-called lattice region; see Kelly and Rival [17] for a +more exact definition. The corresponding geometric area, which is bordered by +C1 and C2, is a geometric region. Note that whenever we define a geometric area +(like a geometric region) or line segment, then (unless otherwise explicitly stated) it +contains its boundaries, that is, it is topologically closed. Minimal non-chain regions +are cells. If a cell contains exactly four lattice elements, then it is a 4-cell. In fact, +4-cells are cover-preserving boolean sublattices with 4 elements. A 4-cell lattice is +a planar lattice in which all cells are 4-cells (in a fixed planar diagram). Gr¨atzer +and Knapp [15, Lemmas 4 and 5] and [16] proved that for a planar lattice L (which +is finite by definition), +if L is a 4-cell lattice, no two distinct 4-cells have the same bot- +tom, and L has exactly two complementary doubly irreducible +elements, then L is a slim rectangular lattice. Conversely, a slim +rectangular lattice is a 4-cell lattice with these properties. +� +� +� +� +� +(3.2) +On the set of prime intervals (i.e., edges) of L, let τ be the smallest equivalence +relation that collapses the opposite sides of every 4-cell. As in Cz´edli and Schmidt +[11], the blocks of τ are called trajectories. Any two consecutive edges of a trajectory +form a 4-cell, which is a 4-cell of a the trajectory in question. Note that a trajectory +never branches out. Sometimes, like in [11], we say that a trajectory goes from left +to right, that is, it starts at the left boundary chain, goes to the northeast direction +(possibly in zero steps) to reach its top edge, which is the unique neon tube5 of the +trajectory, and then it goes to the southeast to reach the right boundary chain and +to stop there; see, for example, Figure 16. In this figure, the double-lined edges form +a trajectory and, as for their color, the 4-cells of this trajectory are “gravel-filled”. +The ascending part of the trajectory is its segment (subsequence consisting of +edges) between its first edge (on the left boundary chain) and the top edge. Simi- +larly, the descending part of the trajectory starts with the top edge and terminates +with the last edge (on the right boundary chain). +5A neon tube is a prime interval with meet-irreducible bottom. +6A part of this figure was presented at SSAOS (conference) in Malenovice, Czech Republic, +2010. + +4 +G. CZ´EDLI +Figure 1. A trajectory goes from left to right +Given a 4-cell H of L and a positive integer k ∈ N+, we obtain the k-fold +multifork extension of L at H by changing H to a copy of S(k) +7 +and proceeding +to the southeast and to the southwest to preserve semimodularity. For the exact +definition, see Cz´edli [2], where this construction was introduced, or see Figure 2, +where the construction is illustrated by performing a 1-fold multifork extension at +H1 of L0 to obtain L1 and performing a 3-fold multifork extension at H2 of L1 to +obtain L2. (To save space, our figures are multi-purpose figures; some ingredients +of Figure 2 will be explained later.) Note in advance that the thick edges of our +lattice diagrams will be called neon tubes. Note also that 1-fold multifork extensions +are also called fork extensions; see Cz´edli and Schmidt [12]; in this case the new +elements form a so-called fork in the new lattice; see (4.1) later. +A grid is (the fixed C1-diagram) of the direct product of two non-singleton finite +chains. A 4-cell H of L is a distributive 4-cell if the principal ideal ↓LPeak(H) is a +distributive lattice. Note that in this case, that is, +when H is a distributive 4-cell, then ↓LPeak(H) is a grid; +(3.3) +this follows, say, from the following lemma and Cz´edli and Schmidt [12]. Most of +the results on slim semimodular lattices were or could have been proved by using, +possibly together with other tools, the following structure theorem. +Lemma 3.3 (Multifork Sequence Lemma [2, Theorem 3.7]). For each slim rectan- +gular lattice L, there exist positive integers m1, . . . , mk, a sequence L0, L1, . . . , Lk +of slim rectangular lattices, and a distributive 4-cell Hi of Li−1 for i ∈ {1, . . . , k} +such that L0 is a grid, Lk = L, and Li is obtained from Li−1 by performing an mi- +fold multifork extension at Hi for i ∈ {1, . . . , k}. Furthermore, any lattice obtained +in this way from a grid is a slim rectangular lattice. +Definition 3.4. +(A) The system (L0, H1, m1, L1, H2, m2, . . . , Lk−1, Hk, mk, L = Lk) with com- +ponents as above is the multifork sequence of L; it is not necessarily unique but we +always fix one. (Note, however, that k is unique.) + +REDUCING THE LENGTHS OF SPS LATTICES +5 +(B) Let n be an edge on the upper boundary of the initial grid L0. The union of +the 4-cells of the trajectory containing n is the original territory of n; it is denoted +by OT(n). +When we obtain Li from �Li−1, then we add several new edges and +exactly mi of these new edges have the same peak as H. Let n be one of these new +edges. In Li, the union of the 4-cells of the trajectory containing n is a geometric +area; we call it the original territory OT(n) of n in L. Note that we have defined +OT(n) if and only if n is an edge of the upper boundary or n is a precipitous edge. +(C) If n is an edge of the upper left boundary chain, then the essential part of the +original territory, denoted by EOT(n), and the right essential part of the original +territory, denoted by REOT(n), of n are OT(n) while the left essential part of the +original territory, denoted by LEOT(n), of n is the empty set. Similarly, for n on +the right upper boundary, EOT(n) = LEOT(n) := OT(n) and REOT(n) = ∅. Next, +let n be a precipitous edge and denote by T the trajectory containing n. The union +of 4-cells of T that do not contain n as an edge is the essential part EOT(n) of the +original territory of n; it is a geometric area and the union of two (geometrically) +connected subsets that are, in a self-explanatory manner, called the left essential +part LEOT(n) of the original territory and the right essential part REOT(n) of the +original territory of n. (The concept of EOT(n) was introduced in Cz´edli [7].) +Some original territories and essential original territories are presented in Figures +2, 3, and 4. Even though the definition of OT(n), . . . , REOT(n) relies on L0 or +Li, we usually use this concept for L, where OT(n), . . . , REOT(n) have not much +connection with the trajectory containing n; this is exemplified by n1 and n2 in L′ +(but not in L) of Figure 3. +It follows from (3.3) that whenever OT(n) is defined, then +OT(n) is bordered by edges of L and all of these edges with +peaks different from Peak(n) are of normal slopes. Further- +more, each of LEOT(n) and REOT(n) is either the empty +set or a rectangle bordered by edges of normal slopes. +� +� +� +� +� +(3.4) +Lamps were introduced in Cz´edli [4], and they proved to be the fundamental tool +to study the congruence lattices of slim rectangular lattices in Cz´edli [5], [7], [8], +and Cz´edli and Gr¨atzer [10]. Originally, lamps are particular intervals I. However, +sometimes we need to consider them pairs (Foot(I), Peak(I)). Based on [4], we +recall the following definition. +Definition 3.5. Let L be a slim rectangular lattice; Convention 3.2 applies. +(A) The prime intervals p of L with Foot(p) ∈ M(L) are called neon tubes. If +Foot(p) ∈ Bnd(L), then p is a boundary neon tube and it is of a normal slope. +Otherwise, p is an internal neon tube and it is precipitous. +(B) Boundary lamps are the same as boundary neon tubes. (However, if I = p is +a boundary lamp, then we sometimes say that p is the neon tube of I). An interval +I is an internal lamp if Peak(I) is the peak of an internal neon tube and Foot(I) +is the meet of the feet of all internal neon tubes with peak Peak(I). (These neon +tubes are called the neon tubes of I.) +(C) In our lattice diagrams (which are C1-diagrams), the neon tubes are exactly +the thick edges and the feet of the lamps are black-filled. Note at this point that we +know from Cz´edli [4, Lemma 3.1] that a lamp is uniquely determined by its foot. +Thus, for a lamp I, we label the black-filled vertex Foot(I) in our figures by I rather +than by Foot(I). (So we follow the convention of the earlier papers.) + +6 +G. CZ´EDLI +3 45 +34543534 +Figure 2. Multifork extensions and some geometric objects +The (fixed) C1-diagram of a slim rectangular lattice L lies in a geometric rectan- +gle; it is bordered (from the left and from the right) by LBnd(L) and RBnd(L), +and it is called the full geometric rectangle FullRect(L) of L. Definition 3.4(B) and +(C) allow us to recall the following definition from Cz´edli [4] in a slightly modified +form. +Definition 3.6 (Some geometric areas and polygons). For a slim rectangular lattice +(diagram) L, let I and J be lamps and p be a neon tube of L. +(A) The illuminated area Lit(I) of I is the union of the original territories of the +neon tubes of I. +(B) The left roof and the left floor of an interval K of L are the line segments +of slope (1, 1) with lower endpoints on the left boundary chain and upper end- +points Peak(K) and Foot(K), respectively. They are denoted by LRoof(K) and +LFloor(K), respectively. +With slope (1, −1), the right roof RRoof(K) and the +right floor RFloor(K) are defined7 analogously. The roof Roof(K) and the floor +Floor(K) of K are LRoof(K)∪RRoof(K) and LFloor(K)∪RFloor(K), respectively. +(C) For a set X of planar points, GInt(X) stands for the geometric (i.e., topo- +logical) interior of X. Let h be a (geometric) polygon with endpoints a and b +7We know from (3.5), or from Cz´edli [4, (2.10) and (2.11)], or from Cz´edli and Gr¨atzer [10, +(3.2)] that each of LRoof(K), LFloor(K), RRoof(K), and RFloor(K) consists of some edges of +the diagram provided its geometric length is positive. See also (4.4). + +REDUCING THE LENGTHS OF SPS LATTICES +7 +such that h \ {a, b} ⊆ TopInt(FullRect(L)), a ∈ LBnd(L), and b ∈ LBnd(L). Then +h cuts FullRect(L) into an upper half ↑gh and a lower half ↓gh; by convention, +h = ↑gh ∩ ↓gh. Note that Lit(I) = ↑gFloor(I) ∩ ↓gRoof(I), and similarly for Lit(p). +(D) The body Body(I) of I is the geometric region determined by I; if I has +only one neon tube, then Body(I) is a line segment. For example, in Figure 2, +C2 ∈ Lamp(L2) and Body(C2) is filled with a “rainbow color”. +(E) If I is a internal lamp, then the circumscribed rectangle CircR(I) is the region +determined by the interval [x, Peak(I)] where x is the meet of the leftmost lower +cover and the rightmost lower cover of Peak(I). (Equivalently, x is the meet of all +lower covers of Peak(I).) +Since the edges occurring in Definition 3.4(B) and (C) are the same as the neon +tubes of L, the following lemma in the present setting is not surprising; we recall +it from [4, (2.10)]. +Lemma 3.7 (Cz´edli [4]). For the fixed multifork sequence of L in Lemma 3.3, +the set of internal lamps of L is of the form {Ij : 1 ≤ j ≤ k} where, for j ∈ +{1, . . . , k}, the “lamp” (Foot(Ij), Peak(Ij)) comes to existence by the j-th multifork +extension, CircR(Ij) in L = Lk is the geometric region determined by Hj in Lj−1, +and Foot(Ij) ∈ Lj \ Lj−1. +Since the multifork extensions in Lemma 3.3 are performed at distributive 4-cells, +it follows easily that, using the notations of Lemma 3.7, for any j ∈ {1, . . . , k}, +the lower covers of Peak(Ij) are the same in Lj as in L = +Lk. In particular, Ij has the same neon tubes in Lj as in L. +Furthermore, if a neon tube n comes to existence in Lj, then +EOT(n), LEOT(n), and REOT(n) are the same in Lj as in L. +(3.5) +Definition 3.8. With the notation used in Lemma 3.7, let Ii and Ij be lamps of +L. If i < j, then we say that Ij is younger than Ii and Ii is older than Ij. (This +concept depends on the multifork sequence, but this sequence is always fixed.) +By an edge segment we mean a geometric line segment g of positive length with +endpoints lying on the same edge e of (the fixed C1-diagram of) L. In this case, we +say that g is an edge segment of e. Based on the fact that the neon tubes of L are +exactly the prime intervals occurring in Definition 3.4 (B) and (C), we can recall a +part of Cz´edli [4, Definition 2.9] and extend it as follows. +Definition 3.9. Let I and J be lamps of a slim rectangular lattice L. +(A) Let (I, J) ∈ ρfoot mean that I ̸= J, Foot(I) ∈ Lit(J), and I is an internal +lamp. +(B) Let (I, J) ∈ ρOTfoot mean that I ̸= J, I is an internal lamp, and J has a +neon tube n such that Foot(I) ∈ GInt(LEOT(n)) or Foot(I) ∈ GInt(REOT(n)). +(C) Let (I, J) ∈ ρOTCR mean that I ̸= J, I is an internal lamp, and J has a +neon tube n such that CircR(I) ⊆ LEOT(n) or CircR(I) ⊆ REOT(n). +(D) Let (I, J) ∈ ρCircR mean that I ̸= J, I is an internal lamp, and CircR(I) ⊆ +Lit(J). +(E) Let Lamp(L) be the set of lamps of L, and let “≤ ” be the reflexive and +transitive closure of the relation ρfoot. The relational structure (Lamp(L); ≤) is +also denoted by Lamp(L). +The congruence generated by a pair (x, y) of elements will be denoted by con(x, y). + +8 +G. CZ´EDLI +Lemma 3.10 (Mostly [4, Lemma 2.11]). If L is a slim rectangular lattice, then +ρfoot = ρCircR = ρOTfoot = ρOTCR, Lamp(L) = (Lamp(L); ≤) is a poset, and +whenever I ≺ J in Lamp(L), then (I, J) ∈ ρfoot. +Furthermore, we have that +(Lamp(L); ≤) ∼= (J(Con L); ≤) and the map +ϕ: (Lamp(L); ≤) → (J(Con L); ≤) defined by I �→ con(Foot(I), Peak(I)) +(3.6) +is an order isomorphism. +The advantage of this lemma over its precursor, [4, Lemma 2.11], is that (I, J) ∈ +ρfoot is a mild condition, which is easy to verify, while (I, J) ∈ ρOTCR is a strong +condition, which gives more chance to draw conclusion from. +Proof. With the exception of “ρfoot = ρOTfoot = ρOTCR ”, the lemma is already +known; see Cz´edli [4, Lemma 2.11]. So we need only to show the just-mentioned +equalities. Clearly, ρOTCR ⊆ ρOTfoot ⊆ ρfoot. Assume that Ii, Ij ∈ Lamp(L) such +that (Ii, Ij) ∈ ρfoot. Since S(mi) +7 +is not distributive, it follows from (3.3) and Lemmas +3.3 and 3.7 that Ii is younger than Ij, that is, i > j. In particular, Ii is an internal +lamp. With m := mj, let n1, . . . , nm be the neon tubes of Ij. As i > j, these neon +tubes are present in Li−1, and so are their original territories OT(n1),. . . , OT(nm) +as well as their essential original territories; see (3.5). By (3.4) applied to Li−1, +these territories are separated by polygons consisting of lattice edges. Thus, by +planarity, these “separating polygons” cannot cross the 4-cell Hi of Li−1; this 4-cell +becomes CircR(Ii) in Li and in L. So CircR(Ii) ⊆ OT(nt) for some t ∈ {1, . . . , m}. +However, the 4-cell Hi in question cannot have the same bottom as Ij since the +opposite case would contradict the distributivity of Hi in Li−1. (Alternatively, [7, +Lemma 6.2] would also lead to a contradiction.) Hence, CircR(Ii) = Hi ⊆ EOT(nt). +Since EOT(nt) is the union of its two connected “components”, LEOT(nt) and +REOT(nt), and these connected components are in a positive geometric distance +from each other (provided none of them is the empty set), the planarity of the +diagram yields that CircR(Ii) = Hi ⊆ LEOT(nt) or CircR(Ii) = Hi ⊆ REOT(nt). +Hence, (Ii, Ij) ∈ ρOTCR, implying that ρOTCR ⊆ ρfoot and completing the proof of +Lemma 3.10. +□ +Since we work with the C1-diagram of our slim rectangular lattice L, the illumi- +nated sets Lit(I) and the Foot(I), and so the relation ρfoot are perfectly described +by the geometric structure +Str(L) := +� +FullRect(L), {(Foot(I), Peak(I)) : I ∈ Lamp(L)} +� +. +(3.7) +In particular, if +if L and L′ are slim rectangular lattices such that Str(L) = +Str(L′), then Lamp(L) ∼= Lamp(L′) and so Con L ∼= Con L′. +� +(3.8) +4. Auxiliary statements +The following definition and its terminology are motivated by the relation ρOTCR +from Definition 3.9 and Lemma 3.10. +Definition 4.1. For a slim rectangular lattice L and J ∈ Lamp(L), let p be a +neon tube of J. We say that the original territory of p is used if there is a lamp +I ∈ Lamp(L) such that I ̸= J and CircR(I) ⊆ LEOT(p) or CircR(I) ⊆ REOT(p). + +REDUCING THE LENGTHS OF SPS LATTICES +9 +Figure 3. Illustrating the proof of Lemma 4.5 by Lamp(L) ∼= P ∼= Lamp(L′) +If I is such, then we say that I uses the original territory of p. If there is no such +an I, then the original territory of p is not used. +The following remark follows from Lemma 3.10. +Remark 4.2. In Definition 4.1, “there is a lamp I ∈ Lamp(L) such that I ̸= J” is +equivalent to that “there is an internal lamp I ∈ Lamp(L) such that I < J”. Note +also that I ̸= J occurs in Definition 4.1 only for emphasis, so it could be omitted; +analogous comments would apply to Lemma 4.3 below. +Lemma 4.3. For p and J as in Definition 4.1, the following four conditions are +equivalent. +(a) The original territory of p is used, that is, there is lamp I such that p is not +a neon tube of I and CircR(I) ⊆ LEOT(p) or CircR(I) ⊆ REOT(p). +(b) There is a lamp I ∈ Lamp(L) \ {J} such that Foot(I) is in GInt(LEOT(p)) +or it is in GInt(REOT(p)). +(c) There is a lamp I ∈ Lamp(L) \ {J} such that Foot(I) is in EOT(p). +(d) There is a precipitous edge segment in EOT(p). +Furthermore, if a lamp I satisfies one of (a), (b), and (c), then it satisfies all +the three. +Proof. Since we never change I to another lamp, the last sentence of the lemma +will automatically follow when the equivalence of (a), (b), and (c) has been proved. +Since Foot(I) ∈ GInt(CircR(I)), (a) implies (b). By the equality EOT(p) = +LEOT(p) ∪ REOT(p), we obtain that (b) implies (c). +Next, assume that (c) holds. Then Foot(I) ∈ EOT(p) ⊆ Lit(J) and so (I, J) ∈ +ρfoot. By Lemma 3.10, (I, J) ∈ ρOTCR and so Body(I) ⊆ CircR(I) ⊆ Lit(I). Thus, +It := I is younger than Ik := J in the sense of Definition 3.8, that is, t > k; indeed, +if I = It was older than J = Ik, then the 4-cell Hk would not have been distributive +in Lk−1. In Lk, each of LEOT(p), REOT(p), and FullRect(Lk) \ EOT(p) were the +unions of 4-cells. Some of these 4-cells could have been divided into smaller ones +later, but even in Lt−1, each of LEOT(p), REOT(p), and FullRect(Lt−1) \ EOT(p) + +10 +G. CZ´EDLI +were unions of 4-cells. Hence, Ht ⊆ LEOT(p), Ht ⊆ REOT(p), or Ht is outside +EOT(p). Since Foot(I) = Foot(It) ∈ GInt(Ht) and Foot(I) ∈ EOT(p), Ht was not +outside EOT(p). Hence, CircR(I) = CircR(It) = Ht ⊆ LEOT(p) or CircR(I) ⊆ +REOT(p), whereby the original territory of p is used. Thus, (c) implies (a), and we +have proved that (a), (b), and (c) are equivalent conditions. +By Remark 4.2, the implication (a) ⇒ (d) is trivial. +Finally, assume that (d) holds. Then we have a precipitous edge segment in +LEOT(p) or in REOT(p), say, in LEOT(p). By the second half of (3.4), we can +assume that a precipitous edge segment lies in GInt(LEOT(p)). This edge segment +lies in a neon tube q of a lamp I. By planarity and (3.4), q cannot cross the four +sides bordering (the geometric rectangle) LEOT(p), so q lies fully in LEOT(p). In +particular, Peak(I) = Peak(q) ∈ LEOT(p) and Foot(q) ∈ LEOT(p). Observe that +Peak(I) cannot lie on the lower boundary of LEOT(p) since otherwise q, going +down from Peak(I) with a precipitous slope, could not include an edge segment +lying in LEOT(p). +Next, let r be an arbitrary neon tube of I. It goes down from Peak(r) = Peak(I) +with a precipitous slope. Thus, since Peak(r) is not on the lower boundary, (3.4) +yields that an edge segment lying on r lies also in GInt(LEOT(p)). So r satisfies +the same condition as q above, and it follows that Foot(r) ∈ LEOT(p). +Now let r′ and r′′ be the leftmost neon tube and the rightmost neon tube of I. +If r′ = r′′, then q is the only neon tube of I, and the required Foot(I) ∈ LEOT(p) +follows from Foot(I) = Foot(q) ∈ LEOT(p). So we can assume that r′ ̸= r′′. Then +Foot(r′) and Foot(r′′), as distinct lower covers of Peak(I), are incomparable; see +(3.5). +By the main result of Cz´edli [6] and Foot(I) = Foot(r′) ∧ Foot(r′′), the +interval [Foot(I), Foot(r′)] is a chain (and so a line segment) of slope (1, −1) while +[Foot(I), Foot(r′′)] is a line segment of slope (1, 1). +The top endpoints Foot(r′) +and Foot(r′′) of these line segments are in LEOT(p), whereby so is their common +bottom Foot(I) by the second half of (3.4). Hence, Foot(I) ∈ LEOT(p), that is, +(a) holds. This completes the proof of Lemma 4.3. +□ +For a neon tube p of a slim rectangular lattice L, the fork determined by p is +F(p) := [lsupp(Foot(p)), Foot(p)] ∪ [rsupp(Foot(p)), Foot(p)] to- +gether with the edges of these two intervals and the edge p. +� +(4.1) +With another terminology, this concept was originally introduced in Cz´edli and +Schmidt [12]. A (very) particular case of the following lemma (namely, the case +when ↓L′ Peak(p) is distributive) occurs implicitly in [12]. +Lemma 4.4. If p is a neon tube of a slim rectangular lattice L and L′ := L \ F(p), +see (4.1), then L′ is meet-subsemilattice of L. +Proof. First, we prove that +[lsupp(Foot(p)), Foot(p)] = {x ∈ L : lsupp(x) = lsupp(Foot(p))}. +(4.2) +Denote Foot(p) by w and lsupp(Foot(p)) by u; so u = lsupp(w) and we need +to show that [u, w] = {x ∈ L : lsupp(x) = u}. +For y ∈ [u, w], we have that +u = lsupp(u) ≤ lsupp(y) ≤ lsupp(w) = u. Hence, y ∈ {x ∈ L : lsupp(x) = u} +and we obtain that [u, w] ⊆ {x ∈ L : lsupp(x) = u}. To exclude that “⊂” holds +here, suppose for contradiction that there is a z ∈ {x ∈ L : lsupp(x) = u} such +that z /∈ [u, w]. If rsupp(z) ≤ rsupp(w), then the last equality in Definition 3.1(C) +gives that z = lsupp(z) ∨ rsupp(z), whence u = lsupp(z) ≤ z = u ∨ rsupp(z) ≤ + +REDUCING THE LENGTHS OF SPS LATTICES +11 +u∨rsupp(w) = lsupp(w)∨rsupp(w) = w yields that z ∈ [u, w], a contradiction. But +rsupp(z) and rsupp(w) belong to the same chain, RBnd(L), so they are comparable, +and we obtain that rsupp(w) < rsupp(z). +Hence, w = lsupp(w) ∨ rsupp(w) = +u ∨ rsupp(w) ≤ u ∨ rsupp(z) = lsupp(z) ∨ rsupp(z) = z. The inequality w ≤ z +and z /∈ [u, w] imply that Foot(p) = w < z. +Taking the meet-irreducibility of +Foot(p) into account, we have that Peak(p) ≤ z. Thus, lsupp(Peak(p)) ≤ lsupp(z). +With the notation used in Lemmas 3.3 and 3.7, let Ii be the lamp to which p +belongs. Then Peak(p) = Peak(Ii), and it is clear in Li that u = lsupp(Foot(p)) < +lsupp(Peak(I)) = lsupp(Peak(p)). +Since Li is a sublattice of L, the inequality +u < lsupp(Peak(p)) also holds in L. Combining this with the already established +lsupp(Peak(p)) ≤ lsupp(z), we obtain that u < lsupp(z). +This contradicts the +assumption z ∈ {x ∈ L : lsupp(x) = u} and proves (4.2). +Next, for the sake of contradiction, suppose that L′ is not meet-closed. Then +there are elements s, c, d ∈ L such that s = c ∧ d, s ∈ F(p) = L \ L′ but +c, d /∈ F(p). The top element Foot(p) of F(p) is meet-irreducible, whereby (4.1) +allows us to assume that s ∈ [lsupp(Foot(p)), Foot(p)]. Note that lsupp(Foot(p)) = +lsupp(lsupp(Foot(p))) ≤ lsupp(s) ≤ lsupp(Foot(p)) gives that s = lsupp(Foot(p)). +Since the function L → LBnd(L) defined by t �→ lsupp(t) is clearly an idempo- +tent meet-endomorphism by Definition 3.1(C), lsupp(s) = lsupp(c) ∧ lsupp(d). But +LBnd(L) is a chain, whence lsupp(c) = lsupp(s) or lsupp(d) = lsupp(s). +Let, +say, lsupp(c) = lsupp(s). So, taking s = lsupp(Foot(p)) into account, lsupp(c) = +lsupp(Foot(p)). +Hence, (4.2) implies that s ∈ [lsupp(Foot(p)), Foot(p)] ⊆ F(p), +which is a contradiction showing that L′ is meet-closed, completing the proof of +Lemma 4.4. +□ +For I ∈ Lamp(L), let NumTube(I) = NumTubeL(I) denote the number of neon +tubes of I. The total number of neon tubes is denoted by NumTubeall(L), so +NumTubeall(L) := +� +I∈Lamp(L) +NumTube(I). +(4.3) +Lemma 4.5 (Sandwiched Neon Tube Lemma). For a slim rectangular lattice L, +let n1, p, and n2 be three consecutive neon tubes of an internal lamp I ∈ Lamp(L) +such that the original territory of p is used but those of n1 and n2 are not used. +Then there is a slim rectangular lattice L′ such that Lamp(L′) ∼= Lamp(L) but +|L′| < |L| and NumTubeall(L′) = NumTubeall(L) − 1; in fact, there is an isomor- +phism ϕ: Lamp(L) → Lamp(L′) such that NumTube(ϕ(I)) = NumTube(I)−1 and +NumTube(ϕ(J)) = NumTube(J) for all J ∈ Lamp(L) \ {I}. +Proof. With reference to (4.1), denote by L′ the subposet of L that we obtain from +L by removing the fork F(p) determined by p; see Figure 3 for an illustration. We +are going to show that L′ does the job. By left-right symmetry, we can assume that +n1 is to the left of p and p is to the left of n2. +First, we prove that L′ is a sublattice. By the main result of Cz´edli [6], +both intervals occurring in (4.1) are chains of normal +slopes. Hence, by (3.1), F(p) = Floor(p). +� +(4.4) +In Figure 3, these chains are [u6, u6 ∨v6] and [v6, u6 ∨v6]. Since none of the original +territories of n1 and n2 are used, we obtain from Lemma 4.3 that +none of REOT(n1) and LEOT(n2) contains a precipitous line segment. +(4.5) + +12 +G. CZ´EDLI +These two areas border F(p) = Floor(p) from below. Thus, for any edge r of L, +if Peak(r) ∈ F(p), then r is of a normal slope. +(4.6) +For the sake of contradiction, suppose that L′ is not join-closed. Then we can pick +x′, y′ ∈ L′ such that z := x′ ∨ y′ /∈ L′, that is, z ∈ F(p). (The join is taken in L.) +By (4.1) and left-right symmetry, we can assume that z ∈ [lsupp(Foot(p)), Foot(p)]. +In Figure 3, the situation is illustrated with z as the (unique) element drawn by a +lying oval. Let T := [lsupp(Foot(n2)), Foot(p)] in L; it is [u5, u6 ∨ v6] in Figure 3. +The (area determined by) T is included in LEOT(n2). Hence, by (4.5), T contains +no precipitous line segment. Furthermore, as a lattice interval, +T is the direct product of a chain and the two-element chain. +(4.7) +Hence, z has only two lower covers, x and y (the standing ovals in the figure), +and the edges [x, z] and [y, z] are of normal slopes. +Let, say, x be to the left +of y. Now x′, y′ ∈ ↓Lz \ {z}, but {x′, y′} ⊈ ↓Ly since otherwise z = x′ ∨ y′ ≤ +y ≺ z would be a contradiction. +Hence, at least one of x′ and y′ is in ↓Lz \ +↓Ly ⊆ [lsupp(Foot(p)), Foot(p)] ⊆ F(p), contradicting that x′, y′ ∈ L′ = L \ F(p). +Therefore, L′ is closed with respect to joins. Since it is also closed with respect to +meets by Lemma 4.4, we have proved that L′ is a sublattice of L. +Let e be an edge in the interval [lsupp(Foot(p)), Foot(p)] distinct from the top +edge of this interval. Using (4.7), it is clear that if we merge the two 4-cells that +share e as a common side, we obtain a 4-cell of L′. The situation is similarly for +the non-top edges of [rsupp(Foot(p)), Foot(p)]. The top edges of these two intervals +disappear when Foot(p) and its two lower covers are omitted and three “old” 4-cells +merge into a “new” 4-cell of L′. Now that we have described the new 4-cells, it +follows from (3.2) that L′ is a slim rectangular lattice. +It is clear by the paragraph above that with the exception of p, only some +edges of normal slopes are removed when passing from L to L′. +The removal +of p does not influence the pair (Foot(I), Peak(I)) since Foot(I) is the meet of +the feet of the leftmost neon tube and the rightmost neon tube of I but p is a +“middle” neon tube of I. +Therefore, Str(L′) = Str(L), see (3.7), and so (3.8) +implies that Lamp(L′) ∼= Lamp(L). +Finally, since only one neon tube, p, has +been removed, NumTubeall(L′) = NumTubeall(L) − 1. The existence of ϕ is clear: +for J ∈ Lamp(L), ϕ(J) is defined by the property (Foot(ϕ(J)), Peak(ϕ(J))) = +(Foot(J), Peak(J)). The proof of Lemma 4.5 is complete. +□ +Lemma 4.6 (No Neighboring Neon Tubes Lemma). Let L be a slim rectangular +lattice. Assume that n1 and n2 are two neighboring neon tubes of an internal lamp +I ∈ Lamp(L) such that none of them uses its original territory. Then there exists a +slim rectangular lattice L′ such that |L′| < |L| and (Lamp(L′); ≤) ∼= (Lamp(L); ≤) +but |NumTubeall(L′)| = |NumTubeall(L)|−1; in fact, there is an order isomorphism +ϕ: (Lamp(L); ≤) → (Lamp(L′); ≤) such that |NumTube(ϕ(I))| = |NumTube(I)|−1 +but |NumTube(ϕ(K))| = |NumTube(K)| for any K ∈ Lamp(L) \ {I}. +Proof. The proof borrows some ideas from Cz´edli [7]. +Note, however, that the +present situation is different from that in [7] since now L′, to be defined below, is +not a quotient lattice of L. + +REDUCING THE LENGTHS OF SPS LATTICES +13 +Figure 4. Illustrating the proof of Lemma 4.6 by Lamp(L) ∼= P ∼= Lamp(L′) +Let, say, n2 be to the right of n1; see Figure 4 for an illustration. Observe that, +by Lemma 4.3 (or see the figure) and the fact that REOT(n1) is not used, +the peak of no precipitous edge of L belongs to RFloor(n2) and, in +particular, Foot(n2) cannot be the peak of a precipitous edge of L. +� +(4.8) +Keeping Convention 3.2 in mind, we define L′ by describing its C1-diagram. From +(the diagram of) L, we remove the fork F(n2) together with all edges that have one +or two endpoints in F(n2). Writing this formally, L′ = L \ F(n2). On the left of +Figure 4, the vertices to be omitted are drawn in blue while the edges to be omitted +are the blue dashed edges. Let L′ be the set of the remaining vertices (drawn in +black). (Note that L′ in Figure 4 is not a sublattice of L since u4, v6 ∈ L′ but +u4 ∨L v6 /∈ L′.) At this stage, L′ with the remaining (black solid) edges is not even +a lattice diagram. +Next, let q denote the right neighbor of n2 among the neon tubes of I or, if n2 +is the rightmost neon tube of I, then let q be the upper right edge of CircR(I). +Actually, it is only Foot(q) that we will need, and it is the right neighbor of Foot(n2) +among the lower covers of Peak(n2) = Peak(I). For each edge r of L, we define or +not define an edge r′ of L′ as follows. +If Foot(r) ∈ Floor(n2), then r′ is undefined and r is called an +omitted old edge. +� +(4.9) +If Foot(r) /∈ Floor(n2) and Peak(r) /∈ Floor(n2), then r′ := r and r +is called a remaining old edge of L′. +� +(4.10) +If Foot(r) +/∈ Floor(n2) and Peak(r) ∈ LFloor(n2), then let +Foot(r′) := Foot(r) and Peak(r′) := Peak(r) ∨L lsupp(Foot(n1)). +� +(4.11) +If Foot(r) +/∈ Floor(n2) and Peak(r) ∈ RFloor(n2), then let +Foot(r′) := Foot(r) and Peak(r′) := Peak(r) ∨L rsupp(Foot(q)). +� +(4.12) +If r is in the scope of (4.11) or (4.12), then r′ and r are called a new edge and a +changing old edge, respectively. In Figure 4, lsupp(Foot(n1)) = u7, rsupp(q) = v9, + +14 +G. CZ´EDLI +and the new edges are the red dashed ones. It follows from (4.8) that each edge r of +L belongs to the scope of exactly one of (4.9)–(4.12). With its new edges and the +remaining old edges, L′ turns into a Hasse diagram of a poset L′ = (L; ≤), which is +a subposet of L = (L; ≤). Actually, we need to verify that the diagram is a poset +diagram, but this is quite easy. Indeed, we only need to show that for every edge +[x, y] of the new diagram L′, there are no edges [x, z1], [z1, z2], . . . , [zk−1, y] of L′ +for some k ≥ 2. This is clear if [x, y] is a new edge, as the only possible z1 ∈ L is +not in L′; the case when [x, y] is a remaining old edge is even more obvious. To +show that the poset L′ is actually (the diagram of) a slim rectangular lattice, we +have to work more. Since none of the original territories OT(n1) and OT(n2) is +used, Lemmas 3.3 and 3.7 imply the following. +Let i ∈ {1, 2}. Then every edge r in LEOT(ni) is either of +(normal) slope (1, 1) and lies on the boundary of LEOT(ni) +or r is of (normal) slope (1, −1). Similarly, every edge r in +REOT(ni) is either of (normal) slope (1, −1) and lies on the +boundary of REOT(ni) or r is of (normal) slope (1, 1). +� +� +� +� +� +� +� +� +� +(4.13) +Hence, even though L can be more complicated in general than in Figure 4, the +original territories indicated by appropriate fill patterns in the figure reflect the +general case well. The new edges of L′, which originate from changing old edges of +L, belong to three categories, which will be discussed separately. +Category 1. We assume that r is a precipitous edge in the scope of (4.11). Then +r is a neon tube of a lamp J ∈ Lamp(L) such that Peak(J) = Peak(r) lies on +LFloor(n2). In Figure 4, J can be J1 or J2. It follows from (4.13) that we obtain +r′ from r by moving the peak of r to the northwest along an edge of slope (1, −1). +Thus, using that r is precipitous, it follows trivially that r′ is also precipitous; for +more details, the reader can (but need not) see [7, (6.8)]. +Since no precipitous +edge will occur in other categories for changing edges, let us summarize for later +references that +if a precipitous old edge h of L is a changing edge, then it changes +to a precipitous new edge h′ and Foot(h′) = Foot(h). +� +(4.14) +A line or an edge is of a slight slope if it is parallel to the vector (1, t) for some +t ∈ R such that |r| < 1. That is, a line or edge is of a slight slope if and only if it is +neither of a normal slope nor precipitous. We know from [7, (6.9)] (and it is easy +to see) that if ℓ is a (geometric) line through two distinct lower covers of Peak(J), +then ℓ is of a slight slope. +Next, let UHCircR(J) stand for the union of the 4-cells whose peaks are Peak(J); +it is a geometric area. (The acronym, taken from [7], comes from “upper half of the +circumscribed rectangle”.) For J ∈ {J1, J2} in Figure 4, UHCircR(J) in L is curl- +filled. Note that on the right of the figure, the curl-filled areas are UHCircR(J1) +and UHCircR(J2) understood in L but not in L′. It follows from Lemmas 3.3 and +3.7 (and, in a different terminology, it is explicitly stated in [7, (6.3)]) that +GInt(UHCircR(J)) contains no edge segment +that is not a part of a neon tube of J. +� +(4.15) +Practically, this means that the curl-filled areas in the figure reflect generality well. +Let h′ be an edge of L′ such that h′ ̸= r′. +Since ℓ mentioned in the previous +paragraph is of a slight slope, it follows that r′ does not cross h′ provided that + +REDUCING THE LENGTHS OF SPS LATTICES +15 +h′ ̸= h and h is another neon tube of (the same) J. Since neither the curl-filled area +GInt(UHCircR(J)) nor the 4-cell of LEOT(n2) that is the upper left neighbor of +CircR(J) contains an edge of L not mentioned in (4.15), r′ does not cross h′ if h is of +a normal slope. In the remaining case when h is precipitous but not a neon tube of +J and Peak(h) ∈ LFloor(n2), then let K denote the lamp having h as a neon tube. +Then K is an internal lamp and K ̸= J. Since an internal lamp is clearly determined +by its peak, Peak(J) ̸= Peak(K), and they are comparable since LFloor(n2) where +they belong is a chain by (4.4). The role of J and K is interchangeable, so let +Peak(K) < Peak(J). Then (the line determined by) RRoof(K) separates J and K, +and we obtain easily again that r′ and h′ do not cross. We have seen that +if r′ originates from a precipitous edge r of L, +then r′ does not cross any other edge of L′. +� +(4.16) +Category 2. We assume that r is of a normal slope and r′ is defined in (4.11). Then +b := Peak(r′) ∈ L even though r′ is not an edge of L. It is clear either by Lemmas +3.3 and 3.7 or by comparing the present situation to (4.7) that Peak(r) ≺L b. Hence, +d := [Peak(r), b] is an edge. This edge lies in LEOT(n2), and we obtain from (4.13) +that d is of slope (1, −1). So is r since it is of a normal slope but does not lie +on LFloor(Foot(n2)). This means that r′ comes to existence by merging r and d, +which are adjacent edges lying on the same line of slope (1, −1). Hence, r′ is also of +slope (1, −1). Therefore, since Category 3 will be analogous to the current one by +left-right symmetry and we are armed with (4.14), we can conclude even now that +if g is a changing old edge of a normal slope, than the edge +g′ of L′ is of the same (normal) slope and, furthermore, g′ is +obtained by merging two collinear adjacent edges of L. +� +(4.17) +It follows from (4.16) and (4.17) that if r′ crossed an edge g′ of L′, then g′ would +be of the other normal slope, (1, 1), and it would come to existence by merging g +to a collinear other edge of L at b. But then g would lie on RFloor(n2) and instead +of merging it to a collinear edge to obtain g′, g would have been omitted. Thus, +if r belongs to Category 2, then r′ does not cross any other edge of L′. +(4.18) +Category 3. We assume that r is in the scope of (4.12). By (4.8), r is of (a normal) +slope (1, 1). Hence, the situation is basically the left-right symmetric counterpart +of the one discussed in Category 2, whereby no details will be given. +Now that the three categories have been investigated, (4.16), (4.18), and the +left-right symmetric counterpart of (4.18) for Category 3 imply that L′ is a planar +Hasse-diagram. We know from Kelly and Rival [17, Corollary 2.4] that planar posets +with 0 and 1 are lattices. Hence, L′ is a planar lattice. By construction, the number +of upper covers of an element x ∈ L′ is the same in L′ as in L. Furthermore, an +element of L′ belongs to the boundary of L′ if and only if it belongs to the boundary +of L. Therefore, (3.2) and the construction of L′ yield in a straightforward but a +bit tedious way that L′ is a slim rectangular lattice. +Since x ∈ L′ has the same number of covers in L′ as in L, we obtain that M(L′) = +L′ ∩M(L). Moreover, we already have (4.14) and (4.17), and it is clear that an edge +r′ of L′ lies on Bnd(L′) if and only if it lies on Bnd(L). Clearly, lc(L), rc(L) ∈ L′. +Therefore, taking the just mentioned facts of the present paragraph and Convention +3.2 (for L) into account, we conclude that L′ is (given by) a C1-diagram. + +16 +G. CZ´EDLI +Since OT(n2) is not used, it follows from (4.4) and Lemma 4.3 that +if h is a neon tube of L and h ̸= n2, then Foot(h) /∈ F(n2) = Floor(n2). +(4.19) +It follows from (4.14), (4.17), and the construction of L′ that +the neon tubes of L′ are exactly the r′ where r is a neon tube +of L and r ̸= n2. +Furthermore, for neon tubes r and h of +L such that r ̸= n2 ̸= h, Peak(r′) = Peak(h′) if and only if +Peak(r) = Peak(h) and Foot(r′) = Foot(r). +� +� +� +� +� +(4.20) +Hence, for a lamp K ∈ Lamp(L) \ {I}, {r′ : r is a neon tube of K} is exactly the +collection of neon tubes of a lamp K′ of L′. Furthermore, {h : h is a neon tube of +I and h ̸= n2} is the set of neon tubes of an internal lamp I′ of L′ — this is the +definition of I′. Note that Lemma 4.4 and (4.20) give that Foot(K′) = Foot(K) +for K ∈ Lamp(L) \ {I}. Now (4.20) and the facts mentioned thereafter allow us to +conclude that the function ϕ: Lamp(L) → Lamp(L′) defined by +K �→ +� +K′ +if K′ ∈ Lamp(L′) such that Foot(K′) = Foot(K), +I′ +if K = I +(4.21) +is bijective. (Remark that if n2 is not the rightmost neon tube of I, then I belongs +to the scope of both lines of (4.21).) Note the rule, which follows from (4.20): for +any K ∈ Lamp(L), we have that Peak(ϕ(K)) = Peak(K). +We know from Lemma 3.10 that, in order to see that ϕ is an order isomorphism, +it suffices to show that, for J, K ∈ Lamp(K), +(J, K) ∈ ρfoot ⇐⇒ (J′, K′) ∈ ρfoot. +(4.22) +Assume that (J, K) ∈ ρfoot and J ̸= I. Since Peak(K′) is to the northwest +(that is, to the (−1, 1) direction) of Peak(K) or Peak(K′) = Peak(K), we have +that Lit(K) ⊆ Lit(K′). Hence, Foot(J′) = Foot(J) ∈ Lit(K) ⊆ Lit(K′) gives the +required (J′, K′) ∈ ρfoot. If (I, K) ∈ ρfoot, then CircR(I′) = CircR(I) ⊆ Lit(K) ⊆ +Lit(K′) by Lemma 3.10, whereby (I′, K′) ∈ ρCircR = ρfoot, as required. This proves +the “⇒” part of (4.22). +Next, assume that (J′, K′) ∈ ρfoot and I /∈ {J, K}. We know that Foot(K′) = +Foot(K) and Foot(J′) = Foot(J). +If Peak(K′) = Peak(K), then Foot(J) = +Foot(J′) ∈ Lit(K′) = Lit(K) gives the required (J, K) ∈ ρfoot. So assume that +Peak(K′) ̸= Peak(K). By construction, Lit(K′) ⊆ Lit(K) ∪ LEOT(n2); see Fig- +ure 4. +Hence, Foot(J) = Foot(J′) ∈ Lit(K′) gives that Foot(J) ∈ Lit(K) or +Foot(J) ∈ LEOT(n2). If the second alternative, Foot(J) ∈ LEOT(n2), holds, then +Foot(J) ⊆ EOT(n2), which contradicts Lemma 4.3 as OT(n2) is not used. Hence, +Foot(J) ∈ Lit(K), which gives that (J, K) ∈ ρfoot, as required. +We are left with the case when one of J and K is I. +Assume that (J′, I′) ∈ ρfoot. Then Foot(J) = Foot(J′) ∈ Lit(I′) ⊆ Lit(I) gives +the required (J, I) ∈ ρfoot. (Note that Lit(I′) ⊂ Lit(I) if n2 is the rightmost neon +tube of I, and Lit(I′) = Lit(I) otherwise.) +Finally, assume that (I′, K′) ∈ ρfoot. Then (I′, K′) ∈ ρCircR by Lemma 3.10. +This fact and CircR(I) = CircR(I′) give that +Peak(I) = Peak(CircR(I)) = Peak(CircR(I′)) ∈ CircR(I′) ⊆ Lit(K′). + +REDUCING THE LENGTHS OF SPS LATTICES +17 +Hence, (Foot(K′), Peak(K′)) = (Foot(K), Peak(K)), and so Lit(K′) = Lit(K). +These facts lead to CircR(I) = CircR(I′) ⊆ Lit(K′) = Lit(K). Thus, (I, K) ∈ +ρCircR = ρfoot, as required. The proof of Lemma 4.6 is complete. +□ +5. An estimate +Our goal is to prove that +Theorem 5.1. Let D be the congruence lattice of a slim semimodular lattice (in +other words, an SPS lattice) K. +Let n := |J(D)|, that is, the number of join- +irreducible congruences of K. If n = 0 or n = 1, then D is the (n + 1)-element +chain and K ∼= D. If n = 2, then the length len(K) of K is 2 and K is either the +three-element chain or the four-element boolean lattice. +If n ≥ 3, then the following two assertions hold. +(A) There is a slim rectangular lattice L such that Con L ∼= D and +len(L) ≤ 2n2 − 10n + 15, +and so +len(L) < 2n2. +(5.1) +(B) For any slim semimodular lattice L′, if Con L′ ∼= D, then len(L′) ≥ n. +Proof. The case n ≤ 2 is trivial. In the rest of the proof, we assume that n ≥ 3. Let +L be a slim rectangular lattice. A trivial induction by Lemmas 3.3 and 3.7 shows +that +len(L) = NumTubeall(L). +(5.2) +Now if Con L ∼= D, then Lamp(L) ∼= J(Con L) ∼= J(D) by Lemma 3.10, and so +(5.2) and the fact that each lamp has at least one neon tube give that len(L) = +NumTubeall(L) ≥ |Lamp(L)| = n. Hence, Part (B) holds for the particular case of +rectangular SPS lattices. +We know from Gr¨atzer and Knapp [16, Theorem 7] and its proof that +for any slim semimodular lattice L′, there is a slim rectangular +lattice L such that Con L ∼= Con L′ and len(L) = len(L′). +� +(5.3) +This statement also follows from Cz´edli and Schmidt [12, Lemma 21] (applied in +the reverse directions) and Cz´edli[1, (Corner) Lemma 5.4]. Therefore, Part (B) +follows from its particular case mentioned above. +Next, we turn our attention to part (A). If J(D) is the n-element antichain, then +any grid G with length n is a slim rectangular lattice such that Con G ∼= D, and +len(G) = n ≤ 2n2 is clear. Therefore, in the rest of the proof, we assume that J(D) +is not an antichain. +Next, take a slim rectangular lattice L of minimal length such that Con L ∼= D. +We know from Lemma 3.10 that Lamp(L) ∼= J(D), and so |Lamp(L)| = n. Let +J ∈ Lamp(L) be an internal lamp. Let t+ +J denote the number of neon tubes of +J whose original territories are used. Similarly, t− +J stands for the number of neon +tubes of J whose original territories are not used; note that t+ +J +t− +j = NumTube(J). +Listing the neon tubes from left to right, let us write a letter u for a used neon +tube and a zero for an unused neon tube. Then we obtain a sequence ⃗s of length +NumTube(J) consisting of t+ +J u’s and t− +J zeros. +Subsequences 0 u 0 and 0 0 are +forbidden by (5.2) and Lemmas 4.5 and 4.6 since len(L) is minimal. For another +look at ⃗s, take the sequence ⃗w := +⋆ u ⋆ u ⋆ u · · · ⋆ u ⋆ u ⋆ u ⋆ +of t+ +J u’s and t+ +J + 1 +stars that alternate. We can obtain ⃗s from ⃗w by removing some stars and replacing +the remaining stars by zeros. Observe that only one zero can replace a star since +0 0 is a forbidden subsequence. Furthermore, for any two consecutive stars (which + +18 +G. CZ´EDLI +occur in a subsequence ⋆ u ⋆), at most one of the two stars can change to 0 and +so the other one should be removed since 0 u 0 cannot be a subsequence. Hence, +at most every second star can turn to 0 and the rest of the stars are removed. +Therefore, the number t− +J of zeros is at most8 ⌈(t+ +J + 1)/2⌉ where ⌈x⌉ denotes the +upper integer part of a real number x. Since ⌈(t+ +J + 1)/2⌉ ≤ t+ +J , we obtain that, for +any J ∈ Lamp(L), +NumTube(J) = t+ +J + t− +j ≤ 2 · t+ +J . +(5.4) +Let m denote the number of boundary lamps, that is, the number of maximal +elements of Lamp(L) (or, equivalently, those of D). Each of LBnd(L) and RBnd(L) +contains at least one boundary lamp, whence m ≥ 2. On the other hand, m < n as +Lamp(L) ∼= J(D) is not an antichain. Thus, 2 ≤ m ≤ n − 1. Let k := n − m, the +number of internal lamps of L. Clearly, k ≥ 1. If p is a neon tube of an internal +lamp J and I uses the original territory of J, then I < J and, in particular, I is +also an internal lamp. Furthermore, if p1,. . . , pt+ +J denote the neon tubes of J whose +original territories are used, then the GInt(LEOT(p1)), . . . , GInt(LEOT(pt+ +J )) are +pairwise disjoint, and so are GInt(REOT(p1)), . . . , GInt(REOT(pt+ +J )). Therefore, +using Lemma 4.3(b), it follows that the lamp I can use the original territories of +at most two of the neon tubes of J. The number of lamps I that use the original +territory of a neon tube of J is at most |↓J\{J}|, whereby J has at most 2·|↓J\{J}| +neon tubes9 whose original territories are used. By (5.4), it has at most twice as +many neon tubes all together. Hence, the total number of neon tubes of the internal +lamps is at most10 +� +internal J∈Lamp(L) +2 · 2 · |↓J \ {J}| = 4 · +� +internal J∈Lamp(L) +|↓J \ {J}|. +(5.5) +Observe that |↓J \ {J}| is the number of pairs (I, I′) of internal lamps subject to +I < I′ and I′ = J. Therefore, the second sum in (5.5) is the number of pairs (I, J) +such that I < J. In other words, the second sum is the number of comparabilities +among internal lamps. So this sum reaches its maximum when for any internal +lamps I ̸= J, we have that I < J or J < I. That is, when the internal lamps form +a chain. Then there are +�k +2 +� += k(k − 1)/2 such pairs, and so the maximum that +(5.5) can take is 2k(k − 1); it might seem to be an upper bound on the number +NumTubeinternal(L) of neon tubes of internal lamps of L. +The only problem with the argument above is that our assertion that “J has at +most 2 · |↓J \ {J}| neon tubes” is true only if ↓J \ {J} is nonempty, that is, if J +is not a minimal lamp. Indeed, if J is a minimal lamp, then 2 · |↓J \ {J}| = 0 but +J must have at least one neon tube. Since NumTubeall(L) is minimal, we obtain +from Lemma 4.5 that a minimal lamp J has exactly one neon tube. Let s be the +number of minimal internal neon tubes; note that 1 ≤ s ≤ k. The number 2k(k−1) +has to be modified in two opposite directions. First, two minimal internal lamps +are incomparable, whereby the number +�k +2 +� += k(k − 1)/2 has to be reduced by +�s +2 +� += s(s − 1)/2. Second, the minimal internal lamps were computed with zero +neon tubes, so we have to add s · 1 = s. So we obtain that +NumTubeinternal(L) ≤ 4 · +� +k(k − 1)/2 − s(s − 1)/2 +� ++ s +8Provided that t+ +J > 0; this correction will be taken into account about nine lines after (5.5). +9For minimal lamps, this will be corrected soon. +10To be improved soon by taking the minimal internal lamps of L into account. + +REDUCING THE LENGTHS OF SPS LATTICES +19 += 2k2 − 2k + 3s − 2s2. +(5.6) +Since 3s − 2s2 is negative for s ≥ 2, we obtain the largest value in (5.6) for s = 1, +when 3s − 2s2 = 1. (Smaller values at s > 1 would not give an upper estimate for +s = 1.) Hence, NumTubeinternal(L) ≤ 2k2 − 2k + 1. Now taking the m boundary +lamps and the equality k = n − m into account, we obtain that +NumTubeall(L) = m + NumTubeinternal(L) +≤ m + 2(n − m)2 − 2(n − m) + 1 += 2n2 − 2n + 1 + 2 · +� +m2 − (2n − 3/2)m +� +� +�� +� +. +(5.7) +Let f(m) = m2 − (2n − 3/2)m denote the under-braced term. By the elementary +theory of quadratic univariate real functions, f(m) decreases in the interval [0, n − +3/4]. This fact and 2 ≤ m ≤ n − 1 imply that the largest value of f(m) is f(2) = +7 − 4n. Substituting this value into (5.7), we obtain that +NumTubeall(L) ≤ 2n2 − 10n + 15 < 2n2. +(5.8) +Finally, len(L) = NumTubeall(L) and (5.8) completes the proof of Theorem 5.1. +□ +Remark 5.2. The first inequality in (5.1) is not sharp. To show this by an example, +let n = |J(D)| = 4. Then 2n2 − 10n + 15 is 7 but, no matter which 4-element +poset J(D) is, there is a slim semimodular lattice L such that |J(Con L)| ∼= D and +len(L) ≤ 5. If J(D) is the four-element “Y poset”11, then len(L) cannot be less +than 5. +Corollary 5.3. For L in Part (A) of Theorem 5.1, |L| ≤ (2n2 −10n+15)2 < 4n4. +Proof. By Theorem 5.1, it suffices to show that if a slim semimodular lattice L is +of length k, then |L| ≤ k2. By (5.3), we can assume that L is a slim rectangular +lattice. By Definition 2.1, there are chains C, U ⊆ J(L) such that J(L) = C ∪ U. +Then 0 /∈ C and, by rectangularity, 1 /∈ C. Hence |C| ≤ k−1. Similarly, |U| ≤ k−1. +Since any element of L \ {0} is of the form c ∨ u with c ∈ C and u ∈ U, L has at +most 1 + |C| · |U| = 1 + (k − 1)2 ≤ k2 elements, completing the proof. +□ +6. Odds and ends +Let P be a poset, and let j ∈ P. We define a new poset P ′ as follows. The +base set of P ′ is (P \ {j}) ∪ {j′, j′′} where P ∩ {j′, j′′} = ∅. The ordering in P ′ +is defined as follows: for a, b ∈ P ′ \ {j′, j′′} = P \ {j}, a ≤P ′ b +⇐⇒ +a ≤P b, +a ≤P ′ j′ ⇐⇒ a ≤P ′ j′′ ⇐⇒ a ≤P j, j′ ≤P ′ b ⇐⇒ j′′ ≤P ′ b ⇐⇒ j ≤P b, and +j′′ ≺P ′ j′. We say that P ′ is obtained from P by doubling the element j of P. For +an example, see P and P ′ in the middle of Figure 5. +Proposition 6.1. Let j be an element of a finite poset P, and let P ′ be the poset +that we obtain from P by doubling j. +If there is a slim semimodular lattice L +such that P ∼= J(Con L) and j is not a maximal element of P, then there ex- +ists a slim rectangular lattice L′ such that P ′ ∼= J(Con L′) and NumTubeall(L′) = +NumTubeall(L) + 2. (Note that (5.2) would allow to rewrite this equality.) +11The diagram of this poset is Y-shaped; this is where the name “Y-poset” comes from. + +20 +G. CZ´EDLI +Figure 5. The construction for Proposition 6.1 without rescaling +Proof. Gr¨atzer and Knapp’s result, quoted here in (5.3), allows us to assume that +L is a rectangular lattice. +Let us agree that whenever we refer to some neon +tube like the m-th neon tube of a lamp, then we mean that the neon tubes of +the lamp in question are listed from left to right and “m-th” refers to this list; +see also Convention 3.2. +We also count on the fixed multifork sequence of L, see +Lemmas 3.3 and 3.7. We know from Lemma 3.10 that there is an order isomorphism +P → Lamp(L); we denote its action by capitalization, that is, x �→ X. The notation +used in Lemma 3.7 is in effect. Since j is not a maximal element of P, J is an internal +lamp; let, say, J = It. In Figure 5, t = 3. +At present, we are only interested in L and P in Figure 5 but note in advance +that L′ in this figure is hardly readable. Indeed, some vertices of L′ are so close to +each other that we cannot see whether there is an edge connecting them or there +is not. Therefore, we also present Figure 6, which is a C1-diagram of the same +lattice L′. The only difference between the two figures is that L′ is rescaled (and so +readable) in Figure 6. Usually, we will simply refer to “the figure” which is Figure +5 but the reader can sometimes check its unreadable details in Figure 6. +Resuming the argument, note that P ∩ P ′ = P \ {j} = P \ {j′, j′′} is a subposet +both in P and in P ′. For any x ∈ P ∩ P ′, the lamp corresponding to x will be +denoted by X both in L and in L′; this should not cause confusion since it will be +clear from the context whether X ∈ Lamp(L) or X ∈ Lamp(L′). This will mean +that the pair (Foot(X), Peak(X)) is the same in L′ as in sublattice L. So, implicitly, +we mostly consider lamps as pairs in the proof. +We define L′ in the following way. Let ϵ ∈ R, ϵ > 0, be the smallest one out of +the geometric lengths of the edges of (the fixed C1-diagram of) L. With reference +to the multifork sequence of L, let L′ +0 := L0, L′ +1 := L1,. . . , L′ +t−1 := Lt−1; these +equations also mean the exact coincidence of the corresponding C1-diagrams in the +plane. As for the forthcoming notation, we will continue the sequence by L′ +t−0.5, +L′ +t, L′ +t+1, . . . , L′ +k =: L′. In L′ +t−1 (which is the same as Lt−1), let H′ +t−0.5 be the +same 4-cell (even geometrically the same) as Ht in Lt−1. +Later, Ht turns into CircR(It) in L; in the figure, CircR(It) = CircR(I3) is the +“3-filled” area in L. In L′, only the “major part” of CircR(I′ +t−0.5) = CircR(I′ +2.5) +is 3-filled; the rest of CircR(I′ +t−0.5) = CircR(I′ +2.5) is yellow-filled. At Ht in Lt−1, + +REDUCING THE LENGTHS OF SPS LATTICES +21 +we perform a NumTube(It)-fold multifork extension, which produces J = It. (In +the figure, where It = I3 = J, NumTube(It) = 4.) However, in L′ +t−1, we add a +2-fold multifork at H′ +t−0.5 to obtain a new lattice L′ +t−0.5. Geometrically (in the C1- +diagram), this new multifork extension and the lamp J′ = It−0.5 it produces look +unusual compared to other figures in the present paper and several earlier papers. +Namely, we require that the 4-cell H′ +t whose peak is the foot of the leftmost neon +tube of J′ should be almost as large as H′ +t−0.5. That is, the width η of the “legs” +of the Λ-shaped difference H′ +t−0.5 \ H′ +t, which is yellow-filled in the figure, should +be very small compared to ϵ. (We may think of η = ϵ/1000.) On the right of the +Figure, H′ +t = H′ +3 in L′ is 3-filled. +Next, we perform a NumTube(It)-fold multifork extension at H′ +t to obtain L′ +t +from Lt−0.5 and to produce the lamp J′′ = It of L′ +t (and of L′). The feet of the +neon tubes of J′′ = It in L′ +t (and in L′) should be the same geometric points as the +feet of the neon tubes of J = It in Lt (and in L). So the geometric shape of J and +that of J′′ are almost the same (and they tend to be the same as η tends to 0). +From L′ +t, we continue the multifork sequence for L′ in the same way as we +continue the sequence from Lt to reach L. Even in geometric sense, we do almost +the same, that is, with very little differences that would diminish if we formed the +limit at η → 0. To be more specific, let us agree that we use the alternative notation +I−1 = A1, I−1.7 = B1, I−2 = A2, I−2.7 = B2, . . . +for the boundary lamps. For +s = t + 1, t + 2, . . . , k, we select H′ +s+1 as follows. In Ls, the trajectory through the +top left edge of the 4-cell Hs+1 contains exactly one neon tube, p. Since the top left +edge of Hs+1 is of slope (1, 1), it is in the descending part of the trajectory. The +neon tube p belongs to exactly one lamp, which is older than or as old as Is; let +Iu denote this lamp. Note that we never use the trajectory through the leftmost +neon tube of It−0.5 (in the figure, the “narrow” trajectory through the yellow-filled +area), whereby u ̸= t − 0.5 and so Iu will also make sense in L′, not only in L. +Among the neon tubes of Iu, let p be the α-th neon tube (from the left). In +L′ +t, let p′ be the α-th neon tube of Iu. By left-right symmetry, the top right edge +of Hs+1 defines a neon tube q of a lamp Iv in Ls and its counterpart q′ in L′ +s. +The top right edge of Hs+1 is in the ascending part of the trajectory in question. +Now we can simply select H′ +s+1 as the unique 4-cell of L′ +s where the descending +part of trajectory through p′ and the ascending part of the trajectory through q′ +cross each other12. Once H′ +s+1 has been selected, we perform a NumTube(Is+1)- +fold multifork extension at this 4-cell of L′ +s to obtain L′ +s+1 and its lamp Is+1. This +multifork extension should almost be the same geometrically as in the passage from +Ls to Ls+1; in particular, the feet of the new neon tubes have to be geometrically +the same in L′ +s+1 as in Ls+1. For later reference, note that +the left upper edge of CircR(Is+1) = Hs+1 belongs to +the trajectory through a neon tube of Iu both in L an +L′, and similarly for the right upper edge and Iv. +� +� +� +(6.1) +Finally, we obtain L′ = L′ +k. +Next, in order to recall Cz´edli [7, Lemma 7.5], we need some notation. Let U be +an internal lamp of a slim rectangular lattice K. Then the top edge of the trajectory +containing the upper left edge of CircR(U) is a neon tube of a lamp, which we denote +by Nwl(U). Left-right symmetrically, Nel(U) stands for the unique lamp that has +12Doubts whether they cross will be dissolved later. + +22 +G. CZ´EDLI +a neon tube whose trajectory contains the upper right edge of CircR(U). For a +poset Q, let Min(Q) stand for the set of minimal elements of Q. Now [7, Lemma +7.5] asserts that if K is a slim rectangular lattice and U, V ∈ Lamp(K), then +U ≺ V in Lamp(K) if and only if V ∈ Min({Nwl(U), Nel(U)}). +(6.2) +Comparing (6.1) and (6.2) and taking into account that only internal lamps, which +all occur in (6.1), can be covered by another lamp, it follows that Lamp(L) \ {J} is +order isomorphic to Lamp(L′)\{J′, J′′}. We obtain from Lemma 3.10 that J′ < J′′ +in Lamp(L′), Lamp(L) ∼= Lamp(L′) \ {J′}, and Lamp(L) ∼= Lamp(L′) \ {J′′}. +Thus, using that P ∼= Lamp(L), we conclude that P ′ ∼= Lamp(L′), as required. +Furthermore, the construction yields that NumTubeall(L′) = NumTubeall(L) + 2. +However, the proof is not ready yet. Indeed, we need to show that the trajectories +mentioned earlier do cross in L′ +s. To be more precise, we need to show that if the +geometric areas REOT(p) and LEOT(q) cross in Ls, than so do REOT(p′) and +LEOT(q′) in L′ +s. +Of course, REOT(p′) and LEOT(q′) are perpendicular if we +disregard their thickness but, in principle, they could avoid each other like the +right leg of the upper ∧∧∧ and the left leg of the lower ∧∧∧ do in +� +� +� +� +� +� +. +(6.3) +Fortunately, it is clear by continuity that whenever η is small enough (compared +to ϵ), then REOT(p′) and REOT(q′) are close enough to REOT(p) and REOT(q), +respectively. Thus, since REOT(p) and REOT(q) cross each other at a rectangle +with sides at least ϵ, REOT(p′) ∩ REOT(q′) is a rectangle of a positive area. Fur- +thermore, in Ls, REOT(p) ∩ REOT(q) is a 4-cell. Since, except when J′′ = It was +created, OT(J′) = OT(It−0.5) is never used, REOT(p′) ∩ REOT(q′) is also a 4-cell. +This shows that the definition of L′ +s+1 and that of L′ make sense, completing the +proof of Proposition 6.1. +□ +Figure 6. The construction for Proposition 6.1, rescaled +In a poset P, where meets and joins need not exist, M(P) and J(P) are defined +in the usual way: x ∈ M(P) means that x has exactly one cover; J(P) is defined +dually. + +REDUCING THE LENGTHS OF SPS LATTICES +23 +Remark 6.2. In most of the cases, the estimate given in (5.1) of Theorem 5.1 is +far from optimal. For example, if J(Con L′) ∼= J(D) ∼= P ′ and P ′ is obtained from a +smaller poset P by doubling a non-maximal element j ∈ P, then, with the notation +of Proposition 6.1, the lamp J′ corresponding to j′ ∈ P ′ has only two neon tubes +and contributes to len(L′) by 2 regardless the size of ↓Lamp(L′)J′. +To present another example of a different nature, let Pn be the n-element poset +consisting of two maximal elements, a and b, n − 3 minimal elements, c1, . . . , cn−3, +and an element u such that u ≺ a, u ≺ b, and ci ≺ u for all i ∈ {1, . . . , n − 3}. +Then there is a slim rectangular lattice L such that Con L ∼= Pn and len(L) = +|NumTubeall(L)| = n + 1, which is much smaller than what the estimate (5.1) +gives. (Note that NumTube(U) = 2; this helps us to draw L.) +In our third example, Qn is the poset with two maximal elements and n − 2 +minimal elements such that every minimal element is covered by both maximal +elements. Then there is a slim rectangular lattice L such that Con L ∼= Pn and +len(L) = |NumTubeall(L)| = n. This example shows that the lower estimate given +in Theorem 5.1(B) cannot be improved. +As Remarks 5.2 and 6.2 allow us to guess, there are many factors that reduce +the number len(L) = |NumTubeall(L)|. This explains that we are far from a sharp +bound instead of (5.1) as well as from a significantly better while still simple one. +Corollary 5.3 is not sharp either. Indeed, in addition to that this corollary is built +on the non-sharp Theorem 5.1, there is another factor explaining this. Namely, if +J(D) ∼= J(Con K) has few non-maximal elements (in particular, if D is Boolean), +then |L| has few internal lamps and |L| is close to len(L)2 but then len(L) is much +smaller than what (5.1) gives. On the other hand, if J(D) ∼= J(Con K) has many +non-maximal elements, then L has many internal lamps and |L| is considerably +smaller than len(L)2. +Remark 6.3. In order to decide whether a given n-element poset P is isomorphic +to J(Con L) of a slim semimodular lattice L, it is not economic and usually it is +not feasible to list all slim rectangular lattices of length at most 2n2 − 10n + 15, +see (5.1), or those of size at most (2n2 − 10n + 15)2, see Corollary 5.3. It is often +much faster to check +(A) the known properties of the posets J(Con L) of slim rectangular lattices L, +see (5.3), Cz´edli [4], [8], and Cz´edli and Gr¨atzer [10] (where two earlier properties +from Gr¨atzer [13] and [14] are also recalled) to see whether these properties exclude +the existence of L; and +(B) the known constructions to see whether they imply the existence of L (and +show how to construct it); see Proposition 6.1, Cz´edli [7, Theorems 3.14 and 3.16], +and Cz´edli and Gr¨atzer [10, Theorem 1.2]. +Furthermore, +(C) in many cases, in particular for small |P|, even if (A) and (B) do not lead +to an answer, the ideas of the proofs in the papers referenced in (A) and (B) above +contain ideas how to construct an L with Lamp(L) ∼= P or how to exclude the +existence of such an L by inspecting much less cases then those suggested by (5.1). +Even though no systematic checking has been done to decide which |P| is “small” +for Part (C) of Remark 6.3, we mention that |P| ≤ 6 is probably small. + +24 +G. CZ´EDLI +References +[1] Cz´edli, G.: Representing homomorphisms of distributive lattices as restrictions of congruences +of rectangular lattices. Algebra Universalis 67 (2012) 313–345. +DOI 10.1007/s00012-012-0190-3 +[2] Cz´edli, G.: Patch extensions and trajectory colorings of slim rectangular lattices. Algebra +Universalis 72, 125–154 (2014) +[3] Cz´edli, G.: Diagrams and rectangular extensions of planar semimodular lattices. Algebra +Universalis 77, 443–498 (2017) +[4] Cz´edli, +G.: +Lamps +in +slim +rectangular +planar +semimodular +lattices. +Acta +Sci. +Math. +(Szeged)13 +87, +381–413 +(2021) +(Open +access +view: +https://doi.org/10.14232/actasm-021-865-y or brows http://www.acta.hu/) +[5] Cz´edli, G.: +Cyclic congruences of slim semimodular lattices and non-finite axiomati- +zability of some finite structures. Archivum Mathematicum Brno 58/1 (2022) 15–33. +https://doi.org/10.5817/AM2022-1-15 +[6] Cz´edli, +G.: +A +property +of +meets +in +slim +semimodular +lattices +and +its +applica- +tion to retracts. Acta Sci. Math. (Szeged), +to appear. For an earlier version, +see +http://arxiv.org/abs/2112.07594 +[7] Cz´edli, G.: Quotient diagrams14; http://arxiv.org/abs/2208.03606 +[8] Cz´edli, G.: Infinitely many new properties of the congruence lattices of slim semimodular +lattices, submitted to Acta. Sci. Math. (Szeged)15 http://arxiv.org/abs/2206.14769 +[9] Cz´edli, G., D´ek´any, Gyenizse, G., Kulin, J.: The number of slim rectangular lattices, Algebra +Universalis 75/1 (2016) 33–50, DOI: 10.1007/s00012-015-0363-y +[10] Cz´edli, G., Gr¨atzer, G.: A new property of congruence lattices of slim, planar, semimodu- +lar lattices. Categories and General Algebraic Structures with Applications 16, 1–28 (2022) +(Open access: https://cgasa.sbu.ac.ir/article 101508.html) +[11] Cz´edli, G., Schmidt, E. T.: +The Jordan-H¨older theorem with uniqueness for groups and +semimodular lattices. Algebra Universalis 66, 69–79 (2011) +[12] Cz´edli, G., Schmidt, E. T.: Slim semimodular lattices. I. A visual approach. Order 29, 481–497 +(2012) +[13] Gr¨atzer, G.: Congruences of fork extensions of slim, planar, semimodular lattices. Algebra +Universalis 76, 139–154 (2016) +[14] Gr¨atzer, G.: Notes on planar semimodular lattices. VIII. Congruence lattices of SPS lattices. +Algebra Universalis 81 (2020), Paper No. 15, 3 pp. +[15] Gr¨atzer, G., Knapp, E.: Notes on planar semimodular lattices. I. Construction. Acta Sci. +Math. (Szeged) 73, 445–462 (2007) +[16] G. Gr¨atzer and E. Knapp, Notes on planar semimodular lattices. III. Rectangular lattices. +Acta Sci. Math. (Szeged) 75 (2009), 29–48. +[17] Kelly, D., Rival, I.: Planar lattices. Canad. J. Math. 27, 636–665 (1975) +Email address: czedli@math.u-szeged.hu +URL: http://www.math.u-szeged.hu/~czedli/ +University of Szeged, Bolyai Institute. Szeged, Aradi v´ertan´uk tere 1, HUNGARY +6720 +13At the time of writing, see also +http://www.acta.hu/acta/ , the good old site of Acta Sci. +Math. (Szeged), where all my papers published not later than 2021 are free to download +14At the time of writing, see https://tinyurl.com/czedli-cde-con-sps +for the most recent +version +15At the time of writing, see also the author’s website. + diff --git a/atAyT4oBgHgl3EQfivhb/content/tmp_files/load_file.txt b/atAyT4oBgHgl3EQfivhb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f171d68c4579dc706bf52d062f0cfb6e2e013f0 --- /dev/null +++ b/atAyT4oBgHgl3EQfivhb/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf,len=1171 +page_content='REDUCING THE LENGTHS OF SLIM PLANAR SEMIMODULAR LATTICES WITHOUT CHANGING THEIR CONGRUENCE LATTICES G´ABOR CZ´EDLI Dedicated to the memory of my paternal grandfather, J´ozsef Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Introduced by Gr¨atzer and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Knapp in 2007, a slim semimodular lattice is a planar semimodular lattice without M3 as a sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We prove that if K is a slim semimodular lattice and n denotes the number of its join- irreducible congruence relations, then there exists a slim semimodular lattice L such that Con L ∼ = Con K, the length of L is at most 2n2, and the number of elements of L is at most 4n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (In fact, we prove slightly more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Also, we present a new construction under which the class of (isomorphism classes of) posets of join-irreducible congruences of slim semimodular lattices is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Goal In 2016, Cz´edli, D´ek´any, Gyenizse, and Kulin [9] proved that for a natural num- ber n, there are asymptotically (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' · e2/2 many slim rectangular lattices1 of length n, where e is the famous mathematical constant �∞ k=0 1/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='71828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' There are several ways, including the one occurring in [9], to construct and list all of these lattices of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' At initial level, this is one of the motivations of our aim to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, which, roughly saying, asserts that if a finite distributive lattice is representable by the congruence lattice of a slim semimodular lattice (equivalently, a slim rectangular lattice) L, then we can find an L with small length for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The secondary aim is to prove a related result, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Basic concepts For the history and the importance of (slim) planar semimodular lattices, see many papers that belong to the more than four dozen papers listed at2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='u-szeged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='hu/~czedli/m/listak/publ-psml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='pdf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As usual, semimodularity is upper semimodularity, that is, the satisfaction of (∀x)(∀y)(x ∧ y ≺ x ⇒ y ≺ x ∨ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The poset (that is, partially ordered set) of join-irreducible elements of a lattice L is denoted by J(L) = (J(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that 0 = 0L /∈ J(L) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The set M(L) of meet-irreducible elements is denoted by M(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' note that 1 /∈ M(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The elements of J(L) ∩ M(L) are called double 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 06C10 January 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Slim rectangular lattice, slim semimodular lattice, planar semimodular lattice, congruence lattice, lattice congruence, lamp, C1-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This research was supported by the National Research, Development and Innovation Fund of Hungary under funding scheme K 138892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 1To be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 2The list the Appendix of https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10202 is no longer up-to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='00401v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='RA] 1 Jan 2023 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The following definition combines the original definitions of Gr¨atzer and Knapp [15] and [16] with Cz´edli and Schmidt [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) A lattice is planar if it is finite and it has a planar (Hasse) diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (All lattices in this paper are assumed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') (B) By [11], a finite lattice L is slim if J(L) is the union of two chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If, in addition, L is semimodular, then we say that it is a slim semimodular lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (C) By [15], a planar semimodular lattice L is slim if M3, the five-element modular lattice of length 2, is not a cover-preserving sublattice of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A slim planar semimodular lattice in this sense is, shortly, an SPS lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (D) If a lattice L is slim in the sense of (B), then it is necessarily planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' There- fore, slim semimodular lattices are the same as SPS lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This paper gives preference to “slim semimodular lattices” to “SPS lattices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (E) Following Gr¨atzer and Knapp [16], a slim rectangular lattice is a slim semi- modular lattice that has exactly two doubly irreducible elements and these two elements are complements of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Concepts, terminology, and tools from earlier papers To denote the bottom and the top of an interval I, we prefer Foot(I) and Peak(I) to 0I and 1I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (In this way, we can avoid subscripts of subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') In a planar (Hasse) diagram of a planar lattice L, the edges p = [Foot(p), Peak(p)] are straight line segments that stand for prime intervals p (that is, for intervals p with Foot(p) ≺ Peak(p));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' we do not allow curves as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We always assume that a usual coordinate system of the plane is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Edges parallel to (1, 1) or (1, −1) are of normal slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Edges parallel to (1, t) for some t ∈ R with |t| > 1 and vertical edges are said to be precipitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As in Cz´edli [4, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1], where a more general concept from Cz´edli [3] is tailored to the peculiarities of slim rectangular lattices, we present the following definition, Part (C) of which is crucial here and in several earlier paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Parts (A) and (B) go after Gr¨atzer and Knapp [15] and [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) Let L♯ be a planar diagram of a slim rectangular lattice L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The left boundary chain and the right boundary chain of L♯ are denoted by LBnd(L) and RBnd(L), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Actually, LBnd(L♯) and RBnd(L♯) would be more precise but we always fix L♯ in a way that we are going to discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This comment applies for several other concepts we are going to define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') The boundary of L is Bnd(L) = LBnd(L) ∪ RBnd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The elements of Bnd(L) and those of L \\ Bnd(L) are called boundary elements and internal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (B) The left corner lc(L) and the right corner rc(L) are the two doubly irre- ducible elements3 of L such that lc(L) ∈ RBnd(L) and rc(L) ∈ RBnd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The upper left boundary and the upper right boundary of L are the principal filters ↑Llc(L) and ↑Lrc(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' note that ↑Llc(L) ⊆ LBnd(L) and ↑Lrc(L) ⊆ RBnd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (C) For x ∈ L, we let lsupp(x) := x ∧ lc(L) and rsupp(x) := x ∧ rc(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' note that lsupp(x) is on the lower left boundary ↓Llc(L), rsupp(x) is on the lower right boundary ↓Llc(L), and x = lsupp(x) ∨ rsupp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4 3We know from Gr¨atzer and Knapp [16] that they are boundary elements 4This follows from Gr¨atzer and Knapp [16, Lemmas 3 and 4] and the equivalence of the two definitions of slimness, see Cz´edli and Schmidt [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 3 (D) The diagram L♯ of L is a C1-diagram if for every edge p = [Foot(p), Peak(p)] of the diagram, p is either precipitous or it is of a normal slope and, furthermore, p is precipitous ⇐⇒ Foot(p) is an internal meet-irreducible element of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Together with each slim rectangular lattice occurring in the pa- per, a C1-diagram of the lattice is fixed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Moreover, even if we do not say it all the times, whenever we construct a lattice (like a sublattice or a larger lattice), then we always construct its fixed C1-diagram as well .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In notation, we rarely dis- tinguish a slim rectangular lattice from its C1-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Complying with Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2, all lattice diagrams in this paper are C1-diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let L denote a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note in advance that quite often, we do not make a distinction be- tween lattice theoretic and geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) If a < b ∈ L and C1, C2 are maximal chains of the interval [a, b] such that C1∩C2 = {a, b} and all elements x of C1 are on the left of C2 (including the possibility of x ∈ C2), then the elements [a, b] that are simultaneously on the right of C1 and on the left of C2 form a so-called lattice region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Kelly and Rival [17] for a more exact definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The corresponding geometric area, which is bordered by C1 and C2, is a geometric region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that whenever we define a geometric area (like a geometric region) or line segment, then (unless otherwise explicitly stated) it contains its boundaries, that is, it is topologically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Minimal non-chain regions are cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If a cell contains exactly four lattice elements, then it is a 4-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In fact, 4-cells are cover-preserving boolean sublattices with 4 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A 4-cell lattice is a planar lattice in which all cells are 4-cells (in a fixed planar diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Gr¨atzer and Knapp [15, Lemmas 4 and 5] and [16] proved that for a planar lattice L (which is finite by definition), if L is a 4-cell lattice, no two distinct 4-cells have the same bot- tom, and L has exactly two complementary doubly irreducible elements, then L is a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Conversely, a slim rectangular lattice is a 4-cell lattice with these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � � � � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) On the set of prime intervals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', edges) of L, let τ be the smallest equivalence relation that collapses the opposite sides of every 4-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As in Cz´edli and Schmidt [11], the blocks of τ are called trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Any two consecutive edges of a trajectory form a 4-cell, which is a 4-cell of a the trajectory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that a trajectory never branches out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Sometimes, like in [11], we say that a trajectory goes from left to right, that is, it starts at the left boundary chain, goes to the northeast direction (possibly in zero steps) to reach its top edge, which is the unique neon tube5 of the trajectory, and then it goes to the southeast to reach the right boundary chain and to stop there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see, for example, Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In this figure, the double-lined edges form a trajectory and, as for their color, the 4-cells of this trajectory are “gravel-filled”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The ascending part of the trajectory is its segment (subsequence consisting of edges) between its first edge (on the left boundary chain) and the top edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Simi- larly, the descending part of the trajectory starts with the top edge and terminates with the last edge (on the right boundary chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 5A neon tube is a prime interval with meet-irreducible bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 6A part of this figure was presented at SSAOS (conference) in Malenovice, Czech Republic, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A trajectory goes from left to right Given a 4-cell H of L and a positive integer k ∈ N+, we obtain the k-fold multifork extension of L at H by changing H to a copy of S(k) 7 and proceeding to the southeast and to the southwest to preserve semimodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For the exact definition, see Cz´edli [2], where this construction was introduced, or see Figure 2, where the construction is illustrated by performing a 1-fold multifork extension at H1 of L0 to obtain L1 and performing a 3-fold multifork extension at H2 of L1 to obtain L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (To save space, our figures are multi-purpose figures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' some ingredients of Figure 2 will be explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Note in advance that the thick edges of our lattice diagrams will be called neon tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note also that 1-fold multifork extensions are also called fork extensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Cz´edli and Schmidt [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' in this case the new elements form a so-called fork in the new lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A grid is (the fixed C1-diagram) of the direct product of two non-singleton finite chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A 4-cell H of L is a distributive 4-cell if the principal ideal ↓LPeak(H) is a distributive lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that in this case, that is, when H is a distributive 4-cell, then ↓LPeak(H) is a grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) this follows, say, from the following lemma and Cz´edli and Schmidt [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Most of the results on slim semimodular lattices were or could have been proved by using, possibly together with other tools, the following structure theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 (Multifork Sequence Lemma [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For each slim rectan- gular lattice L, there exist positive integers m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , mk, a sequence L0, L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , Lk of slim rectangular lattices, and a distributive 4-cell Hi of Li−1 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , k} such that L0 is a grid, Lk = L, and Li is obtained from Li−1 by performing an mi- fold multifork extension at Hi for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, any lattice obtained in this way from a grid is a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) The system (L0, H1, m1, L1, H2, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , Lk−1, Hk, mk, L = Lk) with com- ponents as above is the multifork sequence of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is not necessarily unique but we always fix one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Note, however, that k is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') REDUCING THE LENGTHS OF SPS LATTICES 5 (B) Let n be an edge on the upper boundary of the initial grid L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The union of the 4-cells of the trajectory containing n is the original territory of n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is denoted by OT(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' When we obtain Li from �Li−1, then we add several new edges and exactly mi of these new edges have the same peak as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let n be one of these new edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Li, the union of the 4-cells of the trajectory containing n is a geometric area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' we call it the original territory OT(n) of n in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that we have defined OT(n) if and only if n is an edge of the upper boundary or n is a precipitous edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (C) If n is an edge of the upper left boundary chain, then the essential part of the original territory, denoted by EOT(n), and the right essential part of the original territory, denoted by REOT(n), of n are OT(n) while the left essential part of the original territory, denoted by LEOT(n), of n is the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Similarly, for n on the right upper boundary, EOT(n) = LEOT(n) := OT(n) and REOT(n) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, let n be a precipitous edge and denote by T the trajectory containing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The union of 4-cells of T that do not contain n as an edge is the essential part EOT(n) of the original territory of n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is a geometric area and the union of two (geometrically) connected subsets that are, in a self-explanatory manner, called the left essential part LEOT(n) of the original territory and the right essential part REOT(n) of the original territory of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (The concept of EOT(n) was introduced in Cz´edli [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Some original territories and essential original territories are presented in Figures 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Even though the definition of OT(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , REOT(n) relies on L0 or Li, we usually use this concept for L, where OT(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , REOT(n) have not much connection with the trajectory containing n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this is exemplified by n1 and n2 in L′ (but not in L) of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) that whenever OT(n) is defined, then OT(n) is bordered by edges of L and all of these edges with peaks different from Peak(n) are of normal slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Further- more, each of LEOT(n) and REOT(n) is either the empty set or a rectangle bordered by edges of normal slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � � � � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) Lamps were introduced in Cz´edli [4], and they proved to be the fundamental tool to study the congruence lattices of slim rectangular lattices in Cz´edli [5], [7], [8], and Cz´edli and Gr¨atzer [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Originally, lamps are particular intervals I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' However, sometimes we need to consider them pairs (Foot(I), Peak(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Based on [4], we recall the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let L be a slim rectangular lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) The prime intervals p of L with Foot(p) ∈ M(L) are called neon tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If Foot(p) ∈ Bnd(L), then p is a boundary neon tube and it is of a normal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Otherwise, p is an internal neon tube and it is precipitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (B) Boundary lamps are the same as boundary neon tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (However, if I = p is a boundary lamp, then we sometimes say that p is the neon tube of I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' An interval I is an internal lamp if Peak(I) is the peak of an internal neon tube and Foot(I) is the meet of the feet of all internal neon tubes with peak Peak(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (These neon tubes are called the neon tubes of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') (C) In our lattice diagrams (which are C1-diagrams), the neon tubes are exactly the thick edges and the feet of the lamps are black-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note at this point that we know from Cz´edli [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1] that a lamp is uniquely determined by its foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, for a lamp I, we label the black-filled vertex Foot(I) in our figures by I rather than by Foot(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (So we follow the convention of the earlier papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI 3 45 34543534 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Multifork extensions and some geometric objects The (fixed) C1-diagram of a slim rectangular lattice L lies in a geometric rectan- gle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is bordered (from the left and from the right) by LBnd(L) and RBnd(L), and it is called the full geometric rectangle FullRect(L) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4(B) and (C) allow us to recall the following definition from Cz´edli [4] in a slightly modified form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6 (Some geometric areas and polygons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For a slim rectangular lattice (diagram) L, let I and J be lamps and p be a neon tube of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) The illuminated area Lit(I) of I is the union of the original territories of the neon tubes of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (B) The left roof and the left floor of an interval K of L are the line segments of slope (1, 1) with lower endpoints on the left boundary chain and upper end- points Peak(K) and Foot(K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' They are denoted by LRoof(K) and LFloor(K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With slope (1, −1), the right roof RRoof(K) and the right floor RFloor(K) are defined7 analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The roof Roof(K) and the floor Floor(K) of K are LRoof(K)∪RRoof(K) and LFloor(K)∪RFloor(K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (C) For a set X of planar points, GInt(X) stands for the geometric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', topo- logical) interior of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let h be a (geometric) polygon with endpoints a and b 7We know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5), or from Cz´edli [4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11)], or from Cz´edli and Gr¨atzer [10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2)] that each of LRoof(K), LFloor(K), RRoof(K), and RFloor(K) consists of some edges of the diagram provided its geometric length is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' See also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 7 such that h \\ {a, b} ⊆ TopInt(FullRect(L)), a ∈ LBnd(L), and b ∈ LBnd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then h cuts FullRect(L) into an upper half ↑gh and a lower half ↓gh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' by convention, h = ↑gh ∩ ↓gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that Lit(I) = ↑gFloor(I) ∩ ↓gRoof(I), and similarly for Lit(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (D) The body Body(I) of I is the geometric region determined by I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' if I has only one neon tube, then Body(I) is a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For example, in Figure 2, C2 ∈ Lamp(L2) and Body(C2) is filled with a “rainbow color”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (E) If I is a internal lamp, then the circumscribed rectangle CircR(I) is the region determined by the interval [x, Peak(I)] where x is the meet of the leftmost lower cover and the rightmost lower cover of Peak(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Equivalently, x is the meet of all lower covers of Peak(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Since the edges occurring in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4(B) and (C) are the same as the neon tubes of L, the following lemma in the present setting is not surprising;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' we recall it from [4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 (Cz´edli [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For the fixed multifork sequence of L in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3, the set of internal lamps of L is of the form {Ij : 1 ≤ j ≤ k} where, for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , k}, the “lamp” (Foot(Ij), Peak(Ij)) comes to existence by the j-th multifork extension, CircR(Ij) in L = Lk is the geometric region determined by Hj in Lj−1, and Foot(Ij) ∈ Lj \\ Lj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since the multifork extensions in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 are performed at distributive 4-cells, it follows easily that, using the notations of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , k}, the lower covers of Peak(Ij) are the same in Lj as in L = Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In particular, Ij has the same neon tubes in Lj as in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, if a neon tube n comes to existence in Lj, then EOT(n), LEOT(n), and REOT(n) are the same in Lj as in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With the notation used in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7, let Ii and Ij be lamps of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If i < j, then we say that Ij is younger than Ii and Ii is older than Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (This concept depends on the multifork sequence, but this sequence is always fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') By an edge segment we mean a geometric line segment g of positive length with endpoints lying on the same edge e of (the fixed C1-diagram of) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In this case, we say that g is an edge segment of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Based on the fact that the neon tubes of L are exactly the prime intervals occurring in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4 (B) and (C), we can recall a part of Cz´edli [4, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9] and extend it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let I and J be lamps of a slim rectangular lattice L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) Let (I, J) ∈ ρfoot mean that I ̸= J, Foot(I) ∈ Lit(J), and I is an internal lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (B) Let (I, J) ∈ ρOTfoot mean that I ̸= J, I is an internal lamp, and J has a neon tube n such that Foot(I) ∈ GInt(LEOT(n)) or Foot(I) ∈ GInt(REOT(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (C) Let (I, J) ∈ ρOTCR mean that I ̸= J, I is an internal lamp, and J has a neon tube n such that CircR(I) ⊆ LEOT(n) or CircR(I) ⊆ REOT(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (D) Let (I, J) ∈ ρCircR mean that I ̸= J, I is an internal lamp, and CircR(I) ⊆ Lit(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (E) Let Lamp(L) be the set of lamps of L, and let “≤ ” be the reflexive and transitive closure of the relation ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The relational structure (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) is also denoted by Lamp(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The congruence generated by a pair (x, y) of elements will be denoted by con(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10 (Mostly [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If L is a slim rectangular lattice, then ρfoot = ρCircR = ρOTfoot = ρOTCR, Lamp(L) = (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) is a poset, and whenever I ≺ J in Lamp(L), then (I, J) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, we have that (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) ∼= (J(Con L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) and the map ϕ: (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) → (J(Con L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) defined by I �→ con(Foot(I), Peak(I)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6) is an order isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The advantage of this lemma over its precursor, [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11], is that (I, J) ∈ ρfoot is a mild condition, which is easy to verify, while (I, J) ∈ ρOTCR is a strong condition, which gives more chance to draw conclusion from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With the exception of “ρfoot = ρOTfoot = ρOTCR ”, the lemma is already known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Cz´edli [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So we need only to show the just-mentioned equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Clearly, ρOTCR ⊆ ρOTfoot ⊆ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Assume that Ii, Ij ∈ Lamp(L) such that (Ii, Ij) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since S(mi) 7 is not distributive, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 that Ii is younger than Ij, that is, i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In particular, Ii is an internal lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With m := mj, let n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , nm be the neon tubes of Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As i > j, these neon tubes are present in Li−1, and so are their original territories OT(n1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , OT(nm) as well as their essential original territories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) applied to Li−1, these territories are separated by polygons consisting of lattice edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, by planarity, these “separating polygons” cannot cross the 4-cell Hi of Li−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this 4-cell becomes CircR(Ii) in Li and in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So CircR(Ii) ⊆ OT(nt) for some t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' However, the 4-cell Hi in question cannot have the same bottom as Ij since the opposite case would contradict the distributivity of Hi in Li−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Alternatively, [7, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2] would also lead to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Hence, CircR(Ii) = Hi ⊆ EOT(nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since EOT(nt) is the union of its two connected “components”, LEOT(nt) and REOT(nt), and these connected components are in a positive geometric distance from each other (provided none of them is the empty set), the planarity of the diagram yields that CircR(Ii) = Hi ⊆ LEOT(nt) or CircR(Ii) = Hi ⊆ REOT(nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, (Ii, Ij) ∈ ρOTCR, implying that ρOTCR ⊆ ρfoot and completing the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ Since we work with the C1-diagram of our slim rectangular lattice L, the illumi- nated sets Lit(I) and the Foot(I), and so the relation ρfoot are perfectly described by the geometric structure Str(L) := � FullRect(L), {(Foot(I), Peak(I)) : I ∈ Lamp(L)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7) In particular, if if L and L′ are slim rectangular lattices such that Str(L) = Str(L′), then Lamp(L) ∼= Lamp(L′) and so Con L ∼= Con L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Auxiliary statements The following definition and its terminology are motivated by the relation ρOTCR from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For a slim rectangular lattice L and J ∈ Lamp(L), let p be a neon tube of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We say that the original territory of p is used if there is a lamp I ∈ Lamp(L) such that I ̸= J and CircR(I) ⊆ LEOT(p) or CircR(I) ⊆ REOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Illustrating the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 by Lamp(L) ∼= P ∼= Lamp(L′) If I is such, then we say that I uses the original territory of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If there is no such an I, then the original territory of p is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The following remark follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, “there is a lamp I ∈ Lamp(L) such that I ̸= J” is equivalent to that “there is an internal lamp I ∈ Lamp(L) such that I < J”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note also that I ̸= J occurs in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1 only for emphasis, so it could be omitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' analogous comments would apply to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For p and J as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, the following four conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (a) The original territory of p is used, that is, there is lamp I such that p is not a neon tube of I and CircR(I) ⊆ LEOT(p) or CircR(I) ⊆ REOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (b) There is a lamp I ∈ Lamp(L) \\ {J} such that Foot(I) is in GInt(LEOT(p)) or it is in GInt(REOT(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (c) There is a lamp I ∈ Lamp(L) \\ {J} such that Foot(I) is in EOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (d) There is a precipitous edge segment in EOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, if a lamp I satisfies one of (a), (b), and (c), then it satisfies all the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since we never change I to another lamp, the last sentence of the lemma will automatically follow when the equivalence of (a), (b), and (c) has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since Foot(I) ∈ GInt(CircR(I)), (a) implies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By the equality EOT(p) = LEOT(p) ∪ REOT(p), we obtain that (b) implies (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, assume that (c) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then Foot(I) ∈ EOT(p) ⊆ Lit(J) and so (I, J) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10, (I, J) ∈ ρOTCR and so Body(I) ⊆ CircR(I) ⊆ Lit(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, It := I is younger than Ik := J in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8, that is, t > k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' indeed, if I = It was older than J = Ik, then the 4-cell Hk would not have been distributive in Lk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Lk, each of LEOT(p), REOT(p), and FullRect(Lk) \\ EOT(p) were the unions of 4-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Some of these 4-cells could have been divided into smaller ones later, but even in Lt−1, each of LEOT(p), REOT(p), and FullRect(Lt−1) \\ EOT(p) 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI were unions of 4-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Ht ⊆ LEOT(p), Ht ⊆ REOT(p), or Ht is outside EOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since Foot(I) = Foot(It) ∈ GInt(Ht) and Foot(I) ∈ EOT(p), Ht was not outside EOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, CircR(I) = CircR(It) = Ht ⊆ LEOT(p) or CircR(I) ⊆ REOT(p), whereby the original territory of p is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, (c) implies (a), and we have proved that (a), (b), and (c) are equivalent conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2, the implication (a) ⇒ (d) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Finally, assume that (d) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then we have a precipitous edge segment in LEOT(p) or in REOT(p), say, in LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By the second half of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4), we can assume that a precipitous edge segment lies in GInt(LEOT(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This edge segment lies in a neon tube q of a lamp I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By planarity and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4), q cannot cross the four sides bordering (the geometric rectangle) LEOT(p), so q lies fully in LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In particular, Peak(I) = Peak(q) ∈ LEOT(p) and Foot(q) ∈ LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Observe that Peak(I) cannot lie on the lower boundary of LEOT(p) since otherwise q, going down from Peak(I) with a precipitous slope, could not include an edge segment lying in LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, let r be an arbitrary neon tube of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It goes down from Peak(r) = Peak(I) with a precipitous slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, since Peak(r) is not on the lower boundary, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) yields that an edge segment lying on r lies also in GInt(LEOT(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So r satisfies the same condition as q above, and it follows that Foot(r) ∈ LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now let r′ and r′′ be the leftmost neon tube and the rightmost neon tube of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If r′ = r′′, then q is the only neon tube of I, and the required Foot(I) ∈ LEOT(p) follows from Foot(I) = Foot(q) ∈ LEOT(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So we can assume that r′ ̸= r′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then Foot(r′) and Foot(r′′), as distinct lower covers of Peak(I), are incomparable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By the main result of Cz´edli [6] and Foot(I) = Foot(r′) ∧ Foot(r′′), the interval [Foot(I), Foot(r′)] is a chain (and so a line segment) of slope (1, −1) while [Foot(I), Foot(r′′)] is a line segment of slope (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The top endpoints Foot(r′) and Foot(r′′) of these line segments are in LEOT(p), whereby so is their common bottom Foot(I) by the second half of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Foot(I) ∈ LEOT(p), that is, (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This completes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ For a neon tube p of a slim rectangular lattice L, the fork determined by p is F(p) := [lsupp(Foot(p)), Foot(p)] ∪ [rsupp(Foot(p)), Foot(p)] to- gether with the edges of these two intervals and the edge p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) With another terminology, this concept was originally introduced in Cz´edli and Schmidt [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A (very) particular case of the following lemma (namely, the case when ↓L′ Peak(p) is distributive) occurs implicitly in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If p is a neon tube of a slim rectangular lattice L and L′ := L \\ F(p), see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1), then L′ is meet-subsemilattice of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' First, we prove that [lsupp(Foot(p)), Foot(p)] = {x ∈ L : lsupp(x) = lsupp(Foot(p))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) Denote Foot(p) by w and lsupp(Foot(p)) by u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' so u = lsupp(w) and we need to show that [u, w] = {x ∈ L : lsupp(x) = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For y ∈ [u, w], we have that u = lsupp(u) ≤ lsupp(y) ≤ lsupp(w) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, y ∈ {x ∈ L : lsupp(x) = u} and we obtain that [u, w] ⊆ {x ∈ L : lsupp(x) = u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To exclude that “⊂” holds here, suppose for contradiction that there is a z ∈ {x ∈ L : lsupp(x) = u} such that z /∈ [u, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If rsupp(z) ≤ rsupp(w), then the last equality in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1(C) gives that z = lsupp(z) ∨ rsupp(z), whence u = lsupp(z) ≤ z = u ∨ rsupp(z) ≤ REDUCING THE LENGTHS OF SPS LATTICES 11 u∨rsupp(w) = lsupp(w)∨rsupp(w) = w yields that z ∈ [u, w], a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' But rsupp(z) and rsupp(w) belong to the same chain, RBnd(L), so they are comparable, and we obtain that rsupp(w) < rsupp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, w = lsupp(w) ∨ rsupp(w) = u ∨ rsupp(w) ≤ u ∨ rsupp(z) = lsupp(z) ∨ rsupp(z) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The inequality w ≤ z and z /∈ [u, w] imply that Foot(p) = w < z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Taking the meet-irreducibility of Foot(p) into account, we have that Peak(p) ≤ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, lsupp(Peak(p)) ≤ lsupp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With the notation used in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7, let Ii be the lamp to which p belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then Peak(p) = Peak(Ii), and it is clear in Li that u = lsupp(Foot(p)) < lsupp(Peak(I)) = lsupp(Peak(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since Li is a sublattice of L, the inequality u < lsupp(Peak(p)) also holds in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Combining this with the already established lsupp(Peak(p)) ≤ lsupp(z), we obtain that u < lsupp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This contradicts the assumption z ∈ {x ∈ L : lsupp(x) = u} and proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, for the sake of contradiction, suppose that L′ is not meet-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there are elements s, c, d ∈ L such that s = c ∧ d, s ∈ F(p) = L \\ L′ but c, d /∈ F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The top element Foot(p) of F(p) is meet-irreducible, whereby (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) allows us to assume that s ∈ [lsupp(Foot(p)), Foot(p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that lsupp(Foot(p)) = lsupp(lsupp(Foot(p))) ≤ lsupp(s) ≤ lsupp(Foot(p)) gives that s = lsupp(Foot(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since the function L → LBnd(L) defined by t �→ lsupp(t) is clearly an idempo- tent meet-endomorphism by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1(C), lsupp(s) = lsupp(c) ∧ lsupp(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' But LBnd(L) is a chain, whence lsupp(c) = lsupp(s) or lsupp(d) = lsupp(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let, say, lsupp(c) = lsupp(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So, taking s = lsupp(Foot(p)) into account, lsupp(c) = lsupp(Foot(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) implies that s ∈ [lsupp(Foot(p)), Foot(p)] ⊆ F(p), which is a contradiction showing that L′ is meet-closed, completing the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ For I ∈ Lamp(L), let NumTube(I) = NumTubeL(I) denote the number of neon tubes of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The total number of neon tubes is denoted by NumTubeall(L), so NumTubeall(L) := � I∈Lamp(L) NumTube(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 (Sandwiched Neon Tube Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For a slim rectangular lattice L, let n1, p, and n2 be three consecutive neon tubes of an internal lamp I ∈ Lamp(L) such that the original territory of p is used but those of n1 and n2 are not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there is a slim rectangular lattice L′ such that Lamp(L′) ∼= Lamp(L) but |L′| < |L| and NumTubeall(L′) = NumTubeall(L) − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' in fact, there is an isomor- phism ϕ: Lamp(L) → Lamp(L′) such that NumTube(ϕ(I)) = NumTube(I)−1 and NumTube(ϕ(J)) = NumTube(J) for all J ∈ Lamp(L) \\ {I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With reference to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1), denote by L′ the subposet of L that we obtain from L by removing the fork F(p) determined by p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We are going to show that L′ does the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By left-right symmetry, we can assume that n1 is to the left of p and p is to the left of n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' First, we prove that L′ is a sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By the main result of Cz´edli [6], both intervals occurring in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) are chains of normal slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1), F(p) = Floor(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) In Figure 3, these chains are [u6, u6 ∨v6] and [v6, u6 ∨v6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since none of the original territories of n1 and n2 are used, we obtain from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 that none of REOT(n1) and LEOT(n2) contains a precipitous line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI These two areas border F(p) = Floor(p) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, for any edge r of L, if Peak(r) ∈ F(p), then r is of a normal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6) For the sake of contradiction, suppose that L′ is not join-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then we can pick x′, y′ ∈ L′ such that z := x′ ∨ y′ /∈ L′, that is, z ∈ F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (The join is taken in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) and left-right symmetry, we can assume that z ∈ [lsupp(Foot(p)), Foot(p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Figure 3, the situation is illustrated with z as the (unique) element drawn by a lying oval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let T := [lsupp(Foot(n2)), Foot(p)] in L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is [u5, u6 ∨ v6] in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The (area determined by) T is included in LEOT(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5), T contains no precipitous line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, as a lattice interval, T is the direct product of a chain and the two-element chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7) Hence, z has only two lower covers, x and y (the standing ovals in the figure), and the edges [x, z] and [y, z] are of normal slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let, say, x be to the left of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now x′, y′ ∈ ↓Lz \\ {z}, but {x′, y′} ⊈ ↓Ly since otherwise z = x′ ∨ y′ ≤ y ≺ z would be a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, at least one of x′ and y′ is in ↓Lz \\ ↓Ly ⊆ [lsupp(Foot(p)), Foot(p)] ⊆ F(p), contradicting that x′, y′ ∈ L′ = L \\ F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, L′ is closed with respect to joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since it is also closed with respect to meets by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4, we have proved that L′ is a sublattice of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let e be an edge in the interval [lsupp(Foot(p)), Foot(p)] distinct from the top edge of this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7), it is clear that if we merge the two 4-cells that share e as a common side, we obtain a 4-cell of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The situation is similarly for the non-top edges of [rsupp(Foot(p)), Foot(p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The top edges of these two intervals disappear when Foot(p) and its two lower covers are omitted and three “old” 4-cells merge into a “new” 4-cell of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now that we have described the new 4-cells, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) that L′ is a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It is clear by the paragraph above that with the exception of p, only some edges of normal slopes are removed when passing from L to L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The removal of p does not influence the pair (Foot(I), Peak(I)) since Foot(I) is the meet of the feet of the leftmost neon tube and the rightmost neon tube of I but p is a “middle” neon tube of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, Str(L′) = Str(L), see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7), and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) implies that Lamp(L′) ∼= Lamp(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Finally, since only one neon tube, p, has been removed, NumTubeall(L′) = NumTubeall(L) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The existence of ϕ is clear: for J ∈ Lamp(L), ϕ(J) is defined by the property (Foot(ϕ(J)), Peak(ϕ(J))) = (Foot(J), Peak(J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6 (No Neighboring Neon Tubes Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let L be a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Assume that n1 and n2 are two neighboring neon tubes of an internal lamp I ∈ Lamp(L) such that none of them uses its original territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there exists a slim rectangular lattice L′ such that |L′| < |L| and (Lamp(L′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) ∼= (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) but |NumTubeall(L′)| = |NumTubeall(L)|−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' in fact, there is an order isomorphism ϕ: (Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) → (Lamp(L′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤) such that |NumTube(ϕ(I))| = |NumTube(I)|−1 but |NumTube(ϕ(K))| = |NumTube(K)| for any K ∈ Lamp(L) \\ {I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The proof borrows some ideas from Cz´edli [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note, however, that the present situation is different from that in [7] since now L′, to be defined below, is not a quotient lattice of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Illustrating the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6 by Lamp(L) ∼= P ∼= Lamp(L′) Let, say, n2 be to the right of n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Observe that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 (or see the figure) and the fact that REOT(n1) is not used, the peak of no precipitous edge of L belongs to RFloor(n2) and, in particular, Foot(n2) cannot be the peak of a precipitous edge of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) Keeping Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2 in mind, we define L′ by describing its C1-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' From (the diagram of) L, we remove the fork F(n2) together with all edges that have one or two endpoints in F(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Writing this formally, L′ = L \\ F(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' On the left of Figure 4, the vertices to be omitted are drawn in blue while the edges to be omitted are the blue dashed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let L′ be the set of the remaining vertices (drawn in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Note that L′ in Figure 4 is not a sublattice of L since u4, v6 ∈ L′ but u4 ∨L v6 /∈ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') At this stage, L′ with the remaining (black solid) edges is not even a lattice diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, let q denote the right neighbor of n2 among the neon tubes of I or, if n2 is the rightmost neon tube of I, then let q be the upper right edge of CircR(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Actually, it is only Foot(q) that we will need, and it is the right neighbor of Foot(n2) among the lower covers of Peak(n2) = Peak(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For each edge r of L, we define or not define an edge r′ of L′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If Foot(r) ∈ Floor(n2), then r′ is undefined and r is called an omitted old edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9) If Foot(r) /∈ Floor(n2) and Peak(r) /∈ Floor(n2), then r′ := r and r is called a remaining old edge of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10) If Foot(r) /∈ Floor(n2) and Peak(r) ∈ LFloor(n2), then let Foot(r′) := Foot(r) and Peak(r′) := Peak(r) ∨L lsupp(Foot(n1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11) If Foot(r) /∈ Floor(n2) and Peak(r) ∈ RFloor(n2), then let Foot(r′) := Foot(r) and Peak(r′) := Peak(r) ∨L rsupp(Foot(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='12) If r is in the scope of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='12), then r′ and r are called a new edge and a changing old edge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Figure 4, lsupp(Foot(n1)) = u7, rsupp(q) = v9, 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI and the new edges are the red dashed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) that each edge r of L belongs to the scope of exactly one of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With its new edges and the remaining old edges, L′ turns into a Hasse diagram of a poset L′ = (L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤), which is a subposet of L = (L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Actually, we need to verify that the diagram is a poset diagram, but this is quite easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Indeed, we only need to show that for every edge [x, y] of the new diagram L′, there are no edges [x, z1], [z1, z2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , [zk−1, y] of L′ for some k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This is clear if [x, y] is a new edge, as the only possible z1 ∈ L is not in L′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' the case when [x, y] is a remaining old edge is even more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To show that the poset L′ is actually (the diagram of) a slim rectangular lattice, we have to work more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since none of the original territories OT(n1) and OT(n2) is used, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 imply the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then every edge r in LEOT(ni) is either of (normal) slope (1, 1) and lies on the boundary of LEOT(ni) or r is of (normal) slope (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Similarly, every edge r in REOT(ni) is either of (normal) slope (1, −1) and lies on the boundary of REOT(ni) or r is of (normal) slope (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � � � � � � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='13) Hence, even though L can be more complicated in general than in Figure 4, the original territories indicated by appropriate fill patterns in the figure reflect the general case well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The new edges of L′, which originate from changing old edges of L, belong to three categories, which will be discussed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Category 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We assume that r is a precipitous edge in the scope of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then r is a neon tube of a lamp J ∈ Lamp(L) such that Peak(J) = Peak(r) lies on LFloor(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Figure 4, J can be J1 or J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='13) that we obtain r′ from r by moving the peak of r to the northwest along an edge of slope (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, using that r is precipitous, it follows trivially that r′ is also precipitous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' for more details, the reader can (but need not) see [7, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since no precipitous edge will occur in other categories for changing edges, let us summarize for later references that if a precipitous old edge h of L is a changing edge, then it changes to a precipitous new edge h′ and Foot(h′) = Foot(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14) A line or an edge is of a slight slope if it is parallel to the vector (1, t) for some t ∈ R such that |r| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' That is, a line or edge is of a slight slope if and only if it is neither of a normal slope nor precipitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from [7, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='9)] (and it is easy to see) that if ℓ is a (geometric) line through two distinct lower covers of Peak(J), then ℓ is of a slight slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, let UHCircR(J) stand for the union of the 4-cells whose peaks are Peak(J);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it is a geometric area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (The acronym, taken from [7], comes from “upper half of the circumscribed rectangle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') For J ∈ {J1, J2} in Figure 4, UHCircR(J) in L is curl- filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that on the right of the figure, the curl-filled areas are UHCircR(J1) and UHCircR(J2) understood in L but not in L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 (and, in a different terminology, it is explicitly stated in [7, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3)]) that GInt(UHCircR(J)) contains no edge segment that is not a part of a neon tube of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='15) Practically, this means that the curl-filled areas in the figure reflect generality well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let h′ be an edge of L′ such that h′ ̸= r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since ℓ mentioned in the previous paragraph is of a slight slope, it follows that r′ does not cross h′ provided that REDUCING THE LENGTHS OF SPS LATTICES 15 h′ ̸= h and h is another neon tube of (the same) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since neither the curl-filled area GInt(UHCircR(J)) nor the 4-cell of LEOT(n2) that is the upper left neighbor of CircR(J) contains an edge of L not mentioned in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='15), r′ does not cross h′ if h is of a normal slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In the remaining case when h is precipitous but not a neon tube of J and Peak(h) ∈ LFloor(n2), then let K denote the lamp having h as a neon tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then K is an internal lamp and K ̸= J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since an internal lamp is clearly determined by its peak, Peak(J) ̸= Peak(K), and they are comparable since LFloor(n2) where they belong is a chain by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The role of J and K is interchangeable, so let Peak(K) < Peak(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then (the line determined by) RRoof(K) separates J and K, and we obtain easily again that r′ and h′ do not cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We have seen that if r′ originates from a precipitous edge r of L, then r′ does not cross any other edge of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='16) Category 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We assume that r is of a normal slope and r′ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then b := Peak(r′) ∈ L even though r′ is not an edge of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It is clear either by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 or by comparing the present situation to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7) that Peak(r) ≺L b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, d := [Peak(r), b] is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This edge lies in LEOT(n2), and we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='13) that d is of slope (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So is r since it is of a normal slope but does not lie on LFloor(Foot(n2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This means that r′ comes to existence by merging r and d, which are adjacent edges lying on the same line of slope (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, r′ is also of slope (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, since Category 3 will be analogous to the current one by left-right symmetry and we are armed with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14), we can conclude even now that if g is a changing old edge of a normal slope, than the edge g′ of L′ is of the same (normal) slope and, furthermore, g′ is obtained by merging two collinear adjacent edges of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='17) It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='17) that if r′ crossed an edge g′ of L′, then g′ would be of the other normal slope, (1, 1), and it would come to existence by merging g to a collinear other edge of L at b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' But then g would lie on RFloor(n2) and instead of merging it to a collinear edge to obtain g′, g would have been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, if r belongs to Category 2, then r′ does not cross any other edge of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='18) Category 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We assume that r is in the scope of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8), r is of (a normal) slope (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, the situation is basically the left-right symmetric counterpart of the one discussed in Category 2, whereby no details will be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now that the three categories have been investigated, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='18), and the left-right symmetric counterpart of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='18) for Category 3 imply that L′ is a planar Hasse-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from Kelly and Rival [17, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4] that planar posets with 0 and 1 are lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, L′ is a planar lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By construction, the number of upper covers of an element x ∈ L′ is the same in L′ as in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, an element of L′ belongs to the boundary of L′ if and only if it belongs to the boundary of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) and the construction of L′ yield in a straightforward but a bit tedious way that L′ is a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since x ∈ L′ has the same number of covers in L′ as in L, we obtain that M(L′) = L′ ∩M(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Moreover, we already have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='17), and it is clear that an edge r′ of L′ lies on Bnd(L′) if and only if it lies on Bnd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Clearly, lc(L), rc(L) ∈ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, taking the just mentioned facts of the present paragraph and Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2 (for L) into account, we conclude that L′ is (given by) a C1-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI Since OT(n2) is not used, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 that if h is a neon tube of L and h ̸= n2, then Foot(h) /∈ F(n2) = Floor(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='19) It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='17), and the construction of L′ that the neon tubes of L′ are exactly the r′ where r is a neon tube of L and r ̸= n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, for neon tubes r and h of L such that r ̸= n2 ̸= h, Peak(r′) = Peak(h′) if and only if Peak(r) = Peak(h) and Foot(r′) = Foot(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � � � � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='20) Hence, for a lamp K ∈ Lamp(L) \\ {I}, {r′ : r is a neon tube of K} is exactly the collection of neon tubes of a lamp K′ of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, {h : h is a neon tube of I and h ̸= n2} is the set of neon tubes of an internal lamp I′ of L′ — this is the definition of I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='20) give that Foot(K′) = Foot(K) for K ∈ Lamp(L) \\ {I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='20) and the facts mentioned thereafter allow us to conclude that the function ϕ: Lamp(L) → Lamp(L′) defined by K �→ � K′ if K′ ∈ Lamp(L′) such that Foot(K′) = Foot(K), I′ if K = I (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='21) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Remark that if n2 is not the rightmost neon tube of I, then I belongs to the scope of both lines of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Note the rule, which follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='20): for any K ∈ Lamp(L), we have that Peak(ϕ(K)) = Peak(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10 that, in order to see that ϕ is an order isomorphism, it suffices to show that, for J, K ∈ Lamp(K), (J, K) ∈ ρfoot ⇐⇒ (J′, K′) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='22) Assume that (J, K) ∈ ρfoot and J ̸= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since Peak(K′) is to the northwest (that is, to the (−1, 1) direction) of Peak(K) or Peak(K′) = Peak(K), we have that Lit(K) ⊆ Lit(K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Foot(J′) = Foot(J) ∈ Lit(K) ⊆ Lit(K′) gives the required (J′, K′) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If (I, K) ∈ ρfoot, then CircR(I′) = CircR(I) ⊆ Lit(K) ⊆ Lit(K′) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10, whereby (I′, K′) ∈ ρCircR = ρfoot, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This proves the “⇒” part of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, assume that (J′, K′) ∈ ρfoot and I /∈ {J, K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know that Foot(K′) = Foot(K) and Foot(J′) = Foot(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If Peak(K′) = Peak(K), then Foot(J) = Foot(J′) ∈ Lit(K′) = Lit(K) gives the required (J, K) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So assume that Peak(K′) ̸= Peak(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By construction, Lit(K′) ⊆ Lit(K) ∪ LEOT(n2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Foot(J) = Foot(J′) ∈ Lit(K′) gives that Foot(J) ∈ Lit(K) or Foot(J) ∈ LEOT(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If the second alternative, Foot(J) ∈ LEOT(n2), holds, then Foot(J) ⊆ EOT(n2), which contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 as OT(n2) is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Foot(J) ∈ Lit(K), which gives that (J, K) ∈ ρfoot, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We are left with the case when one of J and K is I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Assume that (J′, I′) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then Foot(J) = Foot(J′) ∈ Lit(I′) ⊆ Lit(I) gives the required (J, I) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Note that Lit(I′) ⊂ Lit(I) if n2 is the rightmost neon tube of I, and Lit(I′) = Lit(I) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Finally, assume that (I′, K′) ∈ ρfoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then (I′, K′) ∈ ρCircR by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This fact and CircR(I) = CircR(I′) give that Peak(I) = Peak(CircR(I)) = Peak(CircR(I′)) ∈ CircR(I′) ⊆ Lit(K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 17 Hence, (Foot(K′), Peak(K′)) = (Foot(K), Peak(K)), and so Lit(K′) = Lit(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' These facts lead to CircR(I) = CircR(I′) ⊆ Lit(K′) = Lit(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, (I, K) ∈ ρCircR = ρfoot, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' An estimate Our goal is to prove that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let D be the congruence lattice of a slim semimodular lattice (in other words, an SPS lattice) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let n := |J(D)|, that is, the number of join- irreducible congruences of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If n = 0 or n = 1, then D is the (n + 1)-element chain and K ∼= D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If n = 2, then the length len(K) of K is 2 and K is either the three-element chain or the four-element boolean lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If n ≥ 3, then the following two assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (A) There is a slim rectangular lattice L such that Con L ∼= D and len(L) ≤ 2n2 − 10n + 15, and so len(L) < 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) (B) For any slim semimodular lattice L′, if Con L′ ∼= D, then len(L′) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The case n ≤ 2 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In the rest of the proof, we assume that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let L be a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A trivial induction by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 shows that len(L) = NumTubeall(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) Now if Con L ∼= D, then Lamp(L) ∼= J(Con L) ∼= J(D) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10, and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) and the fact that each lamp has at least one neon tube give that len(L) = NumTubeall(L) ≥ |Lamp(L)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, Part (B) holds for the particular case of rectangular SPS lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from Gr¨atzer and Knapp [16, Theorem 7] and its proof that for any slim semimodular lattice L′, there is a slim rectangular lattice L such that Con L ∼= Con L′ and len(L) = len(L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) This statement also follows from Cz´edli and Schmidt [12, Lemma 21] (applied in the reverse directions) and Cz´edli[1, (Corner) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, Part (B) follows from its particular case mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, we turn our attention to part (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If J(D) is the n-element antichain, then any grid G with length n is a slim rectangular lattice such that Con G ∼= D, and len(G) = n ≤ 2n2 is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, in the rest of the proof, we assume that J(D) is not an antichain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, take a slim rectangular lattice L of minimal length such that Con L ∼= D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10 that Lamp(L) ∼= J(D), and so |Lamp(L)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let J ∈ Lamp(L) be an internal lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let t+ J denote the number of neon tubes of J whose original territories are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Similarly, t− J stands for the number of neon tubes of J whose original territories are not used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' note that t+ J +t− j = NumTube(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Listing the neon tubes from left to right, let us write a letter u for a used neon tube and a zero for an unused neon tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then we obtain a sequence ⃗s of length NumTube(J) consisting of t+ J u’s and t− J zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Subsequences 0 u 0 and 0 0 are forbidden by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6 since len(L) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For another look at ⃗s, take the sequence ⃗w := ⋆ u ⋆ u ⋆ u · · · ⋆ u ⋆ u ⋆ u ⋆ of t+ J u’s and t+ J + 1 stars that alternate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We can obtain ⃗s from ⃗w by removing some stars and replacing the remaining stars by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Observe that only one zero can replace a star since 0 0 is a forbidden subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, for any two consecutive stars (which 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI occur in a subsequence ⋆ u ⋆), at most one of the two stars can change to 0 and so the other one should be removed since 0 u 0 cannot be a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, at most every second star can turn to 0 and the rest of the stars are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, the number t− J of zeros is at most8 ⌈(t+ J + 1)/2⌉ where ⌈x⌉ denotes the upper integer part of a real number x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since ⌈(t+ J + 1)/2⌉ ≤ t+ J , we obtain that, for any J ∈ Lamp(L), NumTube(J) = t+ J + t− j ≤ 2 · t+ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4) Let m denote the number of boundary lamps, that is, the number of maximal elements of Lamp(L) (or, equivalently, those of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Each of LBnd(L) and RBnd(L) contains at least one boundary lamp, whence m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' On the other hand, m < n as Lamp(L) ∼= J(D) is not an antichain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, 2 ≤ m ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let k := n − m, the number of internal lamps of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Clearly, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If p is a neon tube of an internal lamp J and I uses the original territory of J, then I < J and, in particular, I is also an internal lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, if p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , pt+ J denote the neon tubes of J whose original territories are used, then the GInt(LEOT(p1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , GInt(LEOT(pt+ J )) are pairwise disjoint, and so are GInt(REOT(p1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , GInt(REOT(pt+ J )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3(b), it follows that the lamp I can use the original territories of at most two of the neon tubes of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The number of lamps I that use the original territory of a neon tube of J is at most |↓J\\{J}|, whereby J has at most 2·|↓J\\{J}| neon tubes9 whose original territories are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='4), it has at most twice as many neon tubes all together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence, the total number of neon tubes of the internal lamps is at most10 � internal J∈Lamp(L) 2 · 2 · |↓J \\ {J}| = 4 · � internal J∈Lamp(L) |↓J \\ {J}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) Observe that |↓J \\ {J}| is the number of pairs (I, I′) of internal lamps subject to I < I′ and I′ = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, the second sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) is the number of pairs (I, J) such that I < J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In other words, the second sum is the number of comparabilities among internal lamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So this sum reaches its maximum when for any internal lamps I ̸= J, we have that I < J or J < I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' That is, when the internal lamps form a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there are �k 2 � = k(k − 1)/2 such pairs, and so the maximum that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) can take is 2k(k − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' it might seem to be an upper bound on the number NumTubeinternal(L) of neon tubes of internal lamps of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The only problem with the argument above is that our assertion that “J has at most 2 · |↓J \\ {J}| neon tubes” is true only if ↓J \\ {J} is nonempty, that is, if J is not a minimal lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Indeed, if J is a minimal lamp, then 2 · |↓J \\ {J}| = 0 but J must have at least one neon tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since NumTubeall(L) is minimal, we obtain from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 that a minimal lamp J has exactly one neon tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let s be the number of minimal internal neon tubes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' note that 1 ≤ s ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The number 2k(k−1) has to be modified in two opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' First, two minimal internal lamps are incomparable, whereby the number �k 2 � = k(k − 1)/2 has to be reduced by �s 2 � = s(s − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Second, the minimal internal lamps were computed with zero neon tubes, so we have to add s · 1 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So we obtain that NumTubeinternal(L) ≤ 4 · � k(k − 1)/2 − s(s − 1)/2 � + s 8Provided that t+ J > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this correction will be taken into account about nine lines after (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 9For minimal lamps, this will be corrected soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 10To be improved soon by taking the minimal internal lamps of L into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 19 = 2k2 − 2k + 3s − 2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6) Since 3s − 2s2 is negative for s ≥ 2, we obtain the largest value in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='6) for s = 1, when 3s − 2s2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Smaller values at s > 1 would not give an upper estimate for s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') Hence, NumTubeinternal(L) ≤ 2k2 − 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now taking the m boundary lamps and the equality k = n − m into account, we obtain that NumTubeall(L) = m + NumTubeinternal(L) ≤ m + 2(n − m)2 − 2(n − m) + 1 = 2n2 − 2n + 1 + 2 · � m2 − (2n − 3/2)m � � �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7) Let f(m) = m2 − (2n − 3/2)m denote the under-braced term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By the elementary theory of quadratic univariate real functions, f(m) decreases in the interval [0, n − 3/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This fact and 2 ≤ m ≤ n − 1 imply that the largest value of f(m) is f(2) = 7 − 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Substituting this value into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7), we obtain that NumTubeall(L) ≤ 2n2 − 10n + 15 < 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) Finally, len(L) = NumTubeall(L) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='8) completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The first inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) is not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To show this by an example, let n = |J(D)| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then 2n2 − 10n + 15 is 7 but, no matter which 4-element poset J(D) is, there is a slim semimodular lattice L such that |J(Con L)| ∼= D and len(L) ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If J(D) is the four-element “Y poset”11, then len(L) cannot be less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For L in Part (A) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, |L| ≤ (2n2 −10n+15)2 < 4n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, it suffices to show that if a slim semimodular lattice L is of length k, then |L| ≤ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3), we can assume that L is a slim rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, there are chains C, U ⊆ J(L) such that J(L) = C ∪ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then 0 /∈ C and, by rectangularity, 1 /∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Hence |C| ≤ k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Similarly, |U| ≤ k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since any element of L \\ {0} is of the form c ∨ u with c ∈ C and u ∈ U, L has at most 1 + |C| · |U| = 1 + (k − 1)2 ≤ k2 elements, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Odds and ends Let P be a poset, and let j ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We define a new poset P ′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The base set of P ′ is (P \\ {j}) ∪ {j′, j′′} where P ∩ {j′, j′′} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The ordering in P ′ is defined as follows: for a, b ∈ P ′ \\ {j′, j′′} = P \\ {j}, a ≤P ′ b ⇐⇒ a ≤P b, a ≤P ′ j′ ⇐⇒ a ≤P ′ j′′ ⇐⇒ a ≤P j, j′ ≤P ′ b ⇐⇒ j′′ ≤P ′ b ⇐⇒ j ≤P b, and j′′ ≺P ′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We say that P ′ is obtained from P by doubling the element j of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For an example, see P and P ′ in the middle of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let j be an element of a finite poset P, and let P ′ be the poset that we obtain from P by doubling j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' If there is a slim semimodular lattice L such that P ∼= J(Con L) and j is not a maximal element of P, then there ex- ists a slim rectangular lattice L′ such that P ′ ∼= J(Con L′) and NumTubeall(L′) = NumTubeall(L) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Note that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) would allow to rewrite this equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') 11The diagram of this poset is Y-shaped;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this is where the name “Y-poset” comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 20 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The construction for Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1 without rescaling Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Gr¨atzer and Knapp’s result, quoted here in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3), allows us to assume that L is a rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let us agree that whenever we refer to some neon tube like the m-th neon tube of a lamp, then we mean that the neon tubes of the lamp in question are listed from left to right and “m-th” refers to this list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see also Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We also count on the fixed multifork sequence of L, see Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10 that there is an order isomorphism P → Lamp(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' we denote its action by capitalization, that is, x �→ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The notation used in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 is in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since j is not a maximal element of P, J is an internal lamp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' let, say, J = It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Figure 5, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' At present, we are only interested in L and P in Figure 5 but note in advance that L′ in this figure is hardly readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Indeed, some vertices of L′ are so close to each other that we cannot see whether there is an edge connecting them or there is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Therefore, we also present Figure 6, which is a C1-diagram of the same lattice L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The only difference between the two figures is that L′ is rescaled (and so readable) in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Usually, we will simply refer to “the figure” which is Figure 5 but the reader can sometimes check its unreadable details in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Resuming the argument, note that P ∩ P ′ = P \\ {j} = P \\ {j′, j′′} is a subposet both in P and in P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For any x ∈ P ∩ P ′, the lamp corresponding to x will be denoted by X both in L and in L′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this should not cause confusion since it will be clear from the context whether X ∈ Lamp(L) or X ∈ Lamp(L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This will mean that the pair (Foot(X), Peak(X)) is the same in L′ as in sublattice L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So, implicitly, we mostly consider lamps as pairs in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We define L′ in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let ϵ ∈ R, ϵ > 0, be the smallest one out of the geometric lengths of the edges of (the fixed C1-diagram of) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' With reference to the multifork sequence of L, let L′ 0 := L0, L′ 1 := L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , L′ t−1 := Lt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' these equations also mean the exact coincidence of the corresponding C1-diagrams in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As for the forthcoming notation, we will continue the sequence by L′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5, L′ t, L′ t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , L′ k =: L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In L′ t−1 (which is the same as Lt−1), let H′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 be the same 4-cell (even geometrically the same) as Ht in Lt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Later, Ht turns into CircR(It) in L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' in the figure, CircR(It) = CircR(I3) is the “3-filled” area in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In L′, only the “major part” of CircR(I′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) = CircR(I′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) is 3-filled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' the rest of CircR(I′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) = CircR(I′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) is yellow-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' At Ht in Lt−1, REDUCING THE LENGTHS OF SPS LATTICES 21 we perform a NumTube(It)-fold multifork extension, which produces J = It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (In the figure, where It = I3 = J, NumTube(It) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') However, in L′ t−1, we add a 2-fold multifork at H′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 to obtain a new lattice L′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Geometrically (in the C1- diagram), this new multifork extension and the lamp J′ = It−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 it produces look unusual compared to other figures in the present paper and several earlier papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Namely, we require that the 4-cell H′ t whose peak is the foot of the leftmost neon tube of J′ should be almost as large as H′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' That is, the width η of the “legs” of the Λ-shaped difference H′ t−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 \\ H′ t, which is yellow-filled in the figure, should be very small compared to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (We may think of η = ϵ/1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') On the right of the Figure, H′ t = H′ 3 in L′ is 3-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, we perform a NumTube(It)-fold multifork extension at H′ t to obtain L′ t from Lt−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 and to produce the lamp J′′ = It of L′ t (and of L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The feet of the neon tubes of J′′ = It in L′ t (and in L′) should be the same geometric points as the feet of the neon tubes of J = It in Lt (and in L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' So the geometric shape of J and that of J′′ are almost the same (and they tend to be the same as η tends to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' From L′ t, we continue the multifork sequence for L′ in the same way as we continue the sequence from Lt to reach L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Even in geometric sense, we do almost the same, that is, with very little differences that would diminish if we formed the limit at η → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To be more specific, let us agree that we use the alternative notation I−1 = A1, I−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 = B1, I−2 = A2, I−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='7 = B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' for the boundary lamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For s = t + 1, t + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , k, we select H′ s+1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In Ls, the trajectory through the top left edge of the 4-cell Hs+1 contains exactly one neon tube, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since the top left edge of Hs+1 is of slope (1, 1), it is in the descending part of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The neon tube p belongs to exactly one lamp, which is older than or as old as Is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' let Iu denote this lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Note that we never use the trajectory through the leftmost neon tube of It−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 (in the figure, the “narrow” trajectory through the yellow-filled area), whereby u ̸= t − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5 and so Iu will also make sense in L′, not only in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Among the neon tubes of Iu, let p be the α-th neon tube (from the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In L′ t, let p′ be the α-th neon tube of Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' By left-right symmetry, the top right edge of Hs+1 defines a neon tube q of a lamp Iv in Ls and its counterpart q′ in L′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The top right edge of Hs+1 is in the ascending part of the trajectory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now we can simply select H′ s+1 as the unique 4-cell of L′ s where the descending part of trajectory through p′ and the ascending part of the trajectory through q′ cross each other12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Once H′ s+1 has been selected, we perform a NumTube(Is+1)- fold multifork extension at this 4-cell of L′ s to obtain L′ s+1 and its lamp Is+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This multifork extension should almost be the same geometrically as in the passage from Ls to Ls+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' in particular, the feet of the new neon tubes have to be geometrically the same in L′ s+1 as in Ls+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For later reference, note that the left upper edge of CircR(Is+1) = Hs+1 belongs to the trajectory through a neon tube of Iu both in L an L′, and similarly for the right upper edge and Iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' � � � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) Finally, we obtain L′ = L′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Next, in order to recall Cz´edli [7, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5], we need some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Let U be an internal lamp of a slim rectangular lattice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then the top edge of the trajectory containing the upper left edge of CircR(U) is a neon tube of a lamp, which we denote by Nwl(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Left-right symmetrically, Nel(U) stands for the unique lamp that has 12Doubts whether they cross will be dissolved later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 22 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI a neon tube whose trajectory contains the upper right edge of CircR(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For a poset Q, let Min(Q) stand for the set of minimal elements of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Now [7, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5] asserts that if K is a slim rectangular lattice and U, V ∈ Lamp(K), then U ≺ V in Lamp(K) if and only if V ∈ Min({Nwl(U), Nel(U)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) Comparing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2) and taking into account that only internal lamps, which all occur in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1), can be covered by another lamp, it follows that Lamp(L) \\ {J} is order isomorphic to Lamp(L′)\\{J′, J′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' We obtain from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='10 that J′ < J′′ in Lamp(L′), Lamp(L) ∼= Lamp(L′) \\ {J′}, and Lamp(L) ∼= Lamp(L′) \\ {J′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, using that P ∼= Lamp(L), we conclude that P ′ ∼= Lamp(L′), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, the construction yields that NumTubeall(L′) = NumTubeall(L) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' However, the proof is not ready yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Indeed, we need to show that the trajectories mentioned earlier do cross in L′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To be more precise, we need to show that if the geometric areas REOT(p) and LEOT(q) cross in Ls, than so do REOT(p′) and LEOT(q′) in L′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Of course, REOT(p′) and LEOT(q′) are perpendicular if we disregard their thickness but, in principle, they could avoid each other like the right leg of the upper ∧∧∧ and the left leg of the lower ∧∧∧ do in � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3) Fortunately, it is clear by continuity that whenever η is small enough (compared to ϵ), then REOT(p′) and REOT(q′) are close enough to REOT(p) and REOT(q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Thus, since REOT(p) and REOT(q) cross each other at a rectangle with sides at least ϵ, REOT(p′) ∩ REOT(q′) is a rectangle of a positive area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Fur- thermore, in Ls, REOT(p) ∩ REOT(q) is a 4-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Since, except when J′′ = It was created, OT(J′) = OT(It−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5) is never used, REOT(p′) ∩ REOT(q′) is also a 4-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This shows that the definition of L′ s+1 and that of L′ make sense, completing the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' □ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' The construction for Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, rescaled In a poset P, where meets and joins need not exist, M(P) and J(P) are defined in the usual way: x ∈ M(P) means that x has exactly one cover;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' J(P) is defined dually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' REDUCING THE LENGTHS OF SPS LATTICES 23 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In most of the cases, the estimate given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1 is far from optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For example, if J(Con L′) ∼= J(D) ∼= P ′ and P ′ is obtained from a smaller poset P by doubling a non-maximal element j ∈ P, then, with the notation of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, the lamp J′ corresponding to j′ ∈ P ′ has only two neon tubes and contributes to len(L′) by 2 regardless the size of ↓Lamp(L′)J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' To present another example of a different nature, let Pn be the n-element poset consisting of two maximal elements, a and b, n − 3 minimal elements, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , cn−3, and an element u such that u ≺ a, u ≺ b, and ci ≺ u for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' , n − 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there is a slim rectangular lattice L such that Con L ∼= Pn and len(L) = |NumTubeall(L)| = n + 1, which is much smaller than what the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) gives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Note that NumTube(U) = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' this helps us to draw L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=') In our third example, Qn is the poset with two maximal elements and n − 2 minimal elements such that every minimal element is covered by both maximal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Then there is a slim rectangular lattice L such that Con L ∼= Pn and len(L) = |NumTubeall(L)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This example shows that the lower estimate given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1(B) cannot be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' As Remarks 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2 allow us to guess, there are many factors that reduce the number len(L) = |NumTubeall(L)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' This explains that we are far from a sharp bound instead of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) as well as from a significantly better while still simple one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3 is not sharp either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Indeed, in addition to that this corollary is built on the non-sharp Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, there is another factor explaining this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Namely, if J(D) ∼= J(Con K) has few non-maximal elements (in particular, if D is Boolean), then |L| has few internal lamps and |L| is close to len(L)2 but then len(L) is much smaller than what (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1) gives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' On the other hand, if J(D) ∼= J(Con K) has many non-maximal elements, then L has many internal lamps and |L| is considerably smaller than len(L)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' In order to decide whether a given n-element poset P is isomorphic to J(Con L) of a slim semimodular lattice L, it is not economic and usually it is not feasible to list all slim rectangular lattices of length at most 2n2 − 10n + 15, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1), or those of size at most (2n2 − 10n + 15)2, see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' It is often much faster to check (A) the known properties of the posets J(Con L) of slim rectangular lattices L, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3), Cz´edli [4], [8], and Cz´edli and Gr¨atzer [10] (where two earlier properties from Gr¨atzer [13] and [14] are also recalled) to see whether these properties exclude the existence of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' and (B) the known constructions to see whether they imply the existence of L (and show how to construct it);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1, Cz´edli [7, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='16], and Cz´edli and Gr¨atzer [10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Furthermore, (C) in many cases, in particular for small |P|, even if (A) and (B) do not lead to an answer, the ideas of the proofs in the papers referenced in (A) and (B) above contain ideas how to construct an L with Lamp(L) ∼= P or how to exclude the existence of such an L by inspecting much less cases then those suggested by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Even though no systematic checking has been done to decide which |P| is “small” for Part (C) of Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='3, we mention that |P| ≤ 6 is probably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 24 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' CZ´EDLI References [1] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 67 (2012) 313–345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1007/s00012-012-0190-3 [2] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Patch extensions and trajectory colorings of slim rectangular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 72, 125–154 (2014) [3] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Diagrams and rectangular extensions of planar semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 77, 443–498 (2017) [4] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Lamps in slim rectangular planar semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged)13 87, 381–413 (2021) (Open access view: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14232/actasm-021-865-y or brows http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='hu/) [5] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Cyclic congruences of slim semimodular lattices and non-finite axiomati- zability of some finite structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Archivum Mathematicum Brno 58/1 (2022) 15–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='5817/AM2022-1-15 [6] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': A property of meets in slim semimodular lattices and its applica- tion to retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged), to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' For an earlier version, see http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/abs/2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='07594 [7] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Quotient diagrams14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='03606 [8] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Infinitely many new properties of the congruence lattices of slim semimodular lattices, submitted to Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged)15 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='org/abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='14769 [9] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', D´ek´any, Gyenizse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Kulin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': The number of slim rectangular lattices, Algebra Universalis 75/1 (2016) 33–50, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='1007/s00012-015-0363-y [10] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Gr¨atzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': A new property of congruence lattices of slim, planar, semimodu- lar lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Categories and General Algebraic Structures with Applications 16, 1–28 (2022) (Open access: https://cgasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='sbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='ir/article 101508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='html) [11] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Schmidt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': The Jordan-H¨older theorem with uniqueness for groups and semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 66, 69–79 (2011) [12] Cz´edli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Schmidt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Slim semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' A visual approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Order 29, 481–497 (2012) [13] Gr¨atzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Congruences of fork extensions of slim, planar, semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 76, 139–154 (2016) [14] Gr¨atzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Notes on planar semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Congruence lattices of SPS lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Algebra Universalis 81 (2020), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 15, 3 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' [15] Gr¨atzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Knapp, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Notes on planar semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged) 73, 445–462 (2007) [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Gr¨atzer and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Knapp, Notes on planar semimodular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Rectangular lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged) 75 (2009), 29–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' [17] Kelly, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=', Rival, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=': Planar lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Canad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' 27, 636–665 (1975) Email address: czedli@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='u-szeged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='hu URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='u-szeged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='hu/~czedli/ University of Szeged, Bolyai Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Szeged, Aradi v´ertan´uk tere 1, HUNGARY 6720 13At the time of writing, see also http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='hu/acta/ , the good old site of Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content=' (Szeged), where all my papers published not later than 2021 are free to download 14At the time of writing, see https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} +page_content='com/czedli-cde-con-sps for the most recent version 15At the time of writing, see also the author’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQfivhb/content/2301.00401v1.pdf'} diff --git a/atE3T4oBgHgl3EQf2wun/content/tmp_files/2301.04758v1.pdf.txt b/atE3T4oBgHgl3EQf2wun/content/tmp_files/2301.04758v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd2e73a191854ceba5b9d09a9f6a7efea1debb5a --- /dev/null +++ b/atE3T4oBgHgl3EQf2wun/content/tmp_files/2301.04758v1.pdf.txt @@ -0,0 +1,5308 @@ +High-Order Mixed Finite Element Variable Eddington Factor +Methods +Samuel Oliviera,b,* and Terry S. Hautc +aUniversity of California, Berkeley, Berkeley, CA, USA; bLos Alamos National Laboratory, +Los Alamos, NM, USA; cLawrence Livermore National Laboratory, Livermore, CA, USA +ARTICLE HISTORY +Compiled January 13, 2023 +ABSTRACT +We apply high-order mixed finite element discretization techniques and their as- +sociated preconditioned iterative solvers to the Variable Eddington Factor (VEF) +equations in two spatial dimensions. The mixed finite element VEF discretizations +are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Dis- +crete Ordinates transport equation to form effective linear transport algorithms +that are compatible with high-order (curved) meshes. This combination of VEF and +transport discretizations is motivated by the use of high-order mixed finite element +methods in hydrodynamics calculations at the Lawrence Livermore National Lab- +oratory. Due to the mathematical structure of the VEF equations, the standard +Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vec- +tor variable in the VEF equations. Instead, we investigate three alternatives based +on the use of continuous finite elements for each vector component, a non-conforming +RT approach where DG-like techniques are used, and a hybridized RT method. We +present numerical results that demonstrate high-order accuracy, compatibility with +curved meshes, and robust and efficient convergence in iteratively solving the cou- +pled transport-VEF system and in the preconditioned linear solvers used to invert +the discretized VEF equations. +KEYWORDS +radiation transport; Variable Eddington Factor; Quasidiffusion; high-order finite +elements; preconditioned iterative solvers +1. +Introduction +The Variable Eddington Factor (VEF) method [1], also known as Quasidiffusion [2], is +an efficient iterative method for solving the Boltzmann transport equation, a crucial +component in the modeling of nuclear reactors, high energy density physics experi- +ments, astrophysical phenomena, and medical physics. In VEF, the transport equa- +tion is iteratively coupled to the VEF equations, a moment-based, reduced-dimensional +model of transport formed through discrete closures. The VEF closures are weak func- +tions of the transport solution allowing the design of rapidly converging and robust +iterative schemes. A key advantage of VEF is that the discretized VEF and transport +*corresponding author. Email: solivier@lanl.gov +arXiv:2301.04758v1 [math.NA] 11 Jan 2023 + +equations do not need to be algebraically consistent to maintain this rapid conver- +gence, even in the thick diffusion limit [3]. These so-called independent VEF methods +[4] then have the flexibility to choose the discretization of the VEF equations to meet +the requirements of the overall algorithm, such as computational efficiency and multi- +physics compatibility. +Mixed finite element methods are a class of discretization techniques for solving the +mixed variational form of a partial differential equation. This variational form is char- +acterized by the inclusion of multiple (typically two) physically disparate quantities +resulting in a saddle point problem. By contrast, primal formulations operate on a +single quantity and produce minimization problems. Mixed methods were invented to +1) allow incorporation of a constraint (e.g. divergence free velocity in fluid flow), 2) +provide direct access to an intermediate variable (e.g. the stress in elasticity), and 3) +allow a weaker formulation than the corresponding primal formulation. In the context +of neutron diffusion, mixed methods are applied to the first-order, or P1, form of ra- +diation diffusion and 1) explicitly include the constraint of particle balance, 2) solve +for the current in addition to the scalar flux, and 3) allow scalar flux solutions with +no continuity requirements at interior element interfaces. Through a process called +mixed finite element hybridization, the resulting block system of equations can be +reduced to a positive definite system that can be efficiently solved with Algebraic +Multigrid (AMG) [5]. The block system can also be directly preconditioned through +block diagonal and lower block triangular preconditioners based on applying AMG to +an approximate Schur complement [6]. +In this paper, we investigate the use of mixed finite elements to solve the VEF equations +in two dimensions. The research goals are to achieve high-order accuracy, compatibil- +ity with high-order (curved) meshes, and scalable preconditioned iterative solvers. The +motivation for this research is that high-order mixed finite element methods on curved +meshes are used in hydrodynamics calculations at the Lawrence Livermore National +Laboratory (LLNL) [7, 8]. In particular, we are interested in designing a discretiza- +tion of the VEF equations that matches as closely as possible to that of Maginot and +Brunner [9], the mixed finite element method used for radiation diffusion at LLNL. +Such a method would 1) have element-local particle balance, 2) solve for the current +directly potentially leading to high accuracy coupling to the hydrodynamics’ momen- +tum equation, and 3) allow the scalar flux to be approximated in the same finite +element space as the hydrodynamics’ thermodynamic variables. In addition, a mixed +finite element VEF discretization could serve as a drop-in replacement for radiation +diffusion at LLNL providing a transport algorithm that allows reuse of the linear and +nonlinear solvers already in place for diffusion. Mixed finite element discretizations of +radiation diffusion have also been used in reactor analysis [10, 11, 12]. +Olivier and Morel [13] developed a lowest-order, hybridized mixed finite element dis- +cretization of the VEF equations in one spatial dimension for the linear transport +problem. Lou et al. [14] and Lou and Morel [15] used this algorithm to form efficient, +VEF-based thermal radiative transfer and radiation-hydrodynamics algorithms, re- +spectively. Multi-dimensional, high-order Discontinuous Galerkin VEF discretizations +with scalable solvers were developed in [16]. However, such methods do not directly +solve for the current, precluding the possibility of high-order coupling to the momen- +tum equation. Furthermore, a mixed finite element VEF discretization has immediate +mathematical and implementational compatibility with the mixed methods used in +the hydrodynamics framework of [7]. +2 + +Here, we extend the lowest-order mixed finite element discretization from Olivier and +Morel [13] to high-order accuracy in two spatial dimensions, prescribe efficient pre- +conditioned iterative solvers for the discretized VEF equations, and compare the per- +formance of the hybridized and unhybridized mixed finite elements techniques. The +extension to multiple dimensions is non-trivial due to the elevation of the VEF closure +from a scalar in one spatial dimension to a symmetric tensor in multiple dimensions. +This introduces complications in approximating the vector-valued neutron current +that prevent a straightforward extension of the one-dimensional discretization. An +early form of this work was presented in Olivier et al. [17]. +The paper proceeds as follows. The VEF algorithm is introduced analytically. We +present background on high-order meshes and the finite element spaces used to solve +the VEF and transport equations numerically. We discuss the algorithmic connec- +tions between the VEF discretizations and a high-order DG discretization of the SN +transport equations (e.g. [18, 19]). We then apply mixed finite element techniques to +the VEF equations. We show that, due to the presence of the Eddington tensor in +the VEF first moment equation, the standard Raviart Thomas (RT) mixed finite el- +ement methods [20, 21] are not appropriate for the VEF equations. We present two +alternatives: a method where each component of the current is approximated with +continuous finite elements and a non-conforming approach where the RT space is used +along with DG-like numerical fluxes. A lower block triangular preconditioner that uses +a combination of classical smoothing and AMG is defined for each of the above VEF +discretizations. We then propose a hybridized version of the RT method that increases +the computational efficiency of the RT method by reducing the number of globally +coupled unknowns. +Next, numerical results are presented. We investigate the accuracy of the methods on +a third-order mesh using the method of manufactured solutions. Convergence of the +fixed-point iteration is tested in the thick diffusion limit on both an orthogonal mesh +and a severely distorted third-order mesh generated using a Lagrangian hydrodynam- +ics code. The robustness of the preconditioned iterative solvers to mesh distortion is +also investigated. The efficiencies of the outer fixed-point iteration and inner precon- +ditioned linear iteration are compared on a challenging, multi-material problem. We +then document the degraded solution quality associated with the method that uses +continuous finite elements for each component of the current on a simple radiation +diffusion eigenvalue problem and provide intuition for AMG’s inability to effectively +precondition the corresponding linear system. A weak scaling study is presented for +the RT and hybridized RT methods showing that the discrete VEF systems can be +solved with similar efficiency to that of an analogous symmetric radiation diffusion +system. Finally, we give conclusions and recommendations for future work. +2. +VEF Algorithm +Here, we describe the VEF method for the steady-state, mono-energetic, fixed-source +transport problem with isotropic scattering. VEF methods simultaneously solve the +coupled transport and VEF equations given by: +Ω · ∇ψ + σtψ = σs +4πϕ + q , +x ∈ D , +(1a) +3 + +ψ(x, Ω) = ¯ψ(x, Ω) , +x ∈ ∂D and Ω · n < 0 , +(1b) +∇ · J + σaϕ = Q0 , +x ∈ D , +(2a) +∇ · (Eϕ) + σtJ = Q1 , +x ∈ D , +(2b) +J · n = Ebϕ + 2Jin , +x ∈ ∂D , +(2c) +where ψ(x, Ω) is the angular flux, ϕ(x) and J(x) the VEF scalar flux and cur- +rent, respectively, Ω ∈ S2 the direction of particle flow, D the spatial domain with +∂D its boundary, σt(x), σs(x), and σa(x) = σt(x) − σs(x) the total, scattering, +and absorption macroscopic cross sections, respectively, q(x, Ω) the fixed source with +Qi = +� +Ωiq dΩ its angular moments, and ¯ψ(x, Ω) the boundary inflow function with +Jin(x) = +� +Ω·n<0 Ω · n ¯ψ(x, Ω) dΩ the inflow partial current. The Eddington tensor and +boundary factor are defined as: +E = +� +Ω ⊗ Ω ψ dΩ +� +ψ dΩ +, +(3a) +Eb = +� +|Ω · n| ψ dΩ +� +ψ dΩ +, +(3b) +respectively. The Miften and Larsen [22] boundary conditions are used for the VEF +equations. Observe that the transport equation (Eqs. 1) is linearly coupled to the VEF +equations (Eqs. 2) through the scattering source and the VEF equations are nonlinearly +coupled to the transport equation through the Eddington tensor and boundary factor. +The VEF closures are weak functions of the transport solution allowing the design of +efficient iterative solution techniques. +The simplest algorithm to solve the coupled transport-VEF system is fixed-point it- +eration. The iteration is: 1) invert the transport equation given a scattering source +from the previous iteration or an initial guess, 2) compute the Eddington tensor and +boundary factor from the angular flux from stage 1), and 3) invert the VEF equations +for a new scalar flux and current. The iteration is repeated until the VEF scalar flux +converges. A discussion of the derivation of the above system and the application of +advanced nonlinear solvers, such as Anderson acceleration, are provided in [16]. +The following sections present methods for efficiently computing the VEF fixed-point +operator numerically. We present the description of high-order meshes and the proce- +dure for integration over arbitrary elements, the finite element spaces used to discretize +the transport and VEF equations, the representation of the VEF data using finite ele- +ment interpolation and angular quadrature, and finally three novel mixed finite element +discretizations for the VEF equations. +4 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +(a) +ξ +η +(0,0) +(1,0) +(0,1) +(1,1) +T(ξ) +T−1(x) +x +y +(b) +Fig. 1.. Depictions of (a) the mesh control points in a quadratic quadrilateral mesh +and (b) the reference transformation used to describe the left element of (a). +3. +Mesh and Finite Element Preliminaries +This section provides background on the representation of high-order meshes and the +transformations used to facilitate numerical integration over arbitrary elements. We +pay particular attention to the transformation of vector-valued functions and their +gradient and divergence on high-order meshes. These transformations are crucial for +the implementation of the approximation techniques used for the current described in +§4. +3.1. +Description of the Mesh +The domain D ⊂ R2 is tesselated into a collection T of quadrilateral elements Ke such +that +D = +� +Ke∈T +Ke . +(4) +Each element Ke is obtained as Ke = Te( ˆK) where ˆK = [0, 1]2 is the reference element. +Let Qm,n( ˆK) be the space of polynomials of degree less than or equal to m and n in +the first and second variables, respectively, with Qm( ˆK) = Qm,m( ˆK). The mapping +Te ∈ [Qm( ˆK)]2 is derived from a set of global control points and an element-local +nodal basis. Figure 1a shows an example mesh where the control points labeled 2, +7, and 12 are shared so that the mesh coordinates are continuous across the interior +interface between the two elements. +A nodal basis for Qm( ˆK) is defined using Lagrange interpolating polynomials. Let ξi +denote the m + 1 Gauss-Lobatto points in the interval [0, 1]. The (m + 1)2 points ξi +on the unit square ˆK = [0, 1]2 are given by the two-fold Cartesian product of the one- +dimensional points. Let ℓi denote the Lagrange interpolating polynomial that satisfies +ℓi(ξj) = δij where δij is the Kronecker delta. The set of functions {ℓi} form a basis for +Qm( ˆK). For each element, the mapping is then +x(ξ) = Te(ξ) = +(m+1)2 +� +i=1 +xe,iℓi(ξ) +(5) +5 + +where x ∈ Ke, ξ ∈ ˆK, and xe,i are the control points corresponding to element Ke. +Figure 1b depicts the mesh transformation used for the left element in Fig. 1a. +We define Γ as the set of unique faces in the mesh with Γ0 = Γ \ ∂D the set of interior +faces and Γb = Γ ∩ ∂D the set of boundary faces so that Γ = Γ0 ∪ Γb. We denote +the outward unit normal to element K as nK. On an interior face F ∈ Γ0 between +elements K1 and K2, we use the convention that n is the unit vector perpendicular to +the shared face K1 ∩ K2 pointing from K1 to K2. On such an interior face, the jump, +�·�, and average, {{·}}, are defined as +�u� = u1 − u2 , +{{u}} = 1 +2(u1 + u2) , +on F ∈ Γ0 , +(6) +where ui = u|∂Ki with analogous definitions for vectors. Note that a continuous func- +tion u satisfies �u� = 0 on each interior face. On boundary faces, the jump and average +are set to +�u� = u , +{{u}} = u , +on F ∈ Γb , +(7) +and likewise for vector-valued functions on the boundary. The tangent vector is denoted +by τ and we use the convection that τ is a 90◦ counter-clockwise rotation of the normal +vector. +Finally, we define the “broken” gradient, denoted by ∇h, obtained by applying the +gradient locally on each element. That is, +(∇hu)|K = ∇(u|K) , +∀K ∈ T . +(8) +This distinction is important for the piecewise polynomial spaces discussed in §4. +3.2. +Integration Transformations +The mesh transformations Te are used to facilitate numerical integration on arbi- +trary elements. Letting ξ = +�ξ +η�T ∈ +ˆK denote the reference coordinates and +x = +�x +y�T ∈ D the physical coordinates such that x(ξ) = Te(ξ), the Jacobian +of the transformation is +Fe = ∂Te +∂ξ = +� ∂x +∂ξ +∂x +∂η +∂y +∂ξ +∂y +∂η +� +, +(9) +with Je = |Fe| its determinant. The partial derivatives of the mesh transformation are +computed by taking derivatives of the nodal basis functions. In other words, +Fe = +(m+1)2 +� +i=1 +xe,i ⊗ ˆ∇ℓi = +(m+1)2 +� +i=1 +� +xe,i ∂ℓi +∂ξ +xe,i ∂ℓi +∂η +ye,i ∂ℓi +∂ξ +ye,i ∂ℓi +∂η +� +, +(10) +where xe,i = +�xe,i +ye,i +�T and ˆ∇ denotes the gradient with respect to ξ. +6 + +A mesh transformation is called affine when it can be written as +T = Aξ + b +(11) +where A ∈ R2×2 and b ∈ R2 are constant with respect to ξ. In such case, the Jacobian +matrix is F = A and the Hessian of the transformation, defined as ∂2T +∂ξ2 , is identically +zero. Quadrilateral elements obtained by scaling, stretching along the ξ or η axes, or +rotating the reference element are all affine while general quadrilateral elements, such +as trapezoidal elements, and curved elements are not affine. +In this document, integration over the domain is implicitly computed in reference space +using the following sum: +� +D +(·) dx = +� +K∈T +� +K +(·) dx = +� +K∈T +� +ˆ +K +(·) Jdξ . +(12) +This provides a systematic way to integrate over arbitrary domains composed of ar- +bitrarily shaped elements as well as the use of numerical quadrature rules defined on +the reference element ˆK. We now discuss the transformations used to represent the +integrand of Eq. 12 in reference space. For a scalar function u : D → R, denote by +ˆu : ˆK → R its representation in reference space. The functions u and ˆu are related by +u(x) = ˆu(T−1(x)) . +(13) +Integration over the physical element is then equivalent to +� +K +u dx = +� +ˆ +K +ˆu Jdξ . +(14) +Using the chain rule, the gradient of a scalar function transforms as +∇ˆu = +� +∂ˆu +∂ξ +∂ξ +∂x + ∂ˆu +∂η +∂η +∂x +∂ˆu +∂ξ +∂ξ +∂y + ∂ˆu +∂η +∂η +∂y +� += F−T ˆ∇ˆu . +(15) +In this way, the gradient in physical space can be computed using the Jacobian of the +mesh transformation and the gradient in reference space. +For vector-valued functions, the basis the vector is defined on must also be considered. +The simplest basis is the canonical basis, ei, corresponding to the x and y axes. In +this case, a vector v : D → R2 is +v = v1e1 + v2e2 +(16) +and each component transforms independently as vi = ˆvi(T−1(x)). Writing +∇v = +� ∂v1 +∂x +∂v1 +∂y +∂v2 +∂x +∂v2 +∂y +� += +� +(∇v1)T +(∇v2)T +� +, +(17) +7 + +the gradient of a vector defined as in Eq. 16 transforms as +∇v = +� +(F−T ˆ∇ˆv1)T +(F−T ˆ∇ˆv2)T +� +. +(18) +Note that defining a vector in this way does not preserve the normal or tangential +components under a rotation. That is, v · n and v · τ are linear combinations of the +vi instead of a single component representing the normal or tangential components, +respectively. +Alternatively, the contravariant Piola transform represents vectors on the so-called +tangent basis so that the normal component can be preserved [23, 24]. Such a trans- +formation is required by the Raviart Thomas space introduced in §4.3 in order to +strongly enforce continuity in the normal component of the current. The contravariant +Piola transform is: +v = 1 +J Fˆv ◦ T−1 . +(19) +Here, ˆv : ˆK → R2 is a vector in reference space. Writing the columns of the Jacobian +matrix as +F = +� +t1 +t2 +� +, +(20) +the contravariant Piola transformation is equivalent to +v = 1 +J (ˆv1t1 + ˆv2t2) . +(21) +Observe that, on the reference canonical basis ˆei, ˆv = ˆv1ˆe1 + ˆv2ˆe2, and thus the +contravariant Piola transform maps the canonical reference basis to the tangent space +spanned by {t1, t2} and scales by 1/J. +When the mesh transformation Te is not affine, the tangent basis is not orthogonal. +In this case, the usual method of selecting components of a vector through the dot +product (e.g. vi = ti · v) is inappropriate since ti · tj ̸= δij. Instead, a dual basis, +referred to as the cotangent basis, is constructed such that +ni · tj = δij . +(22) +Vectors the satisfy Eq. 22 are called bi-orthonormal. Since the ti are the columns of +the Jacobian matrix, defining the cotangent basis as the rows of the inverse of the +Jacobian matrix satisfies the bi-orthonormality condition since F−1F = I. In other +words, the cotangent basis is defined such that +F−1 = +� +nT +1 +nT +2 +� +. +(23) +For a contravariant vector, the usual method of selecting a component is now replaced +with vi = ni·v. The cotangent space is associated with vectors normal to the faces. By +representing a vector on the tangent space, the contravariant Piola transform allows +8 + +ˆe1 +ˆe2 +T(ξ) +T−1(x) +t1 +n1 +t2 +n2 +Fig. 2.. A depiction of the tangent and cotangent bases at the point ξ = (0, 0) under +a non-affine mesh transformation. +selection of the component representing the normal component through n · v. Note +that for non-affine meshes, F depends on ξ and thus the tangent and cotangent bases +also depend on ξ. +Figure 2 depicts an example non-affine mesh transformation and the tangent and +cotangent bases evaluated at the point ξ = (0, 0). Observe that the pairs (t1, n2) and +(t2, n1) are perpendicular. The pairs (t1, n1) and (t2, n2) do not point in the same +direction but their magnitudes and directions balance so that ti · ni = 1. Thus, the +bi-orthonormality condition ni · tj = δij is satisfied. In addition, the tangent vectors +and cotangent vectors are tangential and normal, respectively, to one of the faces +connecting at the point ξ = (0, 0). +For a contravariant vector, +� +K +∇u · v dx = +� +ˆ +K +F−T ˆ∇ˆu · 1 +J Fˆv Jdξ = +� +ˆ +K +ˆ∇ˆu · ˆv dξ . +(24) +The gradient transforms as +∇v = ∇ +� 1 +J Fˆv ◦ T−1 +� += 1 +J F +� +ˆ∇ˆv − ˆB +� +F−1 +(25) +where +ˆB = 1 +J +ˆ∇ +� +JF−1� +Fˆv . +(26) +This result is derived by direct computation in Appendix A along with the details +required to implement this transformation using the machinery commonly provided +in finite element libraries. It is also shown that ˆB = 0 when the mesh transformation +is affine and that trace( ˆB) = 0 for any transformation. This last result is known as +the Piola identity [23]. Using the Piola identity, the linearity of the trace, and the +invariance of the trace under similarity transformations, the divergence transforms as +∇ · v = trace (∇v) = 1 +J trace +� +F +� +ˆ∇ˆv − ˆB +� +F−1� += 1 +J +ˆ∇ · ˆv . +(27) +Thus, +� +K +u ∇ · v dx = +� +ˆ +K +ˆu ˆ∇ · ˆv dξ . +(28) +9 + +Combining the results from Eqs. 24 and 28 yields: +� +∂K +u v · n ds = +� +∂ ˆ +K +ˆu ˆv · ˆn dˆs , +(29) +where ˆn is the normal vector in reference space corresponding to the physical space +normal n. In other words, the contravariant Piola transformation preserves the normal +component. +In this document, integration is implicitly computed using numerical quadrature on +the reference element. Integration over surfaces is performed over the one-dimensional +reference element using the transformed element of length. +4. +Finite Element Spaces +In this section, we define the finite element spaces used to approximate the VEF equa- +tions. These finite element spaces are defined on the mesh T or the interior skeleton of +the mesh Γ0 and consist of an element-local function space and a set of inter-element +matching conditions. The inter-element matching conditions enforce various types of +continuity of the solution between elements. The combination of a locally smooth func- +tion space and suitable matching conditions allows finite element spaces to be discrete +subspaces of Sobolev spaces such as L2(D), H1(D), and H(div; D). The following sub- +sections define the element-local function space and matching conditions used for the +scalar flux, current, and the interface variable used later in hybridization of the mixed +finite element method. +4.1. +Discontinuous Galerkin +The Discontinuous Galerkin (DG) space is a discrete subspace of L2(D), the space of +square-integrable functions. In other words, if u is an element of the DG space, +� +u2 dx < ∞ . +(30) +Since only square integrability is required, functions in L2(D), and thus DG spaces, do +not need to be continuous. DG functions are represented using piecewise-discontinuous +polynomials that are defined on the reference element and mapped to the physical +element using the inverse mesh transformation T−1 +e +: Ke → ˆK. In other words, on +each element, the solution belongs to: +Qp(Ke) = {u = ˆu ◦ T−1 +e +: ˆu ∈ Qp( ˆK)} . +(31) +The distinction between Qp( ˆK) and Qp(Ke) is important for non-affine mesh trans- +formations. In such case, the inverse mesh transformation is generally non-polynomial +so that the composition u = ˆu ◦ T−1 +e +is also non-polynomial. +The degree-p DG space is +Yp = {u ∈ L2(D) : u|K ∈ Qp(K) , +∀K ∈ T } . +(32) +10 + +Fig. 3.. A depiction of the distribution of degrees of freedom in the linear DG space. +The Legendre nodes are used to illustrate that degrees of freedom are not shared +between elements. +An example of the distribution of the degrees of freedom in a linear DG space on a +3 × 3 mesh is shown in Fig. 3. Note that degrees of freedom are not shared between +elements. Since there are no continuity requirements in the DG space, the basis for +the local polynomials can use either open or closed points. That is, a nodal basis can +be formed with Lagrange interpolating polynomials through the two-fold Cartesian +product of either the closed Gauss-Lobatto points or the open Gauss-Legendre points. +4.2. +[H1(D)]2 +Here, we define a discrete subspace of H1(D), the space of functions in L2(D) with +square-integrable gradient. Let the degree-p, scalar continuous finite element space be +Vp = {u ∈ C0(D) : u|K ∈ Qp(K) , +∀K ∈ T } +(33) +so that each function u ∈ Vp is a piecewise-continuous polynomial mapped from the +reference element. Since u ∈ Vp is locally smooth and Vp ⊂ C0(D), it can be shown +that Vp ⊂ H1(D) and, in particular, that u ∈ Vp satisfies ∇u = ∇hu ∈ [L2(D)]2 [25, +Prop. 3.2.1]. The distribution of degrees of freedom for the space V2 is shown in Fig. 4. +Here, continuity is enforced by sharing degrees of freedom between elements. Due to +this, a nodal basis using closed points, such as the Gauss-Lobatto points, must be +used. +The vector-valued analog +Wp = {v : v1 ∈ Vp and v2 ∈ Vp} +(34) +uses the scalar continuous finite element space for each component. In this way, v ∈ +Wp ⊂ [H1(D)]2 and thus ∇v = ∇hv ∈ [L2(D)]2×2. Since each component is defined +independently using the scalar space, vectors v ∈ Wp transform according to Eq. 16. +11 + +Fig. 4.. A depiction of the distribution of degrees of freedom for the quadratic +continuous finite element space. Continuity of members of the finite element space is +enforced by sharing degrees of freedom across neighboring elements. +4.3. +Raviart Thomas +The Raviart Thomas (RT) space is a discrete subspace of H(div; D), the space of +vector-valued functions with square-integrable divergence. That is, +H(div; D) = {v ∈ [L2(D)]2 : ∇ · v ∈ L2(D)} . +(35) +The requirements of a discrete subspace are codified in the following proposition. +Proposition 1 (Cf. Quarteroni and Valli [25], Proposition 3.2.2). Let v : D → R2 be +such that +(a) v|K ∈ [H1(K)]2 for each K ∈ T +(b) �v · n� = 0 for each F ∈ Γ0 +then v ∈ H(div; D). Conversely, if v ∈ H(div; D) and (a) is satisfied, then (b) holds. +Proof. It must be shown that, given (a) and (b), ∇ · v ∈ L2(D). We proceed by +leveraging the fact that ∇h · v ∈ L2(D) (since v|K ∈ [H1(K)]2) and then show that +∇ · v = ∇h · v, proving the claim. Let C∞ +0 (D) be the space of infinitely differentiable +functions that are zero on the boundary of the domain. Using Green’s identity, we +have that for each u ∈ C∞ +0 (D): +� +u ∇ · v dx = − +� +∇u · v dx += − +� +K∈T +� +K +∇u · v|K dx , +(36) +where we have used that u = 0 for each x ∈ ∂D. Since v|K ∈ [H1(K)]2 for each K, +12 + +we can integrate by parts locally on each element to give: +� +u ∇ · v dx = +� +K∈T +�� +K +u ∇ · v|K dx − +� +∂K +u v · n ds +� += +� +u ∇h · v dx − +� +Γ0 +u �v · n� ds += +� +u ∇h · v dx , +∀u ∈ C∞ +0 (D) , +(37) +since u ∈ C∞ +0 (D) satisfies is continuous such that �u� = 0 and �v · n� = 0 from (b). +Therefore, ∇ · v = ∇h · v ∈ L2(D). +On the other hand, if v ∈ H(div; D) then ∇ · v = ∇h · v and, given v|K ∈ [H1(K)]2, +we obtain +� +Γ0 +u �v · n� ds = 0 , +∀u ∈ C∞ +0 (D) , +(38) +hence, (b) holds. +Thus, a discrete subspace of H(div; D) must (a) have a smooth function space on each +element and (b) have suitable matching conditions so that the normal component is +continuous across interior mesh interfaces. +The RT space uses the local polynomial space Qp+1,p( ˆK) × Qp,p+1( ˆK). This choice +can be motivated by the discrete de Rham complex [26] in that +Qp+1( ˆK) +ˆ∇× +−−→ Qp+1,p( ˆK) × Qp,p+1( ˆK) +ˆ∇· +−→ Qp( ˆK) . +(39) +As an example, the lowest-order polynomial space is +Q1,0( ˆK) × Q0,1( ˆK) = span +�� +1 +0 +� +, +� +ξ +0 +� +, +� +0 +1 +� +, +� +0 +η +�� +, +(40) +and thus we have that: +ˆ∇ · Q1,0( ˆK) × Q0,1( ˆK) = span{1} = Q0( ˆK) . +(41) +The nodal basis for Qp+1,p( ˆK) × Qp,p+1( ˆK) uses the closed Gauss-Lobatto points in +the normal direction and the open Gauss-Legendre points in the tangential direction. +The interpolating points for the first three orders are shown in Fig. 5. The circles +denote degrees of freedom corresponding to the ξ component while squares denote the +η component. +The contravariant Piola transformation is used to allow sharing the degrees of free- +dom associated with the normal component with neighboring elements. Note that this +transformation is still required even when continuity in the normal component is re- +laxed and enforced weakly with Lagrange multipliers as in the hybridization procedure +discussed in §7. This is due to the use of anisotropic polynomial interpolation (i.e. dif- +ferent degree polynomials are used in each variable) which requires the vector’s basis +13 + +(a) +(b) +(c) +Fig. 5.. The interpolating points used for the nodal basis of the space +Qp+1,p × Qp,p+1 for (a) p = 0, (b) p = 1, and (c) p = 2. Gauss-Legendre points are +used in the tangential direction and Gauss-Lobatto in the normal direction for each +component of the vector. Circles denote the degrees of freedom associated with the ξ +component and squares the η component. +to be rotated along with the transformation to properly orient the interpolating poly- +nomials in physical space. Combining the local function space Qp+1,p( ˆK) × Qp,p+1( ˆK) +with the contravariant Piola transform yields: +Dp(K) = {v = 1 +J Fˆv ◦ T−1 : ˆv ∈ Qp+1,p( ˆK) × Qp,p+1( ˆK)} . +(42) +Here, both the inverse mesh transformation and 1/J are generally non-polynomial +when T is non-affine. +We now define the degree-p RT space as: +RT p = {v ∈ [L2(D)]2 : v|K ∈ Dp(K) , ∀K ∈ T and �v · n� = 0 , ∀F ∈ Γ0} . +(43) +Note that since the contravariant Piola transform is used, functions in RT transform +according to Eqs. 19, 25, and 27. The location of the degrees of freedom for RT 1 are +shown on a 3 × 3 mesh in Fig. 6. Continuity in the normal component is enforced +by sharing the degrees of freedom corresponding to the normal component on interior +faces. From Proposition 1, v ∈ RT p satisfies ∇ · v = ∇h · v ∈ L2(D). However, the +RT space does not have the continuity to allow a square-integrable gradient. In other +words, ∇v /∈ [L2(D)]2×2 and ∇v ̸= ∇hv. +4.4. +Raviart Thomas Trace Space +The normal trace of the RT space is required for the hybridization procedure discussed +in §7. This space is defined on the interior skeleton of the mesh Γ0 and represents the +normal component of the RT space along the interior mesh faces. Let Pp be the space +of univariate polynomials with degree at most p and +Pp(F) = {u = ˆu ◦ T−1 : ˆu ∈ Pp( ˆF)} +(44) +14 + +Fig. 6.. The distribution of degrees of freedom corresponding to the first degree +Raviart Thomas space. Continuity of the normal component is enforced by sharing +the degrees of freedom corresponding to the normal component along the interior +face between neighboring elements. The circles and squares denote degrees of +freedom in the x and y directions, respectively. +Fig. 7.. The distribution of degrees of freedom corresponding to Λ1, the space defined +as the normal trace of the first degree Raviart Thomas space, on a 3 × 3 mesh. +the space of univariate polynomials mapped from the reference line, ˆF = [0, 1]. The +RT trace space is then +Λp = {µ ∈ L2(Γ0) : µ|F ∈ Pp(F) , +∀F ∈ Γ0} . +(45) +The degrees of freedom in Λ1 are depicted in Fig. 7. Note that these degrees of freedom +are exactly the degrees of freedom corresponding to the normal component of RT 1 on +the interior faces of the mesh. +5. +Transport Discretizations +We assume the transport equation is discretized with the Discrete Ordinates (SN) +angular model and an arbitrary-order DG spatial discretization compatible with curved +meshes (e.g. [18, 19]). The transport equation is collocated at discrete angles Ωd and +integration over the unit sphere is numerically approximated with a suitable angular +quadrature rule, {wd, Ωd}NΩ +d=1. Let ψd(x) = ψ(x, Ωd) be the angular flux in the discrete +direction Ωd. The DG discretization uses ψd ∈ Yp for each discrete angle. Through +finite element interpolation, ψd(x) can be evaluated at any point in the mesh. The +VEF data are computed with SN angular quadrature and finite element interpolation +15 + +as: +E(x) = +�NΩ +d=1 wd Ωd ⊗ Ωd ψd(x) +�NΩ +d=1 wdψd(x) +, +(46a) +Eb(x) = +�NΩ +d=1 wd |Ωd · n| ψd(x) +�NΩ +d=1 wdψd(x) +. +(46b) +Note that we represent the VEF data as ratios of DG grid functions. That is, on each +element K, each component of the Eddington tensor and the boundary factor can be +written as q/p where q, p ∈ Qp(K) and are thus improper rational polynomials mapped +from the reference element. Note that Ω is defined on the canonical basis ei. Thus, +each component of the Eddington tensor transforms independently as a scalar and the +Piola transform is not required to map the Eddington tensor between reference and +physical space. +Since ψd ∈ Yp can be discontinuous across interior mesh interfaces, the VEF data can +also be discontinuous. Thus, global derivatives of the VEF data are not well defined +and, in particular, the Eddington tensor is not single-valued on interior mesh interfaces. +The VEF discretizations presented here are designed to avoid the need for derivatives +of the Eddington tensor and, when needed, use the average to provide a single-valued +approximation of the Eddington tensor on interior mesh faces. The boundary factor +is only needed on the boundary of the domain (i.e. x ∈ Γb) and is thus always single +valued. +We consider problems where ψ ≥ δ in the domain for some δ > 0. This assumption +is reasonable for our applications but may not apply in shielding or deep penetration +problems. Where necessary, negative flux fixups are used to ensure that ψ > 0 numer- +ically since positive angular fluxes are crucial for generating physically realistic VEF +data. Note that the Eddington tensor and boundary factor are angular flux-weighted +averages of Ω ⊗ Ω and |Ω · n|. Combined with the positivity assumption, this means +the each component of the Eddington tensor and the boundary factor are bounded +functions in space such that E ∈ [L∞(D)]2×2 and Eb ∈ L∞(D). +The VEF scalar flux is coupled to the transport equation in the scattering source. +A mixed-space scattering mass matrix is used to support the use of differing finite +element spaces for the angular flux and VEF scalar flux. That is, the scattering source +is built using test functions and trial functions from the spaces corresponding to the +angular and VEF scalar fluxes, respectively. +6. +Mixed Finite Element Discretizations +We now derive mixed finite element discretizations of the VEF equations with Miften- +Larsen boundary conditions. We seek approximations to the scalar flux and current +in the finite-dimensional spaces E and V, respectively, and test the zeroth and first +moments with functions in the spaces E′ and V′, respectively. We consider Galerkin +discretizations so that the test and trial spaces for the scalar flux and current are the +same. In other words, we restrict ourselves to the case that E′ = E and V′ = V. We +proceed by first informally deriving the weak form assuming the spaces E and V have +16 + +the requisite regularity to allow the resulting weak form to be well defined. We will see +that there is no ambiguity in the choice E = Yp ⊂ L2(D). However, due to the presence +of the Eddington tensor, the standard Raviart Thomas methods are inappropriate and +so two choices for V are presented: a method with V = Wp+1 ⊂ [H1(D)]2 and a non- +conforming method where V = RT p ⊂ H(div; D). +6.1. +Weak Form +Multiplying the zeroth and first moments with sufficiently smooth functions u and v, +respectively, and integrating over the domain yields: +� +u ∇ · J dx + +� +σa uϕ dx = +� +u Q0 dx , +(47a) +� +v · ∇ · (Eϕ) dx + +� +σt v · J dx = +� +v · Q1 dx . +(47b) +The differentiability requirements on the Eddington tensor and the VEF scalar flux +can be reduced by integrating the first moment equation by parts: +� +u ∇ · J dx + +� +σa uϕ dx = +� +u Q0 dx , +(48a) +� +∂D +v · En ¯ϕ ds − +� +∇v : Eϕ dx + +� +σt v · J dx = +� +v · Q1 dx , +(48b) +where ϕ = ¯ϕ on the boundary of the domain. We have used Green’s identity for a +tensor multiplied by a vector: +� +∇ · (v · P) dx = +� +v · ∇ · P dx + +� +∇v : P dx = +� +v · Pn ds , +(49) +where +A : B = +2 +� +i=1 +2 +� +j=1 +AijBij , +A, B ∈ R2×2 . +(50) +Integrating by parts moves derivatives from the Eddington tensor and VEF scalar flux +to the test function v allowing weaker requirements for both E and ϕ. In addition, we +assume J ∈ V has enough regularity to allow ∇ · J ∈ L2(D) (i.e. V ⊂ H(div; D)) so +that +� +u ∇ · J dx is well defined. Thus, we can unambiguously take u, ϕ ∈ E ⊂ L2(D). +However, the test function v now has increased regularity requirements. Namely, we +must have ∇v : E ∈ L2(D) instead of the typical requirement that ∇ · v = ∇v : +I ∈ L2(D). In the thick diffusion limit, E = +1 +3I and this requirement reduces to +∇v : E = 1 +3∇ · v ∈ L2(D). In this case, RT methods apply directly for both v and J. +However, for a general Eddington tensor, the RT space does not have the continuity +17 + +requirements to allow the term +� +∇v : Eϕ dx < ∞. This requirement is investigated +in the following proposition. +Proposition 2. For a tensor S ∈ [L∞(D)]2×2 satisfying ∇·S ∈ [L2(D)]2, let v : D → +R2 be such that +(a) v|K ∈ [H1(K)]2 for each K ∈ T +(b) �v · Sn� = 0 for each F ∈ Γ0 +then ∇v : S ∈ L2(D). Conversely, if ∇v : S ∈ L2(D) and (a) is satisfied, then (b) +holds. +Proof. Let C∞ +0 (D) denote the space of infinitely differentiable functions that are zero +on the boundary of the domain. Using Green’s identity, the following holds for each +u ∈ C∞ +0 (D): +� +∇v : S u dx = − +� +v · ∇ · (Su) dx = − +� +K∈T +� +K +v|K · ∇ · (Su) dx += +� +K∈T +�� +K +∇v|K : S u dx − +� +∂K +v · Sn u ds +� += +� +∇hv : S u dx − +� +Γ0 +�v · Sn� u ds += +� +∇hv : S u dx , +(51) +where we have used (b) to cancel the integration over Γ0. The above identifies ∇v : S +with ∇hv : S. Given S ∈ [L∞(D)]2×2 and (a), ∇hv : S ∈ L2(D) and thus we have that +∇v : S = ∇hv : S ∈ L2(D). +On the other hand, if ∇v : S ∈ L2(D), then ∇v : S = ∇hv : S and, given v|K ∈ +[H1(K)]2, we obtain +� +Γ0 +�v · Sn� u ds = 0 , +∀u ∈ C∞ +0 (D) , +(52) +hence, (b) holds. +Observe that Proposition 2 reduces to Proposition 1 when S = I. Due to the DG +interpolation used to approximate the angular flux, the Eddington tensor does not +satisfy ∇ · E ∈ L2(D) and thus Proposition 2 does not apply directly. However, we +can consider approximating the Eddington tensor by projecting it onto a space that +satisfies this requirement. In such case, Proposition 2 implies that +∇v : E = ∇hv : E ⇐⇒ �v · En� = 0 , +∀F ∈ Γ0 . +(53) +Figure 8 depicts an example of the Eddington tensor rotating and scaling the normal +vector, altering the continuity requirement of the space. Note that since the Edding- +ton tensor is symmetric positive definite, n · En > 0 and thus θ ∈ (−π/2, π/2). In +18 + +n +En +θ +K1 +K2 +Fig. 8.. A depiction of the rotation and scaling of the normal vector induced by the +Eddington tensor. Since the Eddington tensor is symmetric positive definite, the +angle θ cannot be larger than ±90◦. Due to the presence of the Eddington tensor in +the VEF first moment equation, continuity of the En component is required. +other words, the Eddington tensor cannot rotate the normal past a direction tangen- +tial to the face. This altered continuity requirement makes standard RT methods an +inappropriate choice for the test function v. +In light of Eq. 53, the weak form in Eq. 48 will hold only when the space V is chosen +so that both �J · n� = 0 and �v · En� = 0 on all interior faces. These conditions can +only be met by using v, J ∈ V ⊂ [H1(D)]2 so that all components of v and J are +continuous. A Petrov-Galerkin discretization where the test space satisfies �v · En� = 0 +and the trial space satisfies �J · n� = 0 may be possible. In this case, the test space +would need to use a more general Piola transform that preserves the En component of a +vector, making the test space dependent on the angular flux. Furthermore, Proposition +2 indicates this approach would require the use of an approximate projection of the +Eddington tensor that satisfies ∇ · E ∈ [L2(D)]2, which could degrade solution quality +on problems with steep solution gradients in parts of the domain. The Petrov-Galerkin +discretization is not considered here due to these complications. Alternatively, non- +conforming, DG-like techniques can be used to allow use of the RT space for both the +test and trial spaces. That is, both v, J ∈ V = RT p ⊂ H(div; D) and the discontinuity +in v · En is handled with numerical fluxes. +6.2. +[H1(D)]2 +Setting v, J ∈ V ⊂ [H1(D)]2 and u, ϕ ∈ E ⊂ L2(D) allows the weak form in Eq. 48 to +hold. The inf-sup condition [26] states that the discretization arising from the pairing +of equal degree interpolation for the scalar flux and current will be singular. That is, +the Yp × Wp discretization does not have a unique solution. The smallest non-singular +pairing of spaces is then Yp × Wp+1. In other words, if the scalar flux is piecewise- +constant, continuous linear finite elements for each component of the current must be +used. Background on the discrete inf-sup condition is provided in Appendix B.1 in the +context of the Poisson equation. +The discretization is complete by supplying boundary conditions. Solving the Miften- +Larsen boundary conditions (Eq. 2c) for ϕ yields +¯ϕ = 1 +Eb +(J · n − 2Jin) . +(54) +The [H1(D)]2 × L2(D) mixed finite element VEF discretization is then: find (ϕ, J) ∈ +19 + +Yp × Wp+1 such that +� +u ∇ · J dx + +� +σa uϕ dx = +� +u Q0 dx , +∀u ∈ Yp , +(55a) +− +� +∇v : Eϕ dx + +� +σt v · J dx + +� +Γb +1 +Eb +(v · En)(J · n) ds += +� +v · Q1 dx + 2 +� +Γb +1 +Eb +v · En Jin ds , +∀v ∈ Wp+1 . +(55b) +Equation 18 is used to compute the gradient and divergence of v , J ∈ Wp+1 in refer- +ence space. +Using V ⊂ [H1(D)]2 is simple to implement in that it relies only on the scalar contin- +uous finite element space and does not require interior face bilinear forms. However, +this choice has been seen to degrade both solution quality and solver performance due +to allowing non-physical, spurious modes. These so-called checkerboard modes are a +well-known issue with [H1(D)]2 × L2(D) discretizations in the context of fluid flow +[27] and are a consequence of the mismatch between the spaces ∇ · Wp+1 and Yp. The +space V ⊂ [H1(D)]2 is either too small with respect to Yp, leading to a singular system +in the case V = Wp or too large, allowing spurious modes for V = Wp+1. The effect of +these modes on solution quality and solver performance is investigated in §8.5 in the +context of radiation diffusion. Furthermore, Appendix B.2 investigates these modes +analytically on a lowest-order, single-element Poisson problem. +6.3. +Raviart Thomas +If v, J ∈ V ⊂ H(div; D), a non-conforming approach must be used for the first moment +equation due to the presence of the Eddington tensor. We proceed by locally integrating +the first moment equation by parts on each element so that the global gradient of +v ∈ H(div; D) is avoided. This requires the introduction of an auxiliary equation, +referred to as the numerical flux, that approximates the product of the Eddington +tensor and VEF scaluar flux on interior mesh interfaces. The local weak form for the +first moment corresponding to each element K is: +� +∂K +v· � +Eϕn ds− +� +K +∇v|K : Eϕ dx+ +� +K +σt v·J dx = +� +K +v·Q1 dx , +∀v ∈ Dp(K) , (56) +where � +Eϕ is the aformentioned numerical flux for the Eddington tensor and scalar +flux. Summing over all elements K ∈ T : +� +Γ +�v� · � +Eϕn ds − +� +∇hv : Eϕ dx + +� +σt v · J dx = +� +v · Q1 dx , +∀v ∈ RT p . (57) +We have used the fact that on a face F = K1 ∩ K2, n = n1 = −n2 and the definitions +of the jump and broken gradient in Eqs. 6 and 8, respectively. In addition, we assume +the use of a so-called conservative numerical flux such that � +Eϕn is single-valued on +20 + +interior faces. In other words, the numerical flux satisfies +� +� +Eϕn +� += 0 , +�� +� +Eϕn +�� += � +Eϕn , +on F ∈ Γ0 , +(58) +where the average is defined in Eq. 6. In this form, only the gradient restricted to each +element is required of the test function, v. Since v ∈ RT p satisfies v|K ∈ D(K) ⊂ +[H1(D)]2 on each K ∈ T , the broken gradient ∇hv ∈ [L2(D)]2×2 is well defined. +We now define define the numerical flux and boundary conditions. The resulting dis- +cretization will provide optimal accuracy if � +Eϕn is an optimal approximation of the +true value of Eϕn on interior faces. A conservative numerical flux that satisfies this +requirement is: +� +Eϕn = {{En}}{{ϕ}} , +on F ∈ Γ0 . +(59) +While many choices of the numerical flux are possible, we show below that this partic- +ular choice of numerical flux has the benefit of limiting to a standard RT discretization +of radiation diffusion in the thick diffusion limit. The Miften-Larsen boundary condi- +tions are applied with +� +Eϕn = En +Eb +(J · n − 2Jin) , +on F ∈ Γb . +(60) +This is derived by solving Eq. 2c for the scalar flux and multiplying by En. The +Yp × RT p discretization is then: find (ϕ, J) ∈ Yp × RT p such that +� +u ∇ · J dx + +� +σa uϕ dx = +� +u Q0 dx , +∀u ∈ Yp , +(61) +� +Γ0 +�v · {{En}}� {{ϕ}} ds − +� +∇hv : Eϕ dx + +� +σt v · J dx + +� +Γb +1 +Eb +(v · En)(J · n) ds += +� +v · Q1 dx + 2 +� +Γb +1 +Eb +v · En Jin ds , +∀v ∈ RT p . +(62) +Since RT vectors use the contravariant Piola transform, we substitute v = 1 +J Fˆv in all +terms involving v and use Eqs. 25 and 27 to evaluate ∇hv and ∇ · J, respectively. +In the thick diffusion limit, E = 1 +3I and +�v · {{En}}� = 1 +3 �v · n� = 0 , +(63) +since v ∈ RT p has a continuous normal component. Furthermore, ∇hv : E = 1 +3∇ · v. +This discretization with this choice of numerical flux is then equivalent to the standard +RT discretization of diffusion in the thick diffusion limit. +The RT space satisfies ∇ · RT p = Yp avoiding the spurious modes seen for the +[H1(D)]2 × L2(D) discretization. This allows superior solution quality and excellent +solver performance. However, the RT method is more complex due to the need for +21 + +interior face bilinear forms, the contravariant Piola transform, and the comparatively +less simple RT space. +6.4. +Solvers +The above discretizations admit the following block system +� +A +G +D +Ma +� �J +ϕ +� += +�g +f +� +, +(64) +where for u, ϕ ∈ E and v, J ∈ V: +vT AJ = +� +σt v · J dx + +� +Γb +1 +Eb +(v · En)(J · n) ds , +(65a) +uT Maϕ = +� +σa uϕ dx , +(65b) +uT DJ = +� +u ∇ · J dx , +(65c) +vT Gϕ = +� +− +� +∇v : Eϕ dx , +V = Wp+1 +� +Γ0 �v · {{En}}�{{ϕ}} ds − +� +∇hv : Eϕ dx , +V = RT p +, +(65d) +vT g = +� +v · Q1 dx + 2 +� +Γb +1 +Eb +v · En Jin ds , +(65e) +uT f = +� +u Q0 dx . +(65f) +Note that the integration transformations described in §3.2 are implicitly used and, in +particular, the contravariant Piola transform is implicitly used when V = RT p. +We use a lower block triangular preconditioner of the form +M = +�A +D +˜S +� +, +(66) +where ˜S is an approximation to the Schur complement S = Ma − DA−1G. Block +preconditioners seek to modify the system such that it has a minimal polynomial +with small degree [6]. Iterative solvers with an optimality condition, such as GMRES, +can then converge in a small number of iterations. However, computing the generally +dense Schur complement and exactly inverting it are impractical. Instead, we use +22 + +an approximate Schur complement formed from a sparse approximation to A−1 and +sparse matrix multiplication. That is, we use +˜S = Ma − D ˜A +−1G +(67) +where ˜A is the lumped mass matrix and boundary term. On elements with no bound- +ary faces (i.e. ∂K ∩ Γb = ∅), the lumping procedure is to sum the rows of the matrix +into the diagonal. This is computed on the element-local matrix as: +˜Ae +ij = +�� +k Ae +ik , +i = j +0 , +i ̸= j , +(68) +where Ae and ˜A +e are the unlumped and lumped matrices associated with the degrees +of freedom corresponding to element Ke, respectively. On elements with a boundary +face, the boundary integral over Γb contributes. Due to the Eddington tensor, v · En +couples degrees of freedom corresponding to the normal and tangential components +of v. We leverage the block structure of the local matrices to lump the boundary +elements. Let +Ae = +� +Ae +11 +Ae +12 +Ae +21 +Ae +22 +� +(69) +where Ae +ij is the sub-block corresponding to the degrees of freedom of the ith and jth +components of the test and trial functions, respectively. We then lump each of these +sub-blocks separately so that: +˜A +e = +� ˜A +e +11 +˜A +e +12 +˜A +e +21 +˜A +e +22 +� +. +(70) +The lumped local matrix ˜A +e is diagonal by vector component. That is, each row has at +most two entries corresponding to the two components of a vector in R2. This lumping +procedure allows approximation of the boundary terms and has an inverse that can +be computed efficiently without fill-in. +For both interior and boundary elements, the local matrices ˜A +e are assembled into the +global matrix ˜A. For rows corresponding to interior degrees of freedom, the lumped +matrix is diagonal and thus the inverse is simply 1/ ˜Aii. For rows corresponding to +boundary degrees of freedom, ˜A is a diagonal matrix for each vector component. The +inverse is computed by gathering the entries corresponding to each vector component +into a 2 × 2 matrix, inverting it, and scattering the inverse back to a sparse matrix +representing ˜A +−1. The above lumping procedure results in a sparse ˜A +−1 that ap- +proximates the true inverse A−1. Finally, the lumped Schur complement is formed +with sparse matrix multiplication according to Eq. 67. Note that computing the Schur +complement is numerically analogous to eliminating the current in the analytic equa- +tions to form a second-order, elliptic partial differential equation. Thus, one Algebraic +Multigrid (AMG) V-cycle is expected to be a spectrally equivalent approximation to +˜S +−1. +The approximate inverse of the block preconditioner in Eq. 66 is applied with forward +23 + +substitution. In other words, we solve +�A +D +˜S +� � +x1 +x2 +� += +� +r1 +r2 +� +(71) +by approximately solving the block problems: +Ax1 = r1 , +(72a) +˜Sx2 = r2 − Dx1 . +(72b) +We stress that the sub-problems in Eqs. 72 do not need to be solved exactly. In +fact, one iteration of Jacobi smoothing and one AMG V-cycle applied to A and ˜S, +respectively, often lead to a scalable preconditioner. More accurately solving the sub- +problems (e.g. using more than one Jacobi/AMG iteration or nested iteration) gen- +erally improves robustness to problem size but typically not to the extent that fewer +Jacobi iterations and AMG V-cycles are performed. This behavior is investigated in +§8.6 where we compare the scaling of solving the RT VEF system using one and three +AMG V-cycles per preconditioner application as well as the effect of using one itera- +tion of Gauss-Seidel, a more expensive and thus more robust smoother, to approximate +A−1. +7. +Hybridization +A hybridized version of the RT mixed method is obtained by relaxing the continuity +requirements of the space RT p and reimposing them weakly. Removing the continuity +requirement from RT p yields the broken space +ˆ +RT p = {v ∈ [L2(D)]2 : v|K ∈ Dk(K) , +∀K ∈ T } . +(73) +This space is equivalent to RT p on each element but +ˆ +RT p does not have the matching +conditions that strongly enforce continuity in the normal component. Note that RT p ⊂ +ˆ +RT p and that v ∈ ˆ +RT p belongs to RT p if and only if �v · n� = 0 on all interior mesh +interfaces. In other words, the mixed problem can be reformulated to use the space +ˆ +RT p instead of RT p by adding the constraint that �J · n� = 0 for each F ∈ Γ0. The +methods presented in this section enforce this constraint with a Lagrange multiplier. +Hybridized methods are attractive for three reasons. First, since J ∈ ˆ +RT p and ϕ ∈ Yp +are both discontinuous, their degrees of freedom are coupled only locally on each +element. It is then possible to locally eliminate the scalar flux and current arriving at +a system of equations for just the Lagrange multiplier. This reduced system is much +smaller than the original 2 × 2 system. Second, the reduced system for the Lagrange +multiplier will be positive definite and AMG can be applied directly, avoiding the +need for block preconditioners. Finally, the Lagrange multiplier provides an additional +approximation for the scalar flux not provided by the original mixed problem. +Since the VEF equations are not symmetric, the variational principles typically used to +derive hybridized mixed finite element methods are not appropriate. We first show the +24 + +derivation of a hybridized method for the symmetric case of radiation diffusion using +variational principles. This method is extended to the VEF equations by emulating +the properties of the symmetric case. Finally, we discuss the details of an efficient +implementation. +7.1. +Derivation for Radiation Diffusion +In this section, we provide background on the dual, mixed, and hybrid variational forms +associated with the symmetric radiation diffusion system with Dirichlet boundary +conditions: +∇ · J + σaϕ = Q0 , +x ∈ D , +(74a) +∇ϕ + 3σtJ = 0 , +x ∈ D , +(74b) +ϕ = 0 , +x ∈ ∂D , +(74c) +where the source has been assumed to be isotropic. This coupled system can be +viewed as the first moment equation (Eq. 74b) with the constraint of particle bal- +ance (Eq. 74a). The RT mixed finite element discretization of radiation diffusion is: +find (ϕ, J) ∈ Yp × RT p such that +� +3σt v · J dx − +� +∇ · v ϕ dx = 0 , +∀v ∈ RT p , +(75a) +� +u ∇ · J dx + +� +σa uϕ dx = +� +u Q0 dx , +∀u ∈ Yp . +(75b) +Our goal is to identify the variational problem associated with this mixed finite element +discretization, modify it to support the use of the broken RT space, +ˆ +RT p, and derive +the hybridized system of equations that solves this modified variational problem. We +follow Quarteroni and Valli [25, Chapter 7] in the presentation of these topics. +The dual formulation is to minimize the so-called complementary energy functional: +I(J) = 1 +2 +� +3σt J · J dx +(76) +under the constraint of particle balance. Quarteroni and Valli [25, Theorem 7.1.1] +shows that this constrained minimization problem and Eqs. 74 are equivalent for- +mulations. Mixed methods enforce the particle balance constraint using a Lagrange +multiplier, ϕ. Let the Lagrangian functional be L : RT p × Yp → R such that +L(J, ϕ) = I(J) − +�� +ϕ ∇ · J dx + 1 +2 +� +σa ϕ2 dx − +� +ϕ Q dx +� +. +(77) +25 + +By minimizing over J and maximizing over ϕ, we can minimize I(J) while enforcing +particle balance. To see this, observe that, for J fixed, +− +�� +ϕ ∇ · J dx + 1 +2 +� +σa ϕ2 dx − +� +ϕ Q dx +� +(78) +is a concave quadratic functional with respect to ϕ and is maximized when particle +balance occurs such that ∇ · J + σaϕ = Q. The resulting mixed variational problem is +then written: +inf +J∈RT p sup +ϕ∈Yp +L(J, ϕ) . +(79) +Such a problem finds the “saddle point” that balances the convex functional I(J) with +the concave particle balance constraint. Since I(J) and the particle balance constraint +are quadratic functionals, the saddle point occurs at ∇L = 0: +∂L +∂J = +� +3σt v · J dx − +� +∇ · v ϕ dx = 0 , +∀v ∈ RT p , +(80a) +∂L +∂ϕ = − +� +u ∇ · J dx − +� +σa uϕ dx + +� +u Q0 dx = 0 , +∀u ∈ Yp . +(80b) +Observe that this system of equations for the saddle point exactly matches the mixed +discretization in Eq. 75. Thus, the solution of the mixed discretization is also the +saddle point of L. +We now wish to modify L to define a variational problem with an equivalent solution +that allows use of the broken RT space +ˆ +RT p. This is accomplished by searching for +J ∈ +ˆ +RT p and adding an additional constraint that the normal component of the +current is continuous such that �J · n� = 0. Let ˆL : ˆ +RT p × Yp → R such that +ˆL(J, ϕ) = I(J) − +�� +ϕ ∇h · J dx + 1 +2 +� +σa ϕ2 dx − +� +ϕ Q0 dx +� +, +(81) +be the broken Lagrangian functional. Since +ˆ +RT p and Yp are piecewise discontinuous, +ˆL is equivalent to L on each element due to the use of the broken divergence. The +constrained variational problem is then: +inf +J∈ ˆ +RT p +sup +ϕ∈Yp +ˆL(J, ϕ) , +such that �J · n� = 0 . +(82) +As with particle balance, the normal continuity constraint is imposed with a Lagrange +multiplier. Defining H : ˆ +RT p × Yp × Λp as +H(J, ϕ, λ) = ˆL(J, ϕ, λ) + +� +Γ0 +λ �J · n� ds , +(83) +26 + +the constrained saddle point problem is equivalent to: +inf +J∈ ˆ +RT p +sup +ϕ∈Yp +sup +λ∈Λp +H(J, ϕ, λ) . +(84) +If (J, ϕ, λ) is a solution of Eq. 84, then �J · n� = 0, otherwise setting λ = λτ = τ �J · n� +for some τ > 0 would give +� +Γ0 +λτ �J · n� ds = τ +� +Γ0 +�J · n�2 ds +(85) +and thus +lim +τ→∞ H(J, ϕ, λτ) = ∞ . +(86) +In other words, the supremum over λ drives the solution towards currents with con- +tinuous normal components. The solution of the hybridized variational form is found +by setting ∇H = 0: +∂H +∂J = +� +3σt v · J dx − +� +∇h · v ϕ dx + +� +Γ0 +�v · n� λ ds = 0 , +∀v ∈ ˆ +RT p , +(87a) +∂H +∂ϕ = − +� +u ∇h · J dx − +� +σa uϕ dx + +� +u Q0 dx = 0 , +∀u ∈ Yp , +(87b) +∂H +∂λ = +� +Γ0 +µ �J · n� ds = 0 , +∀µ ∈ Λp . +(87c) +Since ˆ +RT p and Yp are discontinuous spaces, the hybridized mixed method is equivalent +to: +� +K +3σt v · J dx − +� +K +∇ · v|K ϕ dx + +� +∂K∩Γ0 +v · nKλ ds = 0 , +∀v ∈ Dp(K) , K ∈ T , +(88a) +� +K +u ∇ · J|K dx + +� +K +σa uϕ dx = +� +K +u Q0 dx , +∀u ∈ Qp(K) , K ∈ T , +(88b) +� +Γ0 +µ �J · n� ds = 0 , +∀µ ∈ Λp . +(88c) +Here, it can be seen that the degrees of freedom for the scalar flux and current are +no longer globally coupled. In fact, if λ were known, the scalar flux and current could +be recovered by solving element-local radiation diffusion problems where λ plays the +role of a weak boundary condition applied on each element. Note that the non-zero +boundary condition ϕ = ¯ϕ for x ∈ Γb can be applied by subtracting +� +Γb v · n ¯ϕ ds from +27 + +the right hand side of Eq. 87a or equivalently by subtracting +� +∂K∩Γb v · nK ¯ϕ from the +right hand side of Eq. 88a. +In hybridization, continuity of the normal component is enforced weakly (e.g. see +Eq. 88c). However, it is well known that the resulting discrete solution will actually +satisfy continuity of the normal component in a strong sense (i.e. independent of the +discretization parameters h and p). In fact, hybridization has also been viewed as +an algebraic technique similar to static condensation in [5]. This behavior is owed to +the fact that the dual, mixed, and hybrid formulations all correspond to equivalent +variational problems. +7.2. +Extension to VEF +The above variational process cannot be applied directly to the VEF equations due +to their lack of symmetry. Without symmetry, it is unclear which potential the weak +VEF equations correspond to or whether it would have a unique, global saddle point +found by setting its gradient to zero. However, we can define a hybrid method for +the VEF equations by mimicking the properties seen above for the symmetric case. In +particular, we use the broken RT space, ˆ +RT p, and a Lagrange multiplier that 1) weakly +enforces continuity of the normal component of the current and 2) provides inter- +element boundary conditions for element-local VEF problems. As in the symmetric +case, this will allow elimination of the scalar flux and current, leading to a smaller +system for just the Lagrange multiplier where AMG can be applied directly. However, +since the resulting method cannot be derived from a variational principle it is unclear +whether the resulting hybrid formulation will be equivalent to the original mixed +formulation. +The hybridized diffusion method can be extended to the VEF equations with Miften- +Larsen boundary conditions by replacing the diffusion first moment with the VEF first +moment equation and using the boundary condition ¯ϕ = +1 +Eb (J · n − 2Jin). This can +be accomplished by using the element-local weak form of the first moment equation +in Eq. 56 and setting +� +Eϕn = {{En}} λ , +on F ∈ Γ0 . +(89) +Here, we are using the Lagrange multiplier λ ∈ Λp as a single-valued approximation +for the scalar flux on interior faces. The numerical flux on the boundary is the same +as in Eq. 60. For each K, the element-local VEF problem is then: +� +∂K∩Γ0 +v·{{EnK}} λ ds− +� +K +∇v|K : Eϕ dx+ +� +K +σt v·J dx+ +� +∂K∩Γb +1 +Eb +(v · EnK)(J · nK) ds += +� +K +v · Q1 dx + 2 +� +∂K∩Γb +1 +Eb +v · EnK Jin ds , +∀v ∈ Dp(K) , +(90a) +� +K +u ∇ · J|K dx + +� +K +σa uϕ dx = +� +K +u Q0 dx , +∀u ∈ Qp(K) . +(90b) +28 + +The resulting hybrid VEF method is: find (J, ϕ, λ) ∈ ˆ +RT p × Yp × Λp such that +� +Γ0 +�v · {{En}}� λ ds − +� +∇hv : Eϕ dx + +� +σt v · J dx + +� +Γb +1 +Eb +(v · En)(J · n) ds += +� +v · Q1 dx + 2 +� +Γb +1 +Eb +v · En Jin ds , +∀v ∈ ˆ +RT p , +(91a) +� +K +u ∇h · J dx + +� +K +σa uϕ dx = +� +K +u Q0 dx , +∀u ∈ Yp , +(91b) +� +Γ0 +µ �J · n� ds = 0 , +∀µ ∈ Λp . +(91c) +Observe that this represents element-local VEF problems where the boundary condi- +tions are provided either by the Miften-Larsen boundary conditions on the boundary +of the domain or by the Lagrange multiplier λ for interior elements. Thus, if λ were +known, the scalar flux and current could be solved for independently on each element. +7.3. +Implementation Details +In matrix form, the hybridized system is +� +� +ˆA +ˆG +C2 +ˆD +Ma +C1 +� +� +� +� +J +ϕ +λ +� +� = +� +� +ˆg +f +0 +� +� , +(92) +where ˆA, ˆD, and ˆg are defined in Eqs. 65a, 65c, and 65e, respectively, but use V = ˆ +RT p +and +ˆG = − +� +∇hv : Eϕ dx +(93) +is the analog of G in Eq. 65d that uses V = +ˆ +RT p and does not include the interior +face bilinear form. The DG absorption mass matrix, Ma, and right hand side, f, are +unchanged from the original mixed form defined in Eqs. 65b and 65f, respectively. The +constraint matrices are defined as: +µT C1J = +� +Γ0 +µ �J · n� ds , +(94a) +vT C2λ = +� +Γ0 +�v · {{En}}� λ ds , +(94b) +where µ, λ ∈ Λp and v, J ∈ ˆ +RT p. +29 + +(a) +(b) +Fig. 9.. Sparsity plots for the block system corresponding to the hybridized Raviart +Thomas discretization for the VEF equations. In (a), the degrees of freedom are +organized as J1, J2, ϕ, λ. In (b), the rows and columns of the matrix in (a) are +permuted to group the currents and scalar fluxes associated with each element +together. With this ordering, it is clear that the scalar flux and current can be +eliminated on each element without fill-in, leaving a system for λ only. Note that in +practice, the elimination of the element-local problems is performed locally with +dense operations and global sparse matrices are used to form the reduced system for +the Lagrange multiplier. +Only the constraint matrices C1 and C2 are globally coupled. The matrices ˆA, ˆG, ˆD, +and Ma are all block diagonal by element and can thus be eliminated on each element +without fill-in. Figure 9a shows the sparsity pattern of the block system in Eq. 92. +Note that this matrix can be permuted to be block diagonal by element by grouping +the current and scalar flux degrees of freedom associated with each element together. +This matrix is shown in Fig. 9b where it is clear that the block system has a structure +amenable to efficient solution via block Gaussian elimination. +Performing block Gaussian elimination on each element, the reduced system for the +Lagrange multiplier reads +Hλ = +�C1 +0� � ˆA +ˆG +ˆD +Ma +�−1 � +C2 +0 +� +λ = +�C1 +0� � ˆA +ˆG +ˆD +Ma +�−1 �ˆg +f +� +. +(95) +The inverse of the local VEF problems is derived by finding the blocks W, X, Y, and +Z that satisfy +� ˆA +ˆG +ˆD +Ma +� � +W +X +Y +Z +� += +� +I +I +� +. +(96) +We assume that ˆA and the Schur complement ˆS = Ma − ˆD ˆA +−1 ˆG are non-singular. +This is justified in non-void regions where σt > 0. However, we do not assume Ma is +non-singular since σa ≥ 0 can be zero. Solving Eq. 96 for the blocks W, X, Y, and Z +30 + +L +:8 +1 +1!II +I +- +II +I +II +11 +- +11 +I +1 +I +- +1 +Iunder these constraints yields: +W = ˆA +−1(I + ˆGˆS +−1 ˆD ˆA +−1) , +(97a) +X = − ˆA +−1 ˆGˆS +−1 , +(97b) +Y = −ˆS +−1 ˆD ˆA +−1 , +(97c) +Z = ˆS +−1 . +(97d) +The reduced system for the Lagrange multiplier is then +Hλ = C1WC2λ = C1 ˆA +−1� +I + ˆGˆS +−1 ˆD ˆA +−1� +C2λ = C1 +� +Wˆg + Xf +� +. +(98) +We can now rewrite the 3 × 3 block system as +� +� +ˆA +ˆG +C2 +ˆD +Ma +H +� +� +� +� +J +ϕ +λ +� +� = +� +� +ˆg +f +C1 +� +Wˆg + Xf +� +� +� . +(99) +This system can be solved with block back substitution. First, solve the globally cou- +pled system +Hλ = C1 +� +Wˆg + Xf +� +(100) +for λ. The element-local inverse can then be used to solve for the scalar flux and +current with +�J +ϕ +� += +� +W +X +Y +Z +� �ˆg − C2λ +f +� += +�W +� +ˆg − C2λ +� ++ Xf +Y +� +ˆg − C2λ +� ++ Zf +� +. +(101) +In this way, only dim(Λp) globally coupled unknowns must be solved for as opposed +to the dim(RT p) + dim(Yp) required by the original mixed formulation. +In practice, the blocks of the inverse W, X, Y, and Z are formed using dense ma- +trix operations applied on the element-local matrices corresponding to the degrees of +freedom of a single element. The local matrices are then broadcast to a global sparse +matrix in order to perform the sparse matrix multiplication required to form the re- +duced system for the Lagrange multiplier. The Lagrange multiplier can be scalably +solved for by preconditioning H with AMG. In addition, recovering the scalar flux +and current is a post-processing step that is independent on each element and thus +scales optimally. +31 + +8. +Results +The VEF algorithms presented here were implemented using the MFEM [28, 29] fi- +nite element framework. The stabilized bi-conjugate gradient (BiCGStab) solver from +MFEM was used to solve the discretized VEF equations. Lower block triangular pre- +conditioners were built using MFEM’s Jacobi smoother and BoomerAMG from the +sparse linear algebra package hypre [30]. KINSOL, from the Sundials package [31], +provided the fixed-point and Anderson-accelerated fixed-point solvers. As described in +Hindmarsh et al. [31, §2], the fixed-point and Anderson-accelerated fixed-point itera- +tion is terminated when the max norm of the difference between successive iterates is +below the iterative tolerance. The parallel implementation of the sparse direct solver +SuperLU [32] is used when preconditioned iterative solvers results are not presented. +The streaming and collision operator is inverted using the transport solver from [19]. +Unless otherwise noted, the angular flux and VEF scalar flux are approximated using +the same degree finite element spaces. However, the positive Bernstein polynomials [33] +are used for the transport discretization’s local polynomial basis whereas the Lagrange +basis through the Gauss-Legendre points is used for the VEF scalar flux. The positive +transport basis facilitates the application of the quadratic programming negative flux +fixup from [34] that is used on the crooked pipe problem in §8.4. +Since all methods produce a VEF scalar flux in Yp, the methods are parameterized by +their choice of space for the current. Thus, we refer to the Yp × Wp+1, Yp × RT p, and +Yp × ˆ +RT p × Λp methods as H1, RT, and HRT, respectively. +8.1. +Method of Manufactured Solutions +The accuracy of the methods are determined with the Method of Manufactured Solu- +tions (MMS). The solution is set to +ψ = 1 +4π[α(x) + Ω · β(x) + Ω ⊗ Ω : Θ(x)] , +(102) +where +α(x) = sin(πx) sin(πy) + δ , +(103a) +β(x) = +� +�sin +� +2π(x+ω) +1+2ω +� +sin +� +2π(y+ω) +1+2ω +� +sin +� +2π(x+ω) +1+2ω +� +sin +� +2π(y+ω) +1+2ω +� +� +� , +(103b) +Θ(x) = +� +� +1 +2 sin +� +3π(x+ζ) +1+2ζ +� +sin +� +3π(y+ζ) +1+2ζ +� +sin +� +2π(x+ω) +1+2ω +� +sin +� +2π(y+ω) +1+2ω +� +sin +� +2π(x+ω) +1+2ω +� +sin +� +2π(y+ω) +1+2ω +� +1 +4 sin +� +3π(x+ζ) +1+2ζ +� +sin +� +3π(y+ζ) +1+2ζ +� +� +� . +(103c) +Here, δ = 1.25 is used to ensure ψ > 0 and ζ = 0.1 and ω = 0.05 are used to +test spatially dependent, non-isotropic inflow boundary conditions. The domain is +32 + +D = [0, 1]2. With this definition: +φ(x) = α(x) + 1 +3 trace Θ(x) , +(104a) +J(x) = 1 +3β(x) , +(104b) +P(x) = α(x) +3 +I + 1 +15 +� +3Θ11(x) + Θ22(x) +Θ12(x) +Θ21(x) +Θ11(x) + 3Θ22(x) +� +. +(104c) +This leads to an exact Eddington tensor E = P/φ that is dense and spatially varying. +The MMS ψ and φ are substituted into the transport equation to solve for the MMS +source q that forces the solution to Eq. 102. +The accuracy of the VEF discretizations are investigated in isolation by computing +the VEF data from the MMS angular flux and setting the sources Q0 and Q1 to the +moments of the MMS source. This is accomplished by projecting the MMS angular +flux onto a degree-p DG finite element space and using Level Symmetric S4 angular +quadrature to compute the VEF data, the moments of the MMS source, and the inflow +boundary function. The VEF equations are then solved as if E, Eb, Q0, Q1, and Jin +are given data. Errors are calculated with the L2(D) norm for scalars and the [L2(D)]2 +norm for vectors given by +∥u∥ = +�� +u2 dx , +(105) +and +∥v∥ = +�� +v · v dx , +(106) +respectively. We also use the L2(D) projection operator Πp : L2(D) → Yp such that +� +u(v − Πpv) dx = 0 , +∀u ∈ Yp , +(107) +for some v ∈ L2(D). In particular, Πp is used to project the exact MMS scalar flux onto +a Yp finite element grid function in order to investigate a superconvergence property +of mixed finite elements. +We use refinements of a third-order mesh created by distorting an orthogonal mesh +according to the velocity field of the Taylor Green vortex. This mesh distortion is +generated by advecting the mesh control points with +x = +� T +0 +v dt , +(108) +33 + +Fig. 10.. A depiction of a third-order mesh generated by distorting an orthogonal +mesh according to the Taylor Green vortex. Refinements of this mesh are used in +calculating the error with the method of manufactured solutions. +where the final time T = 0.3π and +v = +� +sin(x) cos(y) +− cos(x) sin(y) +� +(109) +is the analytic solution of the Taylor Green vortex. 300 forward Euler steps were used +to advect the mesh. An example mesh is shown in Fig. 10. Logarithmic regression is +used to fit the constant and order of accuracy according to +E = Ch˜p +(110) +where E is the error, C the constant, and ˜p the order of accuracy. Four values of h +were used for each MMS problem considered in this section. The raw error values for +each of the MMS problems presented in this section are provided in Appendix C. +We first show the accuracy of the three methods on a simple radiation diffusion prob- +lem. The above process is used with Θ = 0 so that the angular flux is linearly +anisotropic. This forces the Eddington tensor and boundary factor to E = 1 +3I and +Eb = 1 +2, mimicking a radiation diffusion problem. Table 1 shows the estimated order +of accuracy and constant for p ∈ [1, 4]. The error in the scalar flux is computed with +two methods: 1) by comparing to the analytic MMS scalar flux solution directly and +2) by projecting the analytic MMS solution onto the corresponding Yp space. For all +orders, the first error measure for the scalar flux converges O(hp+1) while the second +converges O(hp+2). This is a mixed finite element superconvergence result that indi- +cates that the nodal values of the scalar flux solution converge one order higher than +the Yp interpolation allows. The current converges as O(hp+1) for all three methods +and all orders. On this diffusive problem, the scalar flux and current solutions from +the unhybridized and hybridized RT methods are equivalent to machine precision. +This test is repeated for the quadratically anisotropic MMS problem (i.e. using Θ(x) +as defined in Eq. 103c) in Table 2. Projecting the MMS angular flux solution onto +Yp induces errors of order O(hp+1) in the calculation of the VEF data. Thus, since +the VEF data are computed from the projected MMS solution, it is expected that +this problem can converge at a maximum of order p + 1. This can be seen in the +loss of the superconvergence property. Here, both error measures for the scalar flux +34 + +TABLE 1. +Estimates of the order of accuracy and constant from an isotropic MMS test +problem. The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and +hybridized Yp × RT p discretizations, respectively. The error in the scalar flux, the +error in the scalar flux when the exact solution is first projected onto Yp, and the +error in the current are presented for each method over a range of values of p. Here, +the VEF data are constant in space and thus are represented exactly. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +Value +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +1 +Order +2.000 +2.000 +2.000 +3.017 +3.053 +3.053 +2.001 +2.000 +2.000 +Constant +0.261 +0.261 +0.261 +0.163 +0.197 +0.197 +0.353 +0.785 +0.785 +2 +Order +3.001 +3.003 +3.003 +4.144 +4.096 +4.096 +3.150 +2.989 +2.989 +Constant +0.070 +0.070 +0.070 +0.090 +0.142 +0.142 +0.202 +0.780 +0.780 +3 +Order +3.995 +4.016 +4.016 +5.098 +5.125 +5.125 +4.018 +4.016 +4.016 +Constant +0.027 +0.030 +0.030 +0.048 +0.132 +0.132 +0.118 +0.928 +0.928 +4 +Order +4.971 +4.971 +4.971 +6.013 +5.964 +5.963 +5.096 +4.675 +4.675 +Constant +0.034 +0.034 +0.034 +0.048 +0.045 +0.045 +0.157 +0.217 +0.217 +TABLE 2. +Estimates of the order of accuracy and constant from a quadratically anisotropic +MMS test problem. The H1, RT, and HRT columns refer to the Yp × Wp+1, +Yp × RT p, and hybridized Yp × RT p discretizations, respectively. The error in the +scalar flux, the error in the scalar flux when the exact solution is first projected onto +Yp, and the error in the current are presented for each method over a range of values +of p. Here, the angular flux used to calculate the VEF data is represented with Yp. +Due to this, the maximum accuracy expected is order p + 1. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +Value +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +1 +Order +2.004 +2.004 +2.004 +2.310 +2.332 +2.318 +1.939 +0.974 +0.980 +Constant +1.200 +1.200 +1.198 +0.430 +0.488 +0.451 +3.831 +0.394 +0.353 +2 +Order +2.958 +2.957 +2.963 +2.995 +2.979 +3.054 +2.486 +2.564 +2.522 +Constant +1.233 +1.225 +1.263 +0.352 +0.329 +0.485 +1.601 +1.605 +1.477 +3 +Order +4.046 +4.045 +4.044 +4.348 +4.313 +4.263 +4.003 +2.857 +2.905 +Constant +2.612 +2.599 +2.592 +0.942 +0.837 +0.710 +12.054 +0.555 +0.584 +4 +Order +4.787 +4.785 +4.783 +5.033 +4.921 +4.845 +4.221 +4.351 +4.454 +Constant +0.931 +0.923 +0.923 +0.421 +0.283 +0.258 +1.011 +1.458 +2.050 +35 + +TABLE 3. +Estimates of the order of accuracy and constant from a quadratically anisotropic +MMS test problem. The H1, RT, and HRT columns refer to the Yp × Wp+1, +Yp × RT p, and hybridized Yp × RT p discretizations, respectively. The error in the +scalar flux, the error in the scalar flux when the exact solution is first projected onto +Yp, and the error in the current are presented for each method over a range of values +of p. Here, the angular flux used to calculate the VEF data is represented with Yp+1. +Due to this, the maximum accuracy expected is order p + 2. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +Value +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +0 +Order +0.999 +0.999 +0.999 +2.019 +2.002 +2.001 +1.477 +1.001 +1.001 +Constant +0.781 +0.780 +0.780 +1.439 +1.338 +1.304 +2.561 +0.517 +0.516 +1 +Order +2.001 +2.001 +2.001 +3.012 +2.954 +2.969 +1.941 +0.987 +1.887 +Constant +1.180 +1.179 +1.178 +1.683 +1.488 +1.390 +2.377 +0.083 +0.583 +2 +Order +2.961 +2.960 +2.960 +3.990 +4.028 +4.006 +3.065 +2.967 +2.903 +Constant +1.208 +1.204 +1.204 +2.383 +2.447 +2.347 +3.312 +1.273 +0.783 +3 +Order +4.042 +4.041 +4.041 +4.965 +4.732 +4.759 +3.931 +2.726 +3.667 +Constant +2.554 +2.545 +2.545 +1.883 +0.896 +0.864 +6.673 +0.102 +0.575 +converge with O(hp+1). On this transport MMS problem, the current convergence is +also reduced. Compared to the diffusion case, the H1 current error is maintained for +p odd but is reduced by 1/2 for p even. The RT and HRT methods lose one order for +p odd but only half an order for p even. In addition, the RT and HRT discretizations +are no longer equivalent to machine precision. This loss of equivalence may be due to +inexact numerical quadrature in terms involving the VEF data – the VEF data are +improper rational polynomials in space and thus cannot be exactly integrated with +numerical quadrature – or may indicate that the hybrid and mixed formulations are +only equivalent for the symmetric case of radiation diffusion. +Finally, we repeat the transport MMS problem in the case where the angular flux +solution is projected onto Yp+1 instead of Yp. This allows a maximum accuracy in the +problem of O(hp+2). The estimated orders of convergence and constants are provided +in Table 3. Convergence rates similar to the diffusion problem are observed: the scalar +flux solutions converge optimally for all methods and superconvergence of the scalar +flux returns. The H1 and HRT methods produce currents that converge at similar rates +as in the diffusion case. However, the unhybridized RT method converges suboptimally +by one order for p even. The difference in convergence rates between the RT and +HRT methods indicates the HRT method is in fact a new discretization for the VEF +equations and not simply an algebraic method to reduce the number of globally coupled +unknowns. +The error behavior of the current on the above three MMS problems is summarized +in Table 4. We stress that the H1, RT, and HRT methods all generated scalar flux +solutions with the optimal error behavior on each of the above MMS problems. The +methods differed only in the error associated with the current. +36 + +TABLE 4. +A summary of the order of accuracies of the current on the three MMS problems for +the H1, RT, and HRT methods grouped into even and odd polynomial degrees. All +methods converged the scalar flux with optimal O(hp+1) accuracy on all problems. +Even p +Odd p +Problem +H1 +RT +HRT +H1 +RT +HRT +Radiation Diffusion +p + 1 +p + 1 +p + 1 +p + 1 +p + 1 +p + 1 +Transport w/ ψMMS ∈ Yp +p + 1/2 +p + 1/2 +p + 1/2 +p + 1 +p +p +Transport w/ ψMMS ∈ Yp+1 +p + 1∗ +p + 1 +p + 1 +p + 1 +p +p + 1 +∗converged O(h3/2) for p = 0 +8.2. +Thick Diffusion Limit +The convergence of the VEF methods are investigated in the thick diffusion limit. The +material data are set to +σt = 1/ϵ , +σa = ϵ , +σs = 1/ϵ − ϵ , +q = ϵ , +(111) +where ϵ ∈ (0, 1] and the thick diffusion limit corresponds to the limit ϵ → 0. We use two +coarse meshes that do not resolve the mean free path to stress the convergence of the +VEF method. The first is an orthogonal 8×8 mesh with D = [0, 1]2. The second is the +triple point mesh shown in Fig. 11, a third-order mesh generated with a Lagrangian +hydrodynamics code where D = [0, 7] × [0, 3]. On the triple point mesh, the streaming +and collision operator cannot be reordered to be lower block triangular by element due +to the presence of reentrant/concave faces. A standard transport sweep can be applied +by iteratively lagging the strictly upper block triangular components of the streaming +and collision operator. The pseudo-optimal element reordering proposed in Haut et al. +[19] is used to minimize the amount of information lagged due to reentrant faces. Since +the angular flux is only approximately inverted at each iteration it is expected that +iterative efficiency will degrade compared to an analogous problem on a straight-edged +mesh. In addition, highly distorted elements have poor approximation properties. We +use Level Symmetric S4 angular quadrature. The three methods are compared when +p = 2. The coupled transport-VEF system is solved with fixed-point iteration. +Table 5 shows the number of fixed-point iterations until convergence to a tolerance of +10−6 for each method on the orthogonal and triple point meshes. Rapid convergence is +seen for all methods on both problems. The three methods converged equivalently on +the orthogonal mesh. On the triple point mesh, all of the methods converged slower +compared to the corresponding problem solved on an orthogonal mesh. The RT and +HRT methods converged in an equivalent number of iterations with H1 converging +a few iterations faster than RT/HRT. Lineouts of the 2D VEF scalar flux solutions +for each method as ϵ → 0 are provided in Figs. 12 and 13 for the orthogonal and +triple point meshes, respectively. In all cases, the non-trivial diffusion limit solution +was found. On the triple point problem, non-physical, non-monotone oscillations are +observed due to the imprinting of the mesh on the solution. The oscillations are larger +in magnitude for RT and HRT methods compared to H1. This may suggest that the +37 + +Fig. 11.. A depiction of the triple point mesh used to stress the VEF algorithms on a +severely distorted, third-order mesh. This mesh was generated with a Lagrangian +hydrodynamics simulation. +TABLE 5. +The number of fixed-point iterations required for convergence as the thick diffusion +limit parameter ϵ → 0. The H1, RT, and HRT columns refer to the Y2 × W3, +Y2 × RT 2, and hybridized Y2 × RT 2 discretizations, respectively. Convergence is +tested on an orthogonal 8 × 8 mesh and on the triple point mesh, a mesh with +re-entrant faces. Due to the re-entrant faces, an inexact transport sweep is used +making convergence slower on the triple point mesh. +Orthogonal +Triple Point +ϵ +H1 +RT +HRT +H1 +RT +HRT +10−1 +8 +8 +8 +20 +21 +21 +10−2 +6 +6 +6 +13 +19 +19 +10−3 +4 +4 +4 +9 +13 +13 +10−4 +3 +3 +3 +6 +8 +8 +quality of the RT and HRT solutions are more sensitive to mesh distortion than H1. +8.3. +Solver Performance on Curved Meshes +Here, we investigate the robustness of the preconditioned iterative solvers for the VEF +linear systems on increasingly distorted meshes. The meshes were created by moving +the interior control points of an initially orthogonal, third-order mesh according to the +sine distortion: +x → x + α +� +sin(2πx) sin(2πy) +sin(2πx) sin(2πy) +� +, +(112) +where α controls the amount of distortion. When α = 0, the mesh is unchanged. The +initial mesh was 16 × 16 with D = [0, 1]2. Meshes corresponding to a range of values +of α are shown in Fig. 14. Solver performance is evaluated on the first iteration of the +thick diffusion limit problem introduced in the previous section. We use ϵ = 10−1. The +number of BiCGStab iterations until convergence to a tolerance of 10−6 are shown for +a range of mesh distortions in Table 6 for the H1, RT, and HRT VEF methods. The H1 +38 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(a) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(c) +Fig. 12.. Lineouts of the 2D solution at y = 1/2 as ϵ → 0 for the (a) H1, (b) RT, and +(c) HRT methods on an orthogonal 8 × 8 mesh. The methods all converge to the +asymptotic solution indicating they preserve the thick diffusion limit. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +y +0 +2 +4 +6 +8 +10 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(a) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +y +0 +2 +4 +6 +8 +10 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +y +0 +2 +4 +6 +8 +10 +ϕ +ε = 10−1 +ε = 10−2 +ε = 10−3 +ε = 10−4 +(c) +Fig. 13.. Lineouts of the 2D solution at x = 3.5 as ϵ → 0 for the (a) H1, (b) RT, and +(c) HRT methods on the triple point mesh. All methods produce non-trivial +solutions even on the severely distorted triple point mesh. Non-monotonic +oscillations are present in the solution due to mesh imprinting. +39 + +TABLE 6. +Number of BiCGStab iterations until convergence on the first iteration of a thick +diffusion limit problem with ϵ = 10−1 as the mesh distortion parameter increases. +Here, H1, RT, and HRT rows refer to the Yp × Wp+1, Yp × RT p, and hybridized +Yp × RT p discretizations, respectively. +p +α +0.000 +0.025 +0.050 +0.060 +0.070 +0.080 +1 +H1 +46 +48 +48 +48 +48 +50 +RT +20 +22 +26 +31 +72 +– +HRT +7 +10 +8 +8 +8 +8 +2 +H1 +59 +61 +52 +55 +54 +57 +RT +28 +27 +31 +– +– +– +HRT +11 +10 +9 +9 +10 +9 +3 +H1 +54 +54 +56 +69 +55 +57 +RT +29 +28 +41 +– +– +– +HRT +9 +9 +8 +8 +9 +9 +4 +H1 +51 +55 +55 +66 +57 +61 +RT +41 +44 +46 +78 +– +– +HRT +7 +9 +10 +10 +10 +11 +– indicates solver did not converge in 250 iterations. +and RT methods use the lower block triangular preconditioner described in §6.4 while +the HRT method is preconditioned with one V-cycle of AMG. The solver for the RT +method did not converge in 250 iterations once the mesh became too distorted. The +H1 discretization converged on all the meshes tested but the iteration counts varied +between 46 and 69 whereas HRT was solved more uniformly, varying only between 7 +and 11 iterations. This indicates the solvers for the RT method are sensitive to mesh +distortion whereas HRT and, to a lesser extent, H1 are robust. Note that the meshes +used in this section are small, allowing the unscalable H1 solver to still converge. +8.4. +Linearized Crooked Pipe +We now show convergence in outer fixed-point iterations and inner preconditioned +linear solver iterations on a more realistic, multi-material problem. The geometry +and materials are shown in Fig. 15. The problem consists of two materials, the wall +and the pipe, which have an 1000x difference in total interaction cross section. Time +dependence is mocked by including artificial absorption and sources that correspond to +backward Euler time integration. The time step is set so that c∆t = 103 and the initial +condition is ψ0 = 10−4. The absorption and source are then σa = 1/c∆t = 10−3 1 +cm +and q = ψ0/c∆t = 10−1 +1 +cm3 · s · str. The boundary conditions are set so that isotropic +inflow of magnitude 1/2π enters on the left entrance of the pipe with vacuum on all +other surfaces. A Level Symmetric S12 angular quadrature set is used. The quadratic +programming negative flux fixup from [34] is used inside the transport sweep to ensure +positivity so that the VEF data are well defined. Timing data is presented as the +minimum time recorded across five repeated runs. +40 + +(a) α = 0.000 +(b) α = 0.025 +(c) α = 0.060 +(d) α = 0.080 +Fig. 14.. A selection of meshes generated by distorting a third-order, orthogonal +16 × 16 mesh according to the sine distortion. The parameter α controls the amount +of distortion. These meshes are used to assess linear solver robustness against mesh +distortion. +Isotropic Inflow +(0,-2) +(7,-2) +(7,2) +(0,2) +Vacuum +σt = 0.2 cm−1 +σa = 10−3 cm−1 +σt = 200 cm−1 +σa = 10−3 cm−1 +2.5 cm +5 mm +1 cm +5 mm +2.5 cm +1.5 cm +1 cm +1.5 cm +Fig. 15.. The geometry, material data, and boundary conditions for the linearized +crooked pipe problem. +41 + +The outer fixed-point and inner linear iterative efficiencies are shown by refining in +h and p on an orthogonal mesh. Anderson acceleration with two Anderson vectors +is used. The previous outer iteration’s solution is used as an initial guess for the +inner solver so that the initial guess becomes progressively more accurate as the outer +iteration converges. The outer tolerance is 10−6 and the inner BiCGStab tolerance is +10−8. The H1 and RT methods use the lower block triangular preconditioner described +in §6.4 with one Jacobi iteration on the total interaction mass matrix and one AMG +V-cycle on the lumped Schur complement. The HRT method is preconditioned using +one V-cycle of AMG. +Table 7 shows the number of Anderson-accelerated fixed-point iterations to conver- +gence and the maximum and average number of inner iterations performed across all +outer iterations for the H1, RT, and HRT methods. The RT and HRT methods had +equivalent convergence in outer iterations except for p = 2 with three refinements +where HRT required one fewer iteration than RT. H1 converged slower than the RT +or HRT methods, requiring on average 159% more iterations. The RT and HRT inner +solvers were scalable in h and p while the H1 solvers were not. On the problems with +two and three refinements in h, the H1 inner solver did not converge within 100 iter- +ations on at least one of the solves for all values of p. The nested H1 iteration is only +able to converge due to the use of the previous outer iteration as the initial guess for +the inner iteration. +Note that the H1 method is slower to converge the fixed-point problem even when +the inner solver converged within the maximum allowed number of inner iterations. +Table 8 shows the average percentage of elements in the space-angle phase space that +required application of the negative flux fixup. RT and HRT had similar reliance on +the fixup. On the coarsest problems in h – where the fixup is needed most due to +lack of numerical resolution – the H1 method produced significantly more negativities +which led to 1.6x times more elements needing the fixup for p = 2 and p = 3. On the +most refined problems, 5-10% more elements were fixed up for the H1 method. This +increased reliance on the negative flux fixup is likely the source of the H1 method’s +reduced efficiency on problems where the inner iteration completed successfully at +each outer iteration. The H1 method’s proclivity for producing negativities within the +transport sweep is indicative of poorer solution quality compared to the RT and HRT +discretizations of the VEF system. This reduced solution quality is investigated in +§8.5. +Table 9 shows the average costs per outer iteration of assembling and solving the VEF +system. The H1 method is the cheapest to assemble but the most expensive to solve. +This is due to the H1 method’s simpler left hand side that does not require assem- +bly over interior faces in the mesh. Since the H1 solvers were not scalable, H1 was +around 4x more expensive than RT on the problems most refined in h. Compared to +RT, the HRT assembly includes the additional cost of forming the reduced system +via element-local, dense matrix operations. Thus, the HRT method has the highest +assembly cost, especially for large p. However, HRT was the cheapest to solve at each +outer iteration since BiCGStab is applied to the reduced system. The HRT precondi- +tioner is cheaper than the lower block triangular preconditioner and the HRT reduced +system is positive definite whereas the RT system is indefinite. These benefits led to +both faster BiCGStab convergence and cheaper cost per iteration resulting in dramat- +ically reduced solve times compared to RT: on the most refined problems in h, RT +was 3x, 8x, and 18x more expensive to solve than HRT for p = 1, p = 2, and p = 3, +42 + +TABLE 7. +The number of outer Anderson-accelerated fixed-point iterations until convergence +along with the maximum and average number of inner BiCGStab iterations until +convergence on the linearized crooked pipe problem. Two Anderson vectors were +used. The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and +hybridized Yp × RT p discretizations, respectively. The H1 and RT methods were +preconditioned with a block lower triangular preconditioner with AMG applied to +the lumped Schur complement. HRT was preconditioned with AMG. The previous +outer iteration’s solution was used as the initial guess for the inner iteration. +Outer +Max Inner +Avg. Inner +Ne +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +p = 1 +112 +16 +13 +13 +37 +16 +6 +22.62 +10.77 +4.69 +448 +20 +13 +13 +66 +18 +7 +38.00 +12.23 +4.92 +1792 +25∗ +15 +15 +100 +19 +8 +54.44 +12.27 +5.27 +7168 +23∗ +16 +16 +100 +20 +8 +71.22 +12.88 +5.38 +p = 2 +112 +23 +13 +13 +47 +27 +10 +26.22 +16.92 +7.15 +448 +25 +15 +15 +75 +26 +10 +39.68 +17.20 +7.20 +1792 +27∗ +16 +16 +100 +26 +11 +55.44 +18.25 +7.44 +7168 +24∗ +16 +15 +100 +29 +14 +69.67 +19.38 +8.67 +p = 3 +112 +23 +14 +14 +50 +24 +10 +26.65 +16.71 +6.50 +448 +25 +15 +15 +73 +29 +10 +42.44 +17.80 +6.33 +1792 +26∗ +16 +16 +100 +28 +10 +55.62 +18.56 +6.44 +7168 +28∗ +17 +17 +100 +31 +11 +65.50 +19.41 +6.53 +∗ indicates at least one inner solve did not converge in 100 iterations. +43 + +TABLE 8. +The average number of elements in the space-angle phase space that required +application of the negative flux fixup in the transport sweep on the linearized +crooked pipe problem. +Ne +H1 +RT +HRT +p = 1 +112 +21.055 +21.063 +21.073 +448 +6.331 +5.043 +5.042 +1792 +2.626 +2.282 +2.283 +7168 +1.432 +1.295 +1.295 +p = 2 +112 +26.123 +16.314 +16.313 +448 +7.974 +5.691 +5.692 +1792 +3.300 +2.796 +2.796 +7168 +1.795 +1.661 +1.646 +p = 3 +112 +29.224 +20.285 +20.285 +448 +9.529 +7.037 +7.037 +1792 +4.150 +3.815 +3.818 +7168 +2.247 +2.132 +2.132 +respectively. +The total time to solve the fixed-point problems for each of the methods is presented in +Table 10 along with the breakdown of the total cost into time spent in the inversion of +the streaming and collision operator and forming and solving the VEF system. Here, +it can be seen that HRT’s higher assembly costs are sufficiently balanced by reduced +solve costs such that HRT spent the least time in the VEF portion of the algorithm for +all values of h and p. In particular, for the most refined problem with p = 3, forming +and solving the HRT system was more than twice as fast as the RT method. Overall, +the HRT method was the fastest to solve the fixed-point problems. However, due to +the high cost of the transport sweep relative to forming and solving the VEF system, +the variance in cost between the RT and HRT methods was less pronounced with RT +being at most 1.18x more expensive despite the HRT inner solve being twice as fast +as the RT solve. H1 was the most expensive method due to its slower outer and inner +iterative efficiency and its higher reliance on the negative flux fixup. +8.5. +Eigenvalue Problem +It was observed that the H1 discretization exhibited poor solution quality in under +resolved problems and could not be scalably solved using block preconditioners. Here, +we investigate the presence of so-called “checkerboard” modes that are allowed by the +H1 discretization. These modes are not physical, contaminate solution quality, and +degrade the effectiveness of AMG. To investigate this issue, we consider the following +eigenvalue problem: +− ∇2u = λu , +x ∈ D , +(113a) +44 + +TABLE 9. +The average time spent per outer iteration assembling and solving the VEF systems +on hp refinements of the linearized crooked pipe problem. Times are presented in +seconds and represent the minimum time achieved across five repeated runs for each +value of h and p. +VEF Assembly Time (s) +VEF Solve Time (s) +Ne +H1 +RT +HRT +H1 +RT +HRT +p = 1 +112 +0.0155 +0.0221 +0.0225 +0.0098 +0.0050 +0.0023 +448 +0.0408 +0.0708 +0.0725 +0.0603 +0.0223 +0.0089 +1792 +0.1191 +0.2483 +0.2548 +0.3366 +0.0923 +0.0363 +7168 +0.3990 +0.9174 +0.9541 +1.7994 +0.3876 +0.1457 +p = 2 +112 +0.0272 +0.0400 +0.0426 +0.0436 +0.0234 +0.0035 +448 +0.0792 +0.1333 +0.1460 +0.2680 +0.0983 +0.0142 +1792 +0.2637 +0.4965 +0.5550 +1.4964 +0.4367 +0.0589 +7168 +1.0182 +1.9546 +2.1516 +8.4629 +2.0172 +0.2510 +p = 3 +112 +0.0500 +0.0764 +0.0947 +0.1244 +0.0653 +0.0052 +448 +0.1568 +0.2764 +0.3598 +0.8199 +0.2960 +0.0213 +1792 +0.5698 +1.0642 +1.3765 +4.9136 +1.3698 +0.0862 +7168 +2.4576 +4.4729 +5.5667 +25.1319 +6.2214 +0.3479 +TABLE 10. +The total runtime along with the total time spent in the transport sweep and VEF +portions of the algorithm. Times are presented in seconds and represent the +minimum time achieved across five repeated runs for each value of h and p. +Total Time (s) +Sweep Time (s) +VEF Time (s) +Ne +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +p = 1 +112 +3.47 +2.88 +2.79 +2.87 +2.36 +2.36 +0.50 +0.41 +0.33 +448 +14.62 +9.69 +9.34 +11.90 +7.92 +7.90 +2.35 +1.39 +1.07 +1792 +69.90 +41.85 +40.16 +55.85 +34.54 +34.26 +12.60 +5.79 +4.44 +7168 +264.52 +175.63 +168.00 +204.34 +145.85 +143.85 +54.50 +23.58 +18.04 +p = 2 +112 +8.62 +4.91 +4.55 +6.57 +3.80 +3.80 +1.90 +0.97 +0.61 +448 +35.90 +20.18 +18.69 +25.77 +15.63 +15.71 +9.59 +4.01 +2.44 +1792 +161.03 +84.59 +77.42 +107.84 +65.20 +65.18 +51.01 +17.13 +10.05 +7168 +634.11 +344.51 +291.24 +385.29 +263.46 +245.23 +239.62 +72.04 +37.10 +p = 3 +112 +18.17 +10.85 +9.87 +13.29 +8.21 +8.17 +4.61 +2.37 +1.41 +448 +82.09 +45.23 +40.63 +54.36 +34.01 +33.68 +26.61 +10.09 +5.81 +1792 +378.46 +191.25 +169.54 +222.79 +141.46 +140.99 +151.09 +45.21 +23.94 +7168 +1828.29 +858.09 +743.84 +997.93 +630.32 +621.21 +810.42 +208.14 +102.94 +45 + +Fig. 16.. A depiction of an eigenmode corresponding to an eigenvalue of 8π2 of the +Poisson eigenvalue problem discretized with the Y1 × W2 discretization’s lumped +Schur complement. For this eigenvalue, the exact solution is sin(2πx) sin(2πy) +meaning this mode is spurious. The presence of high-frequency spurious modes in the +Yp × Wp+1 discretization’s lumped Schur complement degrades the effectiveness of +AMG and thus the performance of the block preconditioners used to solve the full +Yp × Wp+1 discretization. +u = 0 , +x ∈ ∂D , +(113b) +with D = [0, 1]2. The exact solutions are +u = sin(kxπx) sin(kyπy) , +λ = π2(k2 +x + k2 +y) . +(114) +The Y1 ×W2 discretization’s lumped Schur complement is used to discretize this prob- +lem as: find u ∈ Y1 such that +˜Su = λMu , +(115) +where M is the Y1 mass matrix and ˜S is the lumped Schur complement defined in +Eq. 67. The Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) +solver from hypre was used to solve for the five smallest eigenvalues and their cor- +responding eigenvectors on this problem. The H1 discretization correctly produced +the first four smallest eigenvalues and associated eigenvectors but found the high- +frequency, checkerboard mode shown in Fig. 16 for the fifth. This checkerboard mode +corresponded to a non-physically degenerate eigenvalue of 8π2. The presence of this +mode indicates the Yp × Wp+1 discretization allows non-physical, spurious modes that +are slowly decaying and high frequency. Such modes are slow to remove with relaxation +and also cannot be accurately represented on a coarser grid, meaning AMG will not +be an effective preconditioner. Furthermore, the presence of these oscillatory modes +can degrade solution quality in underresolved problems leading to increased negativ- +ities in the VEF scalar flux. We note that the lumped Schur complement associated +with the RT discretization does not contain these non-physical modes and can thus +be effectively preconditioned by AMG. +46 + +6.0e+00 +2 +.3 + -4 +-6.0e+008.6. +Weak Scaling +Finally, we show that the RT and HRT methods weak scale in parallel on the first iter- +ation of the linearized crooked pipe problem from §8.4. The inversion of the streaming +and collision operator is approximated with one iteration of a parallel block Jacobi +sweep where each processor performs a transport sweep on its processor-local domain +using angular fluxes on inflow processor boundaries that are iteratively lagged. Due +to this, the transport sweep is not exact when more than one processor is used. Uni- +form refinements are used in tandem with increasing the processor count by four so +that the number of unknowns per processor remains constant. Since the sizes of the +systems corresponding to the RT and HRT methods differ, the data are tabulated in +terms of the number of scalar flux degrees of freedom. We stress that the linear system +for the RT method additionally includes the degrees of freedom associated with each +component of the current and that the HRT system solves the reduced problem for +the Lagrange multiplier unknowns only. The results were generated on 29 nodes of the +rztopaz machine at LLNL which has two 18-core Intel Xeon E5-2695 CPUs per node. +Timing data is presented as the minimum time achieved across three repeated runs. +We compare the efficiency and performance of the inner solvers for the RT and HRT +discretizations with p = 2 when 1) the parallel block Jacobi transport sweep is used +to compute the VEF data and 2) the VEF data are set to their asymptotic, diffusive +values of E = 1/3I and Eb = 1/2. The BiCGStab tolerance was 10−8. +Table 11 compares the number of iterations to convergence and solve times for solving +the RT VEF and RT diffusion linear systems. For the VEF system, a range of precon- +ditioner options are presented. The preconditioners are parameterized by the use of +one iteration of Jacobi (J) or Gauss-Seidel (GS) for approximating the inverse of the +total interaction mass matrix and the number of AMG V-cycles applied to the lumped +Schur complement at each preconditioner application. We consider the permutations +of using Jacobi, Gauss-Seidel, one AMG V-cycle, and three AMG V-cycles. The RT +diffusion system is preconditioned by one iteration of Jacobi on the total interaction +mass matrix and one AMG V-cycle on the lumped Schur complement (i.e. the “J-1” +preconditioner). The J-1 preconditioner led to a scalable solver for the diffusion sys- +tem with an increase of only 6 iterations when the problem size was increased by a +factor of 1024. However, the J-1 preconditioner did not perform as uniformly when +applied to the VEF system showing an increase of 14 iterations over the same range of +problem sizes. Increasing the number of AMG V-cycles per preconditioner application +to three led to an algorithm that scaled more closely to the diffusion case. Switching +to Gauss-Seidel for approximating the inverse of the total interaction mass matrix did +reduce the total number of iterations to convergence but did not alter the scaling of it- +erations with problem size seen with the J-1 preconditioner. The GS-3 option required +the least iterations to converge and iteration counts scaled similarly to that of the J-1 +preconditioner applied to the diffusion system. These results suggest that scalability +can be improved by performing more AMG V-cycles on the lumped Schur comple- +ment. However, using a more expensive approximation of the inverse of the total mass +matrix was less effective. In terms of solve time, the J-1 preconditioner was the fastest. +In other words, the increased robustness in iteration count to problem size provided +by the more expensive preconditioners did not adequately balance their increased cost +per iteration. For the largest problem size, the J-1 preconditioner applied to the VEF +system was only 1.23x more expensive than solving the diffusion system with the J-1 +preconditioner. +47 + +TABLE 11. +A weak scaling study of the first iteration of the linearized crooked pipe problem +using p = 2 for the RT discretization preconditioned with a lower block triangular +preconditioner. The number of BiCGStab iterations to converge to a tolerance of +10−8 and the total solve time are presented. Solving the linear systems corresponding +to RT VEF and RT diffusion are compared with the RT VEF system preconditioned +using a range of preconditioners corresponding to the use of Jacobi (J) or +Gauss-Seidel (GS) to approximate the inverse of the total interaction mass matrix +and the application of one or three AMG V-cycles per iteration to approximate the +inverse of the lumped Schur complement. The diffusion system is solved using the J-1 +preconditioner. Times are provided in seconds and represent the minimum time +achieved over three repeated runs. +Iterations +Solve Time (s) +Processors +DOF +J-1 +J-3 +GS-1 +GS-3 +Diffusion +J-1 +J-3 +GS-1 +GS-3 +Diffusion +1 +42 588 +29 +18 +24 +14 +27 +1.82 +2.51 +1.83 +2.22 +1.68 +4 +170 352 +31 +20 +24 +15 +29 +2.18 +3.24 +2.08 +2.74 +2.03 +16 +681 408 +34 +20 +27 +17 +29 +3.31 +4.41 +3.11 +4.15 +2.83 +64 +2 725 632 +37 +21 +33 +19 +30 +5.67 +7.10 +5.98 +6.75 +4.72 +256 +10 902 528 +38 +23 +33 +21 +31 +5.98 +7.87 +6.50 +8.19 +4.91 +1024 +43 610 112 +43 +25 +38 +22 +33 +7.69 +9.66 +9.52 +9.46 +6.22 +This comparison is repeated for the HRT method in Table 12. Here, we compare only +the use of one AMG V-cycle to precondition the reduced system corresponding to +the VEF and diffusion problems. Solving the VEF system required at most 4 more +iterations compared to solving diffusion. Solving the largest VEF problem was 1.19x +more expensive than the largest diffusion problem. Compared to RT, the HRT VEF +solve was up to 13x faster. +Figures 17a and 17b show the weak scaling efficiency for the RT and HRT discretiza- +tions, respectively. Weak scaling efficiency is defined as +εn = solve time with one processor +solve time with n processors , +(116) +where ideal scaling is εn = 1. Ideal scaling is not expected since solving these linear +systems requires parallel communication. For both the RT and HRT methods, the VEF +system can be scalably solved with comparable efficiency to solving the corresponding +diffusion problem. In particular, for the problem size solved with 1024 processors, the +RT VEF preconditioners scaled with efficiencies of 23.6%, 26.0%, 19.1%, and 23.4% for +the J-1, J-3, GS-1, and GS-3 preconditioners, respectively, while J-1 applied to diffusion +scaled with an efficiency of 27.0%. This suggests that the J-3 preconditioner may scale +more robustly despite being more expensive than J-1. For HRT, the efficiencies were +22.9% and 24.0% for solving the VEF and diffusion systems, respectively. +48 + +TABLE 12. +Weak scaling the HRT discretization of VEF and diffusion on the linearized crooked +pipe with p = 2. Both problems used one AMG V-cycle to precondition the HRT +reduced system. The number of BiCGStab iterations to converge to a tolerance of +10−8 and the total solve times in seconds are presented. The timing data is +represents the minimum time recorded over three repeated runs. +Iterations +Solve Time (s) +Processors +DOF +VEF +Diffusion +VEF +Diffusion +1 +42 588 +11 +10 +0.14 +0.12 +4 +170 352 +13 +12 +0.18 +0.16 +16 +681 408 +14 +12 +0.28 +0.25 +64 +2 725 632 +16 +12 +0.47 +0.36 +256 +10 902 528 +15 +12 +0.48 +0.39 +1024 +43 610 112 +16 +13 +0.61 +0.51 +100 +101 +102 +103 +Number of Processors +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Weak Scaling Efficiency +J-1 +J-3 +GS-1 +GS-3 +Diffusion +(a) +100 +101 +102 +103 +Number of Processors +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Weak Scaling Efficiency +VEF +Diffusion +(b) +Fig. 17.. Weak scaling efficiency for solving the (a) RT and (b) HRT VEF systems on +the first iteration of the linearized crooked pipe problem. For the RT discretization, a +range of preconditioners are presented corresponding to the use of Jacobi (J) or +Gauss-Seidel (GS) and the number of AMG V-cycles applied per iteration. In both +cases, the scaling of the VEF solve is compared to solving radiation diffusion. The +RT diffusion system is preconditioned with the J-1 preconditioner. Both HRT VEF +and HRT diffusion use one AMG V-cycle to precondition the reduced system. +49 + +9. +Conclusions +We have developed high-order mixed finite element discretizations for the Variable Ed- +dington Factor (VEF) equations that are compatible with curved meshes. The methods +were designed to have element-local particle balance and immediate multiphysics com- +patibility with the mixed finite element techniques used for hydrodynamics calcula- +tions in [7]. Each method produces a scalar flux solution in the Discontinuous Galerkin +(DG) space to match the approximation space used for the thermodynamic variables +in [7]. We considered three choices for the finite element space that approximates the +current: a discrete subspace of [H1(D)]2 where each component is represented with +continuous finite elements, a method that uses the Raviart Thomas (RT) space along +with DG-like numerical fluxes to treat the discontinuities arising from the presence of +the Eddington tensor in the VEF first moment equation, and a hybrid RT method +where continuity of the normal component of the current is enforced weakly with a +Lagrange multiplier. These methods are referred to as H1, RT, and HRT, respectively. +The VEF discretizations were paired with a high-order DG discretization of the SN +transport equations to solve problems from linear transport. +On manufactured solutions problems, the H1, RT, and HRT methods all had opti- +mal O(hp+1) convergence for the VEF scalar flux on refinements of a curved mesh. +However, optimal convergence for the VEF current was observed only on diffusive +problems. This result suggests that the VEF scalar flux and the zeroth angular mo- +ment of the discrete angular flux converge at equivalent rates. However, on transport +problems, the VEF current could not be solved to the same accuracy as the first +angular moment of the discrete angular flux. In addition, the mixed finite element +superconvergence property for the VEF scalar flux was lost on the standard use case +of equal degree interpolation for the VEF scalar and angular fluxes on a quadratically +anisotropic transport problem. This suggests that post-processing techniques, such as +the method proposed by Stenberg [35], that leverage the mixed finite element super- +convergence property to produce more accurate solutions would not be effective on +transport problems where equal degree interpolation is used for the VEF scalar and +angular fluxes. +All three methods showed rapid and robust convergence on a single-material thick +diffusion limit test problem on both a simple orthogonal mesh and a severely distorted +third-order mesh generated with a Lagrangian hydrodynamics code. The methods +were tested on mesh and polynomial order refinements of a two-material linearized +crooked pipe problem that had a 1000x difference in total cross section. Fixed-point +convergence was robust for all three methods with RT and HRT converging equiva- +lently. The H1 method converged slower, requiring ≈1.75x more iterations than RT +and HRT on the largest mesh for each polynomial order. It was observed that the H1 +VEF discretization produced scattering sources more likely to induce negativity in the +transport sweep leading to an increased reliance on the negative flux fixup compared +to the RT and HRT methods. +We also investigated preconditioned iterative solvers for the H1, RT, and HRT meth- +ods. Lower block triangular preconditioners were used for the H1 and RT methods +that employ Jacobi smoothing on the total interaction mass matrix and Algebraic +Multigrid (AMG) on the lumped Schur complement. The solvers for the HRT method +leverage the element-by-element block structure generated by discontinuous approxi- +mations to form a reduced system for the Lagrange multiplier only, leading to fewer +50 + +globally coupled unknowns than in the H1 or RT methods. AMG is applied directly +to the reduced problem. The preconditioned iterative solvers were tested on a series +of increasingly distorted meshes to test their robustness. The H1 and HRT methods +converged for all distortions but the RT method failed to converge once the mesh be- +came too distorted. The RT and HRT methods were shown to have scalable solvers in +both h and p on the linearized crooked pipe problem. However, the solvers for H1 were +not scalable. It was found that AMG was struggling to adequately precondition the +lumped Schur complement due to the presence of highly oscillatory, slowly decaying +modes. These modes are a consequence of the mismatch between the finite element +spaces used to approximate the VEF scalar flux and current and were present even on +a simple Poisson eigenvalue problem. Finally, a weak scaling study demonstrated that +the RT and HRT methods can be scalably solved out to 1024 processors and over 40 +million VEF scalar flux unknowns. Compared to solving the symmetric positive def- +inite radiation diffusion system, solving the non-symmetric VEF equations was only +1.23x and 1.19x more expensive for the RT and HRT methods, respectively. +The primary takeaway from this work is that the combination of a DG SN discretiza- +tion and the RT or HRT VEF discretizations form an effective high-order method for +linear transport problems. Both the RT and HRT discretizations of the VEF equations +have high-order accuracy, compatibility with curved meshes, and robust and scalable +convergence in both outer fixed-point iterations and inner preconditioned linear solver +iterations. The performance of the methods was differentiated only in the presence +of severely distorted meshes. In such case, the preconditioned iterative solver for the +HRT method was robust to mesh distortion whereas the solver for the RT method was +not. The H1 method is not recommended for use in a production code due to the lack +of scalable iterative solvers. In addition, the H1 method had lower fixed-point iteration +efficiency and higher reliance on the negative flux fixup on the linearized crooked pipe +problem when compared to the RT and HRT methods. +In radiation-hydrodynamics calculations, the scalar flux and current are coupled to +the hydrodynamics’ energy balance and momentum equations, respectively. Due to +the sub-optimal accuracy of the VEF current on transport problems, it is unclear +whether the mixed finite element methods presented here would yield improvements +in physics fidelity commensurate with the increased cost of solving for both the VEF +scalar flux and current. 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Maginot, “A +quadratic programming flux correction method for high-order DG discretizations +of SN transport,” Journal of Computational Physics, vol. 419, p. 109696, 2020. +[35] R. Stenberg, “Postprocessing schemes for some mixed finite elements,” ESAIM: +Mathematical Modelling and Numerical Analysis - Mod´elisation Math´ematique +et Analyse Num´erique, vol. 25, no. 1, pp. 151–167, 1991. [Online]. Available: +http://eudml.org/doc/193618 +[36] F. Brezzi, Stability of Saddle-Points in Finite Dimensions. +Berlin, Heidelberg: +Springer +Berlin +Heidelberg, +2003, +pp. +17–61. +[Online]. +Available: +https: +//doi.org/10.1007/978-3-642-55692-0 2 +Appendix A. The Gradient of the Piola Transform +The goal of this section section is to derive a formula for the transformation of the +gradient of a vector defined under the contravariant Piola transformation. For the +contravariant Piola transform v = 1 +J Fˆv ◦ T−1 the inverse transform is: +ˆv = JF−1v ◦ T . +(A1) +Here, we seek to derive +ˆ∇ˆv = ˆ∇ +� +JF−1v +� +, +(A2) +so that we can solve for ∇v. The goal is to derive the functional form of the trans- +formation in terms of functionality commonly implemented in finite element codes. +That is, we cast the computation in terms of the Jacobian matrix and Hessian of the +transformation. +54 + +Through their connection to the Jacobian matrix and the inverse of the Jacobian +matrix, the tangent and cotangent spaces are related by +n1 = t2 × ˆe3 , +n2 = ˆe3 × t1 , +(A3) +where ˆe3 points out of the page. In other words, n1 is a 90 degree clockwise rotation +of t2 and n2 is a 90 degree counterclockwise rotation of t1 (see Fig. 2). Thus, we can +write +ˆ∇ˆv = ˆ∇ +� +Jn1 · v +Jn2 · v +� += +� +∂ +∂ξ(Jn1 · v) +∂ +∂η(Jn1 · v) +∂ +∂ξ(Jn2 · v) +∂ +∂η(Jn2 · v) +� += +� +∂ +∂ξ(Jn1) · v +∂ +∂η(Jn1) · v +∂ +∂ξ(Jn2) · v +∂ +∂η(Jn2) · v +� ++ +� +Jn1 · ∂v +∂ξ +Jn1 · ∂v +∂η +Jn2 · ∂v +∂ξ +Jn2 · ∂v +∂η +� +. +(A4) +The second term can be written as +� +Jn1 · ∂v +∂ξ +Jn1 · ∂v +∂η +Jn2 · ∂v +∂ξ +Jn2 · ∂v +∂η +� += JF−1 ˆ∇v = JF−1∇vF , +(A5) +where ˆ∇v = ∇vF transforms the reference gradient to the physical gradient. The first +term is a third-order tensor contracted with a vector to yield a second-order tensor. +By expanding the dot products, we can emulate this contraction as a sum of two +second-order tensors: +� +∂ +∂ξ(Jn1) · v +∂ +∂η(Jn1) · v +∂ +∂ξ(Jn2) · v +∂ +∂η(Jn2) · v +� += +� +∂ +∂ξ(Jn11)v1 + ∂ +∂ξ(Jn12)v2 +∂ +∂η(Jn11)v1 + ∂ +∂η(Jn12)v2 +∂ +∂ξ(Jn21)v1 + ∂ +∂ξ(Jn22)v2 +∂ +∂η(Jn21)v1 + ∂ +∂η(Jn22)v2 +� += +� +∂ +∂ξ(Jn11) +∂ +∂η(Jn11) +∂ +∂ξ(Jn21) +∂ +∂η(Jn21) +� +v1 + +� +∂ +∂ξ(Jn12) +∂ +∂η(Jn12) +∂ +∂ξ(Jn22) +∂ +∂η(Jn22) +� +v2 += ˆ∇(JF−1 +1 )v1 + ˆ∇(JF−1 +2 )v2 +(A6) +where F−1 +i +are the columns of F−1. Typically, finite element codes provide the Hes- +sian matrix of the forward map but not the inverse map. Thus, to leverage existing +functionality, we must write the above matrices in terms of H = ˆ∇F instead of ˆ∇F−1. +Assume that the code computes the Hessian matrix in flattened and symmetric form +as: +⟨H⟩ = +� ∂2x +∂ξ2 +∂2x +∂ξ∂η +∂2x +∂η2 +∂2y +∂ξ2 +∂2y +∂ξ∂η +∂2y +∂η2 +� +. +(A7) +55 + +Then the above can be rewritten as +ˆ∇(JF−1 +1 ) = ˆ∇ +� +F22 +−F21 +� += ˆ∇ +� ∂y +∂η +− ∂y +∂ξ +� += +� ∂2y +∂ξ∂η +∂2y +∂η2 +− ∂2y +∂ξ2 +− ∂2y +∂ξ∂η +� += +� +H22 +H23 +−H21 +−H22 +� +, +(A8) +ˆ∇(JF−1 +2 ) = ˆ∇ +� +−F12 +F11 +� += ˆ∇ +� +− ∂x +∂η +∂x +∂ξ +� += +� +− ∂2x +∂ξ∂η +− ∂2x +∂η2 +∂2x +∂ξ2 +∂2x +∂ξ∂η +� += +� +−H12 +−H13 +H11 +H12 +� +. +(A9) +We can define the matrix +ˆB = ˆ∇(JF−1)v = +� +H22 +H23 +−H21 +−H22 +� +v1 + +� +−H12 +−H13 +H11 +H12 +� +v2 . +(A10) +This is computed in flattened form as +⟨ ˆB⟩ = +� +⟨ ˆ∇(JF−1 +1 )⟩ +⟨ ˆ∇(JF−1 +2 )⟩ +� +v += +� +��� +H22 +−H12 +H23 +−H13 +−H21 +H11 +−H22 +H12 +� +��� +1 +J Fˆv +(A11) +where v = 1 +J Fˆv was used. Finally, we have that +ˆ∇ˆv = ˆB + JF−1∇vF ⇐⇒ ∇v = 1 +J F +� +ˆ∇ˆv − ˆB +� +F−1 . +(A12) +We can then say that +∇v : E dx = 1 +J F +� +ˆ∇ˆv − ˆB +� +F−1 : E Jdξ += +� +ˆ∇ˆv − ˆB +� +: FT EF−T dξ . +(A13) +56 + +Here, we use the fact that A : B = trace(ABT ) and apply the cyclic property of the +trace to permute F and F−1. In this form, we can implement the gradient calculation +as a matrix-vector product of the flattened referential gradient and the coefficients of +ˆv. +When the mesh transformation is affine, ˆB = 0 since the Hessian of an affine trans- +formation is zero. In addition, the Piola identity states that trace ˆB = 0. This can be +most easily seen in Eq. A10 where +trace ˆB = (H22 − H22)v1 + (−H12 + H12)v2 = 0 . +(A14) +Using the Piola identity and Eq. A13, we have that +∇ · v dx = ∇v : I dx += +� +ˆ∇ˆv − ˆB +� +: FT IF−T dξ += trace +� +ˆ∇ˆv − ˆB +� +dξ += ˆ∇ · ˆv dξ . +(A15) +Thus, in the thick diffusion limit when E ∝ I, ∇v : E simplifies to the standard +transformation for the divergence of a contravariant vector. +Appendix B. Discrete Inf-Sup Condition +Here, we discuss the the inf-sup condition that governs the solvability of the 2 × 2 +block systems arising in mixed finite element discretizations. Two excellent references +for this topic are Brezzi [36] and Benzi et al. [6]. We present an analysis for Poisson’s +equation since methods not effective for this simpler problem have no hope of being +effective for the VEF equations. +B.1. +Conditions for Solvability +Consider the linear system: +� +M +−DT +D +� �q +u +� += +�0 +f +� +, +(B1) +which corresponds to the mixed finite element discretization of +q + ∇u = 0 , +(B2a) +∇ · q = f . +(B2b) +Note that the above is Poisson’s equation, −∇2u = f, in mixed form. The matrices +are of the form: +vT Mq = +� +v · q dx , +wT Dq = +� +w ∇ · q dx . +(B3) +57 + +We wish to demonstrate the conditions for when the block system in Eq. B1 is non- +singular. To show the solution is unique, it must be verified that f = 0 implies that +u = 0 and q = 0. When f = 0, we have that +Dq = 0 ⇐⇒ q ∈ N(D) , +(B4) +where N(D) denotes the nullspace of D such that +N(D) = {v : Dv = 0} . +(B5) +For some v ∈ N(D), the first equation reads +vT Mq − vT DT u = 0 . +(B6) +Since v ∈ N(D), vT DT = 0. Thus, we have that +vT Mq = 0 , +∀v ∈ N(D) . +(B7) +Since M is a mass matrix, it is symmetric positive definite and thus, vT Mq = 0 ⇐⇒ +q = 0. In other words, we have shown that f = 0 ⇒ q = 0. Setting q = 0 in the first +row of Eq. B1 yields: +DT u = 0 . +(B8) +For the block system to be non-singular, we must have that +DT u = 0 ⇐⇒ u = 0 . +(B9) +Equivalently, we require that the nullspace of DT has only the trivial nullspace of zero +(i.e. N(DT ) = {0}). The discrete inf-sup condition is precisely this condition on the +matrix D that N(DT ) = {0}. +B.2. +Characterization for a Single Element +We now particularize the matrix D for a single element, K = [0, 1]2, of the [H1(D)]2 × +L2(D) and H(div; D) × L2(D) discretizations of the Poisson problem. We consider +the W1 × Y0 and W1 × Y1 discretizations and show that the former and latter have +non-trivial nullspaces for D and DT , respectively. In light of the inf-sup requirement +established above, the W1 × Y1 discretization will lead to a singular block system. We +also show that the pairing W1 and Y0 is solvable but is imbalanced in another way that +allows the presence of non-physical modes. By contrast, the RT 0 × Y0 discretization +is non-singular and does not allow these non-physical modes. +On the single element K, the lowest-order Raviart Thomas and [H1(D)]2 finite element +spaces are given by: +RT 0 = span{ +� +1 +0 +� +, +� +x +0 +� +, +� +0 +1 +� +, +� +0 +y +� +} , +(B10a) +58 + +W1 = span{ +� +1 +0 +� +, +� +x +0 +� +, +� +y +0 +� +, +� +xy +0 +� +, +� +0 +1 +� +, +� +0 +x +� +, +� +0 +y +� +, +� +0 +xy +� +} . +(B10b) +Further, the constant and linear DG spaces are: +Y0 = span{1} , +Y1 = span{1, x, y, xy} . +(B11) +Observe that the divergence of RT 0 is exactly the constant polynomial space, Y0: +∇ · RT 0 = span{0, 1, 0, 1} = span{1} = Y0 , +(B12) +while the divergence of the [H1(D)]2 space is: +∇ · W1 = span{0, 1, 0, y, 0, 0, 1, x} = span{1, x, y} , +(B13) +which is a space larger than Y0 but smaller than Y1. The nullspaces of the divergence +of the RT and [H1(D)]2 local polynomial spaces are spanned by +N(∇ · RT 0) = span{ +� +−x +y +� +, +� +0 +1 +� +, +� +1 +0 +� +} , +(B14a) +N(∇ · W1) = span{ +� +−x +y +� +, +� +0 +1 +� +, +� +1 +0 +� +, +� +0 +x +� +, +� +y +0 +� +} . +(B14b) +Here, we can already see an issue forming: the nullspace for W1 is larger than the +nullspace for RT 0. +We are interested in the bilinear form +D(u, v) = +� +K +u ∇ · v dx , +u ∈ Y , v ∈ X , +(B15) +where Y is either Y0 or Y1 and X is either W1 or RT 0. This bilinear form admits the +matrix D through +uT Dv = D(u, v) , +∀u ∈ Y . +(B16) +D has a nullspace corresponding to the nullspace of the divergence operator and vectors +v such that w = ∇ · v ̸= 0 where +M(u, w) = +� +uw dx = 0 , +(B17) +for each u ∈ Y . We consider elements of the nullspace corresponding to this second +condition to be non-physical since they arise from the mismatch between the spaces +Y and X and not the divergence operator itself. For the case Y = Y1 and X = W1 +corresponding to the W1 × Y1 discretization, the space Y1 is larger than ∇ · W1 and +thus there does not exist w ̸= 0 such that M(u, w) = 0 for all u ∈ Y1. Thus, M has +only the trivial nullspace and N(D) = N(∇ · W1). However, there does exist u ̸= 0 +such that M(u, w) = 0 for each w ∈ ∇ · W1. In particular, u = α +� 1 +4 − 1 +2x − 1 +2y + xy +� +59 + +for any α ̸= 0 satisfies M(u, w) = 0 for all w ∈ ∇ · W1. Note that this particular form +for the nullspace arises from our choice of the domain D = K = [0, 1]2. This means DT +has a non-trivial nullspace and thus the resulting 2 × 2 block system will be singular +by the inf-sup requirement in Eq. B9. +For the case Y = Y0 and X = W1 corresponding to the W1 × Y0 discretization, +there exists w ∈ ∇ · W1 such that w ̸= 0 but M(u, w) = 0. In particular, N(M) = +span{x−1/2, y−1/2} and thus vectors with divergence in N(M) will also be in N(D). +In other words, N(D) = N(∇ · W1) ∪ Nspurious where +Nspurious = span{ +� +x(y − 1/2) +0 +� +, +� +0 +(x − 1/2)y +� +} , +(B18) +is the space of vectors whose divergence belongs to N(M). Observe that for v ∈ +Nspurious, Dv = 0 but ∇ · v ̸= 0. On the other hand, since ∇ · W1 is larger than Y0, +DT has only the trivial nullspace. Thus, the W1 × Y0 discretization is solvable but D +has a non-physically enlarged nullspace that allows spurious solutions. +By contrast, with Y = Y0 and X = RT 0, ∇·RT 0 = Y0 meaning M is the L2(K) inner +product of functions in Y0. In such case, M is symmetric and positive definite and thus +has only the trivial nullspace. This means N(D) = N(∇ · RT 0) and N(DT ) = {0}. +The RT 0×Y0 discretization is then non-singular and D has only the physical nullspace +associated with the divergence operator. +The takeaway is that for the pairing W1 × Y0, W1 is rich enough to ensure non- +singularity but it is too rich with respect to Y0 such that spurious modes are allowed. +The W1 × Y1 discretization has the opposite problem: while ∇ · W1 is small enough +with respect to Y1 to avoid spurious modes it is too small to allow the block system to +be non-singular. In general, Yp ⊂ ∇·Wp ⊂ Yp+1 and thus one must always compromise +between solvability and avoiding spurious modes. On the other hand, ∇ · RT p = Yp so +that the RT p×Yp discretization is both solvable and does not allow spurious solutions. +Appendix C. Method of Manufactured Solutions Supplemental Data +60 + +TABLE C1. +Error values from an isotropic MMS test problem. The H1, RT, and HRT columns +refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, +respectively. The error in the scalar flux, the error in the scalar flux when the exact +solution is first projected onto Yp, and the error in the current are presented for each +method over a range of values of p. Here, the VEF data are constant in space and +thus are represented exactly. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +h +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +1 +3.994 × 10−2 +4.160 × 10−4 +4.161 × 10−4 +4.161 × 10−4 +9.853 × 10−6 +1.067 × 10−5 +1.067 × 10−5 +5.605 × 10−4 +1.251 × 10−3 +1.251 × 10−3 +1.997 × 10−2 +1.040 × 10−4 +1.040 × 10−4 +1.040 × 10−4 +1.209 × 10−6 +1.260 × 10−6 +1.260 × 10−6 +1.399 × 10−4 +3.125 × 10−4 +3.125 × 10−4 +1.331 × 10−2 +4.624 × 10−5 +4.624 × 10−5 +4.624 × 10−5 +3.570 × 10−7 +3.693 × 10−7 +3.693 × 10−7 +6.217 × 10−5 +1.389 × 10−4 +1.389 × 10−4 +9.985 × 10−3 +2.601 × 10−5 +2.601 × 10−5 +2.601 × 10−5 +1.505 × 10−7 +1.552 × 10−7 +1.552 × 10−7 +3.497 × 10−5 +7.812 × 10−5 +7.812 × 10−5 +2 +5.874 × 10−2 +1.407 × 10−5 +1.411 × 10−5 +1.411 × 10−5 +7.222 × 10−7 +1.301 × 10−6 +1.301 × 10−6 +2.718 × 10−5 +1.629 × 10−4 +1.629 × 10−4 +3.026 × 10−2 +1.921 × 10−6 +1.922 × 10−6 +1.922 × 10−6 +4.412 × 10−8 +8.331 × 10−8 +8.331 × 10−8 +3.236 × 10−6 +2.254 × 10−5 +2.254 × 10−5 +1.997 × 10−2 +5.521 × 10−7 +5.523 × 10−7 +5.523 × 10−7 +8.065 × 10−9 +1.542 × 10−8 +1.542 × 10−8 +8.901 × 10−7 +6.495 × 10−6 +6.495 × 10−6 +1.490 × 10−2 +2.295 × 10−7 +2.295 × 10−7 +2.295 × 10−7 +2.466 × 10−9 +4.740 × 10−9 +4.740 × 10−9 +3.626 × 10−7 +2.701 × 10−6 +2.701 × 10−6 +3 +7.681 × 10−2 +9.628 × 10−7 +9.905 × 10−7 +9.905 × 10−7 +9.972 × 10−8 +2.604 × 10−7 +2.604 × 10−7 +3.951 × 10−6 +3.112 × 10−5 +3.112 × 10−5 +3.994 × 10−2 +7.071 × 10−8 +7.112 × 10−8 +7.112 × 10−8 +3.471 × 10−9 +8.727 × 10−9 +8.727 × 10−9 +2.818 × 10−7 +2.231 × 10−6 +2.231 × 10−6 +2.628 × 10−2 +1.326 × 10−8 +1.329 × 10−8 +1.329 × 10−8 +4.149 × 10−10 +1.043 × 10−9 +1.043 × 10−9 +5.275 × 10−8 +4.175 × 10−7 +4.175 × 10−7 +1.997 × 10−2 +4.426 × 10−9 +4.432 × 10−9 +4.432 × 10−9 +1.041 × 10−10 +2.619 × 10−10 +2.619 × 10−10 +1.762 × 10−8 +1.393 × 10−7 +1.393 × 10−7 +4 +9.986 × 10−2 +3.563 × 10−7 +3.566 × 10−7 +3.566 × 10−7 +4.665 × 10−8 +4.712 × 10−8 +4.712 × 10−8 +1.261 × 10−6 +4.258 × 10−6 +4.258 × 10−6 +4.993 × 10−2 +1.155 × 10−8 +1.157 × 10−8 +1.157 × 10−8 +7.186 × 10−10 +8.170 × 10−10 +8.170 × 10−10 +3.640 × 10−8 +2.027 × 10−7 +2.027 × 10−7 +3.328 × 10−2 +1.524 × 10−9 +1.525 × 10−9 +1.525 × 10−9 +6.290 × 10−11 +6.896 × 10−11 +6.896 × 10−11 +4.554 × 10−9 +2.712 × 10−8 +2.712 × 10−8 +2.496 × 10−2 +3.619 × 10−10 +3.619 × 10−10 +3.619 × 10−10 +1.119 × 10−11 +1.209 × 10−11 +1.210 × 10−11 +1.089 × 10−9 +6.473 × 10−9 +6.473 × 10−9 +TABLE C2. +Error values from a quadratically anisotropic MMS test problem. The H1, RT, and +HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p +discretizations, respectively. The error in the scalar flux, the error in the scalar flux +when the exact solution is first projected onto Yp, and the error in the current are +presented for each method over a range of values of p. Here, the angular flux used to +calculate the VEF data is represented with Yp. Due to this, the maximum accuracy +expected is order p + 1. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +h +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +1 +3.994 × 10−2 +1.891 × 10−3 +1.891 × 10−3 +1.890 × 10−3 +2.589 × 10−4 +2.725 × 10−4 +2.638 × 10−4 +7.454 × 10−3 +1.703 × 10−2 +1.499 × 10−2 +1.997 × 10−2 +4.707 × 10−4 +4.708 × 10−4 +4.707 × 10−4 +4.915 × 10−5 +5.110 × 10−5 +4.979 × 10−5 +1.925 × 10−3 +8.746 × 10−3 +7.651 × 10−3 +1.331 × 10−2 +2.090 × 10−4 +2.090 × 10−4 +2.090 × 10−4 +1.980 × 10−5 +2.035 × 10−5 +2.004 × 10−5 +8.786 × 10−4 +5.866 × 10−3 +5.124 × 10−3 +9.985 × 10−3 +1.175 × 10−4 +1.175 × 10−4 +1.175 × 10−4 +1.061 × 10−5 +1.082 × 10−5 +1.066 × 10−5 +5.077 × 10−4 +4.410 × 10−3 +3.850 × 10−3 +2 +5.874 × 10−2 +2.793 × 10−4 +2.787 × 10−4 +2.826 × 10−4 +7.262 × 10−5 +7.092 × 10−5 +8.536 × 10−5 +1.373 × 10−3 +1.125 × 10−3 +1.165 × 10−3 +3.026 × 10−2 +3.994 × 10−5 +3.992 × 10−5 +4.028 × 10−5 +9.827 × 10−6 +9.746 × 10−6 +1.114 × 10−5 +2.745 × 10−4 +2.032 × 10−4 +2.165 × 10−4 +1.997 × 10−2 +1.158 × 10−5 +1.157 × 10−5 +1.160 × 10−5 +2.853 × 10−6 +2.843 × 10−6 +2.931 × 10−6 +9.552 × 10−5 +7.032 × 10−5 +7.532 × 10−5 +1.490 × 10−2 +4.826 × 10−6 +4.825 × 10−6 +4.864 × 10−6 +1.194 × 10−6 +1.192 × 10−6 +1.343 × 10−6 +4.534 × 10−5 +3.348 × 10−5 +3.690 × 10−5 +3 +7.681 × 10−2 +8.143 × 10−5 +8.129 × 10−5 +8.124 × 10−5 +1.357 × 10−5 +1.308 × 10−5 +1.261 × 10−5 +4.150 × 10−4 +3.580 × 10−4 +3.349 × 10−4 +3.994 × 10−2 +5.639 × 10−6 +5.638 × 10−6 +5.638 × 10−6 +7.684 × 10−7 +7.765 × 10−7 +7.752 × 10−7 +3.053 × 10−5 +5.762 × 10−5 +5.128 × 10−5 +2.628 × 10−2 +1.050 × 10−6 +1.050 × 10−6 +1.051 × 10−6 +1.256 × 10−7 +1.270 × 10−7 +1.310 × 10−7 +5.681 × 10−6 +1.708 × 10−5 +1.505 × 10−5 +1.997 × 10−2 +3.498 × 10−7 +3.498 × 10−7 +3.499 × 10−7 +3.895 × 10−8 +3.934 × 10−8 +4.032 × 10−8 +1.888 × 10−6 +7.609 × 10−6 +6.672 × 10−6 +4 +9.986 × 10−2 +1.468 × 10−5 +1.464 × 10−5 +1.473 × 10−5 +4.023 × 10−6 +3.433 × 10−6 +3.797 × 10−6 +5.720 × 10−5 +5.987 × 10−5 +6.984 × 10−5 +4.993 × 10−2 +5.743 × 10−7 +5.738 × 10−7 +5.753 × 10−7 +1.097 × 10−7 +1.077 × 10−7 +1.163 × 10−7 +3.569 × 10−6 +3.667 × 10−6 +3.399 × 10−6 +3.328 × 10−2 +7.940 × 10−8 +7.937 × 10−8 +8.009 × 10−8 +1.530 × 10−8 +1.514 × 10−8 +1.858 × 10−8 +5.995 × 10−7 +5.478 × 10−7 +5.495 × 10−7 +2.496 × 10−2 +1.911 × 10−8 +1.910 × 10−8 +1.926 × 10−8 +3.758 × 10−9 +3.737 × 10−9 +4.489 × 10−9 +1.618 × 10−7 +1.428 × 10−7 +1.432 × 10−7 +61 + +TABLE C3. +Error values from a quadratically anisotropic MMS test problem. The H1, RT, and +HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p +discretizations, respectively. The error in the scalar flux, the error in the scalar flux +when the exact solution is first projected onto Yp, and the error in the current are +presented for each method over a range of values of p. Here, the angular flux used to +calculate the VEF data is represented with Yp+1. Due to this, the maximum +accuracy expected is order p + 2. +∥ϕ − ϕex∥ +∥ϕ − Πϕex∥ +∥J − Jex∥ +p +h +H1 +RT +HRT +H1 +RT +HRT +H1 +RT +HRT +0 +1.997 × 10−2 +1.564 × 10−2 +1.564 × 10−2 +1.564 × 10−2 +5.329 × 10−4 +5.301 × 10−4 +5.172 × 10−4 +7.910 × 10−3 +1.028 × 10−2 +1.028 × 10−2 +9.985 × 10−3 +7.827 × 10−3 +7.827 × 10−3 +7.827 × 10−3 +1.309 × 10−4 +1.323 × 10−4 +1.291 × 10−4 +2.849 × 10−3 +5.138 × 10−3 +5.138 × 10−3 +6.657 × 10−3 +5.219 × 10−3 +5.219 × 10−3 +5.219 × 10−3 +5.784 × 10−5 +5.878 × 10−5 +5.737 × 10−5 +1.564 × 10−3 +3.425 × 10−3 +3.424 × 10−3 +4.993 × 10−3 +3.915 × 10−3 +3.914 × 10−3 +3.914 × 10−3 +3.244 × 10−5 +3.305 × 10−5 +3.226 × 10−5 +1.021 × 10−3 +2.568 × 10−3 +2.568 × 10−3 +1 +3.994 × 10−2 +1.876 × 10−3 +1.875 × 10−3 +1.875 × 10−3 +1.032 × 10−4 +1.097 × 10−4 +9.773 × 10−5 +4.561 × 10−3 +3.443 × 10−3 +1.350 × 10−3 +1.997 × 10−2 +4.683 × 10−4 +4.683 × 10−4 +4.683 × 10−4 +1.275 × 10−5 +1.422 × 10−5 +1.256 × 10−5 +1.210 × 10−3 +1.741 × 10−3 +3.569 × 10−4 +1.331 × 10−2 +2.081 × 10−4 +2.081 × 10−4 +2.081 × 10−4 +3.764 × 10−6 +4.275 × 10−6 +3.754 × 10−6 +5.459 × 10−4 +1.166 × 10−3 +1.675 × 10−4 +9.985 × 10−3 +1.171 × 10−4 +1.171 × 10−4 +1.171 × 10−4 +1.586 × 10−6 +1.826 × 10−6 +1.594 × 10−6 +3.090 × 10−4 +8.758 × 10−4 +9.902 × 10−5 +2 +5.874 × 10−2 +2.717 × 10−4 +2.714 × 10−4 +2.714 × 10−4 +2.914 × 10−5 +2.706 × 10−5 +2.750 × 10−5 +5.506 × 10−4 +2.855 × 10−4 +2.109 × 10−4 +3.026 × 10−2 +3.878 × 10−5 +3.877 × 10−5 +3.877 × 10−5 +2.075 × 10−6 +1.847 × 10−6 +1.917 × 10−6 +7.526 × 10−5 +3.918 × 10−5 +3.010 × 10−5 +1.997 × 10−2 +1.123 × 10−5 +1.123 × 10−5 +1.123 × 10−5 +3.945 × 10−7 +3.488 × 10−7 +3.639 × 10−7 +2.050 × 10−5 +1.148 × 10−5 +9.074 × 10−6 +1.490 × 10−2 +4.677 × 10−6 +4.677 × 10−6 +4.677 × 10−6 +1.224 × 10−7 +1.081 × 10−7 +1.130 × 10−7 +8.233 × 10−6 +4.894 × 10−6 +3.947 × 10−6 +3 +7.681 × 10−2 +8.049 × 10−5 +8.039 × 10−5 +8.038 × 10−5 +5.467 × 10−6 +4.567 × 10−6 +4.109 × 10−6 +2.767 × 10−4 +9.063 × 10−5 +4.817 × 10−5 +3.994 × 10−2 +5.591 × 10−6 +5.589 × 10−6 +5.589 × 10−6 +2.175 × 10−7 +2.337 × 10−7 +2.064 × 10−7 +2.136 × 10−5 +1.636 × 10−5 +4.094 × 10−6 +2.628 × 10−2 +1.043 × 10−6 +1.043 × 10−6 +1.043 × 10−6 +2.684 × 10−8 +3.006 × 10−8 +2.624 × 10−8 +4.109 × 10−6 +5.061 × 10−6 +9.155 × 10−7 +1.997 × 10−2 +3.477 × 10−7 +3.477 × 10−7 +3.477 × 10−7 +6.806 × 10−9 +7.758 × 10−9 +6.731 × 10−9 +1.386 × 10−6 +2.288 × 10−6 +3.453 × 10−7 +62 + diff --git a/atE3T4oBgHgl3EQf2wun/content/tmp_files/load_file.txt b/atE3T4oBgHgl3EQf2wun/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc346f837068f3d4b5ee0d2f752580ffcc990d9b --- /dev/null +++ b/atE3T4oBgHgl3EQf2wun/content/tmp_files/load_file.txt @@ -0,0 +1,2504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf,len=2503 +page_content='High-Order Mixed Finite Element Variable Eddington Factor Methods Samuel Oliviera,b,* and Terry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Hautc aUniversity of California, Berkeley, Berkeley, CA, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' bLos Alamos National Laboratory, Los Alamos, NM, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' cLawrence Livermore National Laboratory, Livermore, CA, USA ARTICLE HISTORY Compiled January 13, 2023 ABSTRACT We apply high-order mixed finite element discretization techniques and their as- sociated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Dis- crete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Lab- oratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vec- tor variable in the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the cou- pled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' KEYWORDS radiation transport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Variable Eddington Factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Quasidiffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' high-order finite elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' preconditioned iterative solvers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Introduction The Variable Eddington Factor (VEF) method [1], also known as Quasidiffusion [2], is an efficient iterative method for solving the Boltzmann transport equation, a crucial component in the modeling of nuclear reactors, high energy density physics experi- ments, astrophysical phenomena, and medical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In VEF, the transport equa- tion is iteratively coupled to the VEF equations, a moment-based, reduced-dimensional model of transport formed through discrete closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF closures are weak func- tions of the transport solution allowing the design of rapidly converging and robust iterative schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A key advantage of VEF is that the discretized VEF and transport corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Email: solivier@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='gov arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='04758v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='NA] 11 Jan 2023 equations do not need to be algebraically consistent to maintain this rapid conver- gence, even in the thick diffusion limit [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These so-called independent VEF methods [4] then have the flexibility to choose the discretization of the VEF equations to meet the requirements of the overall algorithm, such as computational efficiency and multi- physics compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mixed finite element methods are a class of discretization techniques for solving the mixed variational form of a partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This variational form is char- acterized by the inclusion of multiple (typically two) physically disparate quantities resulting in a saddle point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' By contrast, primal formulations operate on a single quantity and produce minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mixed methods were invented to 1) allow incorporation of a constraint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' divergence free velocity in fluid flow), 2) provide direct access to an intermediate variable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' the stress in elasticity), and 3) allow a weaker formulation than the corresponding primal formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In the context of neutron diffusion, mixed methods are applied to the first-order, or P1, form of ra- diation diffusion and 1) explicitly include the constraint of particle balance, 2) solve for the current in addition to the scalar flux, and 3) allow scalar flux solutions with no continuity requirements at interior element interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Through a process called mixed finite element hybridization, the resulting block system of equations can be reduced to a positive definite system that can be efficiently solved with Algebraic Multigrid (AMG) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The block system can also be directly preconditioned through block diagonal and lower block triangular preconditioners based on applying AMG to an approximate Schur complement [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this paper, we investigate the use of mixed finite elements to solve the VEF equations in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The research goals are to achieve high-order accuracy, compatibil- ity with high-order (curved) meshes, and scalable preconditioned iterative solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The motivation for this research is that high-order mixed finite element methods on curved meshes are used in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL) [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, we are interested in designing a discretiza- tion of the VEF equations that matches as closely as possible to that of Maginot and Brunner [9], the mixed finite element method used for radiation diffusion at LLNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Such a method would 1) have element-local particle balance, 2) solve for the current directly potentially leading to high accuracy coupling to the hydrodynamics’ momen- tum equation, and 3) allow the scalar flux to be approximated in the same finite element space as the hydrodynamics’ thermodynamic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, a mixed finite element VEF discretization could serve as a drop-in replacement for radiation diffusion at LLNL providing a transport algorithm that allows reuse of the linear and nonlinear solvers already in place for diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mixed finite element discretizations of radiation diffusion have also been used in reactor analysis [10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Olivier and Morel [13] developed a lowest-order, hybridized mixed finite element dis- cretization of the VEF equations in one spatial dimension for the linear transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [14] and Lou and Morel [15] used this algorithm to form efficient, VEF-based thermal radiative transfer and radiation-hydrodynamics algorithms, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Multi-dimensional, high-order Discontinuous Galerkin VEF discretizations with scalable solvers were developed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, such methods do not directly solve for the current, precluding the possibility of high-order coupling to the momen- tum equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Furthermore, a mixed finite element VEF discretization has immediate mathematical and implementational compatibility with the mixed methods used in the hydrodynamics framework of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2 Here, we extend the lowest-order mixed finite element discretization from Olivier and Morel [13] to high-order accuracy in two spatial dimensions, prescribe efficient pre- conditioned iterative solvers for the discretized VEF equations, and compare the per- formance of the hybridized and unhybridized mixed finite elements techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The extension to multiple dimensions is non-trivial due to the elevation of the VEF closure from a scalar in one spatial dimension to a symmetric tensor in multiple dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This introduces complications in approximating the vector-valued neutron current that prevent a straightforward extension of the one-dimensional discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' An early form of this work was presented in Olivier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF algorithm is introduced analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We present background on high-order meshes and the finite element spaces used to solve the VEF and transport equations numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We discuss the algorithmic connec- tions between the VEF discretizations and a high-order DG discretization of the SN transport equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We then apply mixed finite element techniques to the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We show that, due to the presence of the Eddington tensor in the VEF first moment equation, the standard Raviart Thomas (RT) mixed finite el- ement methods [20, 21] are not appropriate for the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We present two alternatives: a method where each component of the current is approximated with continuous finite elements and a non-conforming approach where the RT space is used along with DG-like numerical fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A lower block triangular preconditioner that uses a combination of classical smoothing and AMG is defined for each of the above VEF discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We then propose a hybridized version of the RT method that increases the computational efficiency of the RT method by reducing the number of globally coupled unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Next, numerical results are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We investigate the accuracy of the methods on a third-order mesh using the method of manufactured solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Convergence of the fixed-point iteration is tested in the thick diffusion limit on both an orthogonal mesh and a severely distorted third-order mesh generated using a Lagrangian hydrodynam- ics code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The robustness of the preconditioned iterative solvers to mesh distortion is also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The efficiencies of the outer fixed-point iteration and inner precon- ditioned linear iteration are compared on a challenging, multi-material problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We then document the degraded solution quality associated with the method that uses continuous finite elements for each component of the current on a simple radiation diffusion eigenvalue problem and provide intuition for AMG’s inability to effectively precondition the corresponding linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A weak scaling study is presented for the RT and hybridized RT methods showing that the discrete VEF systems can be solved with similar efficiency to that of an analogous symmetric radiation diffusion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, we give conclusions and recommendations for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' VEF Algorithm Here, we describe the VEF method for the steady-state, mono-energetic, fixed-source transport problem with isotropic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' VEF methods simultaneously solve the coupled transport and VEF equations given by: Ω · ∇ψ + σtψ = σs 4πϕ + q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (1a) 3 ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) = ¯ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ ∂D and Ω · n < 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (1b) ∇ · J + σaϕ = Q0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (2a) ∇ · (Eϕ) + σtJ = Q1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (2b) J · n = Ebϕ + 2Jin ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ ∂D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (2c) where ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) is the angular flux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ϕ(x) and J(x) the VEF scalar flux and cur- rent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω ∈ S2 the direction of particle flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D the spatial domain with ∂D its boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' σt(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' σs(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' and σa(x) = σt(x) − σs(x) the total,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' scattering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' and absorption macroscopic cross sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) the fixed source with Qi = � Ωiq dΩ its angular moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' and ¯ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) the boundary inflow function with Jin(x) = � Ω·n<0 Ω · n ¯ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ω) dΩ the inflow partial current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Eddington tensor and boundary factor are defined as: E = � Ω ⊗ Ω ψ dΩ � ψ dΩ , (3a) Eb = � |Ω · n| ψ dΩ � ψ dΩ , (3b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Miften and Larsen [22] boundary conditions are used for the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Observe that the transport equation (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 1) is linearly coupled to the VEF equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2) through the scattering source and the VEF equations are nonlinearly coupled to the transport equation through the Eddington tensor and boundary factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF closures are weak functions of the transport solution allowing the design of efficient iterative solution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The simplest algorithm to solve the coupled transport-VEF system is fixed-point it- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The iteration is: 1) invert the transport equation given a scattering source from the previous iteration or an initial guess, 2) compute the Eddington tensor and boundary factor from the angular flux from stage 1), and 3) invert the VEF equations for a new scalar flux and current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The iteration is repeated until the VEF scalar flux converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A discussion of the derivation of the above system and the application of advanced nonlinear solvers, such as Anderson acceleration, are provided in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The following sections present methods for efficiently computing the VEF fixed-point operator numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We present the description of high-order meshes and the proce- dure for integration over arbitrary elements, the finite element spaces used to discretize the transport and VEF equations, the representation of the VEF data using finite ele- ment interpolation and angular quadrature, and finally three novel mixed finite element discretizations for the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 (a) ξ η (0,0) (1,0) (0,1) (1,1) T(ξ) T−1(x) x y (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. Depictions of (a) the mesh control points in a quadratic quadrilateral mesh and (b) the reference transformation used to describe the left element of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mesh and Finite Element Preliminaries This section provides background on the representation of high-order meshes and the transformations used to facilitate numerical integration over arbitrary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We pay particular attention to the transformation of vector-valued functions and their gradient and divergence on high-order meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These transformations are crucial for the implementation of the approximation techniques used for the current described in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Description of the Mesh The domain D ⊂ R2 is tesselated into a collection T of quadrilateral elements Ke such that D = � Ke∈T Ke .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (4) Each element Ke is obtained as Ke = Te( ˆK) where ˆK = [0, 1]2 is the reference element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let Qm,n( ˆK) be the space of polynomials of degree less than or equal to m and n in the first and second variables, respectively, with Qm( ˆK) = Qm,m( ˆK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The mapping Te ∈ [Qm( ˆK)]2 is derived from a set of global control points and an element-local nodal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Figure 1a shows an example mesh where the control points labeled 2, 7, and 12 are shared so that the mesh coordinates are continuous across the interior interface between the two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A nodal basis for Qm( ˆK) is defined using Lagrange interpolating polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let ξi denote the m + 1 Gauss-Lobatto points in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The (m + 1)2 points ξi on the unit square ˆK = [0, 1]2 are given by the two-fold Cartesian product of the one- dimensional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let ℓi denote the Lagrange interpolating polynomial that satisfies ℓi(ξj) = δij where δij is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The set of functions {ℓi} form a basis for Qm( ˆK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For each element, the mapping is then x(ξ) = Te(ξ) = (m+1)2 � i=1 xe,iℓi(ξ) (5) 5 where x ∈ Ke, ξ ∈ ˆK, and xe,i are the control points corresponding to element Ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Figure 1b depicts the mesh transformation used for the left element in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We define Γ as the set of unique faces in the mesh with Γ0 = Γ \\ ∂D the set of interior faces and Γb = Γ ∩ ∂D the set of boundary faces so that Γ = Γ0 ∪ Γb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We denote the outward unit normal to element K as nK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On an interior face F ∈ Γ0 between elements K1 and K2, we use the convention that n is the unit vector perpendicular to the shared face K1 ∩ K2 pointing from K1 to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On such an interior face, the jump, �·�, and average, {{·}}, are defined as �u� = u1 − u2 , {{u}} = 1 2(u1 + u2) , on F ∈ Γ0 , (6) where ui = u|∂Ki with analogous definitions for vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that a continuous func- tion u satisfies �u� = 0 on each interior face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On boundary faces, the jump and average are set to �u� = u , {{u}} = u , on F ∈ Γb , (7) and likewise for vector-valued functions on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The tangent vector is denoted by τ and we use the convection that τ is a 90◦ counter-clockwise rotation of the normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, we define the “broken” gradient, denoted by ∇h, obtained by applying the gradient locally on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, (∇hu)|K = ∇(u|K) , ∀K ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (8) This distinction is important for the piecewise polynomial spaces discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Integration Transformations The mesh transformations Te are used to facilitate numerical integration on arbi- trary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Letting ξ = �ξ η�T ∈ ˆK denote the reference coordinates and x = �x y�T ∈ D the physical coordinates such that x(ξ) = Te(ξ), the Jacobian of the transformation is Fe = ∂Te ∂ξ = � ∂x ∂ξ ∂x ∂η ∂y ∂ξ ∂y ∂η � , (9) with Je = |Fe| its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The partial derivatives of the mesh transformation are computed by taking derivatives of the nodal basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, Fe = (m+1)2 � i=1 xe,i ⊗ ˆ∇ℓi = (m+1)2 � i=1 � xe,i ∂ℓi ∂ξ xe,i ∂ℓi ∂η ye,i ∂ℓi ∂ξ ye,i ∂ℓi ∂η � , (10) where xe,i = �xe,i ye,i �T and ˆ∇ denotes the gradient with respect to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6 A mesh transformation is called affine when it can be written as T = Aξ + b (11) where A ∈ R2×2 and b ∈ R2 are constant with respect to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In such case, the Jacobian matrix is F = A and the Hessian of the transformation, defined as ∂2T ∂ξ2 , is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Quadrilateral elements obtained by scaling, stretching along the ξ or η axes, or rotating the reference element are all affine while general quadrilateral elements, such as trapezoidal elements, and curved elements are not affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this document, integration over the domain is implicitly computed in reference space using the following sum: � D (·) dx = � K∈T � K (·) dx = � K∈T � ˆ K (·) Jdξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (12) This provides a systematic way to integrate over arbitrary domains composed of ar- bitrarily shaped elements as well as the use of numerical quadrature rules defined on the reference element ˆK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We now discuss the transformations used to represent the integrand of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 12 in reference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For a scalar function u : D → R, denote by ˆu : ˆK → R its representation in reference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The functions u and ˆu are related by u(x) = ˆu(T−1(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (13) Integration over the physical element is then equivalent to � K u dx = � ˆ K ˆu Jdξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (14) Using the chain rule, the gradient of a scalar function transforms as ∇ˆu = � ∂ˆu ∂ξ ∂ξ ∂x + ∂ˆu ∂η ∂η ∂x ∂ˆu ∂ξ ∂ξ ∂y + ∂ˆu ∂η ∂η ∂y � = F−T ˆ∇ˆu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (15) In this way, the gradient in physical space can be computed using the Jacobian of the mesh transformation and the gradient in reference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For vector-valued functions, the basis the vector is defined on must also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The simplest basis is the canonical basis, ei, corresponding to the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this case, a vector v : D → R2 is v = v1e1 + v2e2 (16) and each component transforms independently as vi = ˆvi(T−1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Writing ∇v = � ∂v1 ∂x ∂v1 ∂y ∂v2 ∂x ∂v2 ∂y � = � (∇v1)T (∇v2)T � , (17) 7 the gradient of a vector defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 16 transforms as ∇v = � (F−T ˆ∇ˆv1)T (F−T ˆ∇ˆv2)T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (18) Note that defining a vector in this way does not preserve the normal or tangential components under a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, v · n and v · τ are linear combinations of the vi instead of a single component representing the normal or tangential components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Alternatively, the contravariant Piola transform represents vectors on the so-called tangent basis so that the normal component can be preserved [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Such a trans- formation is required by the Raviart Thomas space introduced in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3 in order to strongly enforce continuity in the normal component of the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The contravariant Piola transform is: v = 1 J Fˆv ◦ T−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (19) Here, ˆv : ˆK → R2 is a vector in reference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Writing the columns of the Jacobian matrix as F = � t1 t2 � , (20) the contravariant Piola transformation is equivalent to v = 1 J (ˆv1t1 + ˆv2t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (21) Observe that, on the reference canonical basis ˆei, ˆv = ˆv1ˆe1 + ˆv2ˆe2, and thus the contravariant Piola transform maps the canonical reference basis to the tangent space spanned by {t1, t2} and scales by 1/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' When the mesh transformation Te is not affine, the tangent basis is not orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this case, the usual method of selecting components of a vector through the dot product (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' vi = ti · v) is inappropriate since ti · tj ̸= δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Instead, a dual basis, referred to as the cotangent basis, is constructed such that ni · tj = δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (22) Vectors the satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 22 are called bi-orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the ti are the columns of the Jacobian matrix, defining the cotangent basis as the rows of the inverse of the Jacobian matrix satisfies the bi-orthonormality condition since F−1F = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, the cotangent basis is defined such that F−1 = � nT 1 nT 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (23) For a contravariant vector, the usual method of selecting a component is now replaced with vi = ni·v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The cotangent space is associated with vectors normal to the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' By representing a vector on the tangent space, the contravariant Piola transform allows 8 ˆe1 ˆe2 T(ξ) T−1(x) t1 n1 t2 n2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of the tangent and cotangent bases at the point ξ = (0, 0) under a non-affine mesh transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' selection of the component representing the normal component through n · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that for non-affine meshes, F depends on ξ and thus the tangent and cotangent bases also depend on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Figure 2 depicts an example non-affine mesh transformation and the tangent and cotangent bases evaluated at the point ξ = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Observe that the pairs (t1, n2) and (t2, n1) are perpendicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The pairs (t1, n1) and (t2, n2) do not point in the same direction but their magnitudes and directions balance so that ti · ni = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, the bi-orthonormality condition ni · tj = δij is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, the tangent vectors and cotangent vectors are tangential and normal, respectively, to one of the faces connecting at the point ξ = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For a contravariant vector, � K ∇u · v dx = � ˆ K F−T ˆ∇ˆu · 1 J Fˆv Jdξ = � ˆ K ˆ∇ˆu · ˆv dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (24) The gradient transforms as ∇v = ∇ � 1 J Fˆv ◦ T−1 � = 1 J F � ˆ∇ˆv − ˆB � F−1 (25) where ˆB = 1 J ˆ∇ � JF−1� Fˆv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (26) This result is derived by direct computation in Appendix A along with the details required to implement this transformation using the machinery commonly provided in finite element libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' It is also shown that ˆB = 0 when the mesh transformation is affine and that trace( ˆB) = 0 for any transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This last result is known as the Piola identity [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Using the Piola identity, the linearity of the trace, and the invariance of the trace under similarity transformations, the divergence transforms as ∇ · v = trace (∇v) = 1 J trace � F � ˆ∇ˆv − ˆB � F−1� = 1 J ˆ∇ · ˆv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (27) Thus, � K u ∇ · v dx = � ˆ K ˆu ˆ∇ · ˆv dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (28) 9 Combining the results from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 24 and 28 yields: � ∂K u v · n ds = � ∂ ˆ K ˆu ˆv · ˆn dˆs , (29) where ˆn is the normal vector in reference space corresponding to the physical space normal n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, the contravariant Piola transformation preserves the normal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this document, integration is implicitly computed using numerical quadrature on the reference element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Integration over surfaces is performed over the one-dimensional reference element using the transformed element of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finite Element Spaces In this section, we define the finite element spaces used to approximate the VEF equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These finite element spaces are defined on the mesh T or the interior skeleton of the mesh Γ0 and consist of an element-local function space and a set of inter-element matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The inter-element matching conditions enforce various types of continuity of the solution between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The combination of a locally smooth func- tion space and suitable matching conditions allows finite element spaces to be discrete subspaces of Sobolev spaces such as L2(D), H1(D), and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The following sub- sections define the element-local function space and matching conditions used for the scalar flux, current, and the interface variable used later in hybridization of the mixed finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Discontinuous Galerkin The Discontinuous Galerkin (DG) space is a discrete subspace of L2(D), the space of square-integrable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, if u is an element of the DG space, � u2 dx < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (30) Since only square integrability is required, functions in L2(D), and thus DG spaces, do not need to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' DG functions are represented using piecewise-discontinuous polynomials that are defined on the reference element and mapped to the physical element using the inverse mesh transformation T−1 e : Ke → ˆK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, on each element, the solution belongs to: Qp(Ke) = {u = ˆu ◦ T−1 e : ˆu ∈ Qp( ˆK)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (31) The distinction between Qp( ˆK) and Qp(Ke) is important for non-affine mesh trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In such case, the inverse mesh transformation is generally non-polynomial so that the composition u = ˆu ◦ T−1 e is also non-polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The degree-p DG space is Yp = {u ∈ L2(D) : u|K ∈ Qp(K) , ∀K ∈ T } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (32) 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of the distribution of degrees of freedom in the linear DG space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Legendre nodes are used to illustrate that degrees of freedom are not shared between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' An example of the distribution of the degrees of freedom in a linear DG space on a 3 × 3 mesh is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that degrees of freedom are not shared between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since there are no continuity requirements in the DG space, the basis for the local polynomials can use either open or closed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, a nodal basis can be formed with Lagrange interpolating polynomials through the two-fold Cartesian product of either the closed Gauss-Lobatto points or the open Gauss-Legendre points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [H1(D)]2 Here, we define a discrete subspace of H1(D), the space of functions in L2(D) with square-integrable gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let the degree-p, scalar continuous finite element space be Vp = {u ∈ C0(D) : u|K ∈ Qp(K) , ∀K ∈ T } (33) so that each function u ∈ Vp is a piecewise-continuous polynomial mapped from the reference element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since u ∈ Vp is locally smooth and Vp ⊂ C0(D), it can be shown that Vp ⊂ H1(D) and, in particular, that u ∈ Vp satisfies ∇u = ∇hu ∈ [L2(D)]2 [25, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The distribution of degrees of freedom for the space V2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, continuity is enforced by sharing degrees of freedom between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, a nodal basis using closed points, such as the Gauss-Lobatto points, must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The vector-valued analog Wp = {v : v1 ∈ Vp and v2 ∈ Vp} (34) uses the scalar continuous finite element space for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this way, v ∈ Wp ⊂ [H1(D)]2 and thus ∇v = ∇hv ∈ [L2(D)]2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since each component is defined independently using the scalar space, vectors v ∈ Wp transform according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of the distribution of degrees of freedom for the quadratic continuous finite element space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Continuity of members of the finite element space is enforced by sharing degrees of freedom across neighboring elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Raviart Thomas The Raviart Thomas (RT) space is a discrete subspace of H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D), the space of vector-valued functions with square-integrable divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) = {v ∈ [L2(D)]2 : ∇ · v ∈ L2(D)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (35) The requirements of a discrete subspace are codified in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Proposition 1 (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Quarteroni and Valli [25], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let v : D → R2 be such that (a) v|K ∈ [H1(K)]2 for each K ∈ T (b) �v · n� = 0 for each F ∈ Γ0 then v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Conversely, if v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) and (a) is satisfied, then (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' It must be shown that, given (a) and (b), ∇ · v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We proceed by leveraging the fact that ∇h · v ∈ L2(D) (since v|K ∈ [H1(K)]2) and then show that ∇ · v = ∇h · v, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let C∞ 0 (D) be the space of infinitely differentiable functions that are zero on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Using Green’s identity, we have that for each u ∈ C∞ 0 (D): � u ∇ · v dx = − � ∇u · v dx = − � K∈T � K ∇u · v|K dx , (36) where we have used that u = 0 for each x ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since v|K ∈ [H1(K)]2 for each K, 12 we can integrate by parts locally on each element to give: � u ∇ · v dx = � K∈T �� K u ∇ · v|K dx − � ∂K u v · n ds � = � u ∇h · v dx − � Γ0 u �v · n� ds = � u ∇h · v dx , ∀u ∈ C∞ 0 (D) , (37) since u ∈ C∞ 0 (D) satisfies is continuous such that �u� = 0 and �v · n� = 0 from (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Therefore, ∇ · v = ∇h · v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the other hand, if v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) then ∇ · v = ∇h · v and, given v|K ∈ [H1(K)]2, we obtain � Γ0 u �v · n� ds = 0 , ∀u ∈ C∞ 0 (D) , (38) hence, (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, a discrete subspace of H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) must (a) have a smooth function space on each element and (b) have suitable matching conditions so that the normal component is continuous across interior mesh interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT space uses the local polynomial space Qp+1,p( ˆK) × Qp,p+1( ˆK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This choice can be motivated by the discrete de Rham complex [26] in that Qp+1( ˆK) ˆ∇× −−→ Qp+1,p( ˆK) × Qp,p+1( ˆK) ˆ∇· −→ Qp( ˆK) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (39) As an example, the lowest-order polynomial space is Q1,0( ˆK) × Q0,1( ˆK) = span �� 1 0 � , � ξ 0 � , � 0 1 � , � 0 η �� , (40) and thus we have that: ˆ∇ · Q1,0( ˆK) × Q0,1( ˆK) = span{1} = Q0( ˆK) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (41) The nodal basis for Qp+1,p( ˆK) × Qp,p+1( ˆK) uses the closed Gauss-Lobatto points in the normal direction and the open Gauss-Legendre points in the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The interpolating points for the first three orders are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The circles denote degrees of freedom corresponding to the ξ component while squares denote the η component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The contravariant Piola transformation is used to allow sharing the degrees of free- dom associated with the normal component with neighboring elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that this transformation is still required even when continuity in the normal component is re- laxed and enforced weakly with Lagrange multipliers as in the hybridization procedure discussed in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is due to the use of anisotropic polynomial interpolation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' dif- ferent degree polynomials are used in each variable) which requires the vector’s basis 13 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. The interpolating points used for the nodal basis of the space Qp+1,p × Qp,p+1 for (a) p = 0, (b) p = 1, and (c) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Gauss-Legendre points are used in the tangential direction and Gauss-Lobatto in the normal direction for each component of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Circles denote the degrees of freedom associated with the ξ component and squares the η component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' to be rotated along with the transformation to properly orient the interpolating poly- nomials in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Combining the local function space Qp+1,p( ˆK) × Qp,p+1( ˆK) with the contravariant Piola transform yields: Dp(K) = {v = 1 J Fˆv ◦ T−1 : ˆv ∈ Qp+1,p( ˆK) × Qp,p+1( ˆK)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (42) Here, both the inverse mesh transformation and 1/J are generally non-polynomial when T is non-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We now define the degree-p RT space as: RT p = {v ∈ [L2(D)]2 : v|K ∈ Dp(K) , ∀K ∈ T and �v · n� = 0 , ∀F ∈ Γ0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (43) Note that since the contravariant Piola transform is used, functions in RT transform according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 19, 25, and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The location of the degrees of freedom for RT 1 are shown on a 3 × 3 mesh in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Continuity in the normal component is enforced by sharing the degrees of freedom corresponding to the normal component on interior faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' From Proposition 1, v ∈ RT p satisfies ∇ · v = ∇h · v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the RT space does not have the continuity to allow a square-integrable gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, ∇v /∈ [L2(D)]2×2 and ∇v ̸= ∇hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Raviart Thomas Trace Space The normal trace of the RT space is required for the hybridization procedure discussed in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This space is defined on the interior skeleton of the mesh Γ0 and represents the normal component of the RT space along the interior mesh faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let Pp be the space of univariate polynomials with degree at most p and Pp(F) = {u = ˆu ◦ T−1 : ˆu ∈ Pp( ˆF)} (44) 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. The distribution of degrees of freedom corresponding to the first degree Raviart Thomas space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Continuity of the normal component is enforced by sharing the degrees of freedom corresponding to the normal component along the interior face between neighboring elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The circles and squares denote degrees of freedom in the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. The distribution of degrees of freedom corresponding to Λ1, the space defined as the normal trace of the first degree Raviart Thomas space, on a 3 × 3 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' the space of univariate polynomials mapped from the reference line, ˆF = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT trace space is then Λp = {µ ∈ L2(Γ0) : µ|F ∈ Pp(F) , ∀F ∈ Γ0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (45) The degrees of freedom in Λ1 are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that these degrees of freedom are exactly the degrees of freedom corresponding to the normal component of RT 1 on the interior faces of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Transport Discretizations We assume the transport equation is discretized with the Discrete Ordinates (SN) angular model and an arbitrary-order DG spatial discretization compatible with curved meshes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The transport equation is collocated at discrete angles Ωd and integration over the unit sphere is numerically approximated with a suitable angular quadrature rule, {wd, Ωd}NΩ d=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let ψd(x) = ψ(x, Ωd) be the angular flux in the discrete direction Ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The DG discretization uses ψd ∈ Yp for each discrete angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Through finite element interpolation, ψd(x) can be evaluated at any point in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF data are computed with SN angular quadrature and finite element interpolation 15 as: E(x) = �NΩ d=1 wd Ωd ⊗ Ωd ψd(x) �NΩ d=1 wdψd(x) , (46a) Eb(x) = �NΩ d=1 wd |Ωd · n| ψd(x) �NΩ d=1 wdψd(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (46b) Note that we represent the VEF data as ratios of DG grid functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, on each element K, each component of the Eddington tensor and the boundary factor can be written as q/p where q, p ∈ Qp(K) and are thus improper rational polynomials mapped from the reference element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that Ω is defined on the canonical basis ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, each component of the Eddington tensor transforms independently as a scalar and the Piola transform is not required to map the Eddington tensor between reference and physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since ψd ∈ Yp can be discontinuous across interior mesh interfaces, the VEF data can also be discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, global derivatives of the VEF data are not well defined and, in particular, the Eddington tensor is not single-valued on interior mesh interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF discretizations presented here are designed to avoid the need for derivatives of the Eddington tensor and, when needed, use the average to provide a single-valued approximation of the Eddington tensor on interior mesh faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The boundary factor is only needed on the boundary of the domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' x ∈ Γb) and is thus always single valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We consider problems where ψ ≥ δ in the domain for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This assumption is reasonable for our applications but may not apply in shielding or deep penetration problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Where necessary, negative flux fixups are used to ensure that ψ > 0 numer- ically since positive angular fluxes are crucial for generating physically realistic VEF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that the Eddington tensor and boundary factor are angular flux-weighted averages of Ω ⊗ Ω and |Ω · n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Combined with the positivity assumption, this means the each component of the Eddington tensor and the boundary factor are bounded functions in space such that E ∈ [L∞(D)]2×2 and Eb ∈ L∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF scalar flux is coupled to the transport equation in the scattering source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A mixed-space scattering mass matrix is used to support the use of differing finite element spaces for the angular flux and VEF scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, the scattering source is built using test functions and trial functions from the spaces corresponding to the angular and VEF scalar fluxes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mixed Finite Element Discretizations We now derive mixed finite element discretizations of the VEF equations with Miften- Larsen boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We seek approximations to the scalar flux and current in the finite-dimensional spaces E and V, respectively, and test the zeroth and first moments with functions in the spaces E′ and V′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We consider Galerkin discretizations so that the test and trial spaces for the scalar flux and current are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, we restrict ourselves to the case that E′ = E and V′ = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We proceed by first informally deriving the weak form assuming the spaces E and V have 16 the requisite regularity to allow the resulting weak form to be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We will see that there is no ambiguity in the choice E = Yp ⊂ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, due to the presence of the Eddington tensor, the standard Raviart Thomas methods are inappropriate and so two choices for V are presented: a method with V = Wp+1 ⊂ [H1(D)]2 and a non- conforming method where V = RT p ⊂ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Weak Form Multiplying the zeroth and first moments with sufficiently smooth functions u and v, respectively, and integrating over the domain yields: � u ∇ · J dx + � σa uϕ dx = � u Q0 dx , (47a) � v · ∇ · (Eϕ) dx + � σt v · J dx = � v · Q1 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (47b) The differentiability requirements on the Eddington tensor and the VEF scalar flux can be reduced by integrating the first moment equation by parts: � u ∇ · J dx + � σa uϕ dx = � u Q0 dx , (48a) � ∂D v · En ¯ϕ ds − � ∇v : Eϕ dx + � σt v · J dx = � v · Q1 dx , (48b) where ϕ = ¯ϕ on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We have used Green’s identity for a tensor multiplied by a vector: � ∇ · (v · P) dx = � v · ∇ · P dx + � ∇v : P dx = � v · Pn ds , (49) where A : B = 2 � i=1 2 � j=1 AijBij , A, B ∈ R2×2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (50) Integrating by parts moves derivatives from the Eddington tensor and VEF scalar flux to the test function v allowing weaker requirements for both E and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, we assume J ∈ V has enough regularity to allow ∇ · J ∈ L2(D) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' V ⊂ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D)) so that � u ∇ · J dx is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, we can unambiguously take u, ϕ ∈ E ⊂ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the test function v now has increased regularity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Namely, we must have ∇v : E ∈ L2(D) instead of the typical requirement that ∇ · v = ∇v : I ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In the thick diffusion limit, E = 1 3I and this requirement reduces to ∇v : E = 1 3∇ · v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this case, RT methods apply directly for both v and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, for a general Eddington tensor, the RT space does not have the continuity 17 requirements to allow the term � ∇v : Eϕ dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This requirement is investigated in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For a tensor S ∈ [L∞(D)]2×2 satisfying ∇·S ∈ [L2(D)]2, let v : D → R2 be such that (a) v|K ∈ [H1(K)]2 for each K ∈ T (b) �v · Sn� = 0 for each F ∈ Γ0 then ∇v : S ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Conversely, if ∇v : S ∈ L2(D) and (a) is satisfied, then (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let C∞ 0 (D) denote the space of infinitely differentiable functions that are zero on the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Using Green’s identity, the following holds for each u ∈ C∞ 0 (D): � ∇v : S u dx = − � v · ∇ · (Su) dx = − � K∈T � K v|K · ∇ · (Su) dx = � K∈T �� K ∇v|K : S u dx − � ∂K v · Sn u ds � = � ∇hv : S u dx − � Γ0 �v · Sn� u ds = � ∇hv : S u dx , (51) where we have used (b) to cancel the integration over Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The above identifies ∇v : S with ∇hv : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Given S ∈ [L∞(D)]2×2 and (a), ∇hv : S ∈ L2(D) and thus we have that ∇v : S = ∇hv : S ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the other hand, if ∇v : S ∈ L2(D), then ∇v : S = ∇hv : S and, given v|K ∈ [H1(K)]2, we obtain � Γ0 �v · Sn� u ds = 0 , ∀u ∈ C∞ 0 (D) , (52) hence, (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Observe that Proposition 2 reduces to Proposition 1 when S = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the DG interpolation used to approximate the angular flux, the Eddington tensor does not satisfy ∇ · E ∈ L2(D) and thus Proposition 2 does not apply directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, we can consider approximating the Eddington tensor by projecting it onto a space that satisfies this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In such case, Proposition 2 implies that ∇v : E = ∇hv : E ⇐⇒ �v · En� = 0 , ∀F ∈ Γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (53) Figure 8 depicts an example of the Eddington tensor rotating and scaling the normal vector, altering the continuity requirement of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that since the Edding- ton tensor is symmetric positive definite, n · En > 0 and thus θ ∈ (−π/2, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In 18 n En θ K1 K2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of the rotation and scaling of the normal vector induced by the Eddington tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the Eddington tensor is symmetric positive definite, the angle θ cannot be larger than ±90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the presence of the Eddington tensor in the VEF first moment equation, continuity of the En component is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' other words, the Eddington tensor cannot rotate the normal past a direction tangen- tial to the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This altered continuity requirement makes standard RT methods an inappropriate choice for the test function v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 53, the weak form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 48 will hold only when the space V is chosen so that both �J · n� = 0 and �v · En� = 0 on all interior faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These conditions can only be met by using v, J ∈ V ⊂ [H1(D)]2 so that all components of v and J are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A Petrov-Galerkin discretization where the test space satisfies �v · En� = 0 and the trial space satisfies �J · n� = 0 may be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this case, the test space would need to use a more general Piola transform that preserves the En component of a vector, making the test space dependent on the angular flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Furthermore, Proposition 2 indicates this approach would require the use of an approximate projection of the Eddington tensor that satisfies ∇ · E ∈ [L2(D)]2, which could degrade solution quality on problems with steep solution gradients in parts of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Petrov-Galerkin discretization is not considered here due to these complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Alternatively, non- conforming, DG-like techniques can be used to allow use of the RT space for both the test and trial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, both v, J ∈ V = RT p ⊂ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) and the discontinuity in v · En is handled with numerical fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [H1(D)]2 Setting v, J ∈ V ⊂ [H1(D)]2 and u, ϕ ∈ E ⊂ L2(D) allows the weak form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 48 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The inf-sup condition [26] states that the discretization arising from the pairing of equal degree interpolation for the scalar flux and current will be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, the Yp × Wp discretization does not have a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The smallest non-singular pairing of spaces is then Yp × Wp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, if the scalar flux is piecewise- constant, continuous linear finite elements for each component of the current must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Background on the discrete inf-sup condition is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1 in the context of the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The discretization is complete by supplying boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solving the Miften- Larsen boundary conditions (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2c) for ϕ yields ¯ϕ = 1 Eb (J · n − 2Jin) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (54) The [H1(D)]2 × L2(D) mixed finite element VEF discretization is then: find (ϕ, J) ∈ 19 Yp × Wp+1 such that � u ∇ · J dx + � σa uϕ dx = � u Q0 dx , ∀u ∈ Yp , (55a) − � ∇v : Eϕ dx + � σt v · J dx + � Γb 1 Eb (v · En)(J · n) ds = � v · Q1 dx + 2 � Γb 1 Eb v · En Jin ds , ∀v ∈ Wp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (55b) Equation 18 is used to compute the gradient and divergence of v , J ∈ Wp+1 in refer- ence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Using V ⊂ [H1(D)]2 is simple to implement in that it relies only on the scalar contin- uous finite element space and does not require interior face bilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, this choice has been seen to degrade both solution quality and solver performance due to allowing non-physical, spurious modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These so-called checkerboard modes are a well-known issue with [H1(D)]2 × L2(D) discretizations in the context of fluid flow [27] and are a consequence of the mismatch between the spaces ∇ · Wp+1 and Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The space V ⊂ [H1(D)]2 is either too small with respect to Yp, leading to a singular system in the case V = Wp or too large, allowing spurious modes for V = Wp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The effect of these modes on solution quality and solver performance is investigated in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 in the context of radiation diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Furthermore, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 investigates these modes analytically on a lowest-order, single-element Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Raviart Thomas If v, J ∈ V ⊂ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D), a non-conforming approach must be used for the first moment equation due to the presence of the Eddington tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We proceed by locally integrating the first moment equation by parts on each element so that the global gradient of v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This requires the introduction of an auxiliary equation, referred to as the numerical flux, that approximates the product of the Eddington tensor and VEF scaluar flux on interior mesh interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The local weak form for the first moment corresponding to each element K is: � ∂K v· � Eϕn ds− � K ∇v|K : Eϕ dx+ � K σt v·J dx = � K v·Q1 dx , ∀v ∈ Dp(K) , (56) where � Eϕ is the aformentioned numerical flux for the Eddington tensor and scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Summing over all elements K ∈ T : � Γ �v� · � Eϕn ds − � ∇hv : Eϕ dx + � σt v · J dx = � v · Q1 dx , ∀v ∈ RT p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (57) We have used the fact that on a face F = K1 ∩ K2, n = n1 = −n2 and the definitions of the jump and broken gradient in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, we assume the use of a so-called conservative numerical flux such that � Eϕn is single-valued on 20 interior faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, the numerical flux satisfies � � Eϕn � = 0 , �� � Eϕn �� = � Eϕn , on F ∈ Γ0 , (58) where the average is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this form, only the gradient restricted to each element is required of the test function, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since v ∈ RT p satisfies v|K ∈ D(K) ⊂ [H1(D)]2 on each K ∈ T , the broken gradient ∇hv ∈ [L2(D)]2×2 is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We now define define the numerical flux and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The resulting dis- cretization will provide optimal accuracy if � Eϕn is an optimal approximation of the true value of Eϕn on interior faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A conservative numerical flux that satisfies this requirement is: � Eϕn = {{En}}{{ϕ}} , on F ∈ Γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (59) While many choices of the numerical flux are possible, we show below that this partic- ular choice of numerical flux has the benefit of limiting to a standard RT discretization of radiation diffusion in the thick diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Miften-Larsen boundary condi- tions are applied with � Eϕn = En Eb (J · n − 2Jin) , on F ∈ Γb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (60) This is derived by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2c for the scalar flux and multiplying by En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Yp × RT p discretization is then: find (ϕ, J) ∈ Yp × RT p such that � u ∇ · J dx + � σa uϕ dx = � u Q0 dx , ∀u ∈ Yp , (61) � Γ0 �v · {{En}}� {{ϕ}} ds − � ∇hv : Eϕ dx + � σt v · J dx + � Γb 1 Eb (v · En)(J · n) ds = � v · Q1 dx + 2 � Γb 1 Eb v · En Jin ds , ∀v ∈ RT p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (62) Since RT vectors use the contravariant Piola transform, we substitute v = 1 J Fˆv in all terms involving v and use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 25 and 27 to evaluate ∇hv and ∇ · J, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In the thick diffusion limit, E = 1 3I and �v · {{En}}� = 1 3 �v · n� = 0 , (63) since v ∈ RT p has a continuous normal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Furthermore, ∇hv : E = 1 3∇ · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This discretization with this choice of numerical flux is then equivalent to the standard RT discretization of diffusion in the thick diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT space satisfies ∇ · RT p = Yp avoiding the spurious modes seen for the [H1(D)]2 × L2(D) discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This allows superior solution quality and excellent solver performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the RT method is more complex due to the need for 21 interior face bilinear forms, the contravariant Piola transform, and the comparatively less simple RT space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solvers The above discretizations admit the following block system � A G D Ma � �J ϕ � = �g f � , (64) where for u, ϕ ∈ E and v, J ∈ V: vT AJ = � σt v · J dx + � Γb 1 Eb (v · En)(J · n) ds , (65a) uT Maϕ = � σa uϕ dx , (65b) uT DJ = � u ∇ · J dx , (65c) vT Gϕ = � − � ∇v : Eϕ dx , V = Wp+1 � Γ0 �v · {{En}}�{{ϕ}} ds − � ∇hv : Eϕ dx , V = RT p , (65d) vT g = � v · Q1 dx + 2 � Γb 1 Eb v · En Jin ds , (65e) uT f = � u Q0 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (65f) Note that the integration transformations described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 are implicitly used and, in particular, the contravariant Piola transform is implicitly used when V = RT p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We use a lower block triangular preconditioner of the form M = �A D ˜S � , (66) where ˜S is an approximation to the Schur complement S = Ma − DA−1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Block preconditioners seek to modify the system such that it has a minimal polynomial with small degree [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Iterative solvers with an optimality condition, such as GMRES, can then converge in a small number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, computing the generally dense Schur complement and exactly inverting it are impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Instead, we use 22 an approximate Schur complement formed from a sparse approximation to A−1 and sparse matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, we use ˜S = Ma − D ˜A −1G (67) where ˜A is the lumped mass matrix and boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On elements with no bound- ary faces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∂K ∩ Γb = ∅), the lumping procedure is to sum the rows of the matrix into the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is computed on the element-local matrix as: ˜Ae ij = �� k Ae ik , i = j 0 , i ̸= j , (68) where Ae and ˜A e are the unlumped and lumped matrices associated with the degrees of freedom corresponding to element Ke, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On elements with a boundary face, the boundary integral over Γb contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the Eddington tensor, v · En couples degrees of freedom corresponding to the normal and tangential components of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We leverage the block structure of the local matrices to lump the boundary elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let Ae = � Ae 11 Ae 12 Ae 21 Ae 22 � (69) where Ae ij is the sub-block corresponding to the degrees of freedom of the ith and jth components of the test and trial functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We then lump each of these sub-blocks separately so that: ˜A e = � ˜A e 11 ˜A e 12 ˜A e 21 ˜A e 22 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (70) The lumped local matrix ˜A e is diagonal by vector component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, each row has at most two entries corresponding to the two components of a vector in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This lumping procedure allows approximation of the boundary terms and has an inverse that can be computed efficiently without fill-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For both interior and boundary elements, the local matrices ˜A e are assembled into the global matrix ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For rows corresponding to interior degrees of freedom, the lumped matrix is diagonal and thus the inverse is simply 1/ ˜Aii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For rows corresponding to boundary degrees of freedom, ˜A is a diagonal matrix for each vector component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The inverse is computed by gathering the entries corresponding to each vector component into a 2 × 2 matrix, inverting it, and scattering the inverse back to a sparse matrix representing ˜A −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The above lumping procedure results in a sparse ˜A −1 that ap- proximates the true inverse A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, the lumped Schur complement is formed with sparse matrix multiplication according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that computing the Schur complement is numerically analogous to eliminating the current in the analytic equa- tions to form a second-order, elliptic partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, one Algebraic Multigrid (AMG) V-cycle is expected to be a spectrally equivalent approximation to ˜S −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The approximate inverse of the block preconditioner in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 66 is applied with forward 23 substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, we solve �A D ˜S � � x1 x2 � = � r1 r2 � (71) by approximately solving the block problems: Ax1 = r1 , (72a) ˜Sx2 = r2 − Dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (72b) We stress that the sub-problems in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 72 do not need to be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In fact, one iteration of Jacobi smoothing and one AMG V-cycle applied to A and ˜S, respectively, often lead to a scalable preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' More accurately solving the sub- problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' using more than one Jacobi/AMG iteration or nested iteration) gen- erally improves robustness to problem size but typically not to the extent that fewer Jacobi iterations and AMG V-cycles are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This behavior is investigated in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 where we compare the scaling of solving the RT VEF system using one and three AMG V-cycles per preconditioner application as well as the effect of using one itera- tion of Gauss-Seidel, a more expensive and thus more robust smoother, to approximate A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Hybridization A hybridized version of the RT mixed method is obtained by relaxing the continuity requirements of the space RT p and reimposing them weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Removing the continuity requirement from RT p yields the broken space ˆ RT p = {v ∈ [L2(D)]2 : v|K ∈ Dk(K) , ∀K ∈ T } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (73) This space is equivalent to RT p on each element but ˆ RT p does not have the matching conditions that strongly enforce continuity in the normal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that RT p ⊂ ˆ RT p and that v ∈ ˆ RT p belongs to RT p if and only if �v · n� = 0 on all interior mesh interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, the mixed problem can be reformulated to use the space ˆ RT p instead of RT p by adding the constraint that �J · n� = 0 for each F ∈ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The methods presented in this section enforce this constraint with a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Hybridized methods are attractive for three reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' First, since J ∈ ˆ RT p and ϕ ∈ Yp are both discontinuous, their degrees of freedom are coupled only locally on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' It is then possible to locally eliminate the scalar flux and current arriving at a system of equations for just the Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This reduced system is much smaller than the original 2 × 2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Second, the reduced system for the Lagrange multiplier will be positive definite and AMG can be applied directly, avoiding the need for block preconditioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, the Lagrange multiplier provides an additional approximation for the scalar flux not provided by the original mixed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the VEF equations are not symmetric, the variational principles typically used to derive hybridized mixed finite element methods are not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We first show the 24 derivation of a hybridized method for the symmetric case of radiation diffusion using variational principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This method is extended to the VEF equations by emulating the properties of the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, we discuss the details of an efficient implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Derivation for Radiation Diffusion In this section, we provide background on the dual, mixed, and hybrid variational forms associated with the symmetric radiation diffusion system with Dirichlet boundary conditions: ∇ · J + σaϕ = Q0 , x ∈ D , (74a) ∇ϕ + 3σtJ = 0 , x ∈ D , (74b) ϕ = 0 , x ∈ ∂D , (74c) where the source has been assumed to be isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This coupled system can be viewed as the first moment equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 74b) with the constraint of particle bal- ance (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 74a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT mixed finite element discretization of radiation diffusion is: find (ϕ, J) ∈ Yp × RT p such that � 3σt v · J dx − � ∇ · v ϕ dx = 0 , ∀v ∈ RT p , (75a) � u ∇ · J dx + � σa uϕ dx = � u Q0 dx , ∀u ∈ Yp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (75b) Our goal is to identify the variational problem associated with this mixed finite element discretization, modify it to support the use of the broken RT space, ˆ RT p, and derive the hybridized system of equations that solves this modified variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We follow Quarteroni and Valli [25, Chapter 7] in the presentation of these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The dual formulation is to minimize the so-called complementary energy functional: I(J) = 1 2 � 3σt J · J dx (76) under the constraint of particle balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Quarteroni and Valli [25, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1] shows that this constrained minimization problem and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 74 are equivalent for- mulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mixed methods enforce the particle balance constraint using a Lagrange multiplier, ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let the Lagrangian functional be L : RT p × Yp → R such that L(J, ϕ) = I(J) − �� ϕ ∇ · J dx + 1 2 � σa ϕ2 dx − � ϕ Q dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (77) 25 By minimizing over J and maximizing over ϕ, we can minimize I(J) while enforcing particle balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' To see this, observe that, for J fixed, − �� ϕ ∇ · J dx + 1 2 � σa ϕ2 dx − � ϕ Q dx � (78) is a concave quadratic functional with respect to ϕ and is maximized when particle balance occurs such that ∇ · J + σaϕ = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The resulting mixed variational problem is then written: inf J∈RT p sup ϕ∈Yp L(J, ϕ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (79) Such a problem finds the “saddle point” that balances the convex functional I(J) with the concave particle balance constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since I(J) and the particle balance constraint are quadratic functionals, the saddle point occurs at ∇L = 0: ∂L ∂J = � 3σt v · J dx − � ∇ · v ϕ dx = 0 , ∀v ∈ RT p , (80a) ∂L ∂ϕ = − � u ∇ · J dx − � σa uϕ dx + � u Q0 dx = 0 , ∀u ∈ Yp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (80b) Observe that this system of equations for the saddle point exactly matches the mixed discretization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, the solution of the mixed discretization is also the saddle point of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We now wish to modify L to define a variational problem with an equivalent solution that allows use of the broken RT space ˆ RT p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is accomplished by searching for J ∈ ˆ RT p and adding an additional constraint that the normal component of the current is continuous such that �J · n� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Let ˆL : ˆ RT p × Yp → R such that ˆL(J, ϕ) = I(J) − �� ϕ ∇h · J dx + 1 2 � σa ϕ2 dx − � ϕ Q0 dx � , (81) be the broken Lagrangian functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since ˆ RT p and Yp are piecewise discontinuous, ˆL is equivalent to L on each element due to the use of the broken divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The constrained variational problem is then: inf J∈ ˆ RT p sup ϕ∈Yp ˆL(J, ϕ) , such that �J · n� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (82) As with particle balance, the normal continuity constraint is imposed with a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Defining H : ˆ RT p × Yp × Λp as H(J, ϕ, λ) = ˆL(J, ϕ, λ) + � Γ0 λ �J · n� ds , (83) 26 the constrained saddle point problem is equivalent to: inf J∈ ˆ RT p sup ϕ∈Yp sup λ∈Λp H(J, ϕ, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (84) If (J, ϕ, λ) is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 84, then �J · n� = 0, otherwise setting λ = λτ = τ �J · n� for some τ > 0 would give � Γ0 λτ �J · n� ds = τ � Γ0 �J · n�2 ds (85) and thus lim τ→∞ H(J, ϕ, λτ) = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (86) In other words, the supremum over λ drives the solution towards currents with con- tinuous normal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The solution of the hybridized variational form is found by setting ∇H = 0: ∂H ∂J = � 3σt v · J dx − � ∇h · v ϕ dx + � Γ0 �v · n� λ ds = 0 , ∀v ∈ ˆ RT p , (87a) ∂H ∂ϕ = − � u ∇h · J dx − � σa uϕ dx + � u Q0 dx = 0 , ∀u ∈ Yp , (87b) ∂H ∂λ = � Γ0 µ �J · n� ds = 0 , ∀µ ∈ Λp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (87c) Since ˆ RT p and Yp are discontinuous spaces, the hybridized mixed method is equivalent to: � K 3σt v · J dx − � K ∇ · v|K ϕ dx + � ∂K∩Γ0 v · nKλ ds = 0 , ∀v ∈ Dp(K) , K ∈ T , (88a) � K u ∇ · J|K dx + � K σa uϕ dx = � K u Q0 dx , ∀u ∈ Qp(K) , K ∈ T , (88b) � Γ0 µ �J · n� ds = 0 , ∀µ ∈ Λp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (88c) Here, it can be seen that the degrees of freedom for the scalar flux and current are no longer globally coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In fact, if λ were known, the scalar flux and current could be recovered by solving element-local radiation diffusion problems where λ plays the role of a weak boundary condition applied on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that the non-zero boundary condition ϕ = ¯ϕ for x ∈ Γb can be applied by subtracting � Γb v · n ¯ϕ ds from 27 the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 87a or equivalently by subtracting � ∂K∩Γb v · nK ¯ϕ from the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 88a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In hybridization, continuity of the normal component is enforced weakly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 88c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, it is well known that the resulting discrete solution will actually satisfy continuity of the normal component in a strong sense (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' independent of the discretization parameters h and p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In fact, hybridization has also been viewed as an algebraic technique similar to static condensation in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This behavior is owed to the fact that the dual, mixed, and hybrid formulations all correspond to equivalent variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Extension to VEF The above variational process cannot be applied directly to the VEF equations due to their lack of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Without symmetry, it is unclear which potential the weak VEF equations correspond to or whether it would have a unique, global saddle point found by setting its gradient to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, we can define a hybrid method for the VEF equations by mimicking the properties seen above for the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, we use the broken RT space, ˆ RT p, and a Lagrange multiplier that 1) weakly enforces continuity of the normal component of the current and 2) provides inter- element boundary conditions for element-local VEF problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' As in the symmetric case, this will allow elimination of the scalar flux and current, leading to a smaller system for just the Lagrange multiplier where AMG can be applied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, since the resulting method cannot be derived from a variational principle it is unclear whether the resulting hybrid formulation will be equivalent to the original mixed formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The hybridized diffusion method can be extended to the VEF equations with Miften- Larsen boundary conditions by replacing the diffusion first moment with the VEF first moment equation and using the boundary condition ¯ϕ = 1 Eb (J · n − 2Jin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This can be accomplished by using the element-local weak form of the first moment equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 56 and setting � Eϕn = {{En}} λ , on F ∈ Γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (89) Here, we are using the Lagrange multiplier λ ∈ Λp as a single-valued approximation for the scalar flux on interior faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The numerical flux on the boundary is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For each K, the element-local VEF problem is then: � ∂K∩Γ0 v·{{EnK}} λ ds− � K ∇v|K : Eϕ dx+ � K σt v·J dx+ � ∂K∩Γb 1 Eb (v · EnK)(J · nK) ds = � K v · Q1 dx + 2 � ∂K∩Γb 1 Eb v · EnK Jin ds , ∀v ∈ Dp(K) , (90a) � K u ∇ · J|K dx + � K σa uϕ dx = � K u Q0 dx , ∀u ∈ Qp(K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (90b) 28 The resulting hybrid VEF method is: find (J, ϕ, λ) ∈ ˆ RT p × Yp × Λp such that � Γ0 �v · {{En}}� λ ds − � ∇hv : Eϕ dx + � σt v · J dx + � Γb 1 Eb (v · En)(J · n) ds = � v · Q1 dx + 2 � Γb 1 Eb v · En Jin ds , ∀v ∈ ˆ RT p , (91a) � K u ∇h · J dx + � K σa uϕ dx = � K u Q0 dx , ∀u ∈ Yp , (91b) � Γ0 µ �J · n� ds = 0 , ∀µ ∈ Λp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (91c) Observe that this represents element-local VEF problems where the boundary condi- tions are provided either by the Miften-Larsen boundary conditions on the boundary of the domain or by the Lagrange multiplier λ for interior elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, if λ were known, the scalar flux and current could be solved for independently on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Implementation Details In matrix form, the hybridized system is � � ˆA ˆG C2 ˆD Ma C1 � � � � J ϕ λ � � = � � ˆg f 0 � � , (92) where ˆA, ˆD, and ˆg are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 65a, 65c, and 65e, respectively, but use V = ˆ RT p and ˆG = − � ∇hv : Eϕ dx (93) is the analog of G in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 65d that uses V = ˆ RT p and does not include the interior face bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The DG absorption mass matrix, Ma, and right hand side, f, are unchanged from the original mixed form defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 65b and 65f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The constraint matrices are defined as: µT C1J = � Γ0 µ �J · n� ds , (94a) vT C2λ = � Γ0 �v · {{En}}� λ ds , (94b) where µ, λ ∈ Λp and v, J ∈ ˆ RT p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 29 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. Sparsity plots for the block system corresponding to the hybridized Raviart Thomas discretization for the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In (a), the degrees of freedom are organized as J1, J2, ϕ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In (b), the rows and columns of the matrix in (a) are permuted to group the currents and scalar fluxes associated with each element together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' With this ordering, it is clear that the scalar flux and current can be eliminated on each element without fill-in, leaving a system for λ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that in practice, the elimination of the element-local problems is performed locally with dense operations and global sparse matrices are used to form the reduced system for the Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Only the constraint matrices C1 and C2 are globally coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The matrices ˆA, ˆG, ˆD, and Ma are all block diagonal by element and can thus be eliminated on each element without fill-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Figure 9a shows the sparsity pattern of the block system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that this matrix can be permuted to be block diagonal by element by grouping the current and scalar flux degrees of freedom associated with each element together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This matrix is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 9b where it is clear that the block system has a structure amenable to efficient solution via block Gaussian elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Performing block Gaussian elimination on each element, the reduced system for the Lagrange multiplier reads Hλ = �C1 0� � ˆA ˆG ˆD Ma �−1 � C2 0 � λ = �C1 0� � ˆA ˆG ˆD Ma �−1 �ˆg f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (95) The inverse of the local VEF problems is derived by finding the blocks W, X, Y, and Z that satisfy � ˆA ˆG ˆD Ma � � W X Y Z � = � I I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (96) We assume that ˆA and the Schur complement ˆS = Ma − ˆD ˆA −1 ˆG are non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is justified in non-void regions where σt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, we do not assume Ma is non-singular since σa ≥ 0 can be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 96 for the blocks W, X, Y, and Z 30 L :8 1 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='II I II I II 11 11 I 1 I 1 Iunder these constraints yields: W = ˆA −1(I + ˆGˆS −1 ˆD ˆA −1) , (97a) X = − ˆA −1 ˆGˆS −1 , (97b) Y = −ˆS −1 ˆD ˆA −1 , (97c) Z = ˆS −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (97d) The reduced system for the Lagrange multiplier is then Hλ = C1WC2λ = C1 ˆA −1� I + ˆGˆS −1 ˆD ˆA −1� C2λ = C1 � Wˆg + Xf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (98) We can now rewrite the 3 × 3 block system as � � ˆA ˆG C2 ˆD Ma H � � � � J ϕ λ � � = � � ˆg f C1 � Wˆg + Xf � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (99) This system can be solved with block back substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' First, solve the globally cou- pled system Hλ = C1 � Wˆg + Xf � (100) for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The element-local inverse can then be used to solve for the scalar flux and current with �J ϕ � = � W X Y Z � �ˆg − C2λ f � = �W � ˆg − C2λ � + Xf Y � ˆg − C2λ � + Zf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (101) In this way, only dim(Λp) globally coupled unknowns must be solved for as opposed to the dim(RT p) + dim(Yp) required by the original mixed formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In practice, the blocks of the inverse W, X, Y, and Z are formed using dense ma- trix operations applied on the element-local matrices corresponding to the degrees of freedom of a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The local matrices are then broadcast to a global sparse matrix in order to perform the sparse matrix multiplication required to form the re- duced system for the Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Lagrange multiplier can be scalably solved for by preconditioning H with AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, recovering the scalar flux and current is a post-processing step that is independent on each element and thus scales optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Results The VEF algorithms presented here were implemented using the MFEM [28, 29] fi- nite element framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The stabilized bi-conjugate gradient (BiCGStab) solver from MFEM was used to solve the discretized VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Lower block triangular pre- conditioners were built using MFEM’s Jacobi smoother and BoomerAMG from the sparse linear algebra package hypre [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' KINSOL, from the Sundials package [31], provided the fixed-point and Anderson-accelerated fixed-point solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' As described in Hindmarsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [31, §2], the fixed-point and Anderson-accelerated fixed-point itera- tion is terminated when the max norm of the difference between successive iterates is below the iterative tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The parallel implementation of the sparse direct solver SuperLU [32] is used when preconditioned iterative solvers results are not presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The streaming and collision operator is inverted using the transport solver from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Unless otherwise noted, the angular flux and VEF scalar flux are approximated using the same degree finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the positive Bernstein polynomials [33] are used for the transport discretization’s local polynomial basis whereas the Lagrange basis through the Gauss-Legendre points is used for the VEF scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The positive transport basis facilitates the application of the quadratic programming negative flux fixup from [34] that is used on the crooked pipe problem in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since all methods produce a VEF scalar flux in Yp, the methods are parameterized by their choice of space for the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, we refer to the Yp × Wp+1, Yp × RT p, and Yp × ˆ RT p × Λp methods as H1, RT, and HRT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Method of Manufactured Solutions The accuracy of the methods are determined with the Method of Manufactured Solu- tions (MMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The solution is set to ψ = 1 4π[α(x) + Ω · β(x) + Ω ⊗ Ω : Θ(x)] , (102) where α(x) = sin(πx) sin(πy) + δ , (103a) β(x) = � �sin � 2π(x+ω) 1+2ω � sin � 2π(y+ω) 1+2ω � sin � 2π(x+ω) 1+2ω � sin � 2π(y+ω) 1+2ω � � � , (103b) Θ(x) = � � 1 2 sin � 3π(x+ζ) 1+2ζ � sin � 3π(y+ζ) 1+2ζ � sin � 2π(x+ω) 1+2ω � sin � 2π(y+ω) 1+2ω � sin � 2π(x+ω) 1+2ω � sin � 2π(y+ω) 1+2ω � 1 4 sin � 3π(x+ζ) 1+2ζ � sin � 3π(y+ζ) 1+2ζ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (103c) Here, δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='25 is used to ensure ψ > 0 and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1 and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='05 are used to test spatially dependent, non-isotropic inflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The domain is 32 D = [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' With this definition: φ(x) = α(x) + 1 3 trace Θ(x) , (104a) J(x) = 1 3β(x) , (104b) P(x) = α(x) 3 I + 1 15 � 3Θ11(x) + Θ22(x) Θ12(x) Θ21(x) Θ11(x) + 3Θ22(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (104c) This leads to an exact Eddington tensor E = P/φ that is dense and spatially varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The MMS ψ and φ are substituted into the transport equation to solve for the MMS source q that forces the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The accuracy of the VEF discretizations are investigated in isolation by computing the VEF data from the MMS angular flux and setting the sources Q0 and Q1 to the moments of the MMS source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is accomplished by projecting the MMS angular flux onto a degree-p DG finite element space and using Level Symmetric S4 angular quadrature to compute the VEF data, the moments of the MMS source, and the inflow boundary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF equations are then solved as if E, Eb, Q0, Q1, and Jin are given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Errors are calculated with the L2(D) norm for scalars and the [L2(D)]2 norm for vectors given by ∥u∥ = �� u2 dx , (105) and ∥v∥ = �� v · v dx , (106) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We also use the L2(D) projection operator Πp : L2(D) → Yp such that � u(v − Πpv) dx = 0 , ∀u ∈ Yp , (107) for some v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, Πp is used to project the exact MMS scalar flux onto a Yp finite element grid function in order to investigate a superconvergence property of mixed finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We use refinements of a third-order mesh created by distorting an orthogonal mesh according to the velocity field of the Taylor Green vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This mesh distortion is generated by advecting the mesh control points with x = � T 0 v dt , (108) 33 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of a third-order mesh generated by distorting an orthogonal mesh according to the Taylor Green vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Refinements of this mesh are used in calculating the error with the method of manufactured solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' where the final time T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3π and v = � sin(x) cos(y) − cos(x) sin(y) � (109) is the analytic solution of the Taylor Green vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 300 forward Euler steps were used to advect the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' An example mesh is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Logarithmic regression is used to fit the constant and order of accuracy according to E = Ch˜p (110) where E is the error, C the constant, and ˜p the order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Four values of h were used for each MMS problem considered in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The raw error values for each of the MMS problems presented in this section are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We first show the accuracy of the three methods on a simple radiation diffusion prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The above process is used with Θ = 0 so that the angular flux is linearly anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This forces the Eddington tensor and boundary factor to E = 1 3I and Eb = 1 2, mimicking a radiation diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 1 shows the estimated order of accuracy and constant for p ∈ [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux is computed with two methods: 1) by comparing to the analytic MMS scalar flux solution directly and 2) by projecting the analytic MMS solution onto the corresponding Yp space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For all orders, the first error measure for the scalar flux converges O(hp+1) while the second converges O(hp+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is a mixed finite element superconvergence result that indi- cates that the nodal values of the scalar flux solution converge one order higher than the Yp interpolation allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The current converges as O(hp+1) for all three methods and all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On this diffusive problem, the scalar flux and current solutions from the unhybridized and hybridized RT methods are equivalent to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This test is repeated for the quadratically anisotropic MMS problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' using Θ(x) as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 103c) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Projecting the MMS angular flux solution onto Yp induces errors of order O(hp+1) in the calculation of the VEF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, since the VEF data are computed from the projected MMS solution, it is expected that this problem can converge at a maximum of order p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This can be seen in the loss of the superconvergence property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, both error measures for the scalar flux 34 TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Estimates of the order of accuracy and constant from an isotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the VEF data are constant in space and thus are represented exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p Value H1 RT HRT H1 RT HRT H1 RT HRT 1 Order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='017 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='053 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='053 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='001 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='785 2 Order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='001 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='003 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='003 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='144 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='096 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='096 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='989 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='989 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='780 3 Order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='995 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='016 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='016 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='098 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='125 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='125 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='018 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='016 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='016 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='928 4 Order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='971 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='971 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='971 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='013 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='964 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='963 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='096 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='675 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='675 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='217 TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Estimates of the order of accuracy and constant from a quadratically anisotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the angular flux used to calculate the VEF data is represented with Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, the maximum accuracy expected is order p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p Value H1 RT HRT H1 RT HRT H1 RT HRT 1 Order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='310 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='332 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='318 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='980 Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='451 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='353 2 Order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='958 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='957 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='963 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='995 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='979 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='054 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='486 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='564 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='522 Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='233 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='225 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='485 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='601 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='605 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='477 3 Order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='046 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='045 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='044 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='348 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='313 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='263 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='003 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='857 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='905 Constant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='612 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='599 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='710 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='584 4 Order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='787 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='785 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='783 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='033 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='921 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='845 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='221 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='351 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='454 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='258 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='458 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='050 35 TABLE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Estimates of the order of accuracy and constant from a quadratically anisotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the angular flux used to calculate the VEF data is represented with Yp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, the maximum accuracy expected is order p + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p Value H1 RT HRT H1 RT HRT H1 RT HRT 0 Order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='999 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='019 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='477 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='001 Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='780 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='439 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='338 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='304 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='561 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='683 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='488 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='390 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='583 2 Order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='961 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='903 Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='208 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='204 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='383 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='447 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='347 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='312 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='783 3 Order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='041 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='041 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='965 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='732 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='759 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='931 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='726 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='667 Constant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='554 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='545 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='545 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='864 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='575 converge with O(hp+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On this transport MMS problem, the current convergence is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Compared to the diffusion case, the H1 current error is maintained for p odd but is reduced by 1/2 for p even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT and HRT methods lose one order for p odd but only half an order for p even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, the RT and HRT discretizations are no longer equivalent to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This loss of equivalence may be due to inexact numerical quadrature in terms involving the VEF data – the VEF data are improper rational polynomials in space and thus cannot be exactly integrated with numerical quadrature – or may indicate that the hybrid and mixed formulations are only equivalent for the symmetric case of radiation diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, we repeat the transport MMS problem in the case where the angular flux solution is projected onto Yp+1 instead of Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This allows a maximum accuracy in the problem of O(hp+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The estimated orders of convergence and constants are provided in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Convergence rates similar to the diffusion problem are observed: the scalar flux solutions converge optimally for all methods and superconvergence of the scalar flux returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 and HRT methods produce currents that converge at similar rates as in the diffusion case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the unhybridized RT method converges suboptimally by one order for p even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The difference in convergence rates between the RT and HRT methods indicates the HRT method is in fact a new discretization for the VEF equations and not simply an algebraic method to reduce the number of globally coupled unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error behavior of the current on the above three MMS problems is summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We stress that the H1, RT, and HRT methods all generated scalar flux solutions with the optimal error behavior on each of the above MMS problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The methods differed only in the error associated with the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 36 TABLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A summary of the order of accuracies of the current on the three MMS problems for the H1, RT, and HRT methods grouped into even and odd polynomial degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' All methods converged the scalar flux with optimal O(hp+1) accuracy on all problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Even p Odd p Problem H1 RT HRT H1 RT HRT Radiation Diffusion p + 1 p + 1 p + 1 p + 1 p + 1 p + 1 Transport w/ ψMMS ∈ Yp p + 1/2 p + 1/2 p + 1/2 p + 1 p p Transport w/ ψMMS ∈ Yp+1 p + 1∗ p + 1 p + 1 p + 1 p p + 1 ∗converged O(h3/2) for p = 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thick Diffusion Limit The convergence of the VEF methods are investigated in the thick diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The material data are set to σt = 1/ϵ , σa = ϵ , σs = 1/ϵ − ϵ , q = ϵ , (111) where ϵ ∈ (0, 1] and the thick diffusion limit corresponds to the limit ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We use two coarse meshes that do not resolve the mean free path to stress the convergence of the VEF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The first is an orthogonal 8×8 mesh with D = [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The second is the triple point mesh shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 11, a third-order mesh generated with a Lagrangian hydrodynamics code where D = [0, 7] × [0, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the triple point mesh, the streaming and collision operator cannot be reordered to be lower block triangular by element due to the presence of reentrant/concave faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A standard transport sweep can be applied by iteratively lagging the strictly upper block triangular components of the streaming and collision operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The pseudo-optimal element reordering proposed in Haut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [19] is used to minimize the amount of information lagged due to reentrant faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the angular flux is only approximately inverted at each iteration it is expected that iterative efficiency will degrade compared to an analogous problem on a straight-edged mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, highly distorted elements have poor approximation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We use Level Symmetric S4 angular quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The three methods are compared when p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The coupled transport-VEF system is solved with fixed-point iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 5 shows the number of fixed-point iterations until convergence to a tolerance of 10−6 for each method on the orthogonal and triple point meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Rapid convergence is seen for all methods on both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The three methods converged equivalently on the orthogonal mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the triple point mesh, all of the methods converged slower compared to the corresponding problem solved on an orthogonal mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT and HRT methods converged in an equivalent number of iterations with H1 converging a few iterations faster than RT/HRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Lineouts of the 2D VEF scalar flux solutions for each method as ϵ → 0 are provided in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 12 and 13 for the orthogonal and triple point meshes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In all cases, the non-trivial diffusion limit solution was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the triple point problem, non-physical, non-monotone oscillations are observed due to the imprinting of the mesh on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The oscillations are larger in magnitude for RT and HRT methods compared to H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This may suggest that the 37 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of the triple point mesh used to stress the VEF algorithms on a severely distorted, third-order mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This mesh was generated with a Lagrangian hydrodynamics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' TABLE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The number of fixed-point iterations required for convergence as the thick diffusion limit parameter ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Y2 × W3, Y2 × RT 2, and hybridized Y2 × RT 2 discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Convergence is tested on an orthogonal 8 × 8 mesh and on the triple point mesh, a mesh with re-entrant faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the re-entrant faces, an inexact transport sweep is used making convergence slower on the triple point mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Orthogonal Triple Point ϵ H1 RT HRT H1 RT HRT 10−1 8 8 8 20 21 21 10−2 6 6 6 13 19 19 10−3 4 4 4 9 13 13 10−4 3 3 3 6 8 8 quality of the RT and HRT solutions are more sensitive to mesh distortion than H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solver Performance on Curved Meshes Here, we investigate the robustness of the preconditioned iterative solvers for the VEF linear systems on increasingly distorted meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The meshes were created by moving the interior control points of an initially orthogonal, third-order mesh according to the sine distortion: x → x + α � sin(2πx) sin(2πy) sin(2πx) sin(2πy) � , (112) where α controls the amount of distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' When α = 0, the mesh is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The initial mesh was 16 × 16 with D = [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Meshes corresponding to a range of values of α are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solver performance is evaluated on the first iteration of the thick diffusion limit problem introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We use ϵ = 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The number of BiCGStab iterations until convergence to a tolerance of 10−6 are shown for a range of mesh distortions in Table 6 for the H1, RT, and HRT VEF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. Lineouts of the 2D solution at y = 1/2 as ϵ → 0 for the (a) H1, (b) RT, and (c) HRT methods on an orthogonal 8 × 8 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The methods all converge to the asymptotic solution indicating they preserve the thick diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 y 0 2 4 6 8 10 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 y 0 2 4 6 8 10 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 y 0 2 4 6 8 10 ϕ ε = 10−1 ε = 10−2 ε = 10−3 ε = 10−4 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. Lineouts of the 2D solution at x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 as ϵ → 0 for the (a) H1, (b) RT, and (c) HRT methods on the triple point mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' All methods produce non-trivial solutions even on the severely distorted triple point mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Non-monotonic oscillations are present in the solution due to mesh imprinting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 39 TABLE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Number of BiCGStab iterations until convergence on the first iteration of a thick diffusion limit problem with ϵ = 10−1 as the mesh distortion parameter increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, H1, RT, and HRT rows refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' p α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='080 1 H1 46 48 48 48 48 50 RT 20 22 26 31 72 – HRT 7 10 8 8 8 8 2 H1 59 61 52 55 54 57 RT 28 27 31 – – – HRT 11 10 9 9 10 9 3 H1 54 54 56 69 55 57 RT 29 28 41 – – – HRT 9 9 8 8 9 9 4 H1 51 55 55 66 57 61 RT 41 44 46 78 – – HRT 7 9 10 10 10 11 – indicates solver did not converge in 250 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' and RT methods use the lower block triangular preconditioner described in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 while the HRT method is preconditioned with one V-cycle of AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The solver for the RT method did not converge in 250 iterations once the mesh became too distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 discretization converged on all the meshes tested but the iteration counts varied between 46 and 69 whereas HRT was solved more uniformly, varying only between 7 and 11 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This indicates the solvers for the RT method are sensitive to mesh distortion whereas HRT and, to a lesser extent, H1 are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that the meshes used in this section are small, allowing the unscalable H1 solver to still converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Linearized Crooked Pipe We now show convergence in outer fixed-point iterations and inner preconditioned linear solver iterations on a more realistic, multi-material problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The geometry and materials are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The problem consists of two materials, the wall and the pipe, which have an 1000x difference in total interaction cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Time dependence is mocked by including artificial absorption and sources that correspond to backward Euler time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The time step is set so that c∆t = 103 and the initial condition is ψ0 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The absorption and source are then σa = 1/c∆t = 10−3 1 cm and q = ψ0/c∆t = 10−1 1 cm3 · s · str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The boundary conditions are set so that isotropic inflow of magnitude 1/2π enters on the left entrance of the pipe with vacuum on all other surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A Level Symmetric S12 angular quadrature set is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The quadratic programming negative flux fixup from [34] is used inside the transport sweep to ensure positivity so that the VEF data are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Timing data is presented as the minimum time recorded across five repeated runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 40 (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='000 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='025 (c) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='060 (d) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='080 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A selection of meshes generated by distorting a third-order, orthogonal 16 × 16 mesh according to the sine distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The parameter α controls the amount of distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These meshes are used to assess linear solver robustness against mesh distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Isotropic Inflow (0,-2) (7,-2) (7,2) (0,2) Vacuum σt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 cm−1 σa = 10−3 cm−1 σt = 200 cm−1 σa = 10−3 cm−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 cm 5 mm 1 cm 5 mm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 cm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 cm 1 cm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5 cm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. The geometry, material data, and boundary conditions for the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 41 The outer fixed-point and inner linear iterative efficiencies are shown by refining in h and p on an orthogonal mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Anderson acceleration with two Anderson vectors is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The previous outer iteration’s solution is used as an initial guess for the inner solver so that the initial guess becomes progressively more accurate as the outer iteration converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The outer tolerance is 10−6 and the inner BiCGStab tolerance is 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 and RT methods use the lower block triangular preconditioner described in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 with one Jacobi iteration on the total interaction mass matrix and one AMG V-cycle on the lumped Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The HRT method is preconditioned using one V-cycle of AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 7 shows the number of Anderson-accelerated fixed-point iterations to conver- gence and the maximum and average number of inner iterations performed across all outer iterations for the H1, RT, and HRT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT and HRT methods had equivalent convergence in outer iterations except for p = 2 with three refinements where HRT required one fewer iteration than RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' H1 converged slower than the RT or HRT methods, requiring on average 159% more iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT and HRT inner solvers were scalable in h and p while the H1 solvers were not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the problems with two and three refinements in h, the H1 inner solver did not converge within 100 iter- ations on at least one of the solves for all values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The nested H1 iteration is only able to converge due to the use of the previous outer iteration as the initial guess for the inner iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that the H1 method is slower to converge the fixed-point problem even when the inner solver converged within the maximum allowed number of inner iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 8 shows the average percentage of elements in the space-angle phase space that required application of the negative flux fixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' RT and HRT had similar reliance on the fixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the coarsest problems in h – where the fixup is needed most due to lack of numerical resolution – the H1 method produced significantly more negativities which led to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6x times more elements needing the fixup for p = 2 and p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the most refined problems, 5-10% more elements were fixed up for the H1 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This increased reliance on the negative flux fixup is likely the source of the H1 method’s reduced efficiency on problems where the inner iteration completed successfully at each outer iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 method’s proclivity for producing negativities within the transport sweep is indicative of poorer solution quality compared to the RT and HRT discretizations of the VEF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This reduced solution quality is investigated in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 9 shows the average costs per outer iteration of assembling and solving the VEF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 method is the cheapest to assemble but the most expensive to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This is due to the H1 method’s simpler left hand side that does not require assem- bly over interior faces in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the H1 solvers were not scalable, H1 was around 4x more expensive than RT on the problems most refined in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Compared to RT, the HRT assembly includes the additional cost of forming the reduced system via element-local, dense matrix operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, the HRT method has the highest assembly cost, especially for large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, HRT was the cheapest to solve at each outer iteration since BiCGStab is applied to the reduced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The HRT precondi- tioner is cheaper than the lower block triangular preconditioner and the HRT reduced system is positive definite whereas the RT system is indefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These benefits led to both faster BiCGStab convergence and cheaper cost per iteration resulting in dramat- ically reduced solve times compared to RT: on the most refined problems in h, RT was 3x, 8x, and 18x more expensive to solve than HRT for p = 1, p = 2, and p = 3, 42 TABLE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The number of outer Anderson-accelerated fixed-point iterations until convergence along with the maximum and average number of inner BiCGStab iterations until convergence on the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Two Anderson vectors were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 and RT methods were preconditioned with a block lower triangular preconditioner with AMG applied to the lumped Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' HRT was preconditioned with AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The previous outer iteration’s solution was used as the initial guess for the inner iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Outer Max Inner Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Inner Ne H1 RT HRT H1 RT HRT H1 RT HRT p = 1 112 16 13 13 37 16 6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='69 448 20 13 13 66 18 7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='92 1792 25∗ 15 15 100 19 8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='27 7168 23∗ 16 16 100 20 8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='22 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='38 p = 2 112 23 13 13 47 27 10 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='22 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='15 448 25 15 15 75 26 10 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='68 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='20 1792 27∗ 16 16 100 26 11 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='44 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='44 7168 24∗ 16 15 100 29 14 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='67 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='67 p = 3 112 23 14 14 50 24 10 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='65 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='71 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='50 448 25 15 15 73 29 10 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='44 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='80 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='33 1792 26∗ 16 16 100 28 10 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='62 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='44 7168 28∗ 17 17 100 31 11 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='53 ∗ indicates at least one inner solve did not converge in 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 43 TABLE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The average number of elements in the space-angle phase space that required application of the negative flux fixup in the transport sweep on the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ne H1 RT HRT p = 1 112 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='055 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='063 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='073 448 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='331 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='043 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='042 1792 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='626 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='282 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='283 7168 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='432 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='295 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='295 p = 2 112 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='123 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='314 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='313 448 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='974 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='691 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='692 1792 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='300 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='796 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='796 7168 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='795 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='661 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='646 p = 3 112 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='224 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='285 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='285 448 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='529 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='037 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='037 1792 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='150 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='815 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='818 7168 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='247 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='132 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='132 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The total time to solve the fixed-point problems for each of the methods is presented in Table 10 along with the breakdown of the total cost into time spent in the inversion of the streaming and collision operator and forming and solving the VEF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, it can be seen that HRT’s higher assembly costs are sufficiently balanced by reduced solve costs such that HRT spent the least time in the VEF portion of the algorithm for all values of h and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, for the most refined problem with p = 3, forming and solving the HRT system was more than twice as fast as the RT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Overall, the HRT method was the fastest to solve the fixed-point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, due to the high cost of the transport sweep relative to forming and solving the VEF system, the variance in cost between the RT and HRT methods was less pronounced with RT being at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='18x more expensive despite the HRT inner solve being twice as fast as the RT solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' H1 was the most expensive method due to its slower outer and inner iterative efficiency and its higher reliance on the negative flux fixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Eigenvalue Problem It was observed that the H1 discretization exhibited poor solution quality in under resolved problems and could not be scalably solved using block preconditioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, we investigate the presence of so-called “checkerboard” modes that are allowed by the H1 discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These modes are not physical, contaminate solution quality, and degrade the effectiveness of AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' To investigate this issue, we consider the following eigenvalue problem: − ∇2u = λu , x ∈ D , (113a) 44 TABLE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The average time spent per outer iteration assembling and solving the VEF systems on hp refinements of the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Times are presented in seconds and represent the minimum time achieved across five repeated runs for each value of h and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' VEF Assembly Time (s) VEF Solve Time (s) Ne H1 RT HRT H1 RT HRT p = 1 112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0023 448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0089 1792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0363 7168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='9174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='9541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='7994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1457 p = 2 112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0035 448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0142 1792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5550 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0589 7168 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0182 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='9546 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1516 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4629 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2510 p = 3 112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0052 448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0213 1792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5698 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3765 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='9136 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0862 7168 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4576 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4729 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='5667 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1319 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3479 TABLE 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The total runtime along with the total time spent in the transport sweep and VEF portions of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Times are presented in seconds and represent the minimum time achieved across five repeated runs for each value of h and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Total Time (s) Sweep Time (s) VEF Time (s) Ne H1 RT HRT H1 RT HRT H1 RT HRT p = 1 112 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='33 448 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='62 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. A depiction of an eigenmode corresponding to an eigenvalue of 8π2 of the Poisson eigenvalue problem discretized with the Y1 × W2 discretization’s lumped Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For this eigenvalue, the exact solution is sin(2πx) sin(2πy) meaning this mode is spurious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The presence of high-frequency spurious modes in the Yp × Wp+1 discretization’s lumped Schur complement degrades the effectiveness of AMG and thus the performance of the block preconditioners used to solve the full Yp × Wp+1 discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' u = 0 , x ∈ ∂D , (113b) with D = [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The exact solutions are u = sin(kxπx) sin(kyπy) , λ = π2(k2 x + k2 y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (114) The Y1 ×W2 discretization’s lumped Schur complement is used to discretize this prob- lem as: find u ∈ Y1 such that ˜Su = λMu , (115) where M is the Y1 mass matrix and ˜S is the lumped Schur complement defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) solver from hypre was used to solve for the five smallest eigenvalues and their cor- responding eigenvectors on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 discretization correctly produced the first four smallest eigenvalues and associated eigenvectors but found the high- frequency, checkerboard mode shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 16 for the fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This checkerboard mode corresponded to a non-physically degenerate eigenvalue of 8π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The presence of this mode indicates the Yp × Wp+1 discretization allows non-physical, spurious modes that are slowly decaying and high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Such modes are slow to remove with relaxation and also cannot be accurately represented on a coarser grid, meaning AMG will not be an effective preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Furthermore, the presence of these oscillatory modes can degrade solution quality in underresolved problems leading to increased negativ- ities in the VEF scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We note that the lumped Schur complement associated with the RT discretization does not contain these non-physical modes and can thus be effectively preconditioned by AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0e+00 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='3 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0e+008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Weak Scaling Finally, we show that the RT and HRT methods weak scale in parallel on the first iter- ation of the linearized crooked pipe problem from §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The inversion of the streaming and collision operator is approximated with one iteration of a parallel block Jacobi sweep where each processor performs a transport sweep on its processor-local domain using angular fluxes on inflow processor boundaries that are iteratively lagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, the transport sweep is not exact when more than one processor is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Uni- form refinements are used in tandem with increasing the processor count by four so that the number of unknowns per processor remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Since the sizes of the systems corresponding to the RT and HRT methods differ, the data are tabulated in terms of the number of scalar flux degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We stress that the linear system for the RT method additionally includes the degrees of freedom associated with each component of the current and that the HRT system solves the reduced problem for the Lagrange multiplier unknowns only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The results were generated on 29 nodes of the rztopaz machine at LLNL which has two 18-core Intel Xeon E5-2695 CPUs per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Timing data is presented as the minimum time achieved across three repeated runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We compare the efficiency and performance of the inner solvers for the RT and HRT discretizations with p = 2 when 1) the parallel block Jacobi transport sweep is used to compute the VEF data and 2) the VEF data are set to their asymptotic, diffusive values of E = 1/3I and Eb = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The BiCGStab tolerance was 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Table 11 compares the number of iterations to convergence and solve times for solving the RT VEF and RT diffusion linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the VEF system, a range of precon- ditioner options are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The preconditioners are parameterized by the use of one iteration of Jacobi (J) or Gauss-Seidel (GS) for approximating the inverse of the total interaction mass matrix and the number of AMG V-cycles applied to the lumped Schur complement at each preconditioner application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We consider the permutations of using Jacobi, Gauss-Seidel, one AMG V-cycle, and three AMG V-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT diffusion system is preconditioned by one iteration of Jacobi on the total interaction mass matrix and one AMG V-cycle on the lumped Schur complement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' the “J-1” preconditioner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The J-1 preconditioner led to a scalable solver for the diffusion sys- tem with an increase of only 6 iterations when the problem size was increased by a factor of 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the J-1 preconditioner did not perform as uniformly when applied to the VEF system showing an increase of 14 iterations over the same range of problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Increasing the number of AMG V-cycles per preconditioner application to three led to an algorithm that scaled more closely to the diffusion case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Switching to Gauss-Seidel for approximating the inverse of the total interaction mass matrix did reduce the total number of iterations to convergence but did not alter the scaling of it- erations with problem size seen with the J-1 preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The GS-3 option required the least iterations to converge and iteration counts scaled similarly to that of the J-1 preconditioner applied to the diffusion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These results suggest that scalability can be improved by performing more AMG V-cycles on the lumped Schur comple- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, using a more expensive approximation of the inverse of the total mass matrix was less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In terms of solve time, the J-1 preconditioner was the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, the increased robustness in iteration count to problem size provided by the more expensive preconditioners did not adequately balance their increased cost per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the largest problem size, the J-1 preconditioner applied to the VEF system was only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='23x more expensive than solving the diffusion system with the J-1 preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 47 TABLE 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A weak scaling study of the first iteration of the linearized crooked pipe problem using p = 2 for the RT discretization preconditioned with a lower block triangular preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The number of BiCGStab iterations to converge to a tolerance of 10−8 and the total solve time are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solving the linear systems corresponding to RT VEF and RT diffusion are compared with the RT VEF system preconditioned using a range of preconditioners corresponding to the use of Jacobi (J) or Gauss-Seidel (GS) to approximate the inverse of the total interaction mass matrix and the application of one or three AMG V-cycles per iteration to approximate the inverse of the lumped Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The diffusion system is solved using the J-1 preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Times are provided in seconds and represent the minimum time achieved over three repeated runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Iterations Solve Time (s) Processors DOF J-1 J-3 GS-1 GS-3 Diffusion J-1 J-3 GS-1 GS-3 Diffusion 1 42 588 29 18 24 14 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='68 4 170 352 31 20 24 15 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='03 16 681 408 34 20 27 17 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='83 64 2 725 632 37 21 33 19 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='67 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='98 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='72 256 10 902 528 38 23 33 21 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='91 1024 43 610 112 43 25 38 22 33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='69 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='66 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='22 This comparison is repeated for the HRT method in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, we compare only the use of one AMG V-cycle to precondition the reduced system corresponding to the VEF and diffusion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solving the VEF system required at most 4 more iterations compared to solving diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Solving the largest VEF problem was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='19x more expensive than the largest diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Compared to RT, the HRT VEF solve was up to 13x faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Figures 17a and 17b show the weak scaling efficiency for the RT and HRT discretiza- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Weak scaling efficiency is defined as εn = solve time with one processor solve time with n processors , (116) where ideal scaling is εn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Ideal scaling is not expected since solving these linear systems requires parallel communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For both the RT and HRT methods, the VEF system can be scalably solved with comparable efficiency to solving the corresponding diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, for the problem size solved with 1024 processors, the RT VEF preconditioners scaled with efficiencies of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6%, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0%, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1%, and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4% for the J-1, J-3, GS-1, and GS-3 preconditioners, respectively, while J-1 applied to diffusion scaled with an efficiency of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This suggests that the J-3 preconditioner may scale more robustly despite being more expensive than J-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For HRT, the efficiencies were 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='9% and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0% for solving the VEF and diffusion systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 48 TABLE 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Weak scaling the HRT discretization of VEF and diffusion on the linearized crooked pipe with p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Both problems used one AMG V-cycle to precondition the HRT reduced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The number of BiCGStab iterations to converge to a tolerance of 10−8 and the total solve times in seconds are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The timing data is represents the minimum time recorded over three repeated runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Iterations Solve Time (s) Processors DOF VEF Diffusion VEF Diffusion 1 42 588 11 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='12 4 170 352 13 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='16 16 681 408 14 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='25 64 2 725 632 16 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='36 256 10 902 528 15 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='39 1024 43 610 112 16 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='51 100 101 102 103 Number of Processors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 Weak Scaling Efficiency J-1 J-3 GS-1 GS-3 Diffusion (a) 100 101 102 103 Number of Processors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='0 Weak Scaling Efficiency VEF Diffusion (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='. Weak scaling efficiency for solving the (a) RT and (b) HRT VEF systems on the first iteration of the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the RT discretization, a range of preconditioners are presented corresponding to the use of Jacobi (J) or Gauss-Seidel (GS) and the number of AMG V-cycles applied per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In both cases, the scaling of the VEF solve is compared to solving radiation diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT diffusion system is preconditioned with the J-1 preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Both HRT VEF and HRT diffusion use one AMG V-cycle to precondition the reduced system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 49 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Conclusions We have developed high-order mixed finite element discretizations for the Variable Ed- dington Factor (VEF) equations that are compatible with curved meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The methods were designed to have element-local particle balance and immediate multiphysics com- patibility with the mixed finite element techniques used for hydrodynamics calcula- tions in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Each method produces a scalar flux solution in the Discontinuous Galerkin (DG) space to match the approximation space used for the thermodynamic variables in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We considered three choices for the finite element space that approximates the current: a discrete subspace of [H1(D)]2 where each component is represented with continuous finite elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' a method that uses the Raviart Thomas (RT) space along with DG-like numerical fluxes to treat the discontinuities arising from the presence of the Eddington tensor in the VEF first moment equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' and a hybrid RT method where continuity of the normal component of the current is enforced weakly with a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These methods are referred to as H1, RT, and HRT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The VEF discretizations were paired with a high-order DG discretization of the SN transport equations to solve problems from linear transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On manufactured solutions problems, the H1, RT, and HRT methods all had opti- mal O(hp+1) convergence for the VEF scalar flux on refinements of a curved mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, optimal convergence for the VEF current was observed only on diffusive problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This result suggests that the VEF scalar flux and the zeroth angular mo- ment of the discrete angular flux converge at equivalent rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, on transport problems, the VEF current could not be solved to the same accuracy as the first angular moment of the discrete angular flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, the mixed finite element superconvergence property for the VEF scalar flux was lost on the standard use case of equal degree interpolation for the VEF scalar and angular fluxes on a quadratically anisotropic transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This suggests that post-processing techniques, such as the method proposed by Stenberg [35], that leverage the mixed finite element super- convergence property to produce more accurate solutions would not be effective on transport problems where equal degree interpolation is used for the VEF scalar and angular fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' All three methods showed rapid and robust convergence on a single-material thick diffusion limit test problem on both a simple orthogonal mesh and a severely distorted third-order mesh generated with a Lagrangian hydrodynamics code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The methods were tested on mesh and polynomial order refinements of a two-material linearized crooked pipe problem that had a 1000x difference in total cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Fixed-point convergence was robust for all three methods with RT and HRT converging equiva- lently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 method converged slower, requiring ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='75x more iterations than RT and HRT on the largest mesh for each polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' It was observed that the H1 VEF discretization produced scattering sources more likely to induce negativity in the transport sweep leading to an increased reliance on the negative flux fixup compared to the RT and HRT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We also investigated preconditioned iterative solvers for the H1, RT, and HRT meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Lower block triangular preconditioners were used for the H1 and RT methods that employ Jacobi smoothing on the total interaction mass matrix and Algebraic Multigrid (AMG) on the lumped Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The solvers for the HRT method leverage the element-by-element block structure generated by discontinuous approxi- mations to form a reduced system for the Lagrange multiplier only, leading to fewer 50 globally coupled unknowns than in the H1 or RT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' AMG is applied directly to the reduced problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The preconditioned iterative solvers were tested on a series of increasingly distorted meshes to test their robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 and HRT methods converged for all distortions but the RT method failed to converge once the mesh be- came too distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT and HRT methods were shown to have scalable solvers in both h and p on the linearized crooked pipe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, the solvers for H1 were not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' It was found that AMG was struggling to adequately precondition the lumped Schur complement due to the presence of highly oscillatory, slowly decaying modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' These modes are a consequence of the mismatch between the finite element spaces used to approximate the VEF scalar flux and current and were present even on a simple Poisson eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, a weak scaling study demonstrated that the RT and HRT methods can be scalably solved out to 1024 processors and over 40 million VEF scalar flux unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Compared to solving the symmetric positive def- inite radiation diffusion system, solving the non-symmetric VEF equations was only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='23x and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='19x more expensive for the RT and HRT methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The primary takeaway from this work is that the combination of a DG SN discretiza- tion and the RT or HRT VEF discretizations form an effective high-order method for linear transport problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Both the RT and HRT discretizations of the VEF equations have high-order accuracy, compatibility with curved meshes, and robust and scalable convergence in both outer fixed-point iterations and inner preconditioned linear solver iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The performance of the methods was differentiated only in the presence of severely distorted meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In such case, the preconditioned iterative solver for the HRT method was robust to mesh distortion whereas the solver for the RT method was not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1 method is not recommended for use in a production code due to the lack of scalable iterative solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, the H1 method had lower fixed-point iteration efficiency and higher reliance on the negative flux fixup on the linearized crooked pipe problem when compared to the RT and HRT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In radiation-hydrodynamics calculations, the scalar flux and current are coupled to the hydrodynamics’ energy balance and momentum equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to the sub-optimal accuracy of the VEF current on transport problems, it is unclear whether the mixed finite element methods presented here would yield improvements in physics fidelity commensurate with the increased cost of solving for both the VEF scalar flux and current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Future work includes investigating the quality of the solution provided by the RT and HRT methods on tightly coupled multiphysics problems as compared to the DG VEF methods presented in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Acknowledgments This work was performed under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-829396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, and the Department of Energy Computational Science Graduate Fellowship under Award Number DE- SC0019323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 51 References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Mihalas, Stellar Atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Freeman and Co, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} 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Analyse Num´erique, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 151–167, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Available: http://eudml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='org/doc/193618 [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Brezzi, Stability of Saddle-Points in Finite Dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 17–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Available: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1007/978-3-642-55692-0 2 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The Gradient of the Piola Transform The goal of this section section is to derive a formula for the transformation of the gradient of a vector defined under the contravariant Piola transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the contravariant Piola transform v = 1 J Fˆv ◦ T−1 the inverse transform is: ˆv = JF−1v ◦ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A1) Here, we seek to derive ˆ∇ˆv = ˆ∇ � JF−1v � , (A2) so that we can solve for ∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The goal is to derive the functional form of the trans- formation in terms of functionality commonly implemented in finite element codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' That is, we cast the computation in terms of the Jacobian matrix and Hessian of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 54 Through their connection to the Jacobian matrix and the inverse of the Jacobian matrix, the tangent and cotangent spaces are related by n1 = t2 × ˆe3 , n2 = ˆe3 × t1 , (A3) where ˆe3 points out of the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, n1 is a 90 degree clockwise rotation of t2 and n2 is a 90 degree counterclockwise rotation of t1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, we can write ˆ∇ˆv = ˆ∇ � Jn1 · v Jn2 · v � = � ∂ ∂ξ(Jn1 · v) ∂ ∂η(Jn1 · v) ∂ ∂ξ(Jn2 · v) ∂ ∂η(Jn2 · v) � = � ∂ ∂ξ(Jn1) · v ∂ ∂η(Jn1) · v ∂ ∂ξ(Jn2) · v ∂ ∂η(Jn2) · v � + � Jn1 · ∂v ∂ξ Jn1 · ∂v ∂η Jn2 · ∂v ∂ξ Jn2 · ∂v ∂η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A4) The second term can be written as � Jn1 · ∂v ∂ξ Jn1 · ∂v ∂η Jn2 · ∂v ∂ξ Jn2 · ∂v ∂η � = JF−1 ˆ∇v = JF−1∇vF , (A5) where ˆ∇v = ∇vF transforms the reference gradient to the physical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The first term is a third-order tensor contracted with a vector to yield a second-order tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' By expanding the dot products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' we can emulate this contraction as a sum of two second-order tensors: � ∂ ∂ξ(Jn1) · v ∂ ∂η(Jn1) · v ∂ ∂ξ(Jn2) · v ∂ ∂η(Jn2) · v � = � ∂ ∂ξ(Jn11)v1 + ∂ ∂ξ(Jn12)v2 ∂ ∂η(Jn11)v1 + ∂ ∂η(Jn12)v2 ∂ ∂ξ(Jn21)v1 + ∂ ∂ξ(Jn22)v2 ∂ ∂η(Jn21)v1 + ∂ ∂η(Jn22)v2 � = � ∂ ∂ξ(Jn11) ∂ ∂η(Jn11) ∂ ∂ξ(Jn21) ∂ ∂η(Jn21) � v1 + � ∂ ∂ξ(Jn12) ∂ ∂η(Jn12) ∂ ∂ξ(Jn22) ∂ ∂η(Jn22) � v2 = ˆ∇(JF−1 1 )v1 + ˆ∇(JF−1 2 )v2 (A6) where F−1 i are the columns of F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Typically, finite element codes provide the Hes- sian matrix of the forward map but not the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, to leverage existing functionality, we must write the above matrices in terms of H = ˆ∇F instead of ˆ∇F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Assume that the code computes the Hessian matrix in flattened and symmetric form as: ⟨H⟩ = � ∂2x ∂ξ2 ∂2x ∂ξ∂η ∂2x ∂η2 ∂2y ∂ξ2 ∂2y ∂ξ∂η ∂2y ∂η2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A7) 55 Then the above can be rewritten as ˆ∇(JF−1 1 ) = ˆ∇ � F22 −F21 � = ˆ∇ � ∂y ∂η − ∂y ∂ξ � = � ∂2y ∂ξ∂η ∂2y ∂η2 − ∂2y ∂ξ2 − ∂2y ∂ξ∂η � = � H22 H23 −H21 −H22 � , (A8) ˆ∇(JF−1 2 ) = ˆ∇ � −F12 F11 � = ˆ∇ � − ∂x ∂η ∂x ∂ξ � = � − ∂2x ∂ξ∂η − ∂2x ∂η2 ∂2x ∂ξ2 ∂2x ∂ξ∂η � = � −H12 −H13 H11 H12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A9) We can define the matrix ˆB = ˆ∇(JF−1)v = � H22 H23 −H21 −H22 � v1 + � −H12 −H13 H11 H12 � v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A10) This is computed in flattened form as ⟨ ˆB⟩ = � ⟨ ˆ∇(JF−1 1 )⟩ ⟨ ˆ∇(JF−1 2 )⟩ � v = � ��� H22 −H12 H23 −H13 −H21 H11 −H22 H12 � ��� 1 J Fˆv (A11) where v = 1 J Fˆv was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Finally, we have that ˆ∇ˆv = ˆB + JF−1∇vF ⇐⇒ ∇v = 1 J F � ˆ∇ˆv − ˆB � F−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A12) We can then say that ∇v : E dx = 1 J F � ˆ∇ˆv − ˆB � F−1 : E Jdξ = � ˆ∇ˆv − ˆB � : FT EF−T dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A13) 56 Here, we use the fact that A : B = trace(ABT ) and apply the cyclic property of the trace to permute F and F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In this form, we can implement the gradient calculation as a matrix-vector product of the flattened referential gradient and the coefficients of ˆv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' When the mesh transformation is affine, ˆB = 0 since the Hessian of an affine trans- formation is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In addition, the Piola identity states that trace ˆB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This can be most easily seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A10 where trace ˆB = (H22 − H22)v1 + (−H12 + H12)v2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A14) Using the Piola identity and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' A13, we have that ∇ · v dx = ∇v : I dx = � ˆ∇ˆv − ˆB � : FT IF−T dξ = trace � ˆ∇ˆv − ˆB � dξ = ˆ∇ · ˆv dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (A15) Thus, in the thick diffusion limit when E ∝ I, ∇v : E simplifies to the standard transformation for the divergence of a contravariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Discrete Inf-Sup Condition Here, we discuss the the inf-sup condition that governs the solvability of the 2 × 2 block systems arising in mixed finite element discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Two excellent references for this topic are Brezzi [36] and Benzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We present an analysis for Poisson’s equation since methods not effective for this simpler problem have no hope of being effective for the VEF equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Conditions for Solvability Consider the linear system: � M −DT D � �q u � = �0 f � , (B1) which corresponds to the mixed finite element discretization of q + ∇u = 0 , (B2a) ∇ · q = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B2b) Note that the above is Poisson’s equation, −∇2u = f, in mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The matrices are of the form: vT Mq = � v · q dx , wT Dq = � w ∇ · q dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B3) 57 We wish to demonstrate the conditions for when the block system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' B1 is non- singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' To show the solution is unique, it must be verified that f = 0 implies that u = 0 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' When f = 0, we have that Dq = 0 ⇐⇒ q ∈ N(D) , (B4) where N(D) denotes the nullspace of D such that N(D) = {v : Dv = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B5) For some v ∈ N(D), the first equation reads vT Mq − vT DT u = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B6) Since v ∈ N(D), vT DT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, we have that vT Mq = 0 , ∀v ∈ N(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B7) Since M is a mass matrix, it is symmetric positive definite and thus, vT Mq = 0 ⇐⇒ q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, we have shown that f = 0 ⇒ q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Setting q = 0 in the first row of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' B1 yields: DT u = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B8) For the block system to be non-singular, we must have that DT u = 0 ⇐⇒ u = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B9) Equivalently, we require that the nullspace of DT has only the trivial nullspace of zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' N(DT ) = {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The discrete inf-sup condition is precisely this condition on the matrix D that N(DT ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Characterization for a Single Element We now particularize the matrix D for a single element, K = [0, 1]2, of the [H1(D)]2 × L2(D) and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' D) × L2(D) discretizations of the Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We consider the W1 × Y0 and W1 × Y1 discretizations and show that the former and latter have non-trivial nullspaces for D and DT , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In light of the inf-sup requirement established above, the W1 × Y1 discretization will lead to a singular block system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We also show that the pairing W1 and Y0 is solvable but is imbalanced in another way that allows the presence of non-physical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' By contrast, the RT 0 × Y0 discretization is non-singular and does not allow these non-physical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the single element K, the lowest-order Raviart Thomas and [H1(D)]2 finite element spaces are given by: RT 0 = span{ � 1 0 � , � x 0 � , � 0 1 � , � 0 y � } , (B10a) 58 W1 = span{ � 1 0 � , � x 0 � , � y 0 � , � xy 0 � , � 0 1 � , � 0 x � , � 0 y � , � 0 xy � } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B10b) Further, the constant and linear DG spaces are: Y0 = span{1} , Y1 = span{1, x, y, xy} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B11) Observe that the divergence of RT 0 is exactly the constant polynomial space, Y0: ∇ · RT 0 = span{0, 1, 0, 1} = span{1} = Y0 , (B12) while the divergence of the [H1(D)]2 space is: ∇ · W1 = span{0, 1, 0, y, 0, 0, 1, x} = span{1, x, y} , (B13) which is a space larger than Y0 but smaller than Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The nullspaces of the divergence of the RT and [H1(D)]2 local polynomial spaces are spanned by N(∇ · RT 0) = span{ � −x y � , � 0 1 � , � 1 0 � } , (B14a) N(∇ · W1) = span{ � −x y � , � 0 1 � , � 1 0 � , � 0 x � , � y 0 � } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B14b) Here, we can already see an issue forming: the nullspace for W1 is larger than the nullspace for RT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We are interested in the bilinear form D(u, v) = � K u ∇ · v dx , u ∈ Y , v ∈ X , (B15) where Y is either Y0 or Y1 and X is either W1 or RT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This bilinear form admits the matrix D through uT Dv = D(u, v) , ∀u ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' (B16) D has a nullspace corresponding to the nullspace of the divergence operator and vectors v such that w = ∇ · v ̸= 0 where M(u, w) = � uw dx = 0 , (B17) for each u ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' We consider elements of the nullspace corresponding to this second condition to be non-physical since they arise from the mismatch between the spaces Y and X and not the divergence operator itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the case Y = Y1 and X = W1 corresponding to the W1 × Y1 discretization, the space Y1 is larger than ∇ · W1 and thus there does not exist w ̸= 0 such that M(u, w) = 0 for all u ∈ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, M has only the trivial nullspace and N(D) = N(∇ · W1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' However, there does exist u ̸= 0 such that M(u, w) = 0 for each w ∈ ∇ · W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, u = α � 1 4 − 1 2x − 1 2y + xy � 59 for any α ̸= 0 satisfies M(u, w) = 0 for all w ∈ ∇ · W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Note that this particular form for the nullspace arises from our choice of the domain D = K = [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This means DT has a non-trivial nullspace and thus the resulting 2 × 2 block system will be singular by the inf-sup requirement in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' For the case Y = Y0 and X = W1 corresponding to the W1 × Y0 discretization, there exists w ∈ ∇ · W1 such that w ̸= 0 but M(u, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In particular, N(M) = span{x−1/2, y−1/2} and thus vectors with divergence in N(M) will also be in N(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In other words, N(D) = N(∇ · W1) ∪ Nspurious where Nspurious = span{ � x(y − 1/2) 0 � , � 0 (x − 1/2)y � } , (B18) is the space of vectors whose divergence belongs to N(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Observe that for v ∈ Nspurious, Dv = 0 but ∇ · v ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the other hand, since ∇ · W1 is larger than Y0, DT has only the trivial nullspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Thus, the W1 × Y0 discretization is solvable but D has a non-physically enlarged nullspace that allows spurious solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' By contrast, with Y = Y0 and X = RT 0, ∇·RT 0 = Y0 meaning M is the L2(K) inner product of functions in Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In such case, M is symmetric and positive definite and thus has only the trivial nullspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' This means N(D) = N(∇ · RT 0) and N(DT ) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The RT 0×Y0 discretization is then non-singular and D has only the physical nullspace associated with the divergence operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The takeaway is that for the pairing W1 × Y0, W1 is rich enough to ensure non- singularity but it is too rich with respect to Y0 such that spurious modes are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The W1 × Y1 discretization has the opposite problem: while ∇ · W1 is small enough with respect to Y1 to avoid spurious modes it is too small to allow the block system to be non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' In general, Yp ⊂ ∇·Wp ⊂ Yp+1 and thus one must always compromise between solvability and avoiding spurious modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' On the other hand, ∇ · RT p = Yp so that the RT p×Yp discretization is both solvable and does not allow spurious solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Method of Manufactured Solutions Supplemental Data 60 TABLE C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Error values from an isotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the VEF data are constant in space and thus are represented exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p h H1 RT HRT H1 RT HRT H1 RT HRT 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='994 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='160 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='161 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='161 × 10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='853 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='067 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='067 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='605 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='251 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='251 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='997 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='040 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='040 × 10−4 1.' metadata={'source': 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× 10−7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='665 × 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='712 × 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='712 × 10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='261 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='258 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='258 × 10−6 4.' metadata={'source': 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× 10−10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='640 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='027 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='027 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='328 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='524 × 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='525 × 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='525 × 10−9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='290 × 10−11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='896 × 10−11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='896 × 10−11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='554 × 10−9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='712 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='712 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='496 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='619 × 10−10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='619 × 10−10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='619 × 10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='119 × 10−11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='209 × 10−11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='210 × 10−11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='089 × 10−9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='473 × 10−9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='473 × 10−9 TABLE C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Error values from a quadratically anisotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the angular flux used to calculate the VEF data is represented with Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, the maximum accuracy expected is order p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p h H1 RT HRT H1 RT HRT H1 RT HRT 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='994 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='891 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='891 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='890 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='589 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='725 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='638 × 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='454 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='703 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='499 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='997 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='707 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='708 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='707 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='915 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='110 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='979 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='925 × 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='746 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='651 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='331 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='090 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='090 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='090 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='980 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='035 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='004 × 10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='786 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='866 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='124 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='985 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='175 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='175 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='175 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='061 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='082 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='066 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='077 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='410 × 10−3 3.' metadata={'source': 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10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='496 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='911 × 10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='910 × 10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='926 × 10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='758 × 10−9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='737 × 10−9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='489 × 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='618 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='428 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='432 × 10−7 61 TABLE C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Error values from a quadratically anisotropic MMS test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The H1, RT, and HRT columns refer to the Yp × Wp+1, Yp × RT p, and hybridized Yp × RT p discretizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' The error in the scalar flux, the error in the scalar flux when the exact solution is first projected onto Yp, and the error in the current are presented for each method over a range of values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Here, the angular flux used to calculate the VEF data is represented with Yp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' Due to this, the maximum accuracy expected is order p + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content=' ∥ϕ − ϕex∥ ∥ϕ − Πϕex∥ ∥J − Jex∥ p h H1 RT HRT H1 RT HRT H1 RT HRT 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='997 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='564 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='564 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='564 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='329 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='301 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='172 × 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='910 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='028 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='028 × 10−2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='985 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='827 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='827 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='827 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='309 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='323 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='291 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='849 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='138 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='138 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='657 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='219 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='219 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='219 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='784 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='878 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='737 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='564 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='425 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='424 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='993 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='915 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='914 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='914 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='244 × 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='305 × 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='226 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='021 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='568 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='568 × 10−3 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='994 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='876 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='875 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='875 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='032 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='097 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='683 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='683 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='275 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='422 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='256 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='210 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='741 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='569 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='331 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='081 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='081 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='081 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='764 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='275 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='754 × 10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='459 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='166 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='675 × 10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='985 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='171 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='171 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='171 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='586 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='826 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='594 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='090 × 10−4 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='945 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='488 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='639 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='050 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='148 × 10−5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='074 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='490 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='677 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='677 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='677 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='224 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='081 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='130 × 10−7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='233 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='894 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='947 × 10−6 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='681 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='049 × 10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='039 × 10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='038 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='467 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='567 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='109 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='767 × 10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='063 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='817 × 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='994 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='591 × 10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='589 × 10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='589 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='175 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='337 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='064 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='136 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='636 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='094 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='628 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='043 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='043 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='043 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='684 × 10−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='006 × 10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='624 × 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='109 × 10−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='061 × 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='155 × 10−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='997 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='477 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='477 × 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='477 × 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='806 × 10−9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQf2wun/content/2301.04758v1.pdf'} +page_content='758 × 10−9 6.' metadata={'source': 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International Center for Quantum-field Measurement Systems for Studies of the Universe and +Particles (QUP), KEK, Tsukuba 305-0801, Japan +Abstract +It has been argued that there exist extragalactic magnetic fields in the range from 10−17 to +10−9 Gauss with a cosmological coherence length. One plausible explanation for the origin of +the extragalactic magnetic fields would be quantum fluctuations of the electromagnetic fields +during inflation. It is also believed that primordial gravitational waves (PGWs) arise out of +quantum fluctuations during inflation. We study the graviton-photon conversion process in +the presence of background magnetic fields and find that the process induces the tachyonic +instability of the PGWs. As a consequence, a peak appears in the power spectrum of PGWs. +It turns out that the peak height depends on the direction of observation. The peak frequency +could be in the range from 10−5 to 103 Hertz for GUT scale inflation. Hence, the observation +of PGWs could provide a new window for probing primordial magnetic fields. +arXiv:2301.13540v1 [hep-th] 31 Jan 2023 + +Contents +1 +Introduction +1 +2 +Graviton-photon interaction in the early universe +3 +2.1 +Primordial GWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Primordial magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.3 +Graviton-photon conversion +. . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.4 +Diagonal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +3 +Generation and evolution of primordial GWs +8 +3.1 +Cosmological setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Bogoliubov Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Tachyonic instability and the PGW power spectrum +12 +5 +Conclusion +15 +1 +Introduction +Since the first direct detection of gravitational waves [1], gravitational waves have been the +most important tool for investigating fundamental physics. Gravitational waves interact with +matter very weakly and travel through the universe virtually unimpeded, so they can be a +probe of the early inflationary universe. The inflationary scenario predicts that primordial +gravitational waves (PGWs) are generated from quantum fluctuations of spacetime. In terms +of the density parameter, the spectrum ΩGW(f) for the GUT scale inflation is independent of +frequency f, that is, of wave number k for the gravitons created at the de Sitter to radiation +dominant transition: +ΩGW(f) = 10−14 +� +H +10−4Mpl +�2 +, +(1.1) +1 + +where H is the Hubble parameter, Mpl is the reduced Planck mass. This scale-invariant +power spectrum comes from the time translation invariance of de Sitter space. There are +several experimental projects for detecting PGWs [2, 3, 4, 5]. If the nonclassicality of the +PGWs is found, it implies the existence of gravitons. Notably, due to the particle creation, +the quantum state of gravitons becomes squeezed during inflation +[6, 7, 8, 9, 10]. Since +the squeezing tends to enhance the observability of gravitons, it is expected that we may +be able to probe quantum gravity through observations of PGWs. There are new ideas for +detecting the quantum nature of PGWs. One method is to utilize the Hanbury Brown- +Twiss interferometry developed in quantum optics for the PGWs, which can distinguish +nonclassical particles from classical ones by measuring intensity-intensity correlations [11, +12, 13]. Moreover, the squeezed state of gravitons can be measured indirectly through their +noise in the interferometers [14, 15, 16, 17] or by measuring the decoherence time of a +quantum object caused by the surrounding primordial gravitons [18]. +On top of the PGWs, there might exist primordial magnetic fields (PMFs). Indeed, it +is argued that observations of GeV gamma-ray from blazars give rise to the lower bound +of the extragalactic magnetic field 10−17 Gauss [19, 20, 21, 22, 23, 24]. We also have the +upper bound 10−9 Gauss from CMB data [25, 26]. It is important to seek the origin of the +extragalactic magnetic fields. If a coherence length of magnetic fields is around mega-parsec +scales, it would be natural to consider magnetic fields originated from quantum fluctuations +during inflation [27, 28]. In the presence of PMFs during inflation, the effect of the PMFs on +PGWs is studied in [29]. In the case of scale-invariant electromagnetic fields, there arises an +entanglement between gravitons and photons [30]. Therefore, the presence of scale-invariant +electromagnetic fields may affect the squeezed state of gravitons. Thus, it is important to +clarify the effect of PMFs on PGWs. +In this paper, we consider the background PMFs during inflation and the scale-invariant +perturbed electromagnetic fields. It is known that the presence of background magnetic +fields causes the conversion of gravitons into photons and vice versa [31, 32]. We focus on +the effect of the graviton-photon conversion process on the PGW power spectrum. We show +that there occurs the tachyonic instability of PGWs due to graviton-photon conversion in +the presence of PMFs. Because of this instability, there arises a peak in the PGW spectrum +2 + +(1.1). Moreover, since the magnetic fields specify a direction in the space, there appears +statistical anisotropy in the power spectrum of PGWs. Remarkably, it turns out that the +peak frequency could be in the range from 10−5 to 103 Hertz. The presence of the peak turns +out to enhance the squeezing of gravitons and makes it easy to observe the PGWs. More +interestingly, we may be able to probe the PMFs by observing the spectrum of PGWs. +The organization of the paper is as follows. +In section 2, we present the setup. +In +section 3, we calculate Bogoliubov coefficients. In section 4, we calculate the PGW power +spectrum. We see the tachyonic instability leads to a peak in the spectrum. We also discuss +the observability of the peak. Section 5 is devoted to the conclusion. +2 +Graviton-photon interaction in the early universe +We consider the Einstein-Hilbert action and the action for a U(1) gauge field coupled with +a scalar field: +S = Sg + Sφ + SA = +� +d4x√−g +�M 2 +pl +2 +R − 1 +2(∂µφ)(∂µφ) − V (φ) − 1 +4f 2(φ)F µνFµν +� +, (2.1) +where Mpl = 1/ +√ +8πG is the Planck mass. The gauge field Aµ represents photons and the +field strength is defined by Fµν = ∂µAν − ∂νAµ. +The background inflationary dynamics is determined by the metric +ds2 = a2(η) +� +−dη2 + δijdxidxj� +, +(2.2) +and the inflaton φ(η). Once the background is given, the coupling function can be regarded +as a function of the conformal time η; f = f(η) . We also assume the presence of constant +magnetic fields Bi = constant . It should be emphasized that the physical magnetic fields +are not Bi but fBi. In the following, we consider the quantum evolution of gravitons and +photons in the above background. +3 + +2.1 +Primordial GWs +We consider gravitons in a spatially flat expanding background represented by tensor mode +perturbations in the three-dimensional metric hij, +ds2 = a2(η) +� +−dη2 + (δij + hij) dxidxj� +, +(2.3) +where hij satisfies the transverse traceless conditions hij,j = hii = 0. The spatial indices +i, j, k, · · · are raised and lowered by δij and δkℓ. In the case of de Sitter space, the scale +factor is given by a(η) = −1/(Hη) where −∞ < η < 0. +Expanding the Einstein-Hilbert action up to the second order in perturbations hij, we +have +δSg = M 2 +pl +8 +� +d4x a2 � +hij′ h′ +ij − hij,khij,k +� +, +(2.4) +where a prime denotes the derivative with respect to the conformal time. At this quadratic +order of the action, it is convenient to expand hij(η, xi) in Fourier modes, +hij(η, xi) = +2 +Mpl +� +P +1 +(2π)3/2 +� +d3k hP +k (η) eP +ij(k) eik·x , +(2.5) +where three-vectors are denoted by bold math type and eP +ij(k) are the polarization tensors +for the k mode normalized as eijP(k)eQ +ij(k) = δPQ with P, Q = +, ×. Then the action (2.4) +in the Fourier space becomes +δSg = 1 +2 +� +P +� +d3k dη a2 � +|hP′ +k |2 − k2|hP +k |2 � +. +(2.6) +2.2 +Primordial magnetic fields +Next, we consider the action for the photon up to the second order in perturbations Ai, +which is given by +δSA = 1 +2 +� +d4x f 2 � +A′ 2 +i − A2 +k,i +� +, +(2.7) +4 + +where the photon field satisfies the Coulomb gauge A0 = 0 and Ai,i = 0. +Expanding the field Ai(η, xi) by the Fourier modes, we find +Ai(η, xi) = +� +P +±i +(2π)3/2 +� +d3k AP +k (η) eP +i (k) eik·x , +(2.8) +where eP +i (k) are the polarization vectors for the k mode normalized as eiP(k)eQ +i (k) = δPQ +with P, Q = +, ×. The sign of ±i corresponds to the P, Q = +, ×. The action (2.7) in terms +of the Fourier modes reads +δSA = 1 +2 +� +P +� +d3k dη f 2 � +|AP′ +k |2 − k2|AP +k |2 � +. +(2.9) +2.3 +Graviton-photon conversion +The action for the interaction between the graviton and the photon up to second order in +perturbations hij, Ai is found to be +δSI = +� +d4x +� +εiℓmf 2Bmhij (∂jAℓ − ∂ℓAj) +� +, +(2.10) +where Bm = εmjℓ ∂jAℓ is a constant background magnetic field. +In terms of the Fourier modes defined in Eqs. (2.5) and (2.8), the interaction term reads +δSI = +2 +Mpl +� +P,Q +� +d3k dη f 2 � +εiℓm Bm hP +k AQ +−k eP +ij(k) +� +ikℓ eQ +j (−k) − ikj eQ +ℓ (−k) +�� +, +(2.11) +where k = |k|. Polarization vectors ei+, ei× and a vector ki/k constitute an orthonormal +basis. Without loss of generality, we assume the constant background magnetic field is in +the (ki, ei×)-plane as depicted in FIG. 1. +5 + +Propagation +ACfXichVHJSgNBEH0Z97gk6kXwEgwRQkdCUY8Bb141MQsEDXMjJ04ZDZmJsEY8gP+gAdBUBC3z/DiD3jIJ4jHCF4ErUwGREW +tprtfv65XdUlmapiO4y1fEJPb1/wOCQf3hkdCwQHJ/I2kbVknlGNlTDykuizVF5xlHcVSeNy0uapLKc1JlvXOfq3HLVgx926mb +fFcTy7pSUmTRIaoYnGjsSFqI7y051xUac4Xg2EWZa6FfoKYB8LwbNMI3mIH+zAgowoNHDocwipE2DQKiIHBJG4XDeIsQop7z9GEn +7RV8uLkIRJbobVMp4LH6nTuxLRdtUyvqDQtUoYQY/shrXZA7tjT+zt1gN0YnlzrtEhZpat0I3CwGjqfSr/9qNdodHyq/szcQ +QkrbsYKVWC6TKcWuauvHZ206upSGOWXbBnquKctdg91aHXuTLZ46hZ/aEPv+6T9BdikaW47Gt+Lh5JrXkEFMYwZz9OsJLGBTW +To3UOc4QrXvnchIiwK0a6r4PM0k/hiQuID1ciRsA= +e+(k) +ACfHichVHNLgNRFP46/uv2EhsGkUkqFtpSqyEjWV/FEnbNDPjYtL5y8y0wugLeAELsSB8Bg2XsCijyCWldiIOJ +1OIjQ4N/fe7373fOec49kqortMFYLCG3tHZ1d3T3B3r7+gcHQ0PCmbZQtmWdlQzWsbUm0uaroPOsojsq3TYuLmqTyLam0 +1rjfqnDLVgx9wzk0eUET93RlV5Fh6hiaMjNS1q4VJ13jz1wXC2GIizKPAu3gpgPIvAtaYTukMcODMgoQwOHDoewChE2jR +xiYDCJK8AlziKkePcVQRJWyYvTh4isSVa9+iU81mdzo2YtqeW6RWVpkXKMCbZE7tldfbI7tkze/81luvFaORySLuEWZpa +MwI3i4Mno5m3f7Ua7Q72v1R/Zu5gF0texgpVYHpMoxa5qa8cndYzy+lJd4pdsheq4oLV2APVoVde5asUT58hSG2I/fz0Vr +C5EI0lovFUPLKy6jekG2MYxzT9+iJWsI4ksvTuAc5xjZvAhzAhzAhzTVch4GtG8M2ExCdVRZH5k/|k| +ACg +nichVHNLkNBFP5c/XTYkFi02gIs1cBELYWOJKhKlufea1qT3L/dOm9D0BbyAhRVi4e8pbLyAhUcQy0psLJze3kQnMnMfOeb8505Z0Z3TeFLx +p4alMam5pbWtvZIR2dXdzTW07vpO0XP4GnDMR1vW9d8bgqbp6WQJt92Pa5Zusm39MJy7XyrxD1fOPaGPHT5rqXlbZEThiaJysb6yxndivO9jBQW9 +yujgVuojGVjCZkgcV/AjUECYS26sSukcE+HBgowgKHDUnYhAafxg5UMLjE7aJMnEdIBOcFURIW6QoThEasQVa8+TthKxNfi2nH6gNusWk6ZEyj +mH2yK5YlT2wG/bM3n/NVQ5y1Go5pF3HBE2rnoG72ejxQOrtX61Fu8TBp+rPyiVymAsqFtSBGzC1Xoy6vnR0Uk3Nrw+XR9g5e6EuztgTu6c+7NKrc +bnG108RoW9Qvz/6T7A5mVRnktNr04nFpfBD2jCIYzSq89iEStYRTq49wK3uFOalHFVabqoUpDqOnDF1MWPgBJlJM2e⇥(k) +ACbnicjVG7SgNBFD1ZXzG+oIgaDYiFhIkHF +KsTGMolGBRXZXSc6ZF/sTgIx+AM2lipWChbqZ9j4Axb5BLERFGwsvLtZEAsfd +5iZe8/cx9zNcQnmSsGVHa2js6u6LdsZ7ev6B+ODQumdXZ2XdNuw3U1N9bg +hLF6SQhp803G5amoG39Aqy/7Ro27nrCtNVl3+I6p7luiLHRVElTa1sxEbje +TKdYImflSRCydvxW2xjDzZ0VGCw4Ik3YAKj9YW0mBwCNtBgzCXNBG8cxwhR +twqeXHyUAmt0LlP1laIWmT7Mb2ArVMWg7ZLzASm2CO7Ya/sgd2xJ/bxY6xGEM +OvpU63hlnaZisCd3YHjkdX3/kmnRLHyxfq1cozFoGJBHTgB4veit/i1w9P +X1aXiVGOaXbFn6uKSNdk9WHV3vTrAi9eIPa/MazPpdLzqUwhk8zmwoFEMYZJz +NCvLyCLFeRorwCJzjDeRFGVHGlYmWqxIJOcP4JsrMJ9eNjF0=B +ACb3ichVHNLjRBFD3T/oePwYJEIhMTX0hkclvaz1gJG0t/MySIdLdC03/prpmEiRewshOsSCz4HuPbeAELjyBWQmJj4U5PJ4LgV +qrq1Kl7bt1b1/BtK5REdwmlprauvqGxKdnc8qe1LdXeUQi9YmCKvOnZXrBs6KGwLVfkpSVtsewHQncMWywZu9OV+6WSCELcxflni/ +WH3LtTYtU5dMFVbltpD6eipD2RzRqDaSZpDLqZrGgCJLqzHILZL3WNVWzAg4kiHAi4kIxt6Ah5rEAFwWduDWXmAkZWdC9wgCRri ++wl2ENndpfXLT6txKzL50rMFKb/IrNM2BlGv10S1f0RDf0j+7p9dtY5ShGJZc93g0M8XSqEYS/3nbYvfDyq9bhXWL7XfVj5hKbGI8 +ytrgCP2IqtZhVfWn/+GlhYr6/Jcu6IGrOKc7+s91uKVn83JOzJ8hyW1QP3/6V1AYzqjW1Oy0xOxQ1pRA/6MC/PoZJzGAWeX53 +B0c4wWniUelSepV01VJxJpOfDBl8A0/XI2F✓ +FIG 1. Configuration of the polarization vector eP (k), wave number k, and background magnetic field B. +The angle between the magnetic field and the wavenumber vector is represented by θ. +The polarization tensors can be written in terms of polarization vectors ei+ and ei× as +e+ +ij(k) = 1 +√ +2 +� +e+ +i (k)e+ +j (k) − e× +i (k)e× +j (k) +� +, +(2.12) +e× +ij(k) = 1 +√ +2 +� +e+ +i (k)e× +j (k) + e× +i (k)e+ +j (k) +� +. +(2.13) +Below, we assume e× +i (−k) = −e× +i (k). The action (2.11) is then written as +δSI += +� +d3k dη f 2 λ k +� +h+ +k (η) A+ +−k(η) + h× +k (η) A× +−k(η) +� +, +(2.14) +where we defined the coupling between graviton and photon as +λ ≡ +√ +2 +Mpl +εiℓm e+ +i +kℓ +k Bm . +(2.15) +Here, the conditions for the graviton and photon to be real are read, h+,× +−k (η) = h∗ +,× +k +(η) +and A+,× +−k (η) = −A∗ +,× +k +(η) . Below, we focus on the plus polarization and omit the index P +unless there may be any confusion. +6 + +2.4 +Diagonal equations +If we use the canonical variable yk(η) = a hk(η) and xk(η) = f Ak(η), the total action of +Eqs. (2.6), (2.9) and (2.14) are written as +δS += +δSy + δSx + δSI += +1 +2 +� +d3k dη +� +|y′ +k|2 − +� +k2 − +�a′ +a +�2� +|yk|2 − a′ +a +� +yk y′ +−k + y−k y′ +k +� +� ++1 +2 +� +d3k dη +� +|x′ +k|2 − +� +k2 − +�f ′ +f +�2� +|xk|2 − f ′ +f +� +xk x′ +−k + x−k x′ +k +� +� ++ +� +d3k dη +� f +a λ k yk x−k +� +. +(2.16) +The variation of the action (2.16) with respect to the graviton and the photon fields gives +y′′ +k + +� +k2 − a′′ +a +� +yk = −λ k f xk +a , +x′′ +k + +� +k2 − f ′′ +f +� +xk = −λ k f yk +a . +(2.17) +We assume the gauge kinetic function in the form +f(η) = a(η)−2c, +(2.18) +where c is a constant parameter. We take c = −1/2 to make the analysis easier. For this +parameter, the power spectrum of the electromagnetic fields Aµ is scale-invariant [33]. While +the spectrum of magnetic fields becomes P(B) ∝ k2. Then Eqs. (2.17) become +y′′ +k + +� +k2 − a′′ +a +� +yk = −λ k xk , +x′′ +k + +� +k2 − a′′ +a +� +xk = −λ k yk . +(2.19) +If we define new variables Xk and Yk such as +Yk = 1 +√ +2 (yk + xk) , +Xk = 1 +√ +2 (yk − xk) . +(2.20) +7 + +Eq. (2.19) are diagonalized in the form +Y ′′ +k + +� +k2 + λ k − a′′ +a +� +Yk = 0 , +X′′ +k + +� +k2 − λ k − a′′ +a +� +Xk = 0 . +(2.21) +3 +Generation and evolution of primordial GWs +In this section, we present a cosmological setup in order to calculate Bogoliubov coefficients. +3.1 +Cosmological setup +We consider instantaneous reheating after inflation approximated by the de Sitter phase +leading to a radiation-dominated phase. The scale factor evolves as follows +a(η) = +� +� +� +� +� +1 +−H(η−2η1) +for +η < η1 , +η +Hη2 +1 +for +0 < η1 < η . +(3.1) +where H is the Hubble parameter during inflation and η1 is the time of reheating. The scale +factor is smoothly connected up to the first order of derivative at the reheating. We assume +that magnetic fields arise instantaneously from the Bunch-Davies vacuum at the time η∗. +Reheating +𝜂 +𝜂! +𝜂∗ +During inflation +Radiation dominated era +Magnetic fields exist +Phase I +Phase II +Phase III +𝜆 = 0 +𝜆 ≠ 0 +𝜆 = 0 +𝑢𝒌 𝜂 +𝑤𝒌 +± 𝜂 +𝑣𝒌 𝜂 +FIG 2. During inflation, magnetic fields Bi arise instantaneously at η∗. Instantaneous reheating at η1 leads +to a radiation-dominated phase. +8 + +Before the time η∗ (phase I), the equations of motion read +Y ′′ +k + +� +k2 − +2 +(η − 2η1)2 +� +Yk = 0 , +X′′ +k + +� +k2 − +2 +(η − 2η1)2 +� +Xk = 0 . +(3.2) +After the η∗, the coupling λ is switched on until η1 (phase II). There, gravitons and photons +are coupled as +Y ′′ +k + +� +k2 + λ k − +2 +(η − 2η1)2 +� +Yk = 0 , +X′′ +k + +� +k2 − λ k − +2 +(η − 2η1)2 +� +Xk = 0 . (3.3) +After the reheating time η1 (phase III), we assume f = const. Then, the equations of motion +become +Y ′′ +k + k2 Yk = 0 , +X′′ +k + k2 Xk = 0 . +(3.4) +We assume the vacuum state can be specified by the Bunch-Davies vacuum which behaves +in the same way as the Minkowski vacuum in the remote past in the de Sitter phase (Phase +I). In the Phase II, the basis is properly normalized but deformed due to the presence of the +magnetic field. In the Phase III, we choose the positive frequency mode for the adiabatic +vacuum as a basis in radiation dominated phase in order to discuss particle creation. Thus +we choose the properly normalized basis in each phase as the solution of Eqs. (3.2), (3.3) +and (3.4). +uk += +1 +√ +2k +� +1 − +i +k (η − 2η1) +� +e−ik(η−2η1) +for +Phase I (3.5) +w+ +k += +1 +� +2 +√ +k2 + λk +� +1 − +i +√ +k2 + λk (η − 2η1) +� +e−i +√ +k2+λk (η−2η1) for +Phase II (3.6) +w− +k += +1 +� +2 +√ +k2 − λk +� +1 − +i +√ +k2 − λk (η − 2η1) +� +e−i +√ +k2−λk (η−2η1) for +Phase II (3.7) +vk += +1 +√ +2k +e−ikη +for +Phase III (3.8) +The above mode functions are normalized as +uku∗′ +k − u∗ +ku′ +k = i, +w± +k w±∗′ +k +− w±∗ +k w±′ +k = i, +vkv∗′ +k − v∗ +kv′ +k = i . +(3.9) +9 + +3.2 +Bogoliubov Transformations +Through the history from Phase I to III, the fields Xk and Yk in Eqs. (2.21) can be expanded +by the mode functions Uk and Vk, respectively such as +Yk = ˆakUk + ˆa† +−kU ∗ +k , +Xk = ˆbkVk + ˆb† +−kV ∗ +k . +(3.10) +The commutation relations [ˆak, ˆa† +k′] = δ(k − k′) and [ˆbk,ˆb† +k′] = δ(k − k′) guarantee the +canonical commutation relations. We write the mode function Uk/Vk that satisfies Eqs. (3.2), +(3.3) and (3.4) as U I +k/V I +k, U II +k /V II +k and U III +k /V III +k . +As the initial state in Phase I, we consider the Bunch-Davies vacuum ˆak|0⟩BD = 0, +ˆbk|0⟩BD = 0. Then the mode functions Uk and Vk are expressed by the positive frequency +mode Eq. (3.5): +U I +k = V I +k = uk(η). +(3.11) +In phase II, the mode function Uk and Vk are written by the solutions Eqs. (3.6) and (3.7) +of Eqs. (3.3) and their complex conjugate, +U II +k += +α+ +IIk w+ +k (η) + β+ +IIk w+∗ +k (η) , +(3.12) +V II +k += +α− +IIk w− +k (η) + β− +IIk w−∗ +k (η) , +(3.13) +where α± +IIk and β± +IIk are integration constants which can be interpreted as the Bogoliubov +coefficients. Similarly, in phase III, the mode functions are written by the solution Eq. (3.8) +of Eq. (3.4): +U III +k += +α+ +IIIk vk(η) + β+ +IIIk v∗ +k(η) , +(3.14) +V III +k += +α− +IIIk vk(η) + β− +IIIk v∗ +k(η) , +(3.15) +where α± +IIIk and β± +IIIk are integration constants interpreted as the Bogoliubov coefficients. +Since the solutions of Yk and Xk must be continuous and continuously differentiable, the +boundary conditions for the above mode functions are required at η∗ and η1 respectively. +10 + +Let us focus on Yk. The boundary conditions at η∗ are given by, +U I +k(η∗) = U II +k (η∗) +(3.16) +U I ′ +k (η) +��� +η=η∗ = U II ′ +k (η) +��� +η=η∗ +(3.17) +The same is true for the V I +k(η0). Plugging Eqs. (3.11) and (3.12) into the above, we have +� +�u(η∗) +u′(η∗) +� +� = +� +� w+ +k (η∗) +w+∗ +k (η∗) +w+ ′ +k (η∗) +w+∗ ′ +k +(η∗) +� +� +� +�α+ +IIk +β+ +IIk +� +� . +(3.18) +Similarly, the boundary conditions at η1 are +U II +k (η1) = U III +k (η1) +(3.19) +U II ′ +k (η) +��� +η=η1 = U III ′ +k +(η) +��� +η=η1 +(3.20) +Plugging Eqs. (3.12) and (3.14) into the above, we have +� +� w+ +k (η1) +w+∗ +k (η1) +w+ ′ +k (η1) +w+∗ ′ +k +(η1) +� +� +� +�α+ +IIk +β+ +IIk +� +� = +� +�vk(η1) +v∗ +k(η1) +v ′ +k(η1) +v∗ ′ +k (η1) +� +� +� +�α+ +IIIk +β+ +IIIk +� +� . +(3.21) +Combining Eqs. (3.18) and (3.21), we find +� +�α+ +IIIk +β+ +IIIk +� +� = +� +�vk(η1) +v∗ +k(η1) +v ′ +k(η1) +v∗ ′ +k (η1) +� +� +−1 � +� w+ +k (η1) +w+∗ +k (η1) +w+ ′ +k (η1) +w+∗ ′ +k +(η1) +� +� +� +� w+ +k (η∗) +w+∗ +k (η∗) +w+ ′ +k (η∗) +w+∗ ′ +k +(η∗) +� +� +−1 � +�uk(η∗) +u′ +k(η∗) +� +� .(3.22) +By using the normalization in Eq. (3.9), the above can be written as +� +�α+ +IIIk +β+ +IIIk +� +� = +� +�−v∗′ +k (η1) +v∗ +k(η1) +v′ +k(η1) +−vk(η1) +� +� +� +�w+ +k (η1) +w+∗ +k (η1) +w+′ +k (η1) +w+∗′ +k (η1) +� +� +� +� w+∗′ +k (η∗) +−w+∗ +k (η∗) +−w+′ +k (η∗) +w+ +k (η∗) +� +� +� +�uk(η∗) +u′ +k(η∗) +� +� .(3.23) +For the Vk, the result is obtained by replacing the superscript + by −. Using the above +results, we calculate the power spectrum of energy density in the next section. +11 + +4 +Tachyonic instability and the PGW power spectrum +Let us discuss the power spectrum of the energy density of gravitons. From Eq. (2.19), the +graviton is given by +yk = 1 +√ +2 (Yk + Xk) . +(4.1) +By using Eq. (3.10), the graviton operator in radiation dominated phase is written as +yk += +1 +√ +2 +� +ˆak U III +k + ˆa† +−kU III∗ +k ++ ˆbkV III +k ++ ˆb† +−kV III∗ +k +� += +1 +√ +2 +�� +α+ +IIIk ak + β+∗ +IIIka† +k + α− +IIIkbk + β−∗ +IIIkb† +k +� +vk + +� +β+ +IIIkak + α+∗ +IIIka† +k + β− +IIIkbk + α−∗ +IIIkb† +k +� +v∗ +k +� +≡ +Akvk + A† +kv∗ +k , +(4.2) +where we defined natural annihilation and creation operators Ak, A† +k in the radiation-dominated +phase. Thus, we can identify the number operator of gravitons as +BD⟨0| A† +kAk |0⟩BD = 1 +2 +� +|β+ +IIIk|2 + |β− +IIIk|2 � +. +(4.3) +Then, the GW spectral density parameter can be calculated as +ΩGW(f) = +k4 +2π2ρc +� +|β+ +IIIk|2 + |β− +IIIk|2 � +, +(4.4) +where ρc is the critical energy density of the universe and f = k/2π is a frequency of GWs. +The GW spectral density parameter is plotted in FIG. 3 where we considered the GUT scale +inflation H = 1014 GeV and B0 = 2 +√ +3 × 10−16 G, η∗ = −1 GeV−1. We see a peak appears +at a particular frequency and no more scale-invariant spectrum as in Eq. (1.1). +12 + +10-9 +10-6 +10-3 +100 +103 +10-14 +10-13 +f [ Hz ] +�g( f ) +ACfnic +hVHLSsNAFD2N7/potRvBTbBUXNQ6FVFxJbrpsq1WC0pSZzW0LxI0kIt/QF/wIULURCsfoYbf8CFnyAuFdy48DYNiIp6h5k5c+aeO/fOVWxdcz3GHkPCwO +DQ8MjoWHh8YnIqEp2e2XethqPygmrplNUZJfrmskLnubpvGg7XDYUnR8o9Z3e/UGTO65mXtey+ZlQ6ZWlVTZY+oSjRWlZIlUq2JcQM8diR0qWK9E4S +zHfxJ8gHYA4Asta0RtIOIQFQ0Y4DhEdYhw6VRQhoMNnFltIlzCGn+PUcHYdI2yIuTh0xsndYanUoBa9K5F9P1Sq9otN0SCkiwR5Yl72we3bLntj7r7H +afoxeLi3aFSRpGv0I3K5ETmZ3/7VGrR7OPpU/Zm5hyo2/Iw1qsD2mV4tal/fPD592d3MJ9oL7JI9UxUX7JHdUR1m81W9yvH8GcLUhvT3T/8J9ldS6bXUam +41vrUdNGQUc5jHIv36OraQRYFereFc1yjK0BYEJaE5b6rEAo0MXwxYeMDiXyQlg=f [ Hz ] +GUT scale inflation +Our model +ACfXichVHJSgNBEH0ZtxiXxHgRvARDR +EFCR4IRT0EPejNRs4CRMDN24pDZmJkEY8gP+AMeBEFB3D7Diz/gwU8Qj +wpeBK1MBkRFra7X7+uV13VJZmqYjuMPfiEnt6+/gH/YGBoeGQ0GBoL5 +2jbsk8JxuqYRUl0eaqovOcozgqL5oWFzVJ5QWptK5LzS4ZSuGvuU0T +b6jiVdqSiy6BVDoVL6xqviuVWydIiq4X2TGW2HIqyOHMt8hMkPBCFZ +xkjdI0SdmFARh0aOHQ4hFWIsGlsIwEGk7gdtIizCnuPUcbAdLWyYuTh +0hsjdYqnbY9VqdzJ6btqmV6RaVpkTKCGLtnV+yZ3bEb9sjefo3VcmN0c +mnSLmGOptaNwM1y8HBi8/VfrUa7g71P1Z+ZO6hg0c1YoQpMl+nUInf1jY +Oj582ljVhrmp2xJ6rilD2wW6pDb7zI51m+cYwAtSHx/dN/gvx8PLEQT2 +aT0fSy1xA/JjGFGfr1FNJYQwY5encfJ7jApe9diAlzQrzrKvg8zTi+mJ +D6ABFjkcw=⌦GW(f) +FIG 3. GW power spectrum is plotted for H = 1014 GeV and B0 = 2 +√ +3 × 10−16 G. Parameters are set as +η∗ = −1 GeV−1, η1 = 10−14 GeV−1, and λ = 2 +√ +3 GeV. The GUT scale inflation predicts a flat spectrum of +the amplitude ΩGW = 10−14. +The origin of the peak comes from the equation for the Xk in Eq. (3.3). If we rewrite it +in the form +X′′ +k + +�� +k − λ +2 +�2 +− λ2 +4 − +2 +(η − 2η1)2 +� +Xk = 0 , +(4.5) +we see that the contribution from the coupling λ causes a tachyonic instability around k ≃ +λ/2. From Eq. (2.15), the strength of the coupling constant can be estimated as +λ = +√ +2 |B| +Mpl +sin θ , +(4.6) +where θ is the angle between the direction of the constant background magnetic field and the +wave number vector of GWs. We assume sin θ > 0 below for simplicity. In the case of sin θ < +0, we just switch the role of Xk to Yk. From the point of view of observation, we consider the +magnitude of current extragalactic magnetic fields of the range 10−17 G < |B0| < 10−9 G. +After inflation, the energy density of radiation decays as ργ ∝ a−4. By combining the Stefan- +Boltzman law ργ = σT 4, we find that the temperature of radiation decays as the Universe +expands such as T ∝ 1/a. Then the ratio of the present scale factor a0 to that of reheating +time a1 is found to be a0/a1 = Treh/T0. Here, Treh is the reheating temperature. In our +model, the physical magnetic fields decay as Bphy ∝ 1/a during inflation and then decay as +13 + +Bphy ∝ 1/a2 after the reheating. Hence, the ratio of the magnetic field at the reheating time +Breh to that at present B0 becomes Breh/B0 = (a0/a1)2 = (Treh/T0)2 ∼ 1054 (Treh/10−4Mpl)2 +where we used T0 ∼ 2.7 K ∼ 10−4 eV. Accordingly, the magnitude of magnetic fields at the +end of the inflation is found to be +1037 +� +Treh +10−4Mpl +�2 +G < Breh < 1045 +� +Treh +10−4Mpl +�2 +G , +(4.7) +where Breh = |B|/a1 = |B| because we used a1 = 1 as a convention. By using Eq. (4.6), +1 G ∼ 10−2 eV2 and Mpl ∼ 1018 GeV, the above range gives the range of maximum coupling +λmax = +√ +2Breh/Mpl at θ = π/2 such as +10−1 +� +Treh +10−4Mpl +�2 +GeV < λmax < 107 +� +Treh +10−4Mpl +�2 +GeV . +(4.8) +The wavenumber at the peak of tachyonic instability k ≃ λ/2 is translated into the peak +frequency observed today and the above lowest value is translated into +f0 = λ +4π +a1 +a0 += 10−5 sin θ +� +B0 +10−17G +� � +Treh +10−4Mpl +� +Hz , +(4.9) +where we used 1 eV = 1015 Hz. Note that the peak frequency depends on the direction of +observation. Thus, in terms of maximum frequency at θ = π/2, the current range 10−17 G < +B0 < 10−9 G corresponds to the frequency range +10−5 +� +Treh +10−4Mpl +� +Hz < f0,max < 103 +� +Treh +10−4Mpl +� +Hz . +(4.10) +Without the magnetic field, the density parameter of PGWs is known to be ΩGW = 10−14 +for the GUT scale inflation as in Eq. (1.1). In the presence of the magnetic field, Eq. (4.4) +tells us that the peak height of the ΩGW is given by +ΩGW(fpeak) = 10−14 +� +H +10−4Mpl +�2 1 +2 exp [−λ η∗] . +(4.11) +We find that the PGWs power spectrum depends on the direction of observation due to the +14 + +dependence of θ in λ. The peak height can be determined by +−λmaxη∗ = − +√ +2B +Mpl +η∗ = −2 +√ +3 B/ +√ +2 +√ +3MplH +η∗ +η1 += 2 +√ +3 +�ρB∗ +ρinf +, +(4.12) +where we used a(η1) = 1/(Hη1) = 1 in the second equality. +In the third equality, the +energy density of the magnetic field at η∗ and that of the inflaton field at η1 are written by +ρB∗ = (a∗/a1)2 |B|2/2 and ρinf = 3M 2 +plH2 respectively. Thus, the extra factor exp[−λ η∗]/2 +could be 16 for ρB∗ ≃ ρinf. We see that the peak height also depends on the direction of +observation. The dependence on the direction of observation appears in the presence of the +PMFs. Hence, our results would offer new vistas to probe the PMFs through the observations +of primordial GWs. +5 +Conclusion +In this paper, we studied the effect of PMFs generated during inflation on the PGW power +spectrum. Assuming that there exist extragalactic magnetic fields in the range from 10−17 +to 10−9 Gauss with a cosmological coherence length, we showed that the graviton photon +conversion process affects the PGW power spectrum. The point is that the graviton photon +conversion process induces tachyonic instability. As a consequence, a peak appeared in the +spectrum and the peak height depends on the direction of observation. Remarkably, it turned +out that the peak frequency lies in the range from 10−5 to 103 Hertz in the case of GUT scale +inflation. Thus, we have a possibility to probe the PMFs by observing the PGW spectrum. +Although we considered the graviton photon conversion process, the same argument applies +to the graviton dark photon conversion [34]. In this case, the PGW power spectrum could +be a probe of vector-type ultralight dark matter. +In [35, 36, 37, 38], the statistical anisotropy in the PGW power spectrum is calculated +in the presence of background electric fields. There, no peak appeared in the spectrum. It +would be interesting to study what happens when both electric and magnetic fields coexist +in the background during inflation. +In this work, we have focused on the coupling parameter c = −1/2. +In the case of +15 + +c = −1/2, the physical magnetic fields decay slowly as fB ∝ 1/a during inflation compared +to the case of constant f where the physical magnetic fields decay as fB ∝ 1/a2. On the +other hand, in the case of c = −1, the physical magnetic fields fB do not decay during +inflation. Hence, we expect a different shape of the PGW power spectrum. We leave the +analysis of this issue for future work. +Acknowledgments +S. K. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI +Grant Number JP22K03621. J. S. was in part supported by JSPS KAKENHI Grant Numbers +JP17H02894, JP17K18778, JP20H01902, JP22H01220. K. U. was supported by the JSPS +KAKENHI Grant Number 20J22946. +References +[1] +B. P. 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D +81 (2010), p. 063528. arXiv: 1001.4088 [astro-ph.CO]. +19 + diff --git a/e9FRT4oBgHgl3EQfVjee/content/tmp_files/load_file.txt b/e9FRT4oBgHgl3EQfVjee/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1e627a2a15480a09b9bbccde1f2300adef3d652 --- /dev/null +++ b/e9FRT4oBgHgl3EQfVjee/content/tmp_files/load_file.txt @@ -0,0 +1,926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf,len=925 +page_content='KOBE-COSMO-23-01 A peak in the power spectrum of primordial gravitational waves induced by primordial magnetic fields Sugumi Kanno∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Ann Nakato♭,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Jiro Soda♭,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Japan ♭ Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Kobe University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Kobe 657-8501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Japan ♯ International Center for Quantum-field Measurement Systems for Studies of the Universe and Particles (QUP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' KEK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Tsukuba 305-0801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Japan Abstract It has been argued that there exist extragalactic magnetic fields in the range from 10−17 to 10−9 Gauss with a cosmological coherence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' One plausible explanation for the origin of the extragalactic magnetic fields would be quantum fluctuations of the electromagnetic fields during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' It is also believed that primordial gravitational waves (PGWs) arise out of quantum fluctuations during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We study the graviton-photon conversion process in the presence of background magnetic fields and find that the process induces the tachyonic instability of the PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' As a consequence, a peak appears in the power spectrum of PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' It turns out that the peak height depends on the direction of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The peak frequency could be in the range from 10−5 to 103 Hertz for GUT scale inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Hence, the observation of PGWs could provide a new window for probing primordial magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='13540v1 [hep-th] 31 Jan 2023 Contents 1 Introduction 1 2 Graviton-photon interaction in the early universe 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1 Primordial GWs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 7 3 Generation and evolution of primordial GWs 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1 Cosmological setup .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='2 Bogoliubov Transformations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 10 4 Tachyonic instability and the PGW power spectrum 12 5 Conclusion 15 1 Introduction Since the first direct detection of gravitational waves [1], gravitational waves have been the most important tool for investigating fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Gravitational waves interact with matter very weakly and travel through the universe virtually unimpeded, so they can be a probe of the early inflationary universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The inflationary scenario predicts that primordial gravitational waves (PGWs) are generated from quantum fluctuations of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In terms of the density parameter, the spectrum ΩGW(f) for the GUT scale inflation is independent of frequency f, that is, of wave number k for the gravitons created at the de Sitter to radiation dominant transition: ΩGW(f) = 10−14 � H 10−4Mpl �2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1) 1 where H is the Hubble parameter, Mpl is the reduced Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' This scale-invariant power spectrum comes from the time translation invariance of de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' There are several experimental projects for detecting PGWs [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' If the nonclassicality of the PGWs is found, it implies the existence of gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Notably, due to the particle creation, the quantum state of gravitons becomes squeezed during inflation [6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Since the squeezing tends to enhance the observability of gravitons, it is expected that we may be able to probe quantum gravity through observations of PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' There are new ideas for detecting the quantum nature of PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' One method is to utilize the Hanbury Brown- Twiss interferometry developed in quantum optics for the PGWs, which can distinguish nonclassical particles from classical ones by measuring intensity-intensity correlations [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Moreover, the squeezed state of gravitons can be measured indirectly through their noise in the interferometers [14, 15, 16, 17] or by measuring the decoherence time of a quantum object caused by the surrounding primordial gravitons [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' On top of the PGWs, there might exist primordial magnetic fields (PMFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Indeed, it is argued that observations of GeV gamma-ray from blazars give rise to the lower bound of the extragalactic magnetic field 10−17 Gauss [19, 20, 21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We also have the upper bound 10−9 Gauss from CMB data [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' It is important to seek the origin of the extragalactic magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' If a coherence length of magnetic fields is around mega-parsec scales, it would be natural to consider magnetic fields originated from quantum fluctuations during inflation [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In the presence of PMFs during inflation, the effect of the PMFs on PGWs is studied in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In the case of scale-invariant electromagnetic fields, there arises an entanglement between gravitons and photons [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Therefore, the presence of scale-invariant electromagnetic fields may affect the squeezed state of gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Thus, it is important to clarify the effect of PMFs on PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In this paper, we consider the background PMFs during inflation and the scale-invariant perturbed electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' It is known that the presence of background magnetic fields causes the conversion of gravitons into photons and vice versa [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We focus on the effect of the graviton-photon conversion process on the PGW power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We show that there occurs the tachyonic instability of PGWs due to graviton-photon conversion in the presence of PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Because of this instability, there arises a peak in the PGW spectrum 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Moreover, since the magnetic fields specify a direction in the space, there appears statistical anisotropy in the power spectrum of PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Remarkably, it turns out that the peak frequency could be in the range from 10−5 to 103 Hertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The presence of the peak turns out to enhance the squeezing of gravitons and makes it easy to observe the PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' More interestingly, we may be able to probe the PMFs by observing the spectrum of PGWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In section 2, we present the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In section 3, we calculate Bogoliubov coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In section 4, we calculate the PGW power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We see the tachyonic instability leads to a peak in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We also discuss the observability of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Section 5 is devoted to the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 2 Graviton-photon interaction in the early universe We consider the Einstein-Hilbert action and the action for a U(1) gauge field coupled with a scalar field: S = Sg + Sφ + SA = � d4x√−g �M 2 pl 2 R − 1 2(∂µφ)(∂µφ) − V (φ) − 1 4f 2(φ)F µνFµν � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1) where Mpl = 1/ √ 8πG is the Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The gauge field Aµ represents photons and the field strength is defined by Fµν = ∂µAν − ∂νAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The background inflationary dynamics is determined by the metric ds2 = a2(η) � −dη2 + δijdxidxj� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='2) and the inflaton φ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Once the background is given, the coupling function can be regarded as a function of the conformal time η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' f = f(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' We also assume the presence of constant magnetic fields Bi = constant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' It should be emphasized that the physical magnetic fields are not Bi but fBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In the following, we consider the quantum evolution of gravitons and photons in the above background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1 Primordial GWs We consider gravitons in a spatially flat expanding background represented by tensor mode perturbations in the three-dimensional metric hij, ds2 = a2(η) � −dη2 + (δij + hij) dxidxj� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='3) where hij satisfies the transverse traceless conditions hij,j = hii = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The spatial indices i, j, k, · · · are raised and lowered by δij and δkℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In the case of de Sitter space, the scale factor is given by a(η) = −1/(Hη) where −∞ < η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Expanding the Einstein-Hilbert action up to the second order in perturbations hij, we have δSg = M 2 pl 8 � d4x a2 � hij′ h′ ij − hij,khij,k � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='4) where a prime denotes the derivative with respect to the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' At this quadratic order of the action, it is convenient to expand hij(η, xi) in Fourier modes, hij(η, xi) = 2 Mpl � P 1 (2π)3/2 � d3k hP k (η) eP ij(k) eik·x , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='5) where three-vectors are denoted by bold math type and eP ij(k) are the polarization tensors for the k mode normalized as eijP(k)eQ ij(k) = δPQ with P, Q = +, ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Then the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='4) in the Fourier space becomes δSg = 1 2 � P � d3k dη a2 � |hP′ k |2 − k2|hP k |2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='2 Primordial magnetic fields Next, we consider the action for the photon up to the second order in perturbations Ai, which is given by δSA = 1 2 � d4x f 2 � A′ 2 i − A2 k,i � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='7) 4 where the photon field satisfies the Coulomb gauge A0 = 0 and Ai,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Expanding the field Ai(η, xi) by the Fourier modes, we find Ai(η, xi) = � P ±i (2π)3/2 � d3k AP k (η) eP i (k) eik·x , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='8) where eP i (k) are the polarization vectors for the k mode normalized as eiP(k)eQ i (k) = δPQ with P, Q = +, ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The sign of ±i corresponds to the P, Q = +, ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' The action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='7) in terms of the Fourier modes reads δSA = 1 2 � P � d3k dη f 2 � |AP′ k |2 − k2|AP k |2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='3 Graviton-photon conversion The action for the interaction between the graviton and the photon up to second order in perturbations hij, Ai is found to be δSI = � d4x � εiℓmf 2Bmhij (∂jAℓ − ∂ℓAj) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='10) where Bm = εmjℓ ∂jAℓ is a constant background magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' In terms of the Fourier modes defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='8), the interaction term reads δSI = 2 Mpl � P,Q � d3k dη f 2 � εiℓm Bm hP k AQ −k eP ij(k) � ikℓ eQ j (−k) − ikj eQ ℓ (−k) �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='11) where k = |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Polarization vectors ei+, ei× and a vector ki/k constitute an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' Without loss of generality, we assume the constant background magnetic field is in the (ki, ei×)-plane as depicted in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='Propagation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9FRT4oBgHgl3EQfVjee/content/2301.13540v1.pdf'} +page_content='1000 m for low cemented +pyroclasts (a=6; b=54). A volcanic +edifice constituted by low cemented +pyroclasts has a cohesion between 0.5 +and 1 MPa at depths of 1500-3000 m +(Figure 3a). At this depth, a theoretic +volcanic edifice constituted by low +cemented pyroclasts has an angle of +friction that ranges between 6° and 10° +(Figure 3b). + + + + +6 + + +FIGURE 3: Effect of depth on mechanical properties. (A) Cohesion for low cemented +pyroclasts (k=0.05; l=0.5) and fresh (k=0.37; l=0.60) and weathered (k=0.04; l=0.41 +and 0.60) massive lavas, (B) angle of friction for low cemented pyroclasts (a=6; b=54) +and fresh (a=5.6; b=82) and weathered (a=7; b=75) massive lavas. + +2.3. Modeling +hydroclimatic +variations +In this study, sea-level loading and pore +pressure variation in relation to climatic +evolution were simulated. Basically, the +water column loading that is considered +has been estimated using a proxy. Sea +level +is +known +to +change +contemporaneously with  +18 +O variation +(Gargani and Rigollet, 2007; Gargani et +al., 2008). To simulate climate forcing, +a  +18 +O curve (Lisiecki and Raymo, +2005) was used. This curve has been +recalibrated to simulate Quaternary sea- +level variations that reach an amplitude +of 120 m. +The hydroclimatic variations are also +caused by Milankovitch astronomic +cycles and can be simulated using +insolation or  +18 +O curves (Gargani et +al., 2006a). In this study, it is + +12 +A +10 +FRESHMASSIVELAVAS +8 +COHESION (MPa) +WEATHERED MASSIVELAVAS +2 +PYROCLASTS +0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +SLOPE HEIGHT (m) +70 +B +60 +50 +FRESH MASSIVELAVAS +ANGLE OF FRICTION +40 +30 +WEATHEREDMASSIVE LAVAS +20 +10 +LOW CEMENTEDPYROCLASTS +0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +SLOPE HEIGHT (m)7 + +considered that more humid conditions +are able to cause an increase in the +effective pore pressure. The aim of this +assumption is to discuss the timing of +potential giant landslides in relation to +Quaternary climatic conditions (i.e., +precipitation rates). The time necessary +for the water to propagate from the +surface to a depth of 4.5 km has been +estimated at ~150 days in Oregon (Saar +and Manga, 2003). In the model, the +propagation +is +considered +instantaneous, which is justified on a +long time scale (>1 kyr). The amplitude +of the variation in the pore pressure P +due to water infiltration into the crust is +0.01 MPa at Mt. Hood (Saar and +Manga, 2003) and 2 MPa on the south +flank of Kilauea volcano (Cervelli et al., +2002). In this study, various values for +the pore pressure variation have been +tested in the case where sea-level +effects are dominant (0 < P < 0.125 +MPa) and in the case where pore +pressure processes are dominant (0 < +P < 0.5 MPa). + +2.4. Pressure variation in the magma +reservoir +The loading variation caused by sea +level variation is LSL(t) = g m Hm(t). +Sea water unloading causing decreased +lithostatic pressure at depth, enhances +the +production +of +magma. +Decompression favor partial mantle +melting and magma release. The +magma production rate DF/Dt at +constant entropy S can be estimated as +(Jull and McKenzie, 1996; Sternai et al., +2017): +DF/Dt = (F/PM)S (dPM/dt –UPM) +where PM the magma reservoir +pressure, F the melt fraction, t the time, +U the mean mantle upwelling rate. +The magma pressure variation PM/dt +can be considered as depending from +the sea water unloading variation +LSL/dt under adiabatic condition and +for very high viscosity (i.e. c=1023 +Pa.s) of the crustal rocks. In this case +PM/dt = - LSL/dt. +When the viscous response is not +negligible (c < 1022 Pa.s; Sternai et al., +2017), the equation is PM/dt +PM +(Ec/c) = - LSL/dt, where Ec is the +elastic modulus, and a delay of ~10 kyr +after the forcing by sea water unloading +is expected. + +3. Results +The current question that we attempt to +answer focuses on the thresholds and +conditions that favor slope instability +for volcanic edifices in relation with +climatic variations. +3.1. Influence of water column loading +on slope stability +Considering a simple geometry for the +expected landslide (Fig. 2A, Cullman +wedge geometry), the role of sea level +variation +on +slope +stability +was +investigated using equation (1). When a +volcanic island subsides, an increase in +the relative sea level and, consequently, +an increase in the water loading on the +slope occur and could be significant. +Under the Cullman wedge geometry +condition, an increase in the loading by +a water column of 120 m cause a slight +decrease in the stability height (Figure +4C). The more the water loading +increases, +the +less +the +stability. +Whatever the geometry of the sliding +mass is, the maximum height of the +stable relief is impacted by the increase +in the loading by a water column of 400 +m. A water column increase of 400 m is +reached in less than 1 Myr for a +moderate +subsidence +rate +(~0.25 +mm/yr) contemporaneously with sea- +level high stand at +120 m during an +interglacial period. In this case, slope +instabilities can be observed on higher +slopes (>25°) but also on smaller slopes +(Figure 4). The role of sea-water +loading could trigger landslides of +smaller dimensions (150-300 m) for +higher slopes (25-30°, Fig. 4C, dark +gray square). + +8 + +3.2. Influence of the pore pressure on +slope stability + +FIGURE 4: (A) Effect of the increase in pore pressure and sea-water loading on the +stability of the relief. The light gray area is the area that changes from a theoretically +stable relief to an unstable relief when submitted to an increase in the sea-water +column loading between 0 and 400 m as well as to a pore-water pressure increase +between 0 MPa and 2 MPa. (B) Schematic representation of the effect of subsidence. +(C) The details are as follows: a relief of ~200 m with a slope of 25° <  < 30° could be +affected by water loading (dark gray square). c = 3000 kgm-3, C = 1 MPa, = 10° and +geometry of the landslide identical to those presented in Figure 2A. + +The model suggests that a significant +effect of the pore pressure increase on +the height of the stable relief could +occur (Fig. 4). The pore pressure +significantly reduces the stability of the +relief. As expected, the higher the pore +pressure is, the lower the maximum +height H before the giant landslide +occurs. If the pore pressure increases by +more than 1 MPa, the slope stability is +significantly decreased (Fig. 4A and B). +A relief of more than 1500 m, with a +slope of ~12°, could be affected by +landslides +under +these +conditions. +Depending on the initial relief H, a pore +pressure increase of 1 MPa could cause +instabilities of more than 1000 m. + +3.3. Competing influence of sea-level +loading vs. pore pressure variations +over time on slope stability +In this study, the influence of sea-level +loading and pore pressure variations +over time, in relation to meteoritic water +infiltration, was modeled. The variation +over time of the maximum height H of +the volcanic edifice before any giant +landslide occurs is shown in Figure 5. +The maximum stable height depends on +the loading of the sea level and on the +value of the pore pressure. Taking into +account that the maximum sea-level +variation during the last million years is +~120 m and that the amplitude and the +timing of the loading are known (Figure +5B), the variation in the maximum +stable height H of a theoretical volcano +before any giant landslide occurs has +been calculated. The absolute value of +the maximum stable height H strongly +depends on mechanical properties of +rocks and slope, and will not be +discussed in this study because this +study focus on the variation of stability +caused by climatic evolutions. + + + +4000 +A +POSSIBLYUNSTABLE +1000 +3500 +C +900 +B +3000 +800 +700 +porepressure +2500 +increase +() ) +HEIGHT (m) +600 +HEIGHT +2000 +500 +UNSTABLE +UNSTABLE +400 +1500 +300 +STABLE +STABLE +200 +1000 +400m +400m +400m +om +100 +U=2MRa +U=1MIa +U=0.4MPa +400m +120m +500 +U=OMPa +10 +15 +20 +25 +30 +35 +SLOPE +U=2MF +U-1MPa..U=0.4MPa +400m +400m +_400m +om +400m +0 +10 +15 +20 +25 +30 +35 +Tahiti +SLOPE9 + + +FIGURE 5: (A) Maximum height H before the landslide occurred. The geometry of +the island is shown in Figure 4. The effects of sea-level loading and pore pressure are +simulated. The sea-water loading depends on the variation in sea level, whereas the +effect of the pore pressure depends on the variations in precipitation. In the first case +(sea-level-dominated), the pore pressure is considered to range between 0 and 125 +kPa depending on the climatic variation. In the second case (pore pressure- + +3700 +3650 +pore pressure dominated +3600 +WA +3550 +(m) +height +3500 +3450 +sea level dominated +3400 +Tahiti- +Guadeloupe +La Réunion-Canary +- +LaReu. +3350 +Canary- +Martinique +Hawaii +Can. +Can. +Haw. +3300 +1000 +800 +600 +400 +200 +0 +time (kyr) +50 +sea level (m) +-50 +-100 +-150 +1000 +800 +600 +400 +200 +0 +time (kyr)10 + +dominated), the pore pressure is considered to range between 0 and 500 kPa +depending on the climatic variation. The absolute values of H is indicative and only +the trends are interpreted in this study. g=9.81 ms-2, C=1 MPa, =10°, =2800 kg/m-3, +=7°, and a subsidence rate of 0.25 mm/yr are considered. (B) Simulated sea-level +variations during the last million years using the 18O curve of Lisiecki and Raymo +(2005). The light blue lines represent the giant landslides on volcanoes during the last +1 Ma of Table 1. + +When seawater loading is the main +process that influences landslides, the +higher the sea level is, the lower the +stability. The maximum height H is +expected to decrease when the sea level +rises (Figure 5). Considering all other +parameters as identical, the maximum +stable height H of a volcanic edifice is +50 m higher during interglacial periods +than during glacial periods if the sea- +water loading is the main parameter +influencing the system. + +In the specific case of Polynesian +climate variation, where the climate is +more humid during glacial maxima +(Saéz et al., 2009), as simulated here, +the giant landslides generated by a pore +pressure increase are expected to occur +during glacial periods (Figure 5A). In +the specific case of the Polynesian +volcanic edifice, the period when the +landslides are expected (i.e., the time +when the maximum stable height H +before a giant landslide is expected to +occur), is different when the dominant +process that triggers the landslide is sea- +level loading or pore pressure loading +(Figure 5A). +The influence of sea level unloading on +magma reservoir pressure increases at +depth is better correlated than the sea +level loading with giant landslides +occurrence (Fig. 5). This could suggest +that in many cases the volcanoes were +actives (i.e. a magma reservoir was +present at depth). If this process was the +cause of the giant landslides, the +viscosity of the crust was very high in +many cases, except in La Réunion (290- +320 kyr; Leunat and Labazuy, 2008), +Canary islands (134 Carracedo et +al.) and Hawaii (127 10; +McMurtry et al., 1999) where a delay is +observed with low sea level. The effect +of sea level unloading influence on +magma +pressure +increase +is +contemporaneous with the effect caused +by +pore +pressure +increases +with +meteoritic +water +infiltration +in +Polynesia and East Africa (i.e. where +precipitation increases during glacial +periods). + +4. Discussion +4.1. Rock +weathering +and +slope +instability +Volcanic edifices are usually composed +of massive lavas at the surface. These +lavas can be highly weathered by the +humid climate when they are exposed +for thousands or millions of years. +Laboratory experiments for weathered +massive lavas show that cohesion can +be low (0.8-1 MPa) at depths ranging +from 1500 m to 3000 m (Figure 3a). +The angle of friction of weathered +massive lavas is estimated to range +between +23° +and +19° +based +on +laboratory experiments (Figure 3b). +Lavas that have been extremely altered +by rain and fluid circulation at depth for +thousands of years are expected to have +mechanical properties lower than those +of fresh volcanic rocks. Furthermore, +the effective mechanical properties (C +and ) of fractured and faulted reliefs +are lower than those used for triaxial +experiments on nonfractured rocks. +Effective mechanical properties take +into account the heterogeneities present +in old and complex volcanic edifices +including low cemented pyroclast and +weathered soils. Soils can be observed + +11 + +sandwiched between volcanic lavas +(Hevia-Cruz et al., 2022). These soils +are +expected +to +have +reduced +mechanical properties. The model has +been calibrated to be compatible with +past giant landslide events (~1500-3000 +m thick, initial slope ~8-12°, cohesion +C of 0.3-1 MPa, Figure 2), such as the +event that occurred in Tahiti, Society +Archipelago (Gargani, 2020 and 2022a) +(Figure 2). The values used here +correspond to highly weathered rocks of +an old and fractured volcanic edifice or +low cemented pyroclasts that could be +sandwiched at depth. +Rock mechanical properties C and  +decrease over time when exposed to +weathering. The more weathering that +occurs, the greater the decrease in C and +. During humid periods, this decrease +is higher than during drier periods +(Figure 6A). In contrast, diagenetic +processes are expected to improve the +mechanical properties of rocks (Figure +6B). When pressure increases at depth, +porosity and connectivity can decrease. +Furthermore, mineralization of fluid at +depth can also improve rock cohesion in +specific cases. + + +FIGURE 6: Influence on the rock mechanical properties C and  of (A) weathering +and (B) diagenesis. + +4.2 Weakness zones and geological +inheritance +The construction of volcanoes is often +polygenic, +and +numerous +heterogeneities +exist +on +volcanic +edifices (Hildenbrand et al. 2006). The +complex and long volcano-tectonic +history of old volcanic edifices may +explain why numerous fractures and +significant weathering are observed. +The highly fractured and heterogeneous +material (basaltic flows and volcanic +breccia) of volcanic edifices is often +complex and difficult to model in detail +at all scales. Strata of weathered soils +sandwiched between lavas can also be +observed (Hevia-Cruz et al., 2022) as +well as pyroclastic formations. Low +values for the angle of friction have +been estimated or used in various +studies +for +weakness +structures, +effective rock properties or faults (Cala +and Flisiak, 2001; Bigot-Cormier and +Montgomery, 2007; Got et al., 2008; +Egholm et al., 2008; Abers, 2009). +Modeling studies are used to simplify +the complexity of the real world. The +effective +mechanical +properties +implemented in modeling are different +than +the +laboratory +experimental +properties. +Preexisting +zones +of +weakness at depth are present in many +volcanic edifices but are different in +each case. A real geometry of a specific + +HUMID +个个个个 +C or Φ +Weathering +A +Cor Φ +B +Diagenesis +DRY +个 +个个个 +TIME +TIME12 + +volcanic edifice or of a specific +landslide is not the aim of this study. A +general case is considered. This model +does not permit us to predict exactly +when a large landslide is expected to +occur but only to discuss the possibility +of discriminating the influence of +meteoritic water infiltration from sea- +water loading/unloading using climatic +correlation +from +a +theoretical +perspective. + +4.3 Sea-level +variation effect on +landslides and volcanic activity +The role of water-level rise in landslide +triggering has been suggested in the +case of the Vajont landslide (1963, +Italy) (Muller-Salzburg, 1987; Kilburn +and Petley, 2003). When sea level rises, +(i) water loading increases and (ii) +water infiltrates into rocks. The loading +caused by sea level rise could cause +slope instability when the slope is +almost +at +equilibrium +immediately +before sea level rise. Furthermore, when +sea level rise, erosion can occur at the +base of the cliff and favor cliff retreat, +rock fall and landslide (Ye et al., 2013). +However, the effect of loading forces on +the base of a slope can also favor the +slope stability. A vertical loading force +at the base of the volcano can cause an +increase of stability by opposing a force +to the potential movements of the relief. +In other words, when the sea-level fall +at the base of a slope, it could favor +instability (Figure 7A). This result is +suggested by safety factor FS decrease +when loading decreases. When a +significant loading is above the center +of mass of the potential landslide, it +cause the destabilization of the slope by +triggering rock failure (Figure 7B). + + + +Figure 7: Influence of the sea-level variation in relation with the position of the center +of mass of the potential sliding area: (A) slope instability caused by sea-level lowering +when the load is located at the base of the slope, (B) slope instability caused by sea- +level rise above the center of mass with marine water infiltration. + +The giant landslides in volcanic areas +seem to be correlated with climatic +variation (Quidelleur et al., 2008). Sea +level loading is one of the potential +causes of giant landslides (Aslan et al., +2021). However, indirect interactions +are also possible in volcanic areas. The +unloading associated with sea level +lowering could modify the pressure on +the magma reservoir (Sternai et al., +2017). +A +pressure +decrease +has +implications for the input and output +rates of magma into and out of the +magma reservoir. If the connection to +the deeper magma source remains open, +one consequence is an increased rate of +replenishment with primitive magma +(Pinel and Jaupart, 2003). Furthermore, +a reduced load allows the eruption of +denser magmas that would otherwise +have been stuck at shallow depths +(Pinel and Jaupart, 2000). +The potential interaction between sea- +level variation and landslides on one + +A +B13 + +side +and +landslides +and +volcanic +eruptions on the other side (Longpré et +al., 2009) could cause also difficulties +in interpretation. If the magma reservoir +is shallow, a landslide could generate +dyke intrusion, but if the magma +reservoir is deep, dyke intrusion is more +difficult after a landslide. + +4.4 Pore pressure variation +Another process that is able to cause +landsliding is pore pressure increase. +Fluids are known to be a triggering and +driving factor for landslides (Cappa et +al., 2014). Various mechanisms could +explain a pore pressure increase of ~1 +MPa. First, it is necessary for a +significant amount of water to enter the +system. For example, unsaturated rocks +of porosity n in an area of volume V +could +become +saturated +when +a +significant water volume n.V infiltrates. +The volume of water could be higher if +a dense fracture network existed. +Meteoritic water infiltration at a depth +of 1000 m under lithostatic pressure +could cause a pore pressure increase. +An increase of more than 1 MPa in the +pore pressure at depth related to +meteoritic water is not unrealistic and +has been proposed in previous studies +(Cervelli et al., 2002). +An alternative mechanism that could +also lead to a significant pore pressure +increase results from the progressive +collapse of the base of the volcanic +edifice. In this case, the slow creep of +the edifice on a preexisting weak +structure +may +trigger +overpressure +conditions locally that reach high +pressures (Veveakis et al., 2007). In a +volcanic context, it may also be +expected that hydrothermal processes +play a role in the pore pressure increase +and rock alteration at depth. +The pore pressure can change in +relation to meteoritic water infiltration +in highly fractured rocks. In this case, +meteoritic water infiltration increases +when the precipitation rate increases. +Consequently, climatic variations are +expected to influence the pore pressure +variation in highly fractured rocks. The +correlation between climatic variation +and giant landslides may be caused by +pore pressure increases at depth. + +4.5 Climate variation and correlation +with giant landslides +Precipitation increased during glacial +maxima in Polynesia (Saez et al., 2009), +Mexico (Ganeshram and Pedersen, +1998), and eastern equatorial Africa +(Chiang, 2009) but decreased in NW +Europe (Guiot et al., 1989), the +Caribbean (Curtis et al., 2001) and +Indonesia (Costa et al., 2015; Russell et +al., 2014). +When +humid +conditions +occurred +during warmer phases, the potential +effect of water infiltration on pore +pressure acted to increase instability +during warmer interglacial periods, +similar to sea-level loading. Interglacial +phases are expected to cause more slope +instability in this case. However, it is +difficult to discriminate the influence of +sea-level loading from meteoritic water +infiltration +at +depth +on +giant +paleolandslides in this case. +In +contrast, +when +precipitation +increased during glacial maxima, the +impacts of sea-level loading and water +infiltration +at +depth +on +giant +paleolandslides +were +not +contemporaneous. In this case, it is +theoretically possible to discriminate +the process that plays the main role in +triggering +giant +landslides. +For +example, +in +Polynesia, +increased +meteoritic water infiltration occurred +during glacial maxima and potentially +increased slope instability during glacial +maxima, whereas sea-level rise caused +slope instability during interglacial +times (Figure 5A). The age of giant +landslides (Table 1) may be a good +argument to discriminate between these +two processes, but only in Polynesia, +Mexico or eastern equatorial Africa and + +14 + +not in NW Europe, the Caribbean or +Indonesia. +This study suggests that if preexisting +weakness zones are present at depth, +causing low effective cohesion and +internal friction, a climatic origin of +landslides is possible even if volcano is +extinct. Seismic or volcanic processes +are not necessarily directly responsible +for giant landslides, even if these +processes may have played a role in +causing weakness zones in the past in +these areas. In 70% of cases, the giant +landslides occurred when sea-level rise, +regardless the precipitation rate is. Due +to potential delay caused by viscous +processes (i.e. 10 kyr of delay for a +viscosity less than 1022 Pa.s), it could be +also correlated to the magma chamber +unloading by sea-level lowering. This +result could be influenced by the set of +data considered and further studies on +giant landslides on volcanic islands +would improve this interpretation. + +Table 1: Giant landslide ages in volcanic areas during the last 1 Ma. +Volcano +Age (ka) +References +Canary Islands +(El Hierro, El Golfo) +10-17 +Gee et al., 2001 +La Réunion +20-68 +Lenat and Labazuy, 2008 +Hawaii (Alika phase 1) +112 +Mc Murtry et al., 1999 +Hawaii (Alika phase 2) +127 10 +Mc Murtry et al., 1999 +Canary Islands +(El Hierro, El Golfo) + +134  +Carracedo et al., 1999 +Hawaii (Southern Lanai) +135 +Rubin et al., 2000 +Canary Islands (Tenerife) +150-170 +Hunt et al., 2011 +Masson et al., 2002 +Hawaii (Southern Lanai) +240 +Rubin et al., 2000 +La Réunion +290-320 +Lenat and Labazuy, 2008 +Martinique +337  +Quidelleur et al., 2004 +Canary Islands (La Palma) +537 8 +Guillou et al., 2001 +Groom et al., 2022 +Guadeloupe +629 13 +Samper et al., 2007 +Hawaii (Haleakala, Hana) +860 +Moore and Clague, 1992 +Tahiti-Nui (north) +872 10 +Hildenbrandt et al., 2004 + +4.6 +Small +landslides +vs. +giant +landslides +The dimension of the expected landslide +depends on the initial slope of the relief. +Locally, as in a deeply incised canyon +of volcanic islands, significant slopes +can be observed. In the case of a +significant slope (>25°), the critical +value for the height of the stable relief +is +less +than +300 +m +(Fig. +3C). +Consequently, smaller landslides may +also +be +triggered +locally. +When +rotational landslides occur, significant +slopes develop in the upstream part. The +resulting relief could be near the +instability conditions that favor the +development of new landslides and +other retrogressive erosion processes. It +can be difficult to discriminate the relief +associated +with +numerous +“small” +landslides from that caused by giant +landslides (Gargani, 2020; Gargani, +2022b). +Erosion +and +repetition +of +small +landslides could progressively reduce +the loading. As previously shown, the +loading influences the maximum height +expected before a landslide occurs. +Reducing +the +load +increases +the +maximum height before landsliding and + +15 + +thus reduces the potential occurrence of +a landslide when volcanic edifice is an +extinct volcano. Consequently, the +potential occurrence of a giant landslide +could be reduced by the occurrence of +small landslides and erosion of the +relief in the case of extinct volcanoes. +Previous studies have shown that the +probability +of +small +landslides +occurring is higher than that of large +landslides (Urgeles and Camerlinghi, +2013). Nevertheless, giant landslides +have also been described, suggesting +that there are specific conditions that +favor giant landslides. + +5. Conclusion + +The probability of a large landslide in +an area where no significant volcanic +and +seismic +activities +have +been +observed during the last thousand years +is not null. Large landslides caused by +nonvolcanic processes could occur on +old +volcanic +islands. +Indeed, our +modeling suggests that a large landslide +may occur due to an increase in pore +pressure and/or sea-level variation. +These processes could be discriminated +in areas where glacial maxima are more +humid than interglacial periods, but not +in the other cases, based only on timing +correlation. This finding is also a +consequence +of +the +mechanical +properties of highly weathered and/or +fractured volcanic rocks in weakness +zones (low cohesion and angle of +friction) at depth in old weathered +volcanic edifices. When volcanoes are +still actives, sea level unloading can be +responsible of giant landslides, causing +magma reservoir pressure increases at +depth and magma release. 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Geomorphology +2013, 186, 162–173. + + diff --git a/g9AyT4oBgHgl3EQfj_jD/content/tmp_files/load_file.txt b/g9AyT4oBgHgl3EQfj_jD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f1a519fbe907f65d3fe8d349951c39e2c7f2014 --- /dev/null +++ b/g9AyT4oBgHgl3EQfj_jD/content/tmp_files/load_file.txt @@ -0,0 +1,1006 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf,len=1005 +page_content='1 Influence of relative sea-level rise, meteoritic water infiltration and rock weathering on giant volcanic landslides: theory and real cases Julien Gargani Université Paris-Saclay, Geops, CNRS, Orsay, France julien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='gargani@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='fr Abstract Recent studies have shown that giant landslides seem to be correlated with climatic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Nevertheless, the precise processes that are involved in this phenomenon need to be better constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this study, the causes of giant landslides are investigated using a modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Here, we show that the effect of meteoritic water infiltration could be discriminated from that of sea-level rise on triggering paleolandslides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' It is possible to identify the cause of coastal paleolandslides based on the age of occurrence and comparison with climatic signals when glacial maxima are more humid than during interglacial times, as in Polynesia and East Equatorial Africa, but not in other cases (Caribbean, Indonesia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The role of pore pressure variations and sea-water loading variations has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The interaction between the relative sea-level rise, preexisting relief and deep weak structure due to the presence of highly weathered lavas may trigger the conditions for a large landslide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Highly weathered lavas have very low friction angles at depth in volcanic islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When volcanoes are still actives, pressure variation of the magma chamber caused by sea-level lowering is expected to play a significant role in destabilization of the relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Competing processes in real cases cause difficulties to discriminate between these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Keywords: landslide, alteration, angle of friction, pore pressure, subsidence, sea level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Introduction Landslides are one of the main processes that destroy relief and displace material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Catastrophic landslides are often expected in areas of active seismicity (Keefer, 1994) or in the context of significant volcanic activity (Carracedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Blahüt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the case of active seismicity, the acceleration of the ground surface and subsurface may generate slope destabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' From a theoretical perspective, the slope stability depends strongly on the mechanical properties of rocks (Rodriguez-Losada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009), on the geometry of weakness zones, on the loading and stress conditions (Keefer, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Cala and Flisiak, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Kilburn and Petley, 2003: Verveakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Urgeles and Camarlinghi, 2013) and on the presence of fluids (Muller- Salzburg, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Crozier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Cappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Aslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In volcanic areas, the interaction between magma reservoirs, magma intrusions, and preexisting faults influences the deformation of volcanic surfaces (Le Corvec and Walter, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Gargani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Hampel and Hetzel, 2008) and could generate landslides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Phreatomagmatic processes (McMurtry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2004), pore pressure increases due to precipitation (Cervelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2002) and rift zone intrusions (Le Corvec and Walter, 2009) are believed to influence slope instability on volcanic edifices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' From a practical perspective, present conditions of stability of volcano slopes are influenced by past volcanic activities and geological history;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' multiple eruptions and landslides, faults and fractures, alteration by fluid circulation, could have generated significant heterogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2 Heterogeneities may hide weaknesses and the potential occurrence of slope instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The long geological history of old volcanic edifices is expected to cause complex geometries that are often difficult to understand in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' On the flanks of volcanoes, the rocks may be constituted by fresh or weathered massive lavas, ignimbrites or low cemented pyroclasts, among other rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The past history of the volcanic flanks, as well as the nature of the rocks that constitute their flanks, must be considered carefully when estimating the slope stability of volcanoes from a geotechnical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' These complex scenarios could cause unfavorable conditions of stability of the present slope and difficulties in taking into account the spatial heterogeneity of the mechanical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' It has been suggested that giant landslides could be correlated with climatic variations (Mc Murtry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A classical and dramatic example of the role of water is the triggering of the catastrophic Vajont landslide in 1963 in Italy in relation to water-level rise and rainfall increase (Muller-Salzburg, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Some studies assume that giant landslides may have occurred during low stand periods (Quidelleur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2007), whereas others suggest that high stands or sea level rise are more favorable for causing slope instability (Gargani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This question remain controversial and poorly studied despite the expected sea- level rise in relation with climate warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The role of climatic conditions on triggering giant landslides on volcanoes is investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The possible occurrence of a giant landslide on a volcano in relation to climatic variation is investigated from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The processes favoring instability that are discussed in this study are (i) sea-water loading with relative sea-level rise, (ii) pore pressure increase in relation to climatic variation, (iii) rock and soil weathering, (iv) weakness zone geometries, and (v) pressure variation of the magma reservoir caused by sea level unloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3 Figure 1: Complex evolution of volcanic edifices and deep weakness zone locations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (A) young volcanic edifice composed of massive lavas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (B) volcanic edifice after a giant landslide caused by a volcanic process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' where pyroclastic flows as well as soil and lava weathering occur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (C) filling of the landslide scar by thick volcanic lavas above the pyroclastic debris and the weathered soils,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (D) deep rooting of the landslide into the weakness zone favored by meteoritic water infiltration or/and sea-level loading in the case of an old and complex volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Slope stability model To analyze the causes of stress that may trigger a giant landslide and to discuss the stability of a given relief, a 2D model has been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Two different geometries of the landslide have been employed to compare the role of the geometry of a weakness zone on the theoretical stability of a volcanic edifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The first geometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2A) is based on the Cullman wedge model (Bigot-Cormier and Montgomery, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Basically, a balance is achieved between the weight of the expected landslide and the force of resistance generated by the rock properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The resistance shear stress along the slipping surface is given by \uf074c\uf020= \uf073n tan\uf066\uf020+ C until the landslide occurs, where \uf073n is the normal stress, \uf066 is the friction angle of the slope-forming material and C is the cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Considering the angle for T2 T1 T3 T4 PYROCLASTICFLOW WEATHEREDSOIL MASSIVE LAVA4 which the effective cohesion is maximized for a value equal to \uf062\uf020\uf02f\uf032\uf020\uf02b\uf020\uf066\uf020\uf02f\uf032 \uf02c\uf020it is assumed that \uf071\uf020\uf03d\uf020\uf062\uf020\uf02f\uf032\uf020\uf02b\uf020\uf066\uf020\uf02f\uf032\uf020(Champel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Bigot-Cornier and Montgomery, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This model allows us to calculate the maximum stable height for a simple geometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2A): H = 4C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' sin\uf062 cos \uf066 / (\uf072r g [1 - cos(\uf062\uf020– \uf020\uf066)]) where H is the maximum stable height of the slope, \uf062 is the hillslope gradient, \uf072r is the bulk density of the rock, and g is the gravitational acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This model has been widely used in geomorphological studies to predict the maximum unfailed height of slopes (Bigot-Cornier and Montgomery, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the theoretical calculation, vertical loading, pore pressure increase and volcanic rock properties with depth are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Including the pore pressure U and considering the load of the sea-water column of thickness Hm with a bulk density of the water \uf072m, the following can be obtained: H2 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' sin\uf062 cos\uf066\uf020\uf02d\uf020U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='sin\uf066\uf020cos\uf062\uf020\uf029 / (\uf072r g [1 - cos(\uf062\uf020–\uf020\uf066)]) + \uf072m Hm2/\uf072r = 0 (1) Equation (1) allows us to estimate the geometry of the maximum stable relief (height and slope) immediately before the collapse, as well as the geometry of the landslide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The thickness of the sea- water column that is considered could include sea-level variation in relation to Quaternary climatic variation but also relative sea-level increase caused by subsidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' FIGURE 2: Effect of the initial slope \uf062 on the maximum stable relief H for two different landslide geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (A) Cullman wedge model, (B) concave geometry model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' \uf072c\uf020= 3000 kgm-3, C = 1 MPa, \uf066\uf020= 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Using the same method, it is also possible to calculate the maximum stable height for a second kind of landslide geometry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' concave geometry model) using the following equation: H=H1+H1 cos\uf062\uf020 sin(\uf062\uf02f\uf032\uf020+\uf020\uf066\uf02f\uf032) / sin(\uf062\uf02f\uf032\uf020–\uf020\uf066\uf02f\uf032) (2) where H1 can be calculated by resolving the following equation: H12 - 4H1 (C cos\uf066\uf020\uf02d\uf020U sin\uf066\uf029 / (\uf072r g [sin\uf062\uf020– sin\uf066\uf05d\uf020)+ \uf072m Hm2 sin(\uf062\uf02f\uf032\uf020–\uf020\uf066\uf02f\uf032) / (\uf072r sin\uf062 cos\uf062\uf02f\uf032\uf020+\uf066\uf02f\uf032 ) = 0 1000 A 800 H 600 HEIGHT (m) 400 UNSTABLE B STABLE 200 H 9 B 0 10 20 30 40 50 60 70 80 90 SLOPE5 This study focuses on the role of pore pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' water loading and mechanical properties of rocks on slope stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Geotechnical approach are also useful to evaluate the safety factor FS = resisting forces / driving forces ≈ [ \uf074 × S ]/[ \uf072r V g sin \uf071\uf020] when the slope is destabilized only by his own weight (Hurliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Cala and Flisiak, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Gargani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Mechanical properties of volcanic rocks There are at least two ways to estimate the effective mechanical properties of rocks in a given geological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The first method is to use experimental data that describe the mechanical behavior of a rock close to that of the studied area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The second is to propose hypotheses on the causes of failure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', gravity, pore pressure, ground acceleration, eruption) and to estimate the effective values of mechanical parameters that triggered the observed failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Here, we verify that the mechanical properties of rocks estimated using a modeling approach are not in contradiction with experimental data for volcanic rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The parameterization of the model was conducted considering that the relief was close to critical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The effective mechanical properties of the rocks of the volcanic edifice considered in this numerical experiment have been calibrated using some characteristics of a giant landslide that occurred 872 kyrs ago in Tahiti (Hildenbrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2004 and 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Gargani, 2020 and 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' However, the investigation presented in this study is purely theoretical, and the geometries as well as the rock heterogeneity do not represent a real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The relief is considered to have a total height of ~4500 m with an initial slope \uf062 of 8-12°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Two different geometries have been tested to estimate the maximum stable relief H as a function of the initial slope (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The effect of the precise geometry (Cullman wedge model vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' concave geometry model) of the landslide on the critical value of height and initial slope of the stable relief is not negligible (Figure 2), but this study will not focus on the precise geometry of landslides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The mechanical properties of volcanic rocks depend strongly on their alteration and stress loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The cohesion and angle of friction have been estimated for different volcanic rocks under various slope heights by Rodriguez- Losada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The cohesion C can be obtained using the equation C=kHl, where H is the slope height and k and l are coefficients that have been experimentally determined for various volcanic rocks (Rodriguez-Losada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' For weathered massive lavas, k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='04 and l varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='41 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' For low cemented pyroclasts, k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='05 and l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The angle of friction \uf066 can also be estimated using the equation \uf066=a ln(H) + b, where a and b are obtained experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Rodriguez- Losada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (2009) show that very low angles of friction <13° are obtained at depths >1000 m for low cemented pyroclasts (a=6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' b=54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A volcanic edifice constituted by low cemented pyroclasts has a cohesion between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5 and 1 MPa at depths of 1500-3000 m (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' At this depth, a theoretic volcanic edifice constituted by low cemented pyroclasts has an angle of friction that ranges between 6° and 10° (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 6 FIGURE 3: Effect of depth on mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (A) Cohesion for low cemented pyroclasts (k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5) and fresh (k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='60) and weathered (k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='04;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' l=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='41 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='60) massive lavas, (B) angle of friction for low cemented pyroclasts (a=6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' b=54) and fresh (a=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' b=82) and weathered (a=7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' b=75) massive lavas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Modeling hydroclimatic variations In this study, sea-level loading and pore pressure variation in relation to climatic evolution were simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Basically, the water column loading that is considered has been estimated using a proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Sea level is known to change contemporaneously with \uf064 18 O variation (Gargani and Rigollet, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Gargani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' To simulate climate forcing, a \uf064 18 O curve (Lisiecki and Raymo, 2005) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This curve has been recalibrated to simulate Quaternary sea- level variations that reach an amplitude of 120 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The hydroclimatic variations are also caused by Milankovitch astronomic cycles and can be simulated using insolation or \uf064 18 O curves (Gargani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this study, it is 12 A 10 FRESHMASSIVELAVAS 8 COHESION (MPa) WEATHERED MASSIVELAVAS 2 PYROCLASTS 0 0 500 1000 1500 2000 2500 3000 3500 4000 SLOPE HEIGHT (m) 70 B 60 50 FRESH MASSIVELAVAS ANGLE OF FRICTION 40 30 WEATHEREDMASSIVE LAVAS 20 10 LOW CEMENTEDPYROCLASTS 0 0 500 1000 1500 2000 2500 3000 3500 4000 SLOPE HEIGHT (m)7 considered that more humid conditions are able to cause an increase in the effective pore pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The aim of this assumption is to discuss the timing of potential giant landslides in relation to Quaternary climatic conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', precipitation rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The time necessary for the water to propagate from the surface to a depth of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5 km has been estimated at ~150 days in Oregon (Saar and Manga, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the model, the propagation is considered instantaneous, which is justified on a long time scale (>1 kyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The amplitude of the variation in the pore pressure \uf044P due to water infiltration into the crust is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='01 MPa at Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Hood (Saar and Manga, 2003) and 2 MPa on the south flank of Kilauea volcano (Cervelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this study, various values for the pore pressure variation have been tested in the case where sea-level effects are dominant (0 < \uf044P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='125 MPa) and in the case where pore pressure processes are dominant (0 < \uf044P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5 MPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Pressure variation in the magma reservoir The loading variation caused by sea level variation is \uf044LSL(t) = g \uf072m Hm(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Sea water unloading causing decreased lithostatic pressure at depth, enhances the production of magma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Decompression favor partial mantle melting and magma release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The magma production rate DF/Dt at constant entropy S can be estimated as (Jull and McKenzie, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Sternai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2017): DF/Dt = (\uf064F/\uf064PM)S (dPM/dt –U\uf020\uf02e\uf0d1PM) where PM the magma reservoir pressure, F the melt fraction, t the time, U the mean mantle upwelling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The magma pressure variation \uf064\uf044PM/dt can be considered as depending from the sea water unloading variation \uf064\uf044LSL/dt under adiabatic condition and for very high viscosity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' \uf068c=1023 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='s) of the crustal rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this case \uf064\uf044PM/dt = - \uf064\uf044LSL/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When the viscous response is not negligible (\uf068c < 1022 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Sternai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2017), the equation is \uf064\uf044PM/dt +\uf044PM (Ec/\uf068c) = - \uf064\uf044LSL/dt, where Ec is the elastic modulus, and a delay of ~10 kyr after the forcing by sea water unloading is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Results The current question that we attempt to answer focuses on the thresholds and conditions that favor slope instability for volcanic edifices in relation with climatic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Influence of water column loading on slope stability Considering a simple geometry for the expected landslide (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2A, Cullman wedge geometry), the role of sea level variation on slope stability was investigated using equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When a volcanic island subsides, an increase in the relative sea level and, consequently, an increase in the water loading on the slope occur and could be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Under the Cullman wedge geometry condition, an increase in the loading by a water column of 120 m cause a slight decrease in the stability height (Figure 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The more the water loading increases, the less the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Whatever the geometry of the sliding mass is, the maximum height of the stable relief is impacted by the increase in the loading by a water column of 400 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A water column increase of 400 m is reached in less than 1 Myr for a moderate subsidence rate (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='25 mm/yr) contemporaneously with sea- level high stand at +120 m during an interglacial period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this case, slope instabilities can be observed on higher slopes (>25°) but also on smaller slopes (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The role of sea-water loading could trigger landslides of smaller dimensions (150-300 m) for higher slopes (25-30°, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4C, dark gray square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Influence of the pore pressure on slope stability FIGURE 4: (A) Effect of the increase in pore pressure and sea-water loading on the stability of the relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The light gray area is the area that changes from a theoretically stable relief to an unstable relief when submitted to an increase in the sea-water column loading between 0 and 400 m as well as to a pore-water pressure increase between 0 MPa and 2 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (B) Schematic representation of the effect of subsidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (C) The details are as follows: a relief of ~200 m with a slope of 25° < \uf062 < 30° could be affected by water loading (dark gray square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' \uf072c = 3000 kgm-3, C = 1 MPa, \uf066\uf020= 10° and geometry of the landslide identical to those presented in Figure 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The model suggests that a significant effect of the pore pressure increase on the height of the stable relief could occur (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The pore pressure significantly reduces the stability of the relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' As expected, the higher the pore pressure is, the lower the maximum height H before the giant landslide occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' If the pore pressure increases by more than 1 MPa, the slope stability is significantly decreased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4A and B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A relief of more than 1500 m, with a slope of ~12°, could be affected by landslides under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Depending on the initial relief H, a pore pressure increase of 1 MPa could cause instabilities of more than 1000 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Competing influence of sea-level loading vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' pore pressure variations over time on slope stability In this study, the influence of sea-level loading and pore pressure variations over time, in relation to meteoritic water infiltration, was modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The variation over time of the maximum height H of the volcanic edifice before any giant landslide occurs is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The maximum stable height depends on the loading of the sea level and on the value of the pore pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Taking into account that the maximum sea-level variation during the last million years is ~120 m and that the amplitude and the timing of the loading are known (Figure 5B), the variation in the maximum stable height H of a theoretical volcano before any giant landslide occurs has been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The absolute value of the maximum stable height H strongly depends on mechanical properties of rocks and slope, and will not be discussed in this study because this study focus on the variation of stability caused by climatic evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4000 A POSSIBLYUNSTABLE 1000 3500 C 900 B 3000 800 700 porepressure 2500 increase () ) HEIGHT (m) 600 HEIGHT 2000 500 UNSTABLE UNSTABLE 400 1500 300 STABLE STABLE 200 1000 400m 400m 400m om 100 U=2MRa U=1MIa U=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='4MPa 400m 120m 500 U=OMPa 10 15 20 25 30 35 SLOPE U=2MF U-1MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='.U=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='4MPa 400m 400m _400m om 400m 0 10 15 20 25 30 35 Tahiti SLOPE9 FIGURE 5: (A) Maximum height H before the landslide occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The geometry of the island is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The effects of sea-level loading and pore pressure are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The sea-water loading depends on the variation in sea level, whereas the effect of the pore pressure depends on the variations in precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the first case (sea-level-dominated), the pore pressure is considered to range between 0 and 125 kPa depending on the climatic variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the second case (pore pressure- 3700 3650 pore pressure dominated 3600 WA 3550 (m) height 3500 3450 sea level dominated 3400 Tahiti- Guadeloupe La Réunion-Canary LaReu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3350 Canary- Martinique Hawaii Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Haw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3300 1000 800 600 400 200 0 time (kyr) 50 sea level (m) 50 100 150 1000 800 600 400 200 0 time (kyr)10 dominated), the pore pressure is considered to range between 0 and 500 kPa depending on the climatic variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The absolute values of H is indicative and only the trends are interpreted in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' g=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='81 ms-2, C=1 MPa, \uf066=10°, \uf072=2800 kg/m-3, \uf062=7°, and a subsidence rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='25 mm/yr are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' (B) Simulated sea-level variations during the last million years using the \uf06418O curve of Lisiecki and Raymo (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The light blue lines represent the giant landslides on volcanoes during the last 1 Ma of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When seawater loading is the main process that influences landslides, the higher the sea level is, the lower the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The maximum height H is expected to decrease when the sea level rises (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Considering all other parameters as identical, the maximum stable height H of a volcanic edifice is 50 m higher during interglacial periods than during glacial periods if the sea- water loading is the main parameter influencing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the specific case of Polynesian climate variation, where the climate is more humid during glacial maxima (Saéz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009), as simulated here, the giant landslides generated by a pore pressure increase are expected to occur during glacial periods (Figure 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the specific case of the Polynesian volcanic edifice, the period when the landslides are expected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', the time when the maximum stable height H before a giant landslide is expected to occur), is different when the dominant process that triggers the landslide is sea- level loading or pore pressure loading (Figure 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The influence of sea level unloading on magma reservoir pressure increases at depth is better correlated than the sea level loading with giant landslides occurrence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This could suggest that in many cases the volcanoes were actives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' a magma reservoir was present at depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' If this process was the cause of the giant landslides, the viscosity of the crust was very high in many cases, except in La Réunion (290- 320 kyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Leunat and Labazuy, 2008), Canary islands (134 \uf0b1\uf020\uf036\uf03b\uf020Carracedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='\uf02c\uf020\uf031\uf039\uf039\uf039) and Hawaii (127 \uf0b1\uf02010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' McMurtry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1999) where a delay is observed with low sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The effect of sea level unloading influence on magma pressure increase is contemporaneous with the effect caused by pore pressure increases with meteoritic water infiltration in Polynesia and East Africa (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' where precipitation increases during glacial periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Rock weathering and slope instability Volcanic edifices are usually composed of massive lavas at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' These lavas can be highly weathered by the humid climate when they are exposed for thousands or millions of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Laboratory experiments for weathered massive lavas show that cohesion can be low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='8-1 MPa) at depths ranging from 1500 m to 3000 m (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The angle of friction of weathered massive lavas is estimated to range between 23° and 19° based on laboratory experiments (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Lavas that have been extremely altered by rain and fluid circulation at depth for thousands of years are expected to have mechanical properties lower than those of fresh volcanic rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Furthermore, the effective mechanical properties (C and \uf066) of fractured and faulted reliefs are lower than those used for triaxial experiments on nonfractured rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Effective mechanical properties take into account the heterogeneities present in old and complex volcanic edifices including low cemented pyroclast and weathered soils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Soils can be observed 11 sandwiched between volcanic lavas (Hevia-Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' These soils are expected to have reduced mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The model has been calibrated to be compatible with past giant landslide events (~1500-3000 m thick, initial slope \uf062\uf020~8-12°, cohesion C of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='3-1 MPa, Figure 2), such as the event that occurred in Tahiti, Society Archipelago (Gargani, 2020 and 2022a) (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The values used here correspond to highly weathered rocks of an old and fractured volcanic edifice or low cemented pyroclasts that could be sandwiched at depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Rock mechanical properties C and \uf066 decrease over time when exposed to weathering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The more weathering that occurs, the greater the decrease in C and \uf066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' During humid periods, this decrease is higher than during drier periods (Figure 6A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In contrast, diagenetic processes are expected to improve the mechanical properties of rocks (Figure 6B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When pressure increases at depth, porosity and connectivity can decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Furthermore, mineralization of fluid at depth can also improve rock cohesion in specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' FIGURE 6: Influence on the rock mechanical properties C and \uf066 of (A) weathering and (B) diagenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='2 Weakness zones and geological inheritance The construction of volcanoes is often polygenic, and numerous heterogeneities exist on volcanic edifices (Hildenbrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The complex and long volcano-tectonic history of old volcanic edifices may explain why numerous fractures and significant weathering are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The highly fractured and heterogeneous material (basaltic flows and volcanic breccia) of volcanic edifices is often complex and difficult to model in detail at all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Strata of weathered soils sandwiched between lavas can also be observed (Hevia-Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2022) as well as pyroclastic formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Low values for the angle of friction have been estimated or used in various studies for weakness structures, effective rock properties or faults (Cala and Flisiak, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Bigot-Cormier and Montgomery, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Got et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Egholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Abers, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Modeling studies are used to simplify the complexity of the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The effective mechanical properties implemented in modeling are different than the laboratory experimental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Preexisting zones of weakness at depth are present in many volcanic edifices but are different in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A real geometry of a specific HUMID 个个个个 C or Φ Weathering A Cor Φ B Diagenesis DRY 个 个个个 TIME TIME12 volcanic edifice or of a specific landslide is not the aim of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A general case is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This model does not permit us to predict exactly when a large landslide is expected to occur but only to discuss the possibility of discriminating the influence of meteoritic water infiltration from sea- water loading/unloading using climatic correlation from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='3 Sea-level variation effect on landslides and volcanic activity The role of water-level rise in landslide triggering has been suggested in the case of the Vajont landslide (1963, Italy) (Muller-Salzburg, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Kilburn and Petley, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When sea level rises, (i) water loading increases and (ii) water infiltrates into rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The loading caused by sea level rise could cause slope instability when the slope is almost at equilibrium immediately before sea level rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Furthermore, when sea level rise, erosion can occur at the base of the cliff and favor cliff retreat, rock fall and landslide (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' However, the effect of loading forces on the base of a slope can also favor the slope stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A vertical loading force at the base of the volcano can cause an increase of stability by opposing a force to the potential movements of the relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In other words, when the sea-level fall at the base of a slope, it could favor instability (Figure 7A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This result is suggested by safety factor FS decrease when loading decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When a significant loading is above the center of mass of the potential landslide, it cause the destabilization of the slope by triggering rock failure (Figure 7B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Figure 7: Influence of the sea-level variation in relation with the position of the center of mass of the potential sliding area: (A) slope instability caused by sea-level lowering when the load is located at the base of the slope, (B) slope instability caused by sea- level rise above the center of mass with marine water infiltration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The giant landslides in volcanic areas seem to be correlated with climatic variation (Quidelleur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Sea level loading is one of the potential causes of giant landslides (Aslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' However, indirect interactions are also possible in volcanic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The unloading associated with sea level lowering could modify the pressure on the magma reservoir (Sternai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' A pressure decrease has implications for the input and output rates of magma into and out of the magma reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' If the connection to the deeper magma source remains open, one consequence is an increased rate of replenishment with primitive magma (Pinel and Jaupart, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Furthermore, a reduced load allows the eruption of denser magmas that would otherwise have been stuck at shallow depths (Pinel and Jaupart, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The potential interaction between sea- level variation and landslides on one A B13 side and landslides and volcanic eruptions on the other side (Longpré et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009) could cause also difficulties in interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' If the magma reservoir is shallow, a landslide could generate dyke intrusion, but if the magma reservoir is deep, dyke intrusion is more difficult after a landslide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='4 Pore pressure variation Another process that is able to cause landsliding is pore pressure increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Fluids are known to be a triggering and driving factor for landslides (Cappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Various mechanisms could explain a pore pressure increase of ~1 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' First, it is necessary for a significant amount of water to enter the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' For example, unsaturated rocks of porosity n in an area of volume V could become saturated when a significant water volume n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='V infiltrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The volume of water could be higher if a dense fracture network existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Meteoritic water infiltration at a depth of 1000 m under lithostatic pressure could cause a pore pressure increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' An increase of more than 1 MPa in the pore pressure at depth related to meteoritic water is not unrealistic and has been proposed in previous studies (Cervelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' An alternative mechanism that could also lead to a significant pore pressure increase results from the progressive collapse of the base of the volcanic edifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this case, the slow creep of the edifice on a preexisting weak structure may trigger overpressure conditions locally that reach high pressures (Veveakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In a volcanic context, it may also be expected that hydrothermal processes play a role in the pore pressure increase and rock alteration at depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The pore pressure can change in relation to meteoritic water infiltration in highly fractured rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this case, meteoritic water infiltration increases when the precipitation rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Consequently, climatic variations are expected to influence the pore pressure variation in highly fractured rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The correlation between climatic variation and giant landslides may be caused by pore pressure increases at depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='5 Climate variation and correlation with giant landslides Precipitation increased during glacial maxima in Polynesia (Saez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009), Mexico (Ganeshram and Pedersen, 1998), and eastern equatorial Africa (Chiang, 2009) but decreased in NW Europe (Guiot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1989), the Caribbean (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2001) and Indonesia (Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When humid conditions occurred during warmer phases, the potential effect of water infiltration on pore pressure acted to increase instability during warmer interglacial periods, similar to sea-level loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Interglacial phases are expected to cause more slope instability in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' However, it is difficult to discriminate the influence of sea-level loading from meteoritic water infiltration at depth on giant paleolandslides in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In contrast, when precipitation increased during glacial maxima, the impacts of sea-level loading and water infiltration at depth on giant paleolandslides were not contemporaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In this case, it is theoretically possible to discriminate the process that plays the main role in triggering giant landslides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' For example, in Polynesia, increased meteoritic water infiltration occurred during glacial maxima and potentially increased slope instability during glacial maxima, whereas sea-level rise caused slope instability during interglacial times (Figure 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The age of giant landslides (Table 1) may be a good argument to discriminate between these two processes, but only in Polynesia, Mexico or eastern equatorial Africa and 14 not in NW Europe, the Caribbean or Indonesia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This study suggests that if preexisting weakness zones are present at depth, causing low effective cohesion and internal friction, a climatic origin of landslides is possible even if volcano is extinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Seismic or volcanic processes are not necessarily directly responsible for giant landslides, even if these processes may have played a role in causing weakness zones in the past in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In 70% of cases, the giant landslides occurred when sea-level rise, regardless the precipitation rate is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Due to potential delay caused by viscous processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 10 kyr of delay for a viscosity less than 1022 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='s), it could be also correlated to the magma chamber unloading by sea-level lowering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This result could be influenced by the set of data considered and further studies on giant landslides on volcanic islands would improve this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Table 1: Giant landslide ages in volcanic areas during the last 1 Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Volcano Age (ka) References Canary Islands (El Hierro, El Golfo) 10-17 Gee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2001 La Réunion 20-68 Lenat and Labazuy, 2008 Hawaii (Alika phase 1) 112 Mc Murtry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1999 Hawaii (Alika phase 2) 127 \uf0b1\uf02010 Mc Murtry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1999 Canary Islands (El Hierro, El Golfo) 134 \uf0b1\uf020\uf036 Carracedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 1999 Hawaii (Southern Lanai) 135 Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2000 Canary Islands (Tenerife) 150-170 Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2011 Masson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2002 Hawaii (Southern Lanai) 240 Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2000 La Réunion 290-320 Lenat and Labazuy, 2008 Martinique 337 \uf0b1\uf020\uf035 Quidelleur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2004 Canary Islands (La Palma) 537 \uf0b1\uf0208 Guillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2001 Groom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2022 Guadeloupe 629 \uf0b1\uf02013 Samper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2007 Hawaii (Haleakala, Hana) 860 Moore and Clague, 1992 Tahiti-Nui (north) 872 \uf0b1\uf02010 Hildenbrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2004 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='6 Small landslides vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' giant landslides The dimension of the expected landslide depends on the initial slope of the relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Locally, as in a deeply incised canyon of volcanic islands, significant slopes can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' In the case of a significant slope (>25°), the critical value for the height of the stable relief is less than 300 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Consequently, smaller landslides may also be triggered locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When rotational landslides occur, significant slopes develop in the upstream part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' The resulting relief could be near the instability conditions that favor the development of new landslides and other retrogressive erosion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' It can be difficult to discriminate the relief associated with numerous “small” landslides from that caused by giant landslides (Gargani, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Gargani, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Erosion and repetition of small landslides could progressively reduce the loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' As previously shown, the loading influences the maximum height expected before a landslide occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Reducing the load increases the maximum height before landsliding and 15 thus reduces the potential occurrence of a landslide when volcanic edifice is an extinct volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Consequently, the potential occurrence of a giant landslide could be reduced by the occurrence of small landslides and erosion of the relief in the case of extinct volcanoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Previous studies have shown that the probability of small landslides occurring is higher than that of large landslides (Urgeles and Camerlinghi, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Nevertheless, giant landslides have also been described, suggesting that there are specific conditions that favor giant landslides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Conclusion The probability of a large landslide in an area where no significant volcanic and seismic activities have been observed during the last thousand years is not null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Large landslides caused by nonvolcanic processes could occur on old volcanic islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Indeed, our modeling suggests that a large landslide may occur due to an increase in pore pressure and/or sea-level variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' These processes could be discriminated in areas where glacial maxima are more humid than interglacial periods, but not in the other cases, based only on timing correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' This finding is also a consequence of the mechanical properties of highly weathered and/or fractured volcanic rocks in weakness zones (low cohesion and angle of friction) at depth in old weathered volcanic edifices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' When volcanoes are still actives, sea level unloading can be responsible of giant landslides, causing magma reservoir pressure increases at depth and magma release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' During the last million years, several giant landslides in tropical areas are correlated with sea level unloading during glacial periods rise suggesting that this effect is a driving mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Acknowledgments: References cited Abers G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Slip on shallow- dipping normal faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content=' Geology, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} +page_content='37, p.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9AyT4oBgHgl3EQfj_jD/content/2301.00426v1.pdf'} diff --git a/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/2301.02767v1.pdf.txt b/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/2301.02767v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3b9be4876fd77e339c9d7eaf690f72857c6ab06 --- /dev/null +++ b/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/2301.02767v1.pdf.txt @@ -0,0 +1,2401 @@ +Health Wearables, Gamification, and Healthful Activity +Muhammad Zia Hydari,a,* Idris Adjerid,b,* Aaron D. Striegelc +aKatz Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania 15260; bPamplin College of Business, Blacksburg, +Virginia 24061; cDepartment of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana 46556 +*Corresponding author +Contact: hydari@alum.mit.edu, +https://orcid.org/0000-0003-4522-326X (MZH); iadjerid@vt.edu, +https://orcid.org/0000-0002-2786-1244 +(IA); striegel@nd.edu, +https://orcid.org/0000-0002-3157-2859 (ADS) +Received: June 19, 2019 +Revised: December 8, 2020; October 7, 2021; +January 30, 2022 +Accepted: February 8, 2022 +Published Online in Articles in Advance: +December 19, 2022 +https://doi.org/10.1287/mnsc.2022.4581 +Copyright: © 2022 The Author(s) +Abstract. Health wearables in combination with gamification enable interventions that +have the potential to increase physical activity—a key determinant of health. However, the +extant literature does not provide conclusive evidence on the benefits of gamification, and +there are persistent concerns that competition-based gamification approaches will only ben- +efit those who are highly active at the expense of those who are sedentary. We investigate +the effect of Fitbit leaderboards on the number of steps taken by the user. Using a unique +data set of Fitbit wearable users, some of whom participate in a leaderboard, we find that +leaderboards lead to a 370 (3.5%) step increase in the users’ daily physical activity. How- +ever, we find that the benefits of leaderboards are highly heterogeneous. Surprisingly, we +find that those who were highly active prior to adoption are hurt by leaderboards and walk +630 fewer steps daily after adoption (a 5% relative decrease). In contrast, those who were +sedentary prior to adoption benefited substantially from leaderboards and walked an addi- +tional 1,300 steps daily after adoption (a 15% relative increase). We find that these effects +emerge because sedentary individuals benefit even when leaderboards are small and when +they do not rank first on them. In contrast, highly active individuals are harmed by smaller +leaderboards and only see benefit when they rank highly on large leaderboards. We posit +that this unexpected divergence in effects could be due to the underappreciated potential of +noncompetition dynamics (e.g., changes in expectations for exercise) to benefit sedentary +users, but harm more active ones. +History: Accepted by Chris Forman, information systems. +Open Access Statement: This work is licensed under a Creative Commons Attribution 4.0 International +License. You are free to copy, distribute, transmit and adapt this work, but you must attribute this +work as “Management Science. Copyright © 2022 The Author(s). https://doi.org/10.1287/mnsc.2022. +4581, used under a Creative Commons Attribution License: https://creativecommons.org/licenses/ +by/4.0/.” +Funding: This work was supported by the National Institutes of Health [Grant 5R01HL117757]. +Supplemental Material: The data and online appendix are available at https://doi.org/10.1287/mnsc.2022.4581. +Keywords: +health wearables • gamification • fitness • physical activity • health • health technology +1. Introduction +The evidence on the health1 benefits of physical activity +is irrefutable (Warburton et al. 2006). Yet, a significant +portion of the world population is not sufficiently ac- +tive.2 This lack of physical activity contributes signifi- +cantly to chronic disease and to most of the leading +causes of death in the United States.3 Prior research sug- +gests that behavioral barriers are one of the most impor- +tant contributing factors to this trend. Mitchell et al. +(2013) suggest that “for many adults, the ‘costs’ of exer- +cise (e.g., time, uncomfortable feelings) loom so large +that they never start” and that the lack of physical activ- +ity is “a problem of both initiation and maintenance” +(p. 658). Recognizing that changing health behaviors +is often challenging and new strategies are needed, +research situated mostly in the health and economics +literature has evaluated a plethora of economic and +non-economic interventions for overcoming motiva- +tional barriers to increasing physical activity (Charness +and Gneezy 2009, Mitchell et al. 2013). The conclusion +from this literature is that, although many interventions +can drive short-run gains in physical activity, these ben- +efits are fleeting and motivating meaningful, and sus- +tained increases in physical activity is elusive. More so, +many of these interventions (e.g., daily payments) are +difficult to implement on a population scale. +One contemporary phenomenon with the potential to +address persistent limitations of prior approaches and +unlock new interventions that can improve the motiva- +tion of individuals to exercise is the rapid consumer +adoption of health wearables (Swan 2013, Lupton 2016). +A health wearable, sometimes referred to as an activity +tracker, is “a wearable device or a computer application +that records a person’s daily physical activity, together +1 +MANAGEMENT SCIENCE +Articles in Advance, pp. 1–19 +ISSN 0025-1909 (print), ISSN 1526-5501 (online) +https://pubsonline.informs.org/journal/mnsc +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +with other data relating to their fitness or health, such as +the number of calories burned, heart rate, etc.”4 Despite +the rapid adoption of health wearables and their poten- +tial for motivating individuals to engage in healthy +activities, scholars suggest it is unlikely that the meas- +urement capabilities that the health wearables provide +would significantly impact health on their own (Patel +et al. 2015, Sullivan and Lachman 2017). Rather, they +suggest that for health wearables to impact health behav- +ior, the information they collect “must be presented back +to the user in a manner that can be understood, that +motivates action, and that sustains that motivation +toward improved health” (Patel et al. 2015, p. 460). +Particularly promising in this regard is combining +granular physical activity data from health wearables +with gamification approaches. Gamification is defined +as the “use of game design elements in nongame con- +texts” (Deterding et al. 2011). Some examples of game +design elements are badges, rules-based competition, +leaderboards, points, ranking, reputation, rewards, +teams, and time pressure (Deterding et al. 2011). Cou- +pling gamification with health wearables has the poten- +tial to improve motivation by converting the usually +mundane action of physical activity into the more enjoy- +able activity of collecting rewards or competing with +other individuals (Hamari et al. 2014a, Johnson et al. +2016). More so, gamification approaches can provide +immediate positive reinforcement that helps individuals +get over the initial hurdles of engaging in exercise and +could also help them sustain higher levels of activity in +the longer term (Mitchell et al. 2013, Shameli et al. 2017). +In addition, the broad adoption of health wearables un- +locks more robust gamification interventions by provid- +ing an objective and common source of measurement +and a form factor that enables real-time feedback while +engaging in physical activity (Johnson et al. 2016). +Although coupling gamification with health wearables +has the potential to generate sustained increases in physical +activity, the evidence on the benefits of gamification is +mixed. Hamari et al. (2014b) reviewed 24 empirical gami- +fication studies, primarily within education contexts, but +reported mixed effects on outcomes. Moreover, these stud- +ies used interviews or surveys to measure outcomes. +Hamari and Koivisto (2013), the only empirical study in a +health context in the aforementioned literature review, used +surveys to measure outcomes and these outcomes were not +health related (e.g., continued use intention for the gamifica- +tion service and the intention to recommend service to +others). Johnson et al. (2016) conducted a systematic litera- +ture review on the impact of gamification on health and +well-being. Of the 19 empirical papers they reviewed, 59% +reported positive results and 41% reported mixed results. +Both Hamari et al. (2014b) and Johnson et al. (2016) also +noted that the quality of evidence was moderate to low. +The significant potential benefits of coupling gamifi- +cation with health wearables and the narrow focus and +lack of evidentiary quality of prior works motivate this +research study. We evaluate the benefits of leaderboards +that allow users to view the performance of others who +also agree to share their activity levels and, in most +cases, to compete with them. We focus on the potential +benefits of leaderboards because they are one of the most +common gamification features available with modern +health wearables. This increases the policy and practical +relevance of our results. Another reason we focus on +leaderboards is that they exemplify the theoretical ten- +sions surrounding gamification interventions; scholars +suggest that gamification features, and leaderboards in +particular, are likely to have heterogeneous effects on +individuals (Deci et al. 1981, Santhanam et al. 2016, Sulli- +van and Lachman 2017). Specifically, a central concern +with competition-based gamification interventions like +leaderboards is that they will lead to motivational bene- +fits only for those who are already highly active (and +need the increased motivation the least), whereas actually +harming the least physically active in the population +(Patel et al. 2015, Wu et al. 2015, Shameli et al. 2017).5 With +these dynamics in mind, our first objective is to evaluate +the average impact on physical activity of leaderboard +adoption by individuals wearing health wearables. Our +second research objective is to evaluate the potential for +leaderboard effect heterogeneity by (i) the activity level of +the focal user prior to adoption, (ii) the number of active +participants on the leaderboard, and (iii) the rank of the +focal user on the leaderboard in the prior period. +We engage in an intensive data collection effort to +estimate the average benefit of leaderboards and the +heterogeneity in these benefits. For approximately 500 +individuals observed over a two-year time period, we +capture leaderboard adoption data and granular meas- +ures of physical activity continuously captured by Fitbit +Charge HR health wearables. For those individuals +with leaderboards, the data set also includes activity +data and rank of all participants in the leaderboard. We +supplement these data from health wearables with peri- +odic surveys (every six months on average) capturing a +rich array of individual characteristics (psychological +attributes, frequency of technology use, etc.). Leverag- +ing variation in physical activity and adoption of leader- +boards over time and between individuals, we use a +difference-in-differences (DID) estimation approach to +evaluate the effect of leaderboards on daily physical +activity as measured by the user’s step count, as well as +heterogeneity in these effects. +We find that leaderboard adoption results in an aver- +age daily increase of 370 steps. This main effect is resil- +ient to various tests for the assumption of common +trends between those who adopt and do not adopt, +estimation of several falsification tests, and other ro- +bustness checks. These initial results, however, mask +important heterogeneity in the benefits of leaderboard +adoption. When we take into account an individual’s +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +2 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +prior activity levels, we find a stark divergence in leader- +board effects. Individuals who were highly active prior to +adoption, instead of benefiting from leaderboards, experi- +enced a significant decrease in their average daily physical +activity after leaderboard adoption (a decrease of 631 steps +daily). Moreover, these negative effects persisted (and +actually increased in magnitude) in the 10 weeks following +leaderboard adoption. In contrast, users who were less +active prior to adoption had large and significant positive +impacts on their daily step counts—On average, their +activity increased by 1,365 steps daily (an approximately +15% increase), and these increases also persisted well after +the adoption decision (10 weeks after adoption). +Examining this trend further, we find significant nu- +ance in how leaderboard size impacts sedentary versus +highly active users. Specifically, the key distinction bet- +ween these groups is that previously sedentary individ- +uals can reap significant benefit from small leaderboards +(only one or two other members) and even if they do +not rank first. In contrast, those who were highly active +(prior to adoption) see the most significant harm when +leaderboards are small. Our interpretation of these +results is that individuals who are already on the high +end of the physical activity distribution can become com- +placent on small leaderboards where, more often than +not, they are paired with those less active than them- +selves.6 In contrast, sedentary individuals who are at the +lower end of the distribution of physical activity often +encounter (even on small leaderboards) peers who are +more active than themselves, who can positively impact +their reference point for exercise, and who can hold +them accountable if their activity levels slump. However, +these benefits for sedentary users diminish if leader- +boards become too large; the marginal benefit of an addi- +tional leaderboard member diminishes at three times the +rate for sedentary users relative to highly active ones. +One explanation for this effect is that the benefits of +social influence that accrue to sedentary users (e.g., posi- +tive impact on their exercise reference points) diminish +as leaderboards become larger and less intimate. +Our research contributes to streams of work at the +intersection of information systems, economics, and +healthcare. Specifically, we contribute to the literature +on the economics of health information technology (IT) +and specifically to the nascent streams of work evaluat- +ing economic and health implications of widespread +adoption of health wearables (Handel and Kolstad +2017) and the broader potential of digital platforms to +unlock interventions that leverage social norms and rec- +iprocity to improve health (Liu et al. 2019a; b; Sun et al. +2019). Currently, the evidence on benefits from health +wearables does not align with their promise. Piwek et al. +(2016, p. 2) suggest that “current empirical evidence is +not supportive” of health benefits from health wear- +ables. Recent studies using large samples and robust +causal approaches find little or no benefit on health +outcomes of using health wearables (Lewis et al. 2015, +Finkelstein et al. 2016, Jakicic et al. 2016). However, +scholars have argued that a limitation of prior works is +that they do not adequately consider the role of innova- +tive technology decision aids and behavioral interven- +tions enabled by broad adoption of health wearables +(Patel et al. 2015). Our study addresses these limitations +of prior work and finds that, on average, leaderboards +promote healthful activity. However, our results also +caution that these benefits may be highly nuanced with +considerable variation in gains. In some cases, individu- +als may opt into variants of these interventions with +either no benefit to them or, in some cases, negative +effects on their physical activity. +We also contribute to the behavioral economics and +information systems (IS) literature on gamification, espe- +cially, within the healthcare context. Despite mixed evi- +dence of benefits and numerous open empirical and +theoretical questions (Liu et al. 2017, Treiblmaier et al. +2018, James et al. 2019), gamification is spreading into a +number of decision contexts. For instance, two recent IS +papers have examined the impact of gamification within +the retail context (Pamuru et al. 2021, Ho et al. 2022). Our +study is differentiated with extant literature in several +ways. First, our study is an individual-level intervention +within healthcare in which the combination of unique +data and rigorous estimation approaches results in more +conservative estimates of average treatment effects of +leaderboards; prior work showing positive effects of simi- +lar gamification interventions has found treatment effects +five times our estimates (Shameli et al. 2017). Second, our +results suggest that the mixed evidence of prior work +may be explained, in part, by significant heterogeneity +in gamification impacts. Not only are we able to provide +more nuance in our study for gamification’s impact (Ho +et al. 2022), we also provide evidence on a substantively +important issue in the medical literature, that is, the im- +pact on the previously less active users (Patel et al. 2015). +In our setting, the relatively conservative estimates of +the average effect of leaderboards mask robust heteroge- +neous effects that are large in magnitude, statistically sig- +nificant, and persistent over time. These heterogeneous +effects support our theoretical conjecture that competi- +tion and social influence are key mechanisms underly- +ing leaderboard effects but also highlight that these +mechanisms can result in unexpected motivational and +de-motivational effects. Specifically, we identify a diver- +gence of benefit for sedentary versus highly active users +that is opposite to the expectation for competition-based +gamification in the literature. These findings point to the +underappreciated role of social influence benefiting sed- +entary users but harming more active ones. These results +not only have significant managerial implications for +firms in the health wearable and gamification spaces, but +also for policy makers, healthcare entities, and employers +interested in improving health. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +3 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +2. Background +Physical activity is a key element of healthful living and +is known to have significant health benefits (Penedo +and Dahn 2005, Warburton et al. 2006). Our main out- +come variable is Fitbit step counts and includes a variety +of these healthful physical activities, such as jogging, +running, walking, playing sports, climbing stairs, and +so on. Moreover, daily step counts are key to Fitbit lead- +erboards, as rankings on leaderboards are determined +exclusively by differences in the step counts of the par- +ticipants of the leaderboard. +2.1. Leaderboards +A leaderboard is “a large board for displaying the rank- +ing of the leaders in a competitive event.”7 In a digital set- +ting, the leaderboard may be displayed on a mobile +application or an online dashboard. In this study, we +utilize health wearables made by Fitbit Inc., which is a +pioneering firm in this market.8 Using Fitbit’s online +dashboard or the mobile application, a Fitbit user can +invite another user (or receive an invitation) to join a lead- +erboard. If there is mutual agreement between the users +to participate, both users will appear on each other’s lead- +erboards. Each leaderboard ranks participating users +based on seven-day running tallies of their steps.9 The +step counts shown on the leaderboard are directly cap- +tured by the Fitbit device and are not manually entered +by the users, thus avoiding the measurement errors that +may result from self-reported activity data. +Figure 1 shows four leaderboards, with the focal user +labeled at the lower left corner. Each leaderboard can have +the same or different user composition. For instance, Ash +and Todd are connected to Mary and to each other. Dave +is only connected to Mary, and Mary is connected to all +other users. The leaderboards also show the seven-day +step count of each participating user. Users are assigned +ranks on participating leaderboards based on their seven- +day step count relative to other users on that leaderboard. +For instance, Mary is ranked second on her own leader- +board, but she is ranked first on Ash’s and Todd’s leader- +boards. Thus, Ash and Todd may be motivated to do +better by seeing their lower rank on the leaderboard rela- +tive to Mary. Users get feedback according to their rank on +their own leaderboard. Although Mary dominates the +highest number of leaderboards, the feedback she gets is +that she is ranked second on her own leaderboard and +must strive harder to achieve a first rank. Leaderboard +adoption is “sticky” on the Fitbit platform. To de-adopt, +users have to go through cumbersome steps and hide +themselves via privacy settings. +3. Effect of Leaderboards on Healthful +Physical Activity +Whether leaderboards will increase or decrease healthful +physical activity is not entirely clear as the effect is un- +likely to be similar for all individuals. Leaderboards can +produce an effect on an individual’s physical activity +primarily by altering this individual’s willingness to en- +gage in physical activity. Specifically, we conjecture that +changes in willingness to engage in physical activity occur +primarily due to the introduction of competitive dynam- +ics, increased individual accountability, and altering an +individual’s reference point for their own activity levels. +3.1. Competition +Social comparison theory suggests that a fundamental +mechanism through which individuals assess their own +ability is through comparison with others (Festinger +1954). Competitiveness is one manifestation of the social +comparison process and drives individuals to increase +Figure 1. (Color online) Fitbit Leaderboard Composition for Four Individuals +Notes. This figure shows leaderboards for Ash, Dave, Mary, and Todd. Ash and Todd are connected to each other and Mary, Dave is only +connected to Mary, and Mary is connected to all other participants. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +4 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +P412:30 +V12:30 +41230 +-12:30 +三 +Friends +Friends +Friends +三 +Friends +7DAYSTEPS +MESSAGES +ZDAYSTEPS +MESSAGES +7DAYSTEPS +MESSAGES +7DAYSTEPS +MESSAGES +Mary +You +Dave +Mary +103.082 +110,193 +110,193 +103,082 +You +Mary +noa +Ash +97.922 +103.082 +103.082 +97.922 +Todd D. +Ash +SHOWINACTIVEFRIENDS +You +95,880 +97.922 +95.880 +A lttlehealthycompetition isagreatthing +Todd D. +SHOW INACTIVEFRIENDS +Tapthe.Addbuttonto find morefriends. +95,880 +SHOWINACTIVEFRIENDS +Alittiehealthy competition is a greathing. +A littiehealthycompetition is a great thing. +TaptheAddbuttontofindmorefriends. +SHOWINACTIVEFRIENDS +TaptheAddbuttonto findmorefriends. +Altehealthy competitionis agreatthing +TaptheAdd buttontofind morefriends. +十 ++ +Ash +Dave +Mary +Todd D.their effort either ex ante to elevate their rank or ex post +to maintain their high rank (Garcia et al. 2013). Thus, the +first and most direct way that leaderboards impact +physical activity is through the competitive dynamic +that ranking a focal user against other users generates. +The tag line on the Fitbit leaderboard (Figure 1)—a little +healthy competition is a great thing—points to the motiva- +tional potential of this competitive mechanism. In addi- +tion, the enjoyment derived from physical activity may +be impacted by the individual’s leaderboard adoption +by converting the mundane activity of walking into the +more exciting activity of competing against others. +Therefore, individuals who may not gain any direct +enjoyment from walking may engage in this activity +because of the indirect enjoyment gained from compet- +ing on the leaderboard. +However, prior work finds that impacts of competi- +tion on motivation and effort are highly heterogeneous +and depend on several factors such as a participant’s +desire to win, whether the competition provides a par- +ticipant the opportunity or reason for improving their +performance, and whether competition motivates a par- +ticipant to put forth greater effort (Deci et al. 1981). +Along this vein, a leaderboard may have minimal +impact on performance if it does not provide sufficient +competition or if the adopting individual is not particu- +larly motivated by competition. More so, prior work has +noted the possibility of competition having negative +impacts on motivation and performance. For instance, +Steinhage et al. (2015) argue that when competition elic- +its excitement, it may foster positive behavior. However, +if competition elicits anxiety, it may foster negative +behavior. Extrapolating this to our context, if the per- +formance of others on the leaderboard elicits anxiety in +the focal user, it would lead to negative outcomes for +them. Reflecting this theoretical tension, the extant liter- +ature has found mixed results regarding competition +with others who significantly surpass the individual in +performance. Rogers and Feller (2016, p. 1) showed that +“exposure to exemplary peer performances can under- +mine motivation and success by causing people to per- +ceive that they cannot attain their peers’ high levels of +performance,” and termed this phenomenon discourage- +ment by peer excellence. However, Uetake and Yang +(2019) find that an individual’s distance from the high- +est achiever has positive motivational effects, whereas +comparison with the average individual has negative +impacts. Thus, competition is likely a focal mechanism +behind leaderboard effects but whether it positively +impacts physical activity is uncertain ex ante. +3.2. Social Influence +Leaderboards involve connecting individuals around +health and the revelation of previously private levels +of physical activity between individuals. These con- +nections and disclosures introduce the potential of social +influence to impact motivation and behavior. We consider +two potential effects in the realm of social influence: indi- +vidual accountability and reference points for exercise.10 +3.2.1. Individual Accountability. Joining a leaderboard +involves the revelation of one’s previously private levels +of physical activity to other users. This self-revelation +allows other leaderboard members to hold the focal +user accountable for lackluster levels of physical activity +and nudge them to do better. In fact, the Fitbit app has a +mechanism for messaging, cheering, and taunting other +users directly from the platform. Some of these interac- +tions may also happen off the Fitbit platform (and are +thus unobserved by us as researchers)—for example, +discussions between family members over dinner. The +potential of group-based interventions to increase +mutual accountability and increase physical activity has +been explored in the literature: Patel et al. (2016) un- +cover benefits of incentive schemes for exercise that are +tied to group versus individual performance targets. +3.2.2. Exercise Reference Points. In addition to the +potential impacts of self-revelation, the revelation by +others of their previously private levels of physical +activity can result in changes to individuals’ reference +points for exercise. Specifically, social comparison +theory suggests that such revelations can lead to an +updated perception of one’s own ability to exercise and +the appropriateness of one’s own level of exercise (Gar- +cia et al. 2013). However, how these comparisons impact +reference points depends on whether individuals +engage in upward comparisons (i.e., comparisons with +those more active than themselves) or downward com- +parisons (i.e., comparisons with those less active than +themselves) (Festinger 1954). In both cases, the literature +suggests that individuals will take action to reduce dis- +crepancies between themselves and similar others (Fes- +tinger 1954, Garcia et al. 2013). Thus, if individuals +compare upward, the revelation of this information +between members of a leaderboard may have a positive +impact on an individual’s reference point for healthful +activity and increase exercise. For instance, a mother +with two young children may aim for a higher level of +healthful activity if she observes another mother with +two young children consistently doing more healthful +activity. Given that these two users may have similar +schedule constraints, the focal user may find the leader- +board information more relatable. If individuals com- +pare downward, the revelation of physical activity +information by others may have unintended negative +impacts on an individual’s reference point for physical +activity. In particular, this informational signal can +work in the opposite direction—that is, focal users may +decrease activity if they see other relatable indivi- +duals on their leaderboards who are less active than +themselves. Related to this point, Schultz et al. (2007) +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +5 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +found that a nudge intended to decrease electricity con- +sumption by revealing the consumption levels of others +in one’s neighborhood had the (opposite) boomerang +effect for those who were underusing electricity (rela- +tive to their neighbors) prior to the intervention. +3.3. Moderating Effects of Prior Activity Levels +and Leaderboard Size +The contradictory effects of competition and social +influence not only make the direction of the average +effect uncertain, they also point to the presence of heter- +ogeneity in the effects. To untangle this heterogeneity, +we consider factors that can impact the propensity for +observing the positive vs. negative dynamics of leader- +boards on physical activity. +3.3.1. Leaderboard Size. First, we consider whether +leaderboard size, that is, the number of other active par- +ticipants on the leaderboard, is an important potential +moderator of leaderboard impacts. Garcia et al. (2013) +suggest that an important situational factor impacting +comparison concerns and competitiveness is the num- +ber of competitors. On the one hand, increasing the +number of active participants is likely to increase the +likelihood of the positive dynamics that leaderboards +introduce. Clearly, the mechanisms of competition, +mutual accountability, and changes in perceived ability +are nonexistent if there are no other active users on a +leaderboard. More so, competitive motives may be +stronger on larger leaderboards because ranking highly +on larger leaderboards can be more motivational than +dominating smaller leaderboards. That said, the effect +of increasing leaderboard size is likely more nuanced. +For instance, it is likely that some benefits of additional +leaderboard participants are diminishing at the margin. +Too many participants can make the leaderboard less +effective because participants get lost in the crowd, +weakening the positive impacts of competition or +mutual accountability (Garcia and Tor 2009, Garcia et al. +2013). The diminishing marginal benefit of an additional +leaderboard member implies nonlinearity in the benefit +of more leaderboard members and may even lead to +harmful effects of leaderboards if they become too large. +3.3.2. Prior Activity Levels. Second, we consider the +physical activity level of an individual prior to leader- +board adoption. +3.3.2.1. Competition. If we consider only the role of +competition (vis-`a-vis prior activity levels), the expecta- +tion in the literature that highly active individuals should +benefit disproportionately from leaderboards is most +plausible (Patel et al. 2015, Wu et al. 2015, Shameli et al. +2017). Individuals with high activity levels prior to leader- +board adoption gain high utility from healthful activity +and thus are likely to perform well on leaderboards. This +positive performance on leaderboards can be motiva- +tional for them and encourage increases in future physical +activity. The impact of competitive dynamics on relatively +more sedentary individuals may be more nebulous. On +the one hand, these individuals may benefit most from +extrinsic motivators such as competition and ranking +themselves against others. On the other hand, the value of +leaderboards for such individuals may be limited by their +lower intrinsic aptitude and motivation for physical activ- +ity. This leaves them prone to de-motivational impacts of +lackluster performance on leaderboards. +3.3.2.2. Accountability and Reference Points. If we +also consider theorized mechanisms related to social +influence, the expectation ex ante is more uncertain. +Because individuals on the low end of the physical +activity distribution are more likely to have other lead- +erboard participants who are more active than they are, +there is increased potential for the leaderboard to act as +a tool that keeps them accountable; individuals who are +more active than the focal user may be more credible in +their attempts to hold the focal user accountable. More +so, individuals at the lower end of the physical activity +distribution are more likely to encounter other users +who facilitate upward comparisons and positively +impact their reference point for exercise and their per- +ceived ability to engage in physical activity. In addition, +individuals with low activity levels prior to leaderboard +adoption may benefit most from leaderboards because +they have more room for improvement and a higher +need for external motivation. The dynamics around +social influence are somewhat reversed for those who +are highly active prior to leaderboard adoption. Follow- +ing the same rationale, individuals who are already +highly active may be less likely to join leaderboards +where other users can hold them accountable (i.e., few +others on their leaderboard can match their physical +activity levels). In addition, these individuals are at ele- +vated risk of leaderboards facilitating downward com- +parisons that negatively impact their exercise reference +points. These comparisons can induce sluggishness if +they highlight the focal user’s disproportionate level of +activity compared with others. Finally, highly active +individuals may suffer from ceiling effects, that is, any +extrinsic intervention is not likely to increase their will- +ingness or ability to increase physical activity. +3.3.3. Leaderboard Mechanisms, Prior Activity Levels, +and Leaderboard Size. The theorized effects of prior +activity levels and leaderboard size can also intersect in +ways that have implications for the diverse mechanisms +through which leaderboards can impact behavior. First, +our theorized mechanisms point to highly active individ- +uals being most likely to be harmed by smaller leader- +boards. Garcia et al. (2013) suggest that competitiveness +emerges when there is a potential for comparisons, up +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +6 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +or down, that credibly threaten the individual’s rank. +With smaller leaderboards (e.g., one other individual), +these high achievers are less likely to interact with +another individual who can credibly compete with them +or hold them accountable (thus nullifying two key mech- +anisms for leaderboard value). At the same time, they +are more likely to be presented with a salient individual +who facilitates downward comparisons that negatively +impact their exercise reference point, induce sluggish- +ness, and diminish their physical activity levels. As lead- +erboard size increases, there is increased potential value +for the highly active because the likelihood increases +of at least one individual joining who can provide a +credible threat to their rank, mutual accountability, and +positive impacts on their reference point for exercise. +Furthermore, it is plausible that individuals who are +highly active are buoyed to perform even better when +part of a relatively large leaderboard. This phenomenon +would be akin to the idea in some sports of a “big match +player,” someone who performs above their average on +big occasions and in front of big crowds. +In contrast, our theorized mechanisms have different +implications for leaderboard size when individuals were +sedentary prior to adoption. Unlike highly active individ- +uals, these individuals can still benefit from adopting +small leaderboards because they are still likely to encoun- +ter other users who are either at a comparable or a higher +level of physical activity. Thus, even small leaderboards +may often provide these individuals with an additional +degree of accountability and the potential for positive +impacts on their exercise reference points. Whether these +individuals benefit from competition with small leader- +boards is less certain, as they may still be dominated on +small leaderboards, leading to de-motivational effects of +competition. Increasing the size of the leaderboards for +lower activity users may still provide some of the benefits +described previously but it is likely that these benefits +diminish faster for this group. Unlike for highly active +individuals, benefits of mutual accountability may be +reduced for these individuals as leaderboard size +increases (via the “getting lost in the crowd” phenomenon +described previously). More so, these users, who are at +the lower end of the distribution of physical activity, may +be more likely to get stuck toward the bottom of larger +leaderboards and this may be more salient with more +users participating. Overall, we conjecture that sedentary +individuals can significantly benefit even when leader- +boards are small. However, increases in leaderboard size +may have diminishing marginal benefit for them. +4. Data and Model +4.1. Data +We use a unique panel data set comprised of 516 under- +graduates at a U.S. university from October 2015 to Sep- +tember 2017.11 This data set consists of granular wearable +device data and periodic survey data. With respect to +wearable device data, the students were offered Fitbit +Charge HR devices, which were then used to record +their physical activity. We access three types of Fitbit +data: (i) step count, accessed on a daily basis; (ii) leader- +board data, which captures if a focal student has a lead- +erboard and, if so, the seven-day average step count of +other leaderboard participants for the determination +of participants’ leaderboard rankings; and (iii) minute- +by-minute heart rate data. Students synchronize their +data with the Fitbit platform either through a dongle +and a desktop application or a smartphone application. +We implemented a client application that invoked the +Fitbit application programming interface (API) to down- +load the synchronized student activity data and store +it locally in a secure database. The client application was +a set of scripts that ran automatically every night. All +study participants explicitly authorized our client appli- +cation to allow access to their data via the Fitbit APIs. +Step measurements only occur if students wear their +Fitbit devices regularly. We will use the term compliance +to refer to the regularity with which students wear their +Fitbit device. We calculate compliance from the heart +rate data by assuming that a student is wearing their Fit- +bit during a particular minute of the day if the reported +heart rate is nonzero. Students were paid $20 for main- +taining at least 40% compliance and synchronizing their +data regularly to Fitbit servers. Fitbit Charge HR could +store up to seven days of data locally, so synchronizing +beyond a seven-day interval would result in lost data +and lower compliance. +Fitbit Charge HR’s step measurements, which we use +as the outcome in this study, are fairly accurate. Valida- +tion studies in laboratory and natural settings have +found Fitbit Charge HR’s mean absolute percent error +(MAPE) for step count to be less than 10%, except for +very light activity (Wahl et al. 2017, Bai et al. 2018). Bai +et al. (2018) also found Fitbit Charge HR’s heart rate +measure to have an MAPE of ≈ 10%, although other +studies have found mixed results. Even if the MAPE for +heart rate were higher, our study is not likely to be nega- +tively impacted. We use heart rate only for measuring +compliance such that any nonzero heart rate measure- +ment is construed as the device being used by the partic- +ipant during that minute. +Participants were also asked to complete an inten- +sive survey at the start of study and were further +asked to take shorter surveys in six-month waves to +refresh key measures. These surveys notably provided +data on demographics (gender, religious affiliation, +parent’s income, etc.), psychological attributes using +validated scales (personality, self-regulation), social in- +teraction and ability (trust, anxiety, etc.), technology +use (social media use, mobile app usage, etc.), and +health state (body mass index, satisfaction with health, +etc.). Although most students took the survey, there +was some nonresponse as these surveys were not +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +7 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +mandatory. On average, students completed three waves +of survey data (approximately six months apart). We use +these data in two ways. Primarily, we use relevant survey +data to model the propensity for opting into a leader- +board and, in conjunction with advanced weighting ap- +proaches, construct a weighted sample that achieves +covariate balance between leaderboard adopters and non- +adopters. Secondarily, we use a subset of the survey data +to generate controls that capture time-varying features of +individuals that may relate to both leaderboard adoption +and physical activity, and check the robustness of our +main results.12 Table 1 provides descriptive statistics +about the outcome, treatment, and some demographic +variables, whereas Online Appendix Table A.1 describes +the relevant portions of the survey. +4.2. Model +The goal of our analysis is to estimate the effect of a +user’s leaderboard adoption on their physical activity as +measured by steps walked, using nonexperimental +data.13 Thus, leaderboard adoption is the treatment in +our observational study. In a randomized experiment, +leaderboards could be randomly assigned to study par- +ticipants, which would make the identification of treat- +ment effect straightforward but would make the study +treatment very different from naturally occurring lead- +erboards. In contrast, any Fitbit user in our study can +opt into and construct their leaderboard, resulting in +more natural leaderboards but making it more difficult +to identify the treatment effect. The main empirical con- +cern in identifying this treatment effect is the confounded- +ness of the leaderboard adoption with respect to users’ +physical activity as measured by their daily step count. +We use a DID research design as Fitbit users are +observed over multiple time periods, with roughly half +of the users adopting leaderboards and the other half +remaining untreated. Although a DID design controls +for any time-invariant user characteristics and common +shocks, it requires the identifying assumption that any +uncontrolled time-varying user characteristics exhibit a +common trend across the treated and untreated individ- +uals. Under this identifying assumption, we can esti- +mate the effect of leaderboards on steps walked for the +Fitbit users who have adopted leaderboards. Our model +specification is given here and its explanation follows: +Stepsit � β0 + β1(Leaderboardit) + θi + λt ++ γi × t + φi × t2 + ɛit: +(1) +Although we observe physical activity data on a daily +basis, the leaderboard data are obtained weekly. Hence, +our unit of analysis is student-week. Stepsit is the aver- +age number of steps walked daily by user i in week t, +and Leaderboardit is a binary indicator for whether a user +i adopted a leaderboard in week t. As stated earlier, the +leaderboard adoption is generally “sticky,” as Fitbit +makes it difficult to de-adopt. We include individual +fixed effects (θi) to account for time-invariant differen- +ces between individuals and time-fixed effects (λt) to +account for any common shocks in our data. Together, +these two-way fixed effects would enable the identi- +fication of the treatment effect in the absence of any +differential trends across the treated and untreated indi- +viduals. Admittedly, the common trends assumption is +very strong, but one way to make it more plausible is to +explicitly control for individual-specific linear time +trend (γi × t) and individual-specific quadratic time +trend (φi × t2). We can then estimate our model under +the weaker assumption that the treatment assignment is +ignorable after controlling for the two-way fixed effects +and the additional individual-specific time trends (Xu +2017).14 In Section 5.1, we will further explore the issue +of time trends across Fitbit users with and without lead- +erboards. For inference, all our analyses use cluster- +robust variance-covariance estimators (VCE), clustered +at the student level, which adjust for heteroskedasticity +and serial correlation. +Table 1. Descriptive Statistics +Variable +Description +Mean +Standard +deviation +Minimum +Maximum +Steps +Number of steps walked daily +10,625.18 +4,247.53 +0 +37,835 +Leaderboard +An indicator if an individual has adopted +a leaderboard +0.47 +0.50 +0 +1 +Age +Age at the start of the study +17.94 +0.72 +17 +26 +Body mass index +Body mass index at the start of the study +22.74 +3.02 +16 +38 +Female +An indicator for whether the individual is +a female +0.50 +0.50 +0 +1 +Leaderboard size +The number of users on the leaderboard +(excluding the focal users) +4.78 +4.46 +1 +25 +Leaderboard size +(active) +The number of users on the leaderboard +that have a nonzero step count +(excluding the focal users) +2.32 +2.64 +0 +17 +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +8 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +5. Estimation and Robustness of the +Main Effects of Leaderboards +In our main analysis, we estimate variants of Specifica- +tion (1), which is a DID specification with flexible user- +specific time trends. Table 2, columns 1 and 2, presents +the estimation results for Specification (1). The first +column presents results for a specification that includes +individual-specific linear time trends only, whereas +the second column additionally includes individual- +specific quadratic time trends. In both columns, we find +a significant (p< 0.05) and meaningful leaderboard +effect of 338–370 steps daily. Column 2 is our preferred +model as it includes more flexible time trends. This +model suggest that the students who adopted leader- +boards have a daily increase of 370 steps, equivalent to a +3.5% increase in physical activity on the average daily +step count of 10,268. These initial results suggest some +support for a main effect of leaderboard adoption on +physical activity. In the Online Appendix, we extend this +analysis to add time-varying survey variables as controls +in Specification (1). The estimated effects with these addi- +tional controls have higher magnitudes, which increases +the plausibility of the main results. However, a number +of concerns commonly arise with analyses using observa- +tional data. In the remainder of this section, we discuss +the robustness of our main results. +5.1. Probing the Common Trends Assumption +Identification of the treatment effect with a DID design +crucially depends on the common trends assumption. +As mentioned earlier, one way to weaken this assump- +tion is to control for individual-specific linear and quad- +ratic time trends, which we have incorporated in our +model estimation. In this section, we will further probe +the plausibility of assuming that no unobserved time- +varying covariates may be confounding our analysis +(i.e., the common trends assumption). +5.1.1. Inverse Probability of Treatment Weighting. A +common concern in a DID design is whether the treated +and control subjects are similar in their baseline charac- +teristics such that the treated and control subjects plausi- +bly follow common trends. As mentioned earlier, we +collected a rich set of baseline characteristics of study +users using a survey instrument. Although the initial +covariate balance did not cause excessive concern, we +use the inverse probability of treatment weighting +(IPTW) method to further improve the covariate balance +of our sample. To estimate the propensity for leader- +board adoption, we use the Toolkit for Weighting and +Analysis of Nonequivalent Groups (TWANG), which +implements a generalized boosted regression model +(GBM). The propensity score estimated by TWANG +optimizes covariate balance across leaderboard adopt- +ers and nonadopters. We observe substantive improve- +ment in the postweighting covariate balance such that +the observed absolute standardized mean difference, +SMD ≤ 0:2, is better than the accepted threshold of +0.25.15 Table 2, column 3, presents the main analysis +using the IPTW sample. The sample size is slightly +smaller (cf. column 2) as a few students did not partici- +pate in the initial study survey. Comparing with the +main result (column 2), we find the effect sizes to be +very similar—370 versus 343 steps. This stability of +effect size boosts our confidence in the main results. +5.1.2. Pretreatment Period Placebo Treatments. Given +that we have multiple pretreatment periods for most users +in our sample, we can probe the plausibility of the com- +mon trends assumption by creating placebo treatments in +Table 2. Fitness Activity and Leaderboard Participation +(1) +Steps +b/se +(2) +Steps +b/se +(3) +Steps +b/se +(4) +Steps +b/se +(5) +Steps +b/se +(6) +Steps +b/se +(7) +Steps +b/se +Leaderboard +338.38** +370.46** +343.02** +383.20** +397.75** +(171.40) +(170.66) +(170.98) +(190.75) +(174.05) +Placebo Leaderboard 4-Weeks +�3.62 +(260.57) +Actual Leaderboard 4-Weeks +419.78† +(271.53) +Leaderboard × Inviter +�98.92 +(387.32) +Individual fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Week fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Individual linear trends +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Individual quadratic trends +No +Yes +Yes +Yes +Yes +Yes +Yes +IPTW +No +No +Yes +No +No +No +No +Observations +27,758 +27,758 +27,409 +14,746 +15,742 +27,758 +27,358 +Individuals +516 +516 +501 +516 +516 +516 +503 +Adjusted R2 +0.3 +0.33 +0.34 +0.34 +0.35 +0.33 +0.33 +VCE +Robust +Robust +Robust +Robust +Robust +Robust +Robust +Notes. Column 7 (cf. column 2) excludes users who hide themselves. Please see Section 5.4 for details. +†p < 0.125; *p < 0.10; **p < 0.05. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +9 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +the pretreatment data alone, that is, by dropping the +posttreatment data and using only the pretreatment data +for this analysis. A failure to reject the null effect for the +placebo treatment would provide support for the com- +mon trends assumption.16 In our study, users opt into +the treatment in different periods. Moreover, our pri- +mary concern is the presence of some unobserved time- +varying factor (e.g., spurts in motivation) that affected +the adopters in the periods closely preceding the treat- +ment. Hence, we implemented our placebo treatment in +the preceding month prior to the actual treatment and +estimated the model in Equation (1) on the altered data. +Table 2, column 4, presents the estimated effect of the +placebo treatment. This estimated effect is small in mag- +nitude, opposite in sign, and statistically insignificant. +However, if we include four weeks of actual treatment +period in our sample, the estimated effect is ≈ 420 steps +(p�0.12) as presented in Table 2, column 5. Thus, for +comparable time periods, the placebo effect is null, +whereas the actual treatment effect is comparable to +our estimated main effect. This null effect in the pretreat- +ment period enhances the plausibility of our common +trends assumption. +5.1.2.1. Leads-Lags Model. The placebo treatment +effect can be further broken into weekly placebo effects +in the pretreatment period and the actual effect in the +posttreatment period using the full data set and a leads- +lags specification: +Stepsit � β0 + +X +9 +τ��9 +βτLi(t+τ) + θi + λt + γi × t ++ φi × t2 + ɛit: +(2) +The dummy variables Li(t+τ) denote the time from adop- +tion; for example, Li(t+τ) would be one for individual “i” +in time period “t” if this time period is τ�weeks from +adoption, where τ � �10,:::,9. If we observe more data +for an individual, we collapse it into the extreme peri- +ods.17 We set Li(t�10) as the baseline period and exclude +it from Equation (2) to avoid the “dummy variable +trap.” Figure 2 (top left panel) plots point estimates and +confidence intervals for βτ�coefficients against the time +from adoption. We find a null effect in the pretreatment +period. In contrast, the effect is positive and statistically +significant in the adoption period (i.e., period 0). The +posttreatment coefficients remain positive but decrease +in magnitude and lose significance in the later periods. +We will explain the reason for this decline in Section +6.1.1. A potential issue with this analysis is that the coef- +ficients are trending upward from period �4 to �1. +However, this trend is not a cause of concern for several +reasons: first, the estimates for periods �4 to �1 are +not statistically significant even at the 75% level (the +lowest p value is 0.29). Second, the estimates for βτ�in the +posttreatment period stay positive (and larger than any +preperiod estimate), whereas the estimated βτ�in the +preperiod oscillate around the zero line. In particular, +the change in coefficient estimates from period �9 to �6 +is roughly the same as the change from �4 to �1, with a +sharp drop to a very small negative value in period �5. +Thus, extrapolating this historical pattern beyond �1 +would plausibly suggest a regression back to an almost +zero value as in period �5, but the adoption breaks that +trend such that we see a large significant effect in the +adoption period and beyond. Finally, the other tests such +as the placebo test presented earlier in this section also +argue against the presence of any pretrend in the month +preceding the adoption. These robustness checks argue +against the presence of any other unmeasured changes +that affect leaderboard adopters in the periods closely +preceding leaderboard adoption, thus enhancing the +plausibility of the common trends assumption. +5.2. Robustness Check for Leaderboard Initiation +As an additional robustness check, we also considered +leaderboard initiation as it may be a proxy for con- +founded leaderboard adoption. Specifically, if the focal +user is the primary inviter to the leaderboard, this lead- +erboard may be more likely to be driven by unobserved +motivation changes. For the purpose of this analysis, we +consider focal users to be of the “inviter-type” if they ini- +tiate most, not necessarily all, of the invitations to other +users on their leaderboard. Although we do not have +access to direct measures of who initiated a leaderboard, +we construct a proxy variable that we argue identifies +users who are more likely to be inviter types. We lever- +age two aspects of leaderboard creation to construct this +proxy variable. First, per the discussion in Section 2.1, +Fitbit does not use a leaderboard that is defined cen- +trally as a group of individuals that others can join or +leave. Rather, each leaderboard is owned by the user +and each user pair must agree to share their step infor- +mation for them to be joined on their individual leader- +boards. In addition, Fitbit does not advertise to the +user’s friends that they have joined the platform. +Based on these aspects of Fitbit leaderboards, we desig- +nated InviterLB using two criteria: (i) whether the leader- +board had three or more individuals when it was first +adopted and (ii) whether the leaderboard was such that +the other users (excluding the focal users) had been on +the Fitbit platform for longer than 90 days. The first crite- +rion is useful because the size of the leaderboard at lead- +erboard initiation can be indicative of the likelihood of +initiation by the focal user. If there are two people when +the leaderboard is started, it is unclear who initiated. +However, if three people (or more) are on the leader- +board at initiation, a leaderboard fully initiated by others +would require that two other users actively searched and +invited the focal user in the same week and that the user +accepted both invitations. However, this criterion may +still include mixed leaderboards that were only partially +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +10 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +initiated by the focal users (e.g., another user initiates but +the focal user invites the third person). Thus, we add the +second criterion that the other users on the leaderboards +have been on the platform for more than 90 days. The +rationale behind this criterion is that users on the platform +for longer periods of time are more settled on the platform +and less likely to be actively scouring the platform for +new connections. The 90-day threshold was chosen based +on data suggesting that Fitbit abandonment happens in +the first few months of adoption.18 Among the leader- +board adopters, 15.3% met these criteria. In Table 2, col- +umn 6, we add an interaction term between leaderboard +and an indicator for InviterLB and identify a negative coef- +ficient that is close to zero and insignificant (p� 0.8). This +result suggests that users who are more likely to have ini- +tiated the leaderboard do not see different treatment +effects and is further evidence that time-varying changes +in motivation are unlikely to be confounding our results. +5.3. Fitbit Compliance +Step measurement through Fitbit only occurs if the par- +ticipants wear their devices regularly. We will use the +term compliance to refer to the regularity with which a +participant wears the Fitbit device. In this subsection, we +will probe two compliance-related concerns that may +cast doubt on the earlier analyses if left unaddressed. For- +tunately, our data include compliance data at a very gran- +ular level, which allows us to construct the participant’s +compliance measure, percent compliance, at the daily and +weekly level, and the participant’s mean compliance for +the study duration. We will exploit these data to probe +compliance-related concerns. +5.3.1. Do Leaderboards Increase Compliance? The +first concern is the possibility that rather than increas- +ing steps, leaderboard adoption increases compliance, +which may lead us to observe higher step count purely +because of better measurement. To address this concern, +we estimated main effects models similar to Equation +(1) and the leads-lags models similar to Equation (2) but +with daily percentage compliance as the dependent var- +iable and student-day as the unit of analysis. We esti- +mate this model for a number of samples—the entire +sample and the subsamples at various mean compliance +levels (ranging from 60% to 95%). The main effect model +estimates are statistically insignificant and have small +magnitudes, ranging from –1.97% to 2.01%. In addition, +the leads-lag model’s coefficient plots do not show any +sharp increase at or after adoption (see the Online +Appendix, Section E.1). These results suggest a null +effect of leaderboard adoption on compliance. +5.3.2. Are Leaderboard Effects Discernible at Higher +Compliance Levels? The second issue is that some of +the participants may have lower compliance and the +full sample estimate includes these participants too. +Figure 2. (Color online) Leaderboard Coefficients by Weeks from Leaderboard Adoption (by Prior Activity Levels) +Note. (a) Please see Equation (2) for charts’ specification, (b) 90% confidence intervals, (c) vertical axes use different scales. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +11 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +Regarding this concern, our empirical analysis would +be more convincing if the leaderboard’s effect on partici- +pant activity was clearly discernible for participants +with high compliance levels. Thus, we estimate impact +of leaderboards for participants with high levels of com- +pliance and find these effects to range from 408 to 598 +steps (see the Online Appendix, Table E.6). The persis- +tence of leaderboard effects at higher levels of compli- +ance supports the claims from our main results. +5.4. Fitbit Attrition, Leaderboard De-Adoption, +and Additional Robustness Checks +Related to the challenge of compliance, we also consider +the role of attrition from the sample due to Fitbit aban- +donment, which has generally been noted in the popu- +lar press for health wearables.18 If sample attrition is +related to leaderboard adoption, it may introduce bias +in our analysis. For example, lower performers may +abandon their Fitbit device after joining leaderboards +because it reveals to them that they are less active than +their peers. We examine this concern extensively and +identify no relationship between leaderboard adoption +and sample attrition for lower performers, and no differ- +ences in physical activity and similar leaderboard effects +for those who eventually leave the sample compared +with those who report data throughout (see the Online +Appendix, Section F). We also consider whether indi- +viduals who eventually hide themselves from the lead- +erboard (the main mechanism for de-adoption) impact +our results. As we mentioned previously, this was rare +for leaderboard adopters (approximately 5%), and +excluding these individuals results in consistent esti- +mates of leaderboard effects (see Table 2, column 7). +5.4.1. Outliers and Falsification with Negative Control +Treatments. We also evaluate the potential for a partic- +ular individual (or time period) in the data to be an out- +lier driving our results. Specifically, we systematically +“leave out one” individual (or time period) and re- +estimate our model (see the Online Appendix, Section +G). We find consistent treatment effects of leaderboards +that are always statistically significant, suggesting mini- +mal risk from outliers in the data. Furthermore, we con- +structed a negative control treatment (NCT), as the focal +user’s leaderboard with no other active users. Such lead- +erboards exist because other users may accept a request +to connect but then become inactive on the platform and +thus neither provide competition nor reference points +(see the Online Appendix, Figure D.II and associated +discussion). Thus, the absence of any other active users +of such leaderboards should result in no effect on the +user’s physical activity. Indeed, we find a null effect of +such leaderboards on steps (see the Online Appendix, +Section D). This falsification test with an NCT strength- +ens the plausibility of the common trends assumption. +6. Heterogeneous Effect of Leaderboards +In this section, we evaluate the potential for heterogene- +ous effects of leaderboards on physical activity focusing +on leaderboard rank, leaderboard size, and prior activ- +ity level. The evaluation of heterogeneous effects of +leaderboards is useful because it can offer additional +insights into the role of competition and social influence +in generating leaderboard value. +We start by evaluating the impact of ranking first on +the leaderboard in the prior period on the activity levels +of individuals in the subsequent period (FirstonLB).19 +Next, we evaluate whether the number of active partici- +pants on a leaderboard (excluding the focal user) modi- +fies the benefit to individuals who adopt leaderboards +(LBActiveUsers) and whether this impact is nonlinear, +by including the square of (LBActiveUsersit).20 Finally, +we consider the interaction of rank and leaderboard +size. Equation (3) provides the specification for this +model. To evaluate heterogeneous impacts by prior +activity levels, we also estimate this specification strati- +fied by preleaderboard activity levels. +Stepsit � β0 + β1(FirstOnLBit�1) + β2(LeaderBoardit) ++ β3(LBActiveUsersit) + β4(LBActiveUsersit)2 ++ β5(FirstOnLBit�1 ∗ LBActiveUsersit) ++ β6(FirstOnLBit�1 ∗ (LBActiveUsersit)2) ++ θi + λt + γi × t + φi × t2 + ɛit +(3) +6.0.1. Impact of Prior Week’s Rank and Leaderboard +Size. We find a substantive impact of prior week’s rank +on the impact of leaderboards in subsequent periods. +Those who were in first place on their leaderboard in +one week, walked 578 steps more a day the following +week (Table 3, column 1). We also find a positive and +significant coefficient on LBActiveUsers (Table 3, column +2), suggesting that an additional person on a leader- +board increases the effect of that leaderboard by 165 +steps (p < 0.05). Moreover, we evaluate whether there +are diminishing benefits from additional active users on +a leaderboard by adding a quadratic term to our estima- +tion (Table 3, column 3). A negative coefficient on the +quadratic term (≈ �21) suggests that the effect peaks at +eight active members, after which the marginal benefit +of an additional member is diminishing. +In Table 3, columns 4 and 5, we also explore the inter- +sectional impact of leaderboard size and prior per- +formance by including FirstOnLB and FirstOnLB × +(LBActiveUsers)k, k � 1,2. We find that the motivational +effects of succeeding in leaderboard competition increase +with the size of the leaderboard. More so, we continue to +identify a positive impact of increased leaderboard size, +suggesting positive impacts of larger leaderboards +when an individual is not first. Finally, we consider +whether the nonlinear impacts of leaderboard size +extend to the interaction with prior week performance +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +12 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +and find no evidence of diminishing impacts (coefficient +for FirstonLB × (LBActiveUsers)2 is near 0 and insignifi- +cant). Overall, we find that leaderboard benefit increases +with leaderboard size (although this benefit is diminish- +ing at the margin), and that ranking highly on a leader- +board is more motivational on larger leaderboards. +6.0.2. Implications of Findings. The positive impact of +prior rank and the increased impact of ranking first on +larger leaderboards point to an important role of com- +petitive dynamics in generating leaderboard value. In +addition, the impact of leaderboard size above and +beyond the impact of rank suggests that social influence +mechanisms also play an important role in observed +leaderboard benefits. However, an insignificant effect of +small leaderboard when the individual is not ranked +first and diminishing marginal benefit of increasing +leaderboard size suggest some nuance around how +social influence mechanisms drive leaderboard benefits. +We explore this nuance further by evaluating how prior +activity levels (which have implications for how other +users on leaderboards exert social influence) moderate +leaderboard impact. +6.1. Heterogeneity by Prior Activity Levels +To evaluate potential heterogeneity in leaderboard +benefit by prior activity, we estimate Specification (1) on +samples stratified by preleaderboard activity levels. +Specifically, we stratify our sample into two groups +based on their preleaderboard activity levels: the top +quartile by daily step count comprise the highly active +group whereas the bottom quartile by daily step count +comprise the sedentary group.21 The differences in step +count between these two groups were meaningful in +terms of magnitude and were statistically significant +(13,000 versus 8,000, p < 0.01)—see the Online Appen- +dix, Table H.9, for summary statistics on these groups. +Before examining the impact of leaderboards on +physical activity levels, we evaluated the correlation +between activity levels and relevant preadoption survey +measures (see the Online Appendix, Table H.10). We +found correlations consistent with our expectations of +the key differences between highly active and sedentary +individuals. Sedentary individuals reported lower lev- +els of self-efficacy and self-regulation for exercise and +reported being more likely to exercise alone. In addition, +sedentary individuals reported higher levels of anxiety +and depression and lower levels of trust. These correla- +tions suggest that sedentary individuals may need these +interventions more than highly active individuals but +that they could also be prone to de-motivational impacts +if these leaderboards exacerbate mental health barriers +to improving health (e.g., increase their anxiety). +Turning to the examination of the effect on steps, we +find stark differences in the effect of leaderboards on the +highly active group vs. the sedentary group. For seden- +tary users, the adoption of leaderboards has large and +significant impacts on their daily step counts (1,365, p < +0.05; see Table 4, column 1). In contrast, we find that the +highly active group, instead of benefiting from leader- +boards, experienced a significant decrease in their daily +physical activity after leaderboard adoption (–631, p < +0.05; see Table 4, column 2). For the sake of completion, +we also estimate the effect for the middle group, i.e., +Table 3. Heterogeneous Effect by Leaderboard Size and User Rank +(1) +Steps +b/se +(2) +Steps +b/se +(3) +Steps +b/se +(4) +Steps +b/se +(5) +Steps +b/se +Leaderboard +219.28 +108.99 +�61.66 +�29.60 +�194.81 +(170.84) +(183.39) +(187.25) +(184.02) +(188.89) +FirstOnLB +577.98** +371.70** +377.58** +(94.80) +(111.31) +(130.19) +LBActiveUsers +164.46** +334.31** +155.28** +315.90** +(31.50) +(57.55) +(32.22) +(56.89) +(LBActiveUsers)2 +�21.36** +�20.00** +(6.72) +(6.47) +FirstOnLB × LBActiveUsers +146.23** +168.20* +(43.49) +(99.07) +FirstOnLB × (LBActiveUsers)2 +�5.91 +(10.96) +Individual fixed effects +Yes +Yes +Yes +Yes +Yes +Week fixed effects +Yes +Yes +Yes +Yes +Yes +Individual linear trends +Yes +Yes +Yes +Yes +Yes +Individual quadratic trends +Yes +Yes +Yes +Yes +Yes +Observations +27,758 +27,758 +27,758 +27,758 +27,758 +Individuals +516 +516 +516 +516 +516 +Adjusted R2 +0.33 +0.33 +0.33 +0.33 +0.34 +VCE +Robust +Robust +Robust +Robust +Robust +*p < 0.10; **p < 0.05. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +13 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +individuals whose activity levels were between the 25th +and 75th percentile. We find that the middle group bene- +fited from leaderboards, but the benefit was less than for +those who were in the bottom quartile (859, p < :05; see +Table 4, column 3). Overall, these results suggest signifi- +cant heterogeneity in the effect of leaderboard based on +prior activity levels, with particularly stark, negative +effects for those who were previously highly active.22 +6.1.1. Leads-Lags Model by Prior Activity. To examine +the presence of any trends before leaderboard adoption, +we plotted the coefficients from a leads-lags model for +the full sample and subsamples by prior activity levels +in Figure 2 (see Section 5.1.2 for details about the specifi- +cation). The first point to note is that the lead coefficients +are relatively small in magnitude and statistically insig- +nificant, which increases the plausibility of the parallel +trends assumption. Second, the lag coefficients shift to +larger magnitudes with the effect persisting for more +than two months after adoption. Third, there is some +attenuation in the effect for the full sample in later time +periods, which can be plausibly explained by the oppo- +site direction of effect within the sedentary and middle +groups versus the highly active subsample. +6.1.2. Implication of Findings. The divergence of lead- +erboard effects for sedentary versus highly active indi- +viduals substantiates our conjecture that leaderboards +can introduce both motivational and de-motivational +dynamics with respect to physical activity. Specifically, +these results are suggestive evidence of different im- +pacts on reference points for sedentary versus highly +active users and differences in the potential of leader- +boards to provide accountability for lapses in physical +activity. We explore this difference and its implications +for mechanisms underlying leaderboard value further +by evaluating the impact of rank and leaderboard size +on the physical activity of sedentary versus highly +active users in the next section. +6.2. Interaction of Leaderboard Size, Rank, and +Prior Activity Levels +Last, we consider the intersection of all the prior factors +in an effort to understand some of the heterogeneity in +leaderboard benefit for sedentary versus active users. +6.2.1. Leaderboard Size and Rank. We start by evaluat- +ing the motivational impact of leaderboard rank and size +separately by prior activity levels. We find that both pre- +viously sedentary and highly active individuals see sub- +stantial increases in physical activity during the week +after they ranked first on their leaderboard (Table 4, col- +umns 4 and 5). For sedentary individuals, being first in +the prior week unlocks even more value for them and +increases the treatment effect for leaderboards by 746 +steps to nearly 2,000 steps. Highly active individuals see +slightly less value (522 steps versus 746 steps) from being +first in the prior week but this benefit counteracts part of +the negative main effect they observe. Extending this +analysis to also include leaderboard size and the interac- +tion of leaderboard size and prior week’s performance +reveals further richness in these results. In column 6, we +observe that previously sedentary individuals still see +substantial benefit when leaderboards are small and +when they are not first (1,016 steps, p < 0.1) and that this +benefit increases further when they rank first and leader- +boards are larger. In column 7, we observe that previously +active individuals are harmed when they are on small +Table 4. Heterogeneous Effect by Leaderboard Size, Rank, and Prior Activity +(1) +Steps +b/se +(2) +Steps +b/se +(3) +Steps +b/se +(4) +Steps +b/se +(5) +Steps +b/se +(6) +Steps +b/se +(7) +Steps +b/se +Leaderboard +1,365.44** +�630.63** +858.60** +1,262.11** +�817.47** +1,016.43** +�1,215.74** +(496.72) +(292.10) +(213.46) +(484.75) +(304.86) +(486.63) +(331.38) +FirstOnLB +745.74* +522.47** +�49.60 +257.18 +(422.44) +(157.44) +(343.07) +(198.09) +LBActiveUsers +214.90** +208.89** +(46.52) +(61.16) +LBActiveUsers × FirstOnLB +462.99** +149.33** +(87.96) +(55.44) +Sample +S +HA +Mid +S +HA +S +HA +Individual fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Week fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Individual linear trends +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Individual quadratic trends +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Observations +5,629 +7,836 +14,293 +5,629 +7,836 +5,629 +7,836 +Individuals +129 +129 +258 +129 +129 +129 +129 +Adjusted R2 +0.38 +0.34 +0.32 +0.38 +0.34 +0.38 +0.34 +VCE +Robust +Robust +Robust +Robust +Robust +Robust +Robust +Note. S, sedentary; Mid, mid 50 percentile; HA, highly active. +*p < 0.10; **p < 0.05. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +14 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +leaderboards; these users only start to see positive impacts +from leaderboards if they rank first on leaderboards with +more than four individuals or if they are on relatively +large leaderboards (more than six active users). +The finding that sedentary individuals observe lead- +erboard benefit despite unsuccessfully competing on +small leaderboards suggests that these individuals ac- +crue significant benefit from noncompetition mecha- +nisms (i.e., social influence). In contrast, the negative +impact of the same type of leaderboard for highly active +individuals suggests that social influence is harming +these individuals, or that its benefit is not sufficient to +overcome any de-motivational effects of not ranking +first. We probe this conclusion further using two addi- +tional analyses that leverage certain leaderboard instan- +ces which could be telling of the impact of these +noncompetition mechanisms on physical activity. The +first analysis seeks to identify instances of leaderboards +where competitive dynamics are arguably weakened +but the potential for mutual accountability and changes +in reference points is still present. Specifically, we create +NoCompetitionLB, which is an indicator of leaderboard +instances where the focal user is sandwiched between +two other users such that there is not a credible threat to +their rank from either user (see the Online Appendix, +Section H.3 for details). Table 5, columns 1 and 2, shows +a positive and significant impact (534 steps, p < 0.05) of +these types of leaderboards for previously sedentary +users but a small and insignificant impact of these same +types of leaderboards for highly active users (�39 steps, +p � 0.9). We examine this conjecture further by identify- +ing instances of leaderboards at the other end of the +competition spectrum. Specifically, we create HighCom- +petitionLB as an indicator for leaderboard instances +where the focal user is regularly being displaced and +then reclaiming the top spot on the leaderboard (see the +Online Appendix, Section H.3 for details). Table 5, col- +umn 3, shows that sedentary users continue to perform +well on leaderboards without high competition intensity, +with no statistically significant benefit to them of being +on a highly competitive leaderboard. In contrast, highly +active users have large negative effects when they are on +leaderboards without intense competition, but there is a +statistically significant offsetting effect of being on leader- +boards with high competition (Table 5, column 4). +6.2.2. Implication of Findings. Although only sugges- +tive evidence, these results lend some credence to the +notion that previously sedentary individuals are bene- +fiting from mutual accountability and positive impacts +on their exercise reference point and do not need com- +petition to benefit. In contrast, highly active individuals +seemed to be harmed by the noncompetition mecha- +nisms, but this harm can be offset if leaderboards are +sufficiently competitive. +6.2.3. Nonlinear Impacts of Leaderboard Size. Finally, +we consider whether nonlinear impacts of leaderboard +size are similar based on prior activity level. In Table 5, +columns 5 and 6, we find that, although both groups +have diminishing returns from additional users, the +negative coefficient on the quadratic term (ActiveLB2) +for the sedentary group is thrice that for the highly +active group. These results suggest that for sedentary +Table 5. Further Heterogeneous Effect Analysis +(1) +Steps +b/se +(2) +Steps +b/se +(3) +Steps +b/se +(4) +Steps +b/se +(5) +Steps +b/se +(6) +Steps +b/se +NoCompetitionLB +533.74** +�38.98 +(261.125) +(283.539) +Leaderboard (LB) +1,283.58** +�1,007.23** +767.49 +�1,268.45** +(560.852) +(343.563) +(516.114) +(316.771) +LB × High Competition +660.56 +1,021.99* +(1,234.148) +(611.764) +LBActiveUsers +610.05** +400.18** +(233.570) +(89.988) +(LBActiveUsers)2 +�57.74* +�19.86** +(31.635) +(7.935) +Sample +S +HA +S +HA +S +HA +Individual fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Week fixed effects +Yes +Yes +Yes +Yes +Yes +Yes +Individual linear trends +Yes +Yes +Yes +Yes +Yes +Yes +Individual quadratic trends +Yes +Yes +Yes +Yes +Yes +Yes +Observations +5,500 +7,707 +5,629 +7,836 +5,629 +7,836 +Individuals +124 +129 +129 +129 +129 +129 +Adjusted R2 +0.38 +0.33 +0.38 +0.34 +0.38 +0.34 +VCE +Robust +Robust +Robust +Robust +Robust +Robust +Note. S, sedentary; HA, highly active. +*p < 0.10; **p < 0.05. +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +15 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +users, the benefits of an additional active leaderboard +member diminish much faster (after 5 users) than they +do for highly active users (after 11 users; Figure 3). +6.2.4. Implications of Findings. The smaller optimal +size for sedentary individuals suggest that they accrue +benefits (e.g., positive impacts on their exercise refer- +ence points) even when leaderboards are small. How- +ever, benefits diminish as leaderboards become larger, +and leaderboards can even be harmful if they become +excessively large (leaderboard sizes that become harm- +ful to sedentary users were uncommon in our data). +In contrast, highly active individuals seem to be de- +motivated by leaderboards with too few individuals +and only start to benefit on larger, more competitive +leaderboards. These dynamics for highly active individ- +uals reinforce the notion that these users require large +leaderboards to derive benefit. +6.3. Summary of Findings from Heterogeneous +Effect Analysis +Ex ante, we theorized that competition and social influ- +ence are key mechanisms underlying leaderboard effects +but that these mechanisms may introduce both motiva- +tional and de-motivational effects of leaderboards. The +heterogeneous effects we identify in this section point to +the importance of these mechanisms as well as the poten- +tial for nuance in their effects. First, we find robust posi- +tive impacts of ranking first on a leaderboard, suggesting +that successfully competing on leaderboards improves +motivation for most users. Interestingly, and contrary to +our theoretical conjecture, the benefits of competition +hold even for sedentary users: they accrue positive effects +from ranking first and are not harmed from leaderboards +when they do not rank first. +We attribute the robust benefit of leaderboards for +sedentary users to the positive impacts of leaderboards +on their exercise reference points and the likelihood of +being held accountable by other users (i.e., social influ- +ence). However, our results also suggest that social +influence enabled by leaderboards can have negative +impacts on motivation for some users (e.g., negative +impacts on exercise reference points for highly active +users). Interestingly, these harms for highly active indi- +viduals are attenuated when leaderboards are highly +competitive. The nuanced impact of social influence is +further demonstrated by the impacts of leaderboard +size on physical activity. While larger leaderboards gen- +erally increase physical activity, sedentary users see +diminishing value from larger leaderboards. This result +is in line with our conjecture that social influence effects +may diminish for sedentary users if they get “lost in the +crowd” of larger leaderboards. In contrast, highly active +individuals thrive in large leaderboards, substantiating +our conjecture that highly active individuals become +more likely to show positive impacts of competition and +social influence as leaderboard size increases. +7. Conclusions and Discussion +The rapid and increasingly broad adoption of health +wearables coupled with the gamification services built +on top of them provides a potentially powerful vehicle +for improving health behaviors at scale. Our results lend +credence to this potential value, particularly when con- +sidered over time. In our data, the average user partici- +pated in a leaderboard for 237 days (conditional on +participating in a leaderboard). Thus, sedentary users +who participated in a leaderboard for at least 237 days +took more than 300,000 additional steps (with a conserva- +tive estimate of 1,300 additional daily steps). To put this +in context, the aggregated benefit of leaderboards (for +these participants) amounts to 150 miles of distance (at +2,000 steps a mile). Importantly, the benefit is not homo- +geneous and there is a decrease in daily steps of nearly +equal measure for those who, prior to adoption, were +highly active. However, this health harm to the highly +active subgroup may not be symmetrical to the gain for +those who were previously sedentary, as this group +remains very active in absolute terms. We also identify +additional heterogeneity in benefit based on the number +of other active participants and leaderboard rank. +This research has some important limitations. First, +we use secondary data in which individuals organically +choose to adopt leaderboards, rather than being ran- +domly assigned to the treatment. Although we put in +significant effort to address potential confounding fac- +tors for our analysis, only a large-scale randomized con- +trol trial can provide a theoretical guarantee that the +treatment is unconfounded. Additionally, we were +missing variables in our data set which can be particu- +larly informative of leaderboard mechanisms and the +heterogeneous value they generate. Specifically, we do +Figure 3. (Color online) Heterogeneous Effect by Active +Users and Prior Activity +-9,000 +-6,000 +-3,000 +0 +3,000 +Effect on Step Counts +0 +5 +10 +15 +20 +Number of Active Users +Sedentary +Highly-Active +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +16 +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +not have deterministic measures of whether the focal +user initiated the leaderboard or whether it was initiated +by another user. A more comprehensive measure of +leaderboard initiation could have provided important +insights into how different types of users are initiating +leaderboards, whether initiating a leaderboard impacts +competitive dynamics, and whether individuals are +selecting into leaderboards of value to them or if others +are prompting them to do so. We hope to address these +questions in subsequent data collection efforts. We also +do not observe granular data on Fitbit app and device +usage. However, this lack of usage data could attenuate +our results (i.e., lack of use could drive our treatment +effects closer to zero) as we are currently assuming all +adopters to be using leaderboards in all periods after +adoption. This makes our results more conservative. +Another potential limitation is that the student popu- +lation in the sample may not be representative of the +general population (e.g., they may be healthier, more +physically active, or have more free time). Although +broad generalization of results to the average popula- +tion may be somewhat uncertain, it is useful to note +that the largest leaderboard benefits accrue to the least +active participants in our sample. These individuals +are more comparable to the average population. More +so, a younger population may be more amenable to +gamification approaches and be more likely to be moti- +vated by such interventions. Finally, we observe in our +data the impact on physical activity and do not observe +other downstream health outcomes (e.g., weight loss). +However, given the documented relationship between +physical exercise and other health outcomes and the +magnitude of our effects, it is likely that individuals +are, on average, healthier after leaderboard adoption. +These limitations notwithstanding, our results have sig- +nificant implications for research and practice. Mitchell +et al. (2013) argue for the potential of wearable technolo- +gies and mobile health more generally to be “leveraged to +more promptly assess and reward behaviors on a popula- +tion scale, further reducing the need for prohibitively +costly incentives” (p. 666); however, there has been lim- +ited research validating this conjecture. We help substan- +tiate this notion by demonstrating the significant potential +for gamification-based interventions to meaningfully im- +pact physical activity levels. Moreover, we find that the +effects of leaderboards persist over time. +Our results also have implications for the general liter- +ature on motivating changes in health behavior. Specifi- +cally, gamification interventions coupled with health +wearables provide notable advantages over other ap- +proaches studied in the literature. First, and unlike most +other behavioral interventions, the widespread adoption +of health wearables allows for much larger scale inter- +ventions. Second, because they are based in digital plat- +forms, the design of these interventions can be tailored +at the individual or group level to maximize benefit to +the population. This customization is particularly valua- +ble given the heterogeneity in benefit between subpopu- +lations in our study and recognition by scholars of the +limits of “one size fits all” approaches typically taken by +most of the extant literature (Rogers et al. 2014, Rogers +and Feller 2016). +Our research also highlights potential areas for future +research. The variation in leaderboard effects points to +the complex interplay between competition and social +influence that underlies leaderboard effects and im- +pacts motivation. Given that gamification approaches +are highly varied, even for the same class of interven- +tions (leaderboards can vary in terms of who can partici- +pate, who on the leaderboard is made salient to users, +etc.), more work needs to be done to rigorously evaluate +the potential benefits of these varied gamification app- +roaches, the key mechanisms which drive their effects, +and how they may have differential impacts across +health contexts and individuals. Finally, there is a need +to explore whether digital gamification interventions +can be used in conjunction with classic interventions +(decision aids, economic incentives, wellness programs, +etc.) to unlock further health benefits. +Our results also have important implications for firms +and policy makers. From the perspective of firms that sell +health wearables and their related platforms, a key ele- +ment of their value proposition is that their products pos- +itively impact health and well-being. Our results help +substantiate the notion that physical activity can be posi- +tively impacted for large swaths of their adopters but also +that some of their adopters (perhaps their earliest and +most enthusiastic ones) can be harmed by some of the +interventions they offer. Our results also suggests that +these interventions need to be designed with careful con- +sideration for how mechanisms intended to be motiva- +tional may not be so for all users. More so, this highlights +the value of customizing gamification interventions to +individuals and the value of nudging individuals toward +a set of features that are likely to benefit them most. Thus, +these firms may need to better incorporate insights from +behavioral research in the design of gamification inter- +ventions available alongside health wearables. +Furthermore, policy makers, employers, and insurers +are experimenting with health wearables and gamifica- +tion as they have significant incentives to encourage +more active lifestyles, which may improve health, im- +prove worker performance, and lower healthcare costs. +Scholars similarly suggest that gamification coupled +with health wearables can be a cost-effective way to +encourage increases in physical activity and to do so at +scale (Mitchell et al. 2013). However, a salient concern +with these approaches is that less healthy individuals, +who are often of primary interest to policy makers and +employers, can be demotivated when competing with +their more active counterparts. Our results highlight that +Hydari, Adjerid, and Striegel: Gamification and Healthful Activity +Management Science, Articles in Advance, pp. 1–19, © 2022 The Author(s) +17 +Downloaded from informs.org by [104.244.78.139] on 06 January 2023, at 06:00 . For personal use only, all rights reserved. + +this concern may be unfounded or at least less salient, as +sedentary individuals may benefit substantially from +these approaches. Although the harm to highly active +individuals is not ideal, some of these harms can be alle- +viated by tailoring leaderboards for these groups, and in +net, these approaches are still likely to be valuable to +employers and policy makers. A lingering challenge +with reaping this value may be encouraging less healthy +individuals to take up these interventions (e.g., join lead- +erboards), and employers and policy makers may need +to invest in incentives to increase uptake for this subset +of individuals to maximize potential value. +Endnotes +1 The World Health Organization (WHO) Constitution defines +health as “a state of complete physical, mental and social well-being +and not merely the absence of disease or infirmity.” +2 See www.who.int/news-room/fact-sheets/detail/physical-activity. +3 See www.cdc.gov/chronicdisease/resources/publications/factsh +eets/physical-activity.htm. +4 See www.lexico.com/en/definition/activity_tracker. +5 See www.wired.com/story/science-says-fitness-trackers-dont-work- +wear-one-anyway and www.fastcompany.com/3031324/why-your- +company-should-think-twice-about-gamification. +6 Descriptive results substantiate this conjecture and suggest that +highly active individuals are much more likely to dominate smaller +leaderboards compared with larger ones, that is, the smaller leader- +boards do not seem to challenge these highly active individuals as +they easily dominate them even with their reduced activity levels. +7 See www.merriam-webster.com/dictionary/leaderboard. +8 Although we chose a particular form factor and a particular ven- +dor, which is arguably the market leader at the time of the study, +the similarity of leaderboards across platforms means our results +may be relevant to other platforms as well. +9 Fitbit previously offered 30-day fixed-time leaderboards that, to +our knowledge, have been discontinued. +10 Mechanisms of individual accountability and changes to exercise +reference points are distinct from competition mechanisms. For +instance, I may have a friend or family member on my leaderboard +who is not credibly competing with me (e.g., because their perform- +ance exceeds my own by a huge margin) but this individual can still +reach out and hold me accountable for my exercise goals or impact +my perception of what is achievable for me. +11 Approximately 600 students were recruited but we are left with +516 students after excluding early dropouts and always adopters of +leaderboards. +12 Due to the coarseness of the survey data relative to the Fitbit data +and some nonresponse in the sample, we use these controls for +robustness checks. +13 The adoption of Fitbit is not a central concern in our study as +every user in our sample is a Fitbit user. +14 Another method to increase the plausibility of the common trends +assumptions is to use propensity score to achieve covariate balance +across the treated and untreated individuals. We perform this propen- +sity score–based adjustment as a robustness check in a later section. +15 In Online Appendix C, we discuss GBM, IPTW, and IPTW use with +DID. Furthermore, Table C.3 shows pre- and post-IPTW covariate bal- +ance, and Table C.4 and Figure C.1 explain the influence of covariates +on leaderboard adoption. +16 Please see Abadie and Cattaneo (2018) for a recent discussion on +such examinations of the common trend assumption and Abadie +and Dermisi (2008) for an example in a two-period setting. +17 Therefore, Li(t+0),:::, Li(t+9) denote the weekly breakdown of the +actual adoption, whereas Li(t�9),:::, Li(t�1) denote the placebo treat- +ments in the weeks prior to actual adoption. Furthermore, Li(t+9) +denotes period 9 and beyond, whereas Li(t�10) denotes period �10 +and before. +18 Please see apnews.com/article/2700956044de4517a471a47c3243078b. +19 Because the average number of participants on leaderboards was +relatively small (2.3), a binary indicator for being first was sufficient +to capture the impact of rank. 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For personal use only, all rights reserved. + diff --git a/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/load_file.txt b/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25aa5f75020b5bb40b7bde30efe55b4052950646 --- /dev/null +++ b/h9E0T4oBgHgl3EQf6wJO/content/tmp_files/load_file.txt @@ -0,0 +1,1321 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf,len=1320 +page_content='Health Wearables, Gamification, and Healthful Activity Muhammad Zia Hydari,a,* Idris Adjerid,b,* Aaron D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Striegelc aKatz Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania 15260;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' bPamplin College of Business, Blacksburg, Virginia 24061;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' cDepartment of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana 46556 Corresponding author Contact: hydari@alum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='edu, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/0000-0003-4522-326X (MZH);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' iadjerid@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='edu, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/0000-0002-2786-1244 (IA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' striegel@nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='edu, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/0000-0002-3157-2859 (ADS) Received: June 19, 2019 Revised: December 8, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' October 7, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' January 30, 2022 Accepted: February 8, 2022 Published Online in Articles in Advance: December 19, 2022 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1287/mnsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4581 Copyright: © 2022 The Author(s) Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Health wearables in combination with gamification enable interventions that have the potential to increase physical activity—a key determinant of health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, the extant literature does not provide conclusive evidence on the benefits of gamification, and there are persistent concerns that competition-based gamification approaches will only ben- efit those who are highly active at the expense of those who are sedentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We investigate the effect of Fitbit leaderboards on the number of steps taken by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Using a unique data set of Fitbit wearable users, some of whom participate in a leaderboard, we find that leaderboards lead to a 370 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='5%) step increase in the users’ daily physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' How- ever, we find that the benefits of leaderboards are highly heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Surprisingly, we find that those who were highly active prior to adoption are hurt by leaderboards and walk 630 fewer steps daily after adoption (a 5% relative decrease).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, those who were sedentary prior to adoption benefited substantially from leaderboards and walked an addi- tional 1,300 steps daily after adoption (a 15% relative increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find that these effects emerge because sedentary individuals benefit even when leaderboards are small and when they do not rank first on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, highly active individuals are harmed by smaller leaderboards and only see benefit when they rank highly on large leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We posit that this unexpected divergence in effects could be due to the underappreciated potential of noncompetition dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', changes in expectations for exercise) to benefit sedentary users, but harm more active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' History: Accepted by Chris Forman, information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Open Access Statement: This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' You are free to copy, distribute, transmit and adapt this work, but you must attribute this work as “Management Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Copyright © 2022 The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1287/mnsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 4581, used under a Creative Commons Attribution License: https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/licenses/ by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='0/.” Funding: This work was supported by the National Institutes of Health [Grant 5R01HL117757].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Supplemental Material: The data and online appendix are available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1287/mnsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Keywords: health wearables • gamification • fitness • physical activity • health • health technology 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Introduction The evidence on the health1 benefits of physical activity is irrefutable (Warburton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Yet, a significant portion of the world population is not sufficiently ac- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2 This lack of physical activity contributes signifi- cantly to chronic disease and to most of the leading causes of death in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3 Prior research sug- gests that behavioral barriers are one of the most impor- tant contributing factors to this trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2013) suggest that “for many adults, the ‘costs’ of exer- cise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', time, uncomfortable feelings) loom so large that they never start” and that the lack of physical activ- ity is “a problem of both initiation and maintenance” (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 658).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Recognizing that changing health behaviors is often challenging and new strategies are needed, research situated mostly in the health and economics literature has evaluated a plethora of economic and non-economic interventions for overcoming motiva- tional barriers to increasing physical activity (Charness and Gneezy 2009, Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The conclusion from this literature is that, although many interventions can drive short-run gains in physical activity, these ben- efits are fleeting and motivating meaningful, and sus- tained increases in physical activity is elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, many of these interventions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', daily payments) are difficult to implement on a population scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' One contemporary phenomenon with the potential to address persistent limitations of prior approaches and unlock new interventions that can improve the motiva- tion of individuals to exercise is the rapid consumer adoption of health wearables (Swan 2013, Lupton 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A health wearable, sometimes referred to as an activity tracker, is “a wearable device or a computer application that records a person’s daily physical activity, together 1 MANAGEMENT SCIENCE Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19 ISSN 0025-1909 (print), ISSN 1526-5501 (online) https://pubsonline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org/journal/mnsc Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' with other data relating to their fitness or health, such as the number of calories burned, heart rate, etc.”4 Despite the rapid adoption of health wearables and their poten- tial for motivating individuals to engage in healthy activities, scholars suggest it is unlikely that the meas- urement capabilities that the health wearables provide would significantly impact health on their own (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Sullivan and Lachman 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Rather, they suggest that for health wearables to impact health behav- ior, the information they collect “must be presented back to the user in a manner that can be understood, that motivates action, and that sustains that motivation toward improved health” (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 460).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Particularly promising in this regard is combining granular physical activity data from health wearables with gamification approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Gamification is defined as the “use of game design elements in nongame con- texts” (Deterding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Some examples of game design elements are badges, rules-based competition, leaderboards, points, ranking, reputation, rewards, teams, and time pressure (Deterding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Cou- pling gamification with health wearables has the poten- tial to improve motivation by converting the usually mundane action of physical activity into the more enjoy- able activity of collecting rewards or competing with other individuals (Hamari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2014a, Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, gamification approaches can provide immediate positive reinforcement that helps individuals get over the initial hurdles of engaging in exercise and could also help them sustain higher levels of activity in the longer term (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013, Shameli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, the broad adoption of health wearables un- locks more robust gamification interventions by provid- ing an objective and common source of measurement and a form factor that enables real-time feedback while engaging in physical activity (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although coupling gamification with health wearables has the potential to generate sustained increases in physical activity, the evidence on the benefits of gamification is mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hamari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2014b) reviewed 24 empirical gami- fication studies, primarily within education contexts, but reported mixed effects on outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, these stud- ies used interviews or surveys to measure outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hamari and Koivisto (2013), the only empirical study in a health context in the aforementioned literature review, used surveys to measure outcomes and these outcomes were not health related (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', continued use intention for the gamifica- tion service and the intention to recommend service to others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2016) conducted a systematic litera- ture review on the impact of gamification on health and well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Of the 19 empirical papers they reviewed, 59% reported positive results and 41% reported mixed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Both Hamari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2014b) and Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2016) also noted that the quality of evidence was moderate to low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The significant potential benefits of coupling gamifi- cation with health wearables and the narrow focus and lack of evidentiary quality of prior works motivate this research study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We evaluate the benefits of leaderboards that allow users to view the performance of others who also agree to share their activity levels and, in most cases, to compete with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We focus on the potential benefits of leaderboards because they are one of the most common gamification features available with modern health wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This increases the policy and practical relevance of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Another reason we focus on leaderboards is that they exemplify the theoretical ten- sions surrounding gamification interventions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' scholars suggest that gamification features, and leaderboards in particular, are likely to have heterogeneous effects on individuals (Deci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1981, Santhanam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2016, Sulli- van and Lachman 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, a central concern with competition-based gamification interventions like leaderboards is that they will lead to motivational bene- fits only for those who are already highly active (and need the increased motivation the least), whereas actually harming the least physically active in the population (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Shameli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='5 With these dynamics in mind, our first objective is to evaluate the average impact on physical activity of leaderboard adoption by individuals wearing health wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our second research objective is to evaluate the potential for leaderboard effect heterogeneity by (i) the activity level of the focal user prior to adoption, (ii) the number of active participants on the leaderboard, and (iii) the rank of the focal user on the leaderboard in the prior period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We engage in an intensive data collection effort to estimate the average benefit of leaderboards and the heterogeneity in these benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For approximately 500 individuals observed over a two-year time period, we capture leaderboard adoption data and granular meas- ures of physical activity continuously captured by Fitbit Charge HR health wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For those individuals with leaderboards, the data set also includes activity data and rank of all participants in the leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We supplement these data from health wearables with peri- odic surveys (every six months on average) capturing a rich array of individual characteristics (psychological attributes, frequency of technology use, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leverag- ing variation in physical activity and adoption of leader- boards over time and between individuals, we use a difference-in-differences (DID) estimation approach to evaluate the effect of leaderboards on daily physical activity as measured by the user’s step count, as well as heterogeneity in these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find that leaderboard adoption results in an aver- age daily increase of 370 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This main effect is resil- ient to various tests for the assumption of common trends between those who adopt and do not adopt, estimation of several falsification tests, and other ro- bustness checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These initial results, however, mask important heterogeneity in the benefits of leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' When we take into account an individual’s Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 2 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' prior activity levels, we find a stark divergence in leader- board effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Individuals who were highly active prior to adoption, instead of benefiting from leaderboards, experi- enced a significant decrease in their average daily physical activity after leaderboard adoption (a decrease of 631 steps daily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, these negative effects persisted (and actually increased in magnitude) in the 10 weeks following leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, users who were less active prior to adoption had large and significant positive impacts on their daily step counts—On average, their activity increased by 1,365 steps daily (an approximately 15% increase), and these increases also persisted well after the adoption decision (10 weeks after adoption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Examining this trend further, we find significant nu- ance in how leaderboard size impacts sedentary versus highly active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, the key distinction bet- ween these groups is that previously sedentary individ- uals can reap significant benefit from small leaderboards (only one or two other members) and even if they do not rank first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, those who were highly active (prior to adoption) see the most significant harm when leaderboards are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our interpretation of these results is that individuals who are already on the high end of the physical activity distribution can become com- placent on small leaderboards where, more often than not, they are paired with those less active than them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='6 In contrast, sedentary individuals who are at the lower end of the distribution of physical activity often encounter (even on small leaderboards) peers who are more active than themselves, who can positively impact their reference point for exercise, and who can hold them accountable if their activity levels slump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, these benefits for sedentary users diminish if leader- boards become too large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' the marginal benefit of an addi- tional leaderboard member diminishes at three times the rate for sedentary users relative to highly active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' One explanation for this effect is that the benefits of social influence that accrue to sedentary users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', posi- tive impact on their exercise reference points) diminish as leaderboards become larger and less intimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our research contributes to streams of work at the intersection of information systems, economics, and healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we contribute to the literature on the economics of health information technology (IT) and specifically to the nascent streams of work evaluat- ing economic and health implications of widespread adoption of health wearables (Handel and Kolstad 2017) and the broader potential of digital platforms to unlock interventions that leverage social norms and rec- iprocity to improve health (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Currently, the evidence on benefits from health wearables does not align with their promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Piwek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2) suggest that “current empirical evidence is not supportive” of health benefits from health wear- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Recent studies using large samples and robust causal approaches find little or no benefit on health outcomes of using health wearables (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2016, Jakicic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, scholars have argued that a limitation of prior works is that they do not adequately consider the role of innova- tive technology decision aids and behavioral interven- tions enabled by broad adoption of health wearables (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our study addresses these limitations of prior work and finds that, on average, leaderboards promote healthful activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, our results also caution that these benefits may be highly nuanced with considerable variation in gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In some cases, individu- als may opt into variants of these interventions with either no benefit to them or, in some cases, negative effects on their physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also contribute to the behavioral economics and information systems (IS) literature on gamification, espe- cially, within the healthcare context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Despite mixed evi- dence of benefits and numerous open empirical and theoretical questions (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017, Treiblmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2018, James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2019), gamification is spreading into a number of decision contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, two recent IS papers have examined the impact of gamification within the retail context (Pamuru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2021, Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our study is differentiated with extant literature in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, our study is an individual-level intervention within healthcare in which the combination of unique data and rigorous estimation approaches results in more conservative estimates of average treatment effects of leaderboards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' prior work showing positive effects of simi- lar gamification interventions has found treatment effects five times our estimates (Shameli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Second, our results suggest that the mixed evidence of prior work may be explained, in part, by significant heterogeneity in gamification impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Not only are we able to provide more nuance in our study for gamification’s impact (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2022), we also provide evidence on a substantively important issue in the medical literature, that is, the im- pact on the previously less active users (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In our setting, the relatively conservative estimates of the average effect of leaderboards mask robust heteroge- neous effects that are large in magnitude, statistically sig- nificant, and persistent over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These heterogeneous effects support our theoretical conjecture that competi- tion and social influence are key mechanisms underly- ing leaderboard effects but also highlight that these mechanisms can result in unexpected motivational and de-motivational effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we identify a diver- gence of benefit for sedentary versus highly active users that is opposite to the expectation for competition-based gamification in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These findings point to the underappreciated role of social influence benefiting sed- entary users but harming more active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These results not only have significant managerial implications for firms in the health wearable and gamification spaces, but also for policy makers, healthcare entities, and employers interested in improving health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 3 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Background Physical activity is a key element of healthful living and is known to have significant health benefits (Penedo and Dahn 2005, Warburton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our main out- come variable is Fitbit step counts and includes a variety of these healthful physical activities, such as jogging, running, walking, playing sports, climbing stairs, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, daily step counts are key to Fitbit lead- erboards, as rankings on leaderboards are determined exclusively by differences in the step counts of the par- ticipants of the leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboards A leaderboard is “a large board for displaying the rank- ing of the leaders in a competitive event.”7 In a digital set- ting, the leaderboard may be displayed on a mobile application or an online dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In this study, we utilize health wearables made by Fitbit Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', which is a pioneering firm in this market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='8 Using Fitbit’s online dashboard or the mobile application, a Fitbit user can invite another user (or receive an invitation) to join a lead- erboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If there is mutual agreement between the users to participate, both users will appear on each other’s lead- erboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Each leaderboard ranks participating users based on seven-day running tallies of their steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='9 The step counts shown on the leaderboard are directly cap- tured by the Fitbit device and are not manually entered by the users, thus avoiding the measurement errors that may result from self-reported activity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Figure 1 shows four leaderboards, with the focal user labeled at the lower left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Each leaderboard can have the same or different user composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, Ash and Todd are connected to Mary and to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Dave is only connected to Mary, and Mary is connected to all other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The leaderboards also show the seven-day step count of each participating user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Users are assigned ranks on participating leaderboards based on their seven- day step count relative to other users on that leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, Mary is ranked second on her own leader- board, but she is ranked first on Ash’s and Todd’s leader- boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, Ash and Todd may be motivated to do better by seeing their lower rank on the leaderboard rela- tive to Mary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Users get feedback according to their rank on their own leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although Mary dominates the highest number of leaderboards, the feedback she gets is that she is ranked second on her own leaderboard and must strive harder to achieve a first rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboard adoption is “sticky” on the Fitbit platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To de-adopt, users have to go through cumbersome steps and hide themselves via privacy settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Effect of Leaderboards on Healthful Physical Activity Whether leaderboards will increase or decrease healthful physical activity is not entirely clear as the effect is un- likely to be similar for all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboards can produce an effect on an individual’s physical activity primarily by altering this individual’s willingness to en- gage in physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we conjecture that changes in willingness to engage in physical activity occur primarily due to the introduction of competitive dynam- ics, increased individual accountability, and altering an individual’s reference point for their own activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Competition Social comparison theory suggests that a fundamental mechanism through which individuals assess their own ability is through comparison with others (Festinger 1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Competitiveness is one manifestation of the social comparison process and drives individuals to increase Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (Color online) Fitbit Leaderboard Composition for Four Individuals Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This figure shows leaderboards for Ash, Dave, Mary, and Todd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Ash and Todd are connected to each other and Mary, Dave is only connected to Mary, and Mary is connected to all other participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 4 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' P412:30 V12:30 41230 12:30 三 Friends Friends Friends 三 Friends 7DAYSTEPS MESSAGES ZDAYSTEPS MESSAGES 7DAYSTEPS MESSAGES 7DAYSTEPS MESSAGES Mary You Dave Mary 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='082 110,193 110,193 103,082 You Mary noa Ash 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='922 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='082 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='082 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='922 Todd D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Ash SHOWINACTIVEFRIENDS You 95,880 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='922 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='880 A lttlehealthycompetition isagreatthing Todd D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' SHOW INACTIVEFRIENDS Tapthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='Addbuttonto find morefriends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 95,880 SHOWINACTIVEFRIENDS Alittiehealthy competition is a greathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A littiehealthycompetition is a great thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' TaptheAddbuttontofindmorefriends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' SHOWINACTIVEFRIENDS TaptheAddbuttonto findmorefriends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Altehealthy competitionis agreatthing TaptheAdd buttontofind morefriends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 十 + Ash Dave Mary Todd D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='their effort either ex ante to elevate their rank or ex post to maintain their high rank (Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, the first and most direct way that leaderboards impact physical activity is through the competitive dynamic that ranking a focal user against other users generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The tag line on the Fitbit leaderboard (Figure 1)—a little healthy competition is a great thing—points to the motiva- tional potential of this competitive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addi- tion, the enjoyment derived from physical activity may be impacted by the individual’s leaderboard adoption by converting the mundane activity of walking into the more exciting activity of competing against others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Therefore, individuals who may not gain any direct enjoyment from walking may engage in this activity because of the indirect enjoyment gained from compet- ing on the leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, prior work finds that impacts of competi- tion on motivation and effort are highly heterogeneous and depend on several factors such as a participant’s desire to win, whether the competition provides a par- ticipant the opportunity or reason for improving their performance, and whether competition motivates a par- ticipant to put forth greater effort (Deci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Along this vein, a leaderboard may have minimal impact on performance if it does not provide sufficient competition or if the adopting individual is not particu- larly motivated by competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, prior work has noted the possibility of competition having negative impacts on motivation and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, Steinhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2015) argue that when competition elic- its excitement, it may foster positive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, if competition elicits anxiety, it may foster negative behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Extrapolating this to our context, if the per- formance of others on the leaderboard elicits anxiety in the focal user, it would lead to negative outcomes for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Reflecting this theoretical tension, the extant liter- ature has found mixed results regarding competition with others who significantly surpass the individual in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Rogers and Feller (2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1) showed that “exposure to exemplary peer performances can under- mine motivation and success by causing people to per- ceive that they cannot attain their peers’ high levels of performance,” and termed this phenomenon discourage- ment by peer excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, Uetake and Yang (2019) find that an individual’s distance from the high- est achiever has positive motivational effects, whereas comparison with the average individual has negative impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, competition is likely a focal mechanism behind leaderboard effects but whether it positively impacts physical activity is uncertain ex ante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Social Influence Leaderboards involve connecting individuals around health and the revelation of previously private levels of physical activity between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These con- nections and disclosures introduce the potential of social influence to impact motivation and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We consider two potential effects in the realm of social influence: indi- vidual accountability and reference points for exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Individual Accountability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Joining a leaderboard involves the revelation of one’s previously private levels of physical activity to other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This self-revelation allows other leaderboard members to hold the focal user accountable for lackluster levels of physical activity and nudge them to do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In fact, the Fitbit app has a mechanism for messaging, cheering, and taunting other users directly from the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Some of these interac- tions may also happen off the Fitbit platform (and are thus unobserved by us as researchers)—for example, discussions between family members over dinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The potential of group-based interventions to increase mutual accountability and increase physical activity has been explored in the literature: Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2016) un- cover benefits of incentive schemes for exercise that are tied to group versus individual performance targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Exercise Reference Points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition to the potential impacts of self-revelation, the revelation by others of their previously private levels of physical activity can result in changes to individuals’ reference points for exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, social comparison theory suggests that such revelations can lead to an updated perception of one’s own ability to exercise and the appropriateness of one’s own level of exercise (Gar- cia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, how these comparisons impact reference points depends on whether individuals engage in upward comparisons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', comparisons with those more active than themselves) or downward com- parisons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', comparisons with those less active than themselves) (Festinger 1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In both cases, the literature suggests that individuals will take action to reduce dis- crepancies between themselves and similar others (Fes- tinger 1954, Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, if individuals compare upward, the revelation of this information between members of a leaderboard may have a positive impact on an individual’s reference point for healthful activity and increase exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, a mother with two young children may aim for a higher level of healthful activity if she observes another mother with two young children consistently doing more healthful activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Given that these two users may have similar schedule constraints, the focal user may find the leader- board information more relatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If individuals com- pare downward, the revelation of physical activity information by others may have unintended negative impacts on an individual’s reference point for physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In particular, this informational signal can work in the opposite direction—that is, focal users may decrease activity if they see other relatable indivi- duals on their leaderboards who are less active than themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Related to this point, Schultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2007) Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 5 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' found that a nudge intended to decrease electricity con- sumption by revealing the consumption levels of others in one’s neighborhood had the (opposite) boomerang effect for those who were underusing electricity (rela- tive to their neighbors) prior to the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moderating Effects of Prior Activity Levels and Leaderboard Size The contradictory effects of competition and social influence not only make the direction of the average effect uncertain, they also point to the presence of heter- ogeneity in the effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To untangle this heterogeneity, we consider factors that can impact the propensity for observing the positive vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' negative dynamics of leader- boards on physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboard Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, we consider whether leaderboard size, that is, the number of other active par- ticipants on the leaderboard, is an important potential moderator of leaderboard impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2013) suggest that an important situational factor impacting comparison concerns and competitiveness is the num- ber of competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' On the one hand, increasing the number of active participants is likely to increase the likelihood of the positive dynamics that leaderboards introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Clearly, the mechanisms of competition, mutual accountability, and changes in perceived ability are nonexistent if there are no other active users on a leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, competitive motives may be stronger on larger leaderboards because ranking highly on larger leaderboards can be more motivational than dominating smaller leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' That said, the effect of increasing leaderboard size is likely more nuanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, it is likely that some benefits of additional leaderboard participants are diminishing at the margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Too many participants can make the leaderboard less effective because participants get lost in the crowd, weakening the positive impacts of competition or mutual accountability (Garcia and Tor 2009, Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The diminishing marginal benefit of an additional leaderboard member implies nonlinearity in the benefit of more leaderboard members and may even lead to harmful effects of leaderboards if they become too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Prior Activity Levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Second, we consider the physical activity level of an individual prior to leader- board adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If we consider only the role of competition (vis-`a-vis prior activity levels), the expecta- tion in the literature that highly active individuals should benefit disproportionately from leaderboards is most plausible (Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2015, Shameli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Individuals with high activity levels prior to leader- board adoption gain high utility from healthful activity and thus are likely to perform well on leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This positive performance on leaderboards can be motiva- tional for them and encourage increases in future physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The impact of competitive dynamics on relatively more sedentary individuals may be more nebulous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' On the one hand, these individuals may benefit most from extrinsic motivators such as competition and ranking themselves against others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' On the other hand, the value of leaderboards for such individuals may be limited by their lower intrinsic aptitude and motivation for physical activ- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This leaves them prone to de-motivational impacts of lackluster performance on leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Accountability and Reference Points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If we also consider theorized mechanisms related to social influence, the expectation ex ante is more uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Because individuals on the low end of the physical activity distribution are more likely to have other lead- erboard participants who are more active than they are, there is increased potential for the leaderboard to act as a tool that keeps them accountable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' individuals who are more active than the focal user may be more credible in their attempts to hold the focal user accountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, individuals at the lower end of the physical activity distribution are more likely to encounter other users who facilitate upward comparisons and positively impact their reference point for exercise and their per- ceived ability to engage in physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, individuals with low activity levels prior to leaderboard adoption may benefit most from leaderboards because they have more room for improvement and a higher need for external motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The dynamics around social influence are somewhat reversed for those who are highly active prior to leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Follow- ing the same rationale, individuals who are already highly active may be less likely to join leaderboards where other users can hold them accountable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', few others on their leaderboard can match their physical activity levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, these individuals are at ele- vated risk of leaderboards facilitating downward com- parisons that negatively impact their exercise reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These comparisons can induce sluggishness if they highlight the focal user’s disproportionate level of activity compared with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, highly active individuals may suffer from ceiling effects, that is, any extrinsic intervention is not likely to increase their will- ingness or ability to increase physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboard Mechanisms, Prior Activity Levels, and Leaderboard Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The theorized effects of prior activity levels and leaderboard size can also intersect in ways that have implications for the diverse mechanisms through which leaderboards can impact behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, our theorized mechanisms point to highly active individ- uals being most likely to be harmed by smaller leader- boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2013) suggest that competitiveness emerges when there is a potential for comparisons, up Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 6 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' or down, that credibly threaten the individual’s rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' With smaller leaderboards (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', one other individual), these high achievers are less likely to interact with another individual who can credibly compete with them or hold them accountable (thus nullifying two key mech- anisms for leaderboard value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' At the same time, they are more likely to be presented with a salient individual who facilitates downward comparisons that negatively impact their exercise reference point, induce sluggish- ness, and diminish their physical activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' As lead- erboard size increases, there is increased potential value for the highly active because the likelihood increases of at least one individual joining who can provide a credible threat to their rank, mutual accountability, and positive impacts on their reference point for exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Furthermore, it is plausible that individuals who are highly active are buoyed to perform even better when part of a relatively large leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This phenomenon would be akin to the idea in some sports of a “big match player,” someone who performs above their average on big occasions and in front of big crowds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, our theorized mechanisms have different implications for leaderboard size when individuals were sedentary prior to adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Unlike highly active individ- uals, these individuals can still benefit from adopting small leaderboards because they are still likely to encoun- ter other users who are either at a comparable or a higher level of physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, even small leaderboards may often provide these individuals with an additional degree of accountability and the potential for positive impacts on their exercise reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Whether these individuals benefit from competition with small leader- boards is less certain, as they may still be dominated on small leaderboards, leading to de-motivational effects of competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Increasing the size of the leaderboards for lower activity users may still provide some of the benefits described previously but it is likely that these benefits diminish faster for this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Unlike for highly active individuals, benefits of mutual accountability may be reduced for these individuals as leaderboard size increases (via the “getting lost in the crowd” phenomenon described previously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, these users, who are at the lower end of the distribution of physical activity, may be more likely to get stuck toward the bottom of larger leaderboards and this may be more salient with more users participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Overall, we conjecture that sedentary individuals can significantly benefit even when leader- boards are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, increases in leaderboard size may have diminishing marginal benefit for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Data and Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Data We use a unique panel data set comprised of 516 under- graduates at a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' university from October 2015 to Sep- tember 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='11 This data set consists of granular wearable device data and periodic survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' With respect to wearable device data, the students were offered Fitbit Charge HR devices, which were then used to record their physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We access three types of Fitbit data: (i) step count, accessed on a daily basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (ii) leader- board data, which captures if a focal student has a lead- erboard and, if so, the seven-day average step count of other leaderboard participants for the determination of participants’ leaderboard rankings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' and (iii) minute- by-minute heart rate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Students synchronize their data with the Fitbit platform either through a dongle and a desktop application or a smartphone application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We implemented a client application that invoked the Fitbit application programming interface (API) to down- load the synchronized student activity data and store it locally in a secure database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The client application was a set of scripts that ran automatically every night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' All study participants explicitly authorized our client appli- cation to allow access to their data via the Fitbit APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Step measurements only occur if students wear their Fitbit devices regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We will use the term compliance to refer to the regularity with which students wear their Fitbit device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We calculate compliance from the heart rate data by assuming that a student is wearing their Fit- bit during a particular minute of the day if the reported heart rate is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Students were paid $20 for main- taining at least 40% compliance and synchronizing their data regularly to Fitbit servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Fitbit Charge HR could store up to seven days of data locally, so synchronizing beyond a seven-day interval would result in lost data and lower compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Fitbit Charge HR’s step measurements, which we use as the outcome in this study, are fairly accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Valida- tion studies in laboratory and natural settings have found Fitbit Charge HR’s mean absolute percent error (MAPE) for step count to be less than 10%, except for very light activity (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2017, Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2018) also found Fitbit Charge HR’s heart rate measure to have an MAPE of ≈ 10%, although other studies have found mixed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Even if the MAPE for heart rate were higher, our study is not likely to be nega- tively impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We use heart rate only for measuring compliance such that any nonzero heart rate measure- ment is construed as the device being used by the partic- ipant during that minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Participants were also asked to complete an inten- sive survey at the start of study and were further asked to take shorter surveys in six-month waves to refresh key measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These surveys notably provided data on demographics (gender, religious affiliation, parent’s income, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' ), psychological attributes using validated scales (personality, self-regulation), social in- teraction and ability (trust, anxiety, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' ), technology use (social media use, mobile app usage, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' ), and health state (body mass index, satisfaction with health, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although most students took the survey, there was some nonresponse as these surveys were not Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 7 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' On average, students completed three waves of survey data (approximately six months apart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We use these data in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Primarily, we use relevant survey data to model the propensity for opting into a leader- board and, in conjunction with advanced weighting ap- proaches, construct a weighted sample that achieves covariate balance between leaderboard adopters and non- adopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Secondarily, we use a subset of the survey data to generate controls that capture time-varying features of individuals that may relate to both leaderboard adoption and physical activity, and check the robustness of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='12 Table 1 provides descriptive statistics about the outcome, treatment, and some demographic variables, whereas Online Appendix Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1 describes the relevant portions of the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Model The goal of our analysis is to estimate the effect of a user’s leaderboard adoption on their physical activity as measured by steps walked, using nonexperimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='13 Thus, leaderboard adoption is the treatment in our observational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In a randomized experiment, leaderboards could be randomly assigned to study par- ticipants, which would make the identification of treat- ment effect straightforward but would make the study treatment very different from naturally occurring lead- erboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, any Fitbit user in our study can opt into and construct their leaderboard, resulting in more natural leaderboards but making it more difficult to identify the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The main empirical con- cern in identifying this treatment effect is the confounded- ness of the leaderboard adoption with respect to users’ physical activity as measured by their daily step count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We use a DID research design as Fitbit users are observed over multiple time periods, with roughly half of the users adopting leaderboards and the other half remaining untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although a DID design controls for any time-invariant user characteristics and common shocks, it requires the identifying assumption that any uncontrolled time-varying user characteristics exhibit a common trend across the treated and untreated individ- uals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Under this identifying assumption, we can esti- mate the effect of leaderboards on steps walked for the Fitbit users who have adopted leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our model specification is given here and its explanation follows: Stepsit � β0 + β1(Leaderboardit) + θi + λt + γi × t + φi × t2 + ɛit: (1) Although we observe physical activity data on a daily basis, the leaderboard data are obtained weekly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hence, our unit of analysis is student-week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Stepsit is the aver- age number of steps walked daily by user i in week t, and Leaderboardit is a binary indicator for whether a user i adopted a leaderboard in week t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' As stated earlier, the leaderboard adoption is generally “sticky,” as Fitbit makes it difficult to de-adopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We include individual fixed effects (θi) to account for time-invariant differen- ces between individuals and time-fixed effects (λt) to account for any common shocks in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Together, these two-way fixed effects would enable the identi- fication of the treatment effect in the absence of any differential trends across the treated and untreated indi- viduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Admittedly, the common trends assumption is very strong, but one way to make it more plausible is to explicitly control for individual-specific linear time trend (γi × t) and individual-specific quadratic time trend (φi × t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We can then estimate our model under the weaker assumption that the treatment assignment is ignorable after controlling for the two-way fixed effects and the additional individual-specific time trends (Xu 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='14 In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1, we will further explore the issue of time trends across Fitbit users with and without lead- erboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For inference, all our analyses use cluster- robust variance-covariance estimators (VCE), clustered at the student level, which adjust for heteroskedasticity and serial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Descriptive Statistics Variable Description Mean Standard deviation Minimum Maximum Steps Number of steps walked daily 10,625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='18 4,247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='53 0 37,835 Leaderboard An indicator if an individual has adopted a leaderboard 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='50 0 1 Age Age at the start of the study 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='72 17 26 Body mass index Body mass index at the start of the study 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='02 16 38 Female An indicator for whether the individual is a female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='50 0 1 Leaderboard size The number of users on the leaderboard (excluding the focal users) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='46 1 25 Leaderboard size (active) The number of users on the leaderboard that have a nonzero step count (excluding the focal users) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='64 0 17 Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 8 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Estimation and Robustness of the Main Effects of Leaderboards In our main analysis, we estimate variants of Specifica- tion (1), which is a DID specification with flexible user- specific time trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Table 2, columns 1 and 2, presents the estimation results for Specification (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The first column presents results for a specification that includes individual-specific linear time trends only, whereas the second column additionally includes individual- specific quadratic time trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In both columns, we find a significant (p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05) and meaningful leaderboard effect of 338–370 steps daily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Column 2 is our preferred model as it includes more flexible time trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This model suggest that the students who adopted leader- boards have a daily increase of 370 steps, equivalent to a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='5% increase in physical activity on the average daily step count of 10,268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These initial results suggest some support for a main effect of leaderboard adoption on physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In the Online Appendix, we extend this analysis to add time-varying survey variables as controls in Specification (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The estimated effects with these addi- tional controls have higher magnitudes, which increases the plausibility of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, a number of concerns commonly arise with analyses using observa- tional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In the remainder of this section, we discuss the robustness of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Probing the Common Trends Assumption Identification of the treatment effect with a DID design crucially depends on the common trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' As mentioned earlier, one way to weaken this assump- tion is to control for individual-specific linear and quad- ratic time trends, which we have incorporated in our model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In this section, we will further probe the plausibility of assuming that no unobserved time- varying covariates may be confounding our analysis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', the common trends assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Inverse Probability of Treatment Weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A common concern in a DID design is whether the treated and control subjects are similar in their baseline charac- teristics such that the treated and control subjects plausi- bly follow common trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' As mentioned earlier, we collected a rich set of baseline characteristics of study users using a survey instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although the initial covariate balance did not cause excessive concern, we use the inverse probability of treatment weighting (IPTW) method to further improve the covariate balance of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To estimate the propensity for leader- board adoption, we use the Toolkit for Weighting and Analysis of Nonequivalent Groups (TWANG), which implements a generalized boosted regression model (GBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The propensity score estimated by TWANG optimizes covariate balance across leaderboard adopt- ers and nonadopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We observe substantive improve- ment in the postweighting covariate balance such that the observed absolute standardized mean difference, SMD ≤ 0:2, is better than the accepted threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='15 Table 2, column 3, presents the main analysis using the IPTW sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The sample size is slightly smaller (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' column 2) as a few students did not partici- pate in the initial study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Comparing with the main result (column 2), we find the effect sizes to be very similar—370 versus 343 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This stability of effect size boosts our confidence in the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Pretreatment Period Placebo Treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Given that we have multiple pretreatment periods for most users in our sample, we can probe the plausibility of the com- mon trends assumption by creating placebo treatments in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Fitness Activity and Leaderboard Participation (1) Steps b/se (2) Steps b/se (3) Steps b/se (4) Steps b/se (5) Steps b/se (6) Steps b/se (7) Steps b/se Leaderboard 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38** 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='46** 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='02** 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='20** 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='75** (171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='40) (170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='66) (170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='98) (190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='75) (174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05) Placebo Leaderboard 4-Weeks �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='62 (260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='57) Actual Leaderboard 4-Weeks 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78† (271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='53) Leaderboard × Inviter �98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='92 (387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='32) Individual fixed effects Yes Yes Yes Yes Yes Yes Yes Week fixed effects Yes Yes Yes Yes Yes Yes Yes Individual linear trends Yes Yes Yes Yes Yes Yes Yes Individual quadratic trends No Yes Yes Yes Yes Yes Yes IPTW No No Yes No No No No Observations 27,758 27,758 27,409 14,746 15,742 27,758 27,358 Individuals 516 516 501 516 516 516 503 Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 VCE Robust Robust Robust Robust Robust Robust Robust Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Column 7 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' column 2) excludes users who hide themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Please see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' †p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='125;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' *p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 9 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' the pretreatment data alone, that is, by dropping the posttreatment data and using only the pretreatment data for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A failure to reject the null effect for the placebo treatment would provide support for the com- mon trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='16 In our study, users opt into the treatment in different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, our pri- mary concern is the presence of some unobserved time- varying factor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', spurts in motivation) that affected the adopters in the periods closely preceding the treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hence, we implemented our placebo treatment in the preceding month prior to the actual treatment and estimated the model in Equation (1) on the altered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Table 2, column 4, presents the estimated effect of the placebo treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This estimated effect is small in mag- nitude, opposite in sign, and statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, if we include four weeks of actual treatment period in our sample, the estimated effect is ≈ 420 steps (p�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='12) as presented in Table 2, column 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, for comparable time periods, the placebo effect is null, whereas the actual treatment effect is comparable to our estimated main effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This null effect in the pretreat- ment period enhances the plausibility of our common trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leads-Lags Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The placebo treatment effect can be further broken into weekly placebo effects in the pretreatment period and the actual effect in the posttreatment period using the full data set and a leads- lags specification: Stepsit � β0 + X 9 τ��9 βτLi(t+τ) + θi + λt + γi × t + φi × t2 + ɛit: (2) The dummy variables Li(t+τ) denote the time from adop- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' for example, Li(t+τ) would be one for individual “i” in time period “t” if this time period is τ�weeks from adoption, where τ � �10,:::,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If we observe more data for an individual, we collapse it into the extreme peri- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='17 We set Li(t�10) as the baseline period and exclude it from Equation (2) to avoid the “dummy variable trap.” Figure 2 (top left panel) plots point estimates and confidence intervals for βτ�coefficients against the time from adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find a null effect in the pretreatment period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, the effect is positive and statistically significant in the adoption period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', period 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The posttreatment coefficients remain positive but decrease in magnitude and lose significance in the later periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We will explain the reason for this decline in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A potential issue with this analysis is that the coef- ficients are trending upward from period �4 to �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, this trend is not a cause of concern for several reasons: first, the estimates for periods �4 to �1 are not statistically significant even at the 75% level (the lowest p value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Second, the estimates for βτ�in the posttreatment period stay positive (and larger than any preperiod estimate), whereas the estimated βτ�in the preperiod oscillate around the zero line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In particular, the change in coefficient estimates from period �9 to �6 is roughly the same as the change from �4 to �1, with a sharp drop to a very small negative value in period �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, extrapolating this historical pattern beyond �1 would plausibly suggest a regression back to an almost zero value as in period �5, but the adoption breaks that trend such that we see a large significant effect in the adoption period and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, the other tests such as the placebo test presented earlier in this section also argue against the presence of any pretrend in the month preceding the adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These robustness checks argue against the presence of any other unmeasured changes that affect leaderboard adopters in the periods closely preceding leaderboard adoption, thus enhancing the plausibility of the common trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Robustness Check for Leaderboard Initiation As an additional robustness check, we also considered leaderboard initiation as it may be a proxy for con- founded leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, if the focal user is the primary inviter to the leaderboard, this lead- erboard may be more likely to be driven by unobserved motivation changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For the purpose of this analysis, we consider focal users to be of the “inviter-type” if they ini- tiate most, not necessarily all, of the invitations to other users on their leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although we do not have access to direct measures of who initiated a leaderboard, we construct a proxy variable that we argue identifies users who are more likely to be inviter types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We lever- age two aspects of leaderboard creation to construct this proxy variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, per the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1, Fitbit does not use a leaderboard that is defined cen- trally as a group of individuals that others can join or leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Rather, each leaderboard is owned by the user and each user pair must agree to share their step infor- mation for them to be joined on their individual leader- boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, Fitbit does not advertise to the user’s friends that they have joined the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Based on these aspects of Fitbit leaderboards, we desig- nated InviterLB using two criteria: (i) whether the leader- board had three or more individuals when it was first adopted and (ii) whether the leaderboard was such that the other users (excluding the focal users) had been on the Fitbit platform for longer than 90 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The first crite- rion is useful because the size of the leaderboard at lead- erboard initiation can be indicative of the likelihood of initiation by the focal user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' If there are two people when the leaderboard is started, it is unclear who initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, if three people (or more) are on the leader- board at initiation, a leaderboard fully initiated by others would require that two other users actively searched and invited the focal user in the same week and that the user accepted both invitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, this criterion may still include mixed leaderboards that were only partially Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 10 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' initiated by the focal users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', another user initiates but the focal user invites the third person).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, we add the second criterion that the other users on the leaderboards have been on the platform for more than 90 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The rationale behind this criterion is that users on the platform for longer periods of time are more settled on the platform and less likely to be actively scouring the platform for new connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The 90-day threshold was chosen based on data suggesting that Fitbit abandonment happens in the first few months of adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='18 Among the leader- board adopters, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3% met these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In Table 2, col- umn 6, we add an interaction term between leaderboard and an indicator for InviterLB and identify a negative coef- ficient that is close to zero and insignificant (p� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This result suggests that users who are more likely to have ini- tiated the leaderboard do not see different treatment effects and is further evidence that time-varying changes in motivation are unlikely to be confounding our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Fitbit Compliance Step measurement through Fitbit only occurs if the par- ticipants wear their devices regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We will use the term compliance to refer to the regularity with which a participant wears the Fitbit device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In this subsection, we will probe two compliance-related concerns that may cast doubt on the earlier analyses if left unaddressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For- tunately, our data include compliance data at a very gran- ular level, which allows us to construct the participant’s compliance measure, percent compliance, at the daily and weekly level, and the participant’s mean compliance for the study duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We will exploit these data to probe compliance-related concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Do Leaderboards Increase Compliance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The first concern is the possibility that rather than increas- ing steps, leaderboard adoption increases compliance, which may lead us to observe higher step count purely because of better measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To address this concern, we estimated main effects models similar to Equation (1) and the leads-lags models similar to Equation (2) but with daily percentage compliance as the dependent var- iable and student-day as the unit of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We esti- mate this model for a number of samples—the entire sample and the subsamples at various mean compliance levels (ranging from 60% to 95%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The main effect model estimates are statistically insignificant and have small magnitudes, ranging from –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='97% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='01%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, the leads-lag model’s coefficient plots do not show any sharp increase at or after adoption (see the Online Appendix, Section E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These results suggest a null effect of leaderboard adoption on compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Are Leaderboard Effects Discernible at Higher Compliance Levels?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The second issue is that some of the participants may have lower compliance and the full sample estimate includes these participants too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (Color online) Leaderboard Coefficients by Weeks from Leaderboard Adoption (by Prior Activity Levels) Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (a) Please see Equation (2) for charts’ specification, (b) 90% confidence intervals, (c) vertical axes use different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 11 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Regarding this concern, our empirical analysis would be more convincing if the leaderboard’s effect on partici- pant activity was clearly discernible for participants with high compliance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, we estimate impact of leaderboards for participants with high levels of com- pliance and find these effects to range from 408 to 598 steps (see the Online Appendix, Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The persis- tence of leaderboard effects at higher levels of compli- ance supports the claims from our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Fitbit Attrition, Leaderboard De-Adoption, and Additional Robustness Checks Related to the challenge of compliance, we also consider the role of attrition from the sample due to Fitbit aban- donment, which has generally been noted in the popu- lar press for health wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='18 If sample attrition is related to leaderboard adoption, it may introduce bias in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For example, lower performers may abandon their Fitbit device after joining leaderboards because it reveals to them that they are less active than their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We examine this concern extensively and identify no relationship between leaderboard adoption and sample attrition for lower performers, and no differ- ences in physical activity and similar leaderboard effects for those who eventually leave the sample compared with those who report data throughout (see the Online Appendix, Section F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also consider whether indi- viduals who eventually hide themselves from the lead- erboard (the main mechanism for de-adoption) impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' As we mentioned previously, this was rare for leaderboard adopters (approximately 5%), and excluding these individuals results in consistent esti- mates of leaderboard effects (see Table 2, column 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Outliers and Falsification with Negative Control Treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also evaluate the potential for a partic- ular individual (or time period) in the data to be an out- lier driving our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we systematically “leave out one” individual (or time period) and re- estimate our model (see the Online Appendix, Section G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find consistent treatment effects of leaderboards that are always statistically significant, suggesting mini- mal risk from outliers in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Furthermore, we con- structed a negative control treatment (NCT), as the focal user’s leaderboard with no other active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Such lead- erboards exist because other users may accept a request to connect but then become inactive on the platform and thus neither provide competition nor reference points (see the Online Appendix, Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='II and associated discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, the absence of any other active users of such leaderboards should result in no effect on the user’s physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Indeed, we find a null effect of such leaderboards on steps (see the Online Appendix, Section D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This falsification test with an NCT strength- ens the plausibility of the common trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Heterogeneous Effect of Leaderboards In this section, we evaluate the potential for heterogene- ous effects of leaderboards on physical activity focusing on leaderboard rank, leaderboard size, and prior activ- ity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The evaluation of heterogeneous effects of leaderboards is useful because it can offer additional insights into the role of competition and social influence in generating leaderboard value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We start by evaluating the impact of ranking first on the leaderboard in the prior period on the activity levels of individuals in the subsequent period (FirstonLB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='19 Next, we evaluate whether the number of active partici- pants on a leaderboard (excluding the focal user) modi- fies the benefit to individuals who adopt leaderboards (LBActiveUsers) and whether this impact is nonlinear, by including the square of (LBActiveUsersit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='20 Finally, we consider the interaction of rank and leaderboard size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Equation (3) provides the specification for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To evaluate heterogeneous impacts by prior activity levels, we also estimate this specification strati- fied by preleaderboard activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Stepsit � β0 + β1(FirstOnLBit�1) + β2(LeaderBoardit) + β3(LBActiveUsersit) + β4(LBActiveUsersit)2 + β5(FirstOnLBit�1 ∗ LBActiveUsersit) + β6(FirstOnLBit�1 ∗ (LBActiveUsersit)2) + θi + λt + γi × t + φi × t2 + ɛit (3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Impact of Prior Week’s Rank and Leaderboard Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find a substantive impact of prior week’s rank on the impact of leaderboards in subsequent periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Those who were in first place on their leaderboard in one week, walked 578 steps more a day the following week (Table 3, column 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also find a positive and significant coefficient on LBActiveUsers (Table 3, column 2), suggesting that an additional person on a leader- board increases the effect of that leaderboard by 165 steps (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, we evaluate whether there are diminishing benefits from additional active users on a leaderboard by adding a quadratic term to our estima- tion (Table 3, column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A negative coefficient on the quadratic term (≈ �21) suggests that the effect peaks at eight active members, after which the marginal benefit of an additional member is diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In Table 3, columns 4 and 5, we also explore the inter- sectional impact of leaderboard size and prior per- formance by including FirstOnLB and FirstOnLB × (LBActiveUsers)k, k � 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find that the motivational effects of succeeding in leaderboard competition increase with the size of the leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, we continue to identify a positive impact of increased leaderboard size, suggesting positive impacts of larger leaderboards when an individual is not first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, we consider whether the nonlinear impacts of leaderboard size extend to the interaction with prior week performance Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 12 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' and find no evidence of diminishing impacts (coefficient for FirstonLB × (LBActiveUsers)2 is near 0 and insignifi- cant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Overall, we find that leaderboard benefit increases with leaderboard size (although this benefit is diminish- ing at the margin), and that ranking highly on a leader- board is more motivational on larger leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Implications of Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The positive impact of prior rank and the increased impact of ranking first on larger leaderboards point to an important role of com- petitive dynamics in generating leaderboard value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, the impact of leaderboard size above and beyond the impact of rank suggests that social influence mechanisms also play an important role in observed leaderboard benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, an insignificant effect of small leaderboard when the individual is not ranked first and diminishing marginal benefit of increasing leaderboard size suggest some nuance around how social influence mechanisms drive leaderboard benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We explore this nuance further by evaluating how prior activity levels (which have implications for how other users on leaderboards exert social influence) moderate leaderboard impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Heterogeneity by Prior Activity Levels To evaluate potential heterogeneity in leaderboard benefit by prior activity, we estimate Specification (1) on samples stratified by preleaderboard activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we stratify our sample into two groups based on their preleaderboard activity levels: the top quartile by daily step count comprise the highly active group whereas the bottom quartile by daily step count comprise the sedentary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='21 The differences in step count between these two groups were meaningful in terms of magnitude and were statistically significant (13,000 versus 8,000, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='01)—see the Online Appen- dix, Table H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='9, for summary statistics on these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Before examining the impact of leaderboards on physical activity levels, we evaluated the correlation between activity levels and relevant preadoption survey measures (see the Online Appendix, Table H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We found correlations consistent with our expectations of the key differences between highly active and sedentary individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Sedentary individuals reported lower lev- els of self-efficacy and self-regulation for exercise and reported being more likely to exercise alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In addition, sedentary individuals reported higher levels of anxiety and depression and lower levels of trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These correla- tions suggest that sedentary individuals may need these interventions more than highly active individuals but that they could also be prone to de-motivational impacts if these leaderboards exacerbate mental health barriers to improving health (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', increase their anxiety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Turning to the examination of the effect on steps, we find stark differences in the effect of leaderboards on the highly active group vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' the sedentary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For seden- tary users, the adoption of leaderboards has large and significant impacts on their daily step counts (1,365, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' see Table 4, column 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, we find that the highly active group, instead of benefiting from leader- boards, experienced a significant decrease in their daily physical activity after leaderboard adoption (–631, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' see Table 4, column 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For the sake of completion, we also estimate the effect for the middle group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Heterogeneous Effect by Leaderboard Size and User Rank (1) Steps b/se (2) Steps b/se (3) Steps b/se (4) Steps b/se (5) Steps b/se Leaderboard 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='28 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='99 �61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='66 �29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='60 �194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='81 (170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='84) (183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='39) (187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='25) (184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='02) (188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='89) FirstOnLB 577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='98** 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='70** 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='58** (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='80) (111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='31) (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='19) LBActiveUsers 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='46** 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='31** 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='28** 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='90** (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='50) (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='55) (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='22) (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='89) (LBActiveUsers)2 �21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='36** �20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='00** (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='72) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='47) FirstOnLB × LBActiveUsers 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='23** 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='20* (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='49) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='07) FirstOnLB × (LBActiveUsers)2 �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='91 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='96) Individual fixed effects Yes Yes Yes Yes Yes Week fixed effects Yes Yes Yes Yes Yes Individual linear trends Yes Yes Yes Yes Yes Individual quadratic trends Yes Yes Yes Yes Yes Observations 27,758 27,758 27,758 27,758 27,758 Individuals 516 516 516 516 516 Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 VCE Robust Robust Robust Robust Robust p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 13 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' individuals whose activity levels were between the 25th and 75th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find that the middle group bene- fited from leaderboards, but the benefit was less than for those who were in the bottom quartile (859, p < :05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' see Table 4, column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Overall, these results suggest signifi- cant heterogeneity in the effect of leaderboard based on prior activity levels, with particularly stark, negative effects for those who were previously highly active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leads-Lags Model by Prior Activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To examine the presence of any trends before leaderboard adoption, we plotted the coefficients from a leads-lags model for the full sample and subsamples by prior activity levels in Figure 2 (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2 for details about the specifi- cation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The first point to note is that the lead coefficients are relatively small in magnitude and statistically insig- nificant, which increases the plausibility of the parallel trends assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Second, the lag coefficients shift to larger magnitudes with the effect persisting for more than two months after adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Third, there is some attenuation in the effect for the full sample in later time periods, which can be plausibly explained by the oppo- site direction of effect within the sedentary and middle groups versus the highly active subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Implication of Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The divergence of lead- erboard effects for sedentary versus highly active indi- viduals substantiates our conjecture that leaderboards can introduce both motivational and de-motivational dynamics with respect to physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, these results are suggestive evidence of different im- pacts on reference points for sedentary versus highly active users and differences in the potential of leader- boards to provide accountability for lapses in physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We explore this difference and its implications for mechanisms underlying leaderboard value further by evaluating the impact of rank and leaderboard size on the physical activity of sedentary versus highly active users in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Interaction of Leaderboard Size, Rank, and Prior Activity Levels Last, we consider the intersection of all the prior factors in an effort to understand some of the heterogeneity in leaderboard benefit for sedentary versus active users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Leaderboard Size and Rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We start by evaluat- ing the motivational impact of leaderboard rank and size separately by prior activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We find that both pre- viously sedentary and highly active individuals see sub- stantial increases in physical activity during the week after they ranked first on their leaderboard (Table 4, col- umns 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For sedentary individuals, being first in the prior week unlocks even more value for them and increases the treatment effect for leaderboards by 746 steps to nearly 2,000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Highly active individuals see slightly less value (522 steps versus 746 steps) from being first in the prior week but this benefit counteracts part of the negative main effect they observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Extending this analysis to also include leaderboard size and the interac- tion of leaderboard size and prior week’s performance reveals further richness in these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In column 6, we observe that previously sedentary individuals still see substantial benefit when leaderboards are small and when they are not first (1,016 steps, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1) and that this benefit increases further when they rank first and leader- boards are larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In column 7, we observe that previously active individuals are harmed when they are on small Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Heterogeneous Effect by Leaderboard Size, Rank, and Prior Activity (1) Steps b/se (2) Steps b/se (3) Steps b/se (4) Steps b/se (5) Steps b/se (6) Steps b/se (7) Steps b/se Leaderboard 1,365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='44** �630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='63** 858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='60** 1,262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='11** �817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='47** 1,016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='43** �1,215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='74** (496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='72) (292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10) (213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='46) (484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='75) (304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='86) (486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='63) (331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38) FirstOnLB 745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='74* 522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='47** �49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='60 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='18 (422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='44) (157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='44) (343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='07) (198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='09) LBActiveUsers 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='90** 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='89** (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='52) (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='16) LBActiveUsers × FirstOnLB 462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='99** 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33** (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='96) (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='44) Sample S HA Mid S HA S HA Individual fixed effects Yes Yes Yes Yes Yes Yes Yes Week fixed effects Yes Yes Yes Yes Yes Yes Yes Individual linear trends Yes Yes Yes Yes Yes Yes Yes Individual quadratic trends Yes Yes Yes Yes Yes Yes Yes Observations 5,629 7,836 14,293 5,629 7,836 5,629 7,836 Individuals 129 129 258 129 129 129 129 Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 VCE Robust Robust Robust Robust Robust Robust Robust Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' S, sedentary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Mid, mid 50 percentile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' HA, highly active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 14 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' leaderboards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' these users only start to see positive impacts from leaderboards if they rank first on leaderboards with more than four individuals or if they are on relatively large leaderboards (more than six active users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The finding that sedentary individuals observe lead- erboard benefit despite unsuccessfully competing on small leaderboards suggests that these individuals ac- crue significant benefit from noncompetition mecha- nisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', social influence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, the negative impact of the same type of leaderboard for highly active individuals suggests that social influence is harming these individuals, or that its benefit is not sufficient to overcome any de-motivational effects of not ranking first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We probe this conclusion further using two addi- tional analyses that leverage certain leaderboard instan- ces which could be telling of the impact of these noncompetition mechanisms on physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The first analysis seeks to identify instances of leaderboards where competitive dynamics are arguably weakened but the potential for mutual accountability and changes in reference points is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we create NoCompetitionLB, which is an indicator of leaderboard instances where the focal user is sandwiched between two other users such that there is not a credible threat to their rank from either user (see the Online Appendix, Section H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Table 5, columns 1 and 2, shows a positive and significant impact (534 steps, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05) of these types of leaderboards for previously sedentary users but a small and insignificant impact of these same types of leaderboards for highly active users (�39 steps, p � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We examine this conjecture further by identify- ing instances of leaderboards at the other end of the competition spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we create HighCom- petitionLB as an indicator for leaderboard instances where the focal user is regularly being displaced and then reclaiming the top spot on the leaderboard (see the Online Appendix, Section H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Table 5, col- umn 3, shows that sedentary users continue to perform well on leaderboards without high competition intensity, with no statistically significant benefit to them of being on a highly competitive leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, highly active users have large negative effects when they are on leaderboards without intense competition, but there is a statistically significant offsetting effect of being on leader- boards with high competition (Table 5, column 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Implication of Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although only sugges- tive evidence, these results lend some credence to the notion that previously sedentary individuals are bene- fiting from mutual accountability and positive impacts on their exercise reference point and do not need com- petition to benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, highly active individuals seemed to be harmed by the noncompetition mecha- nisms, but this harm can be offset if leaderboards are sufficiently competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Nonlinear Impacts of Leaderboard Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, we consider whether nonlinear impacts of leaderboard size are similar based on prior activity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In Table 5, columns 5 and 6, we find that, although both groups have diminishing returns from additional users, the negative coefficient on the quadratic term (ActiveLB2) for the sedentary group is thrice that for the highly active group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These results suggest that for sedentary Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Further Heterogeneous Effect Analysis (1) Steps b/se (2) Steps b/se (3) Steps b/se (4) Steps b/se (5) Steps b/se (6) Steps b/se NoCompetitionLB 533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='74** �38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='98 (261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='125) (283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='539) Leaderboard (LB) 1,283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='58** �1,007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='23** 767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='49 �1,268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='45** (560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='852) (343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='563) (516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='114) (316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='771) LB × High Competition 660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='56 1,021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='99* (1,234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='148) (611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='764) LBActiveUsers 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05** 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='18** (233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='570) (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='988) (LBActiveUsers)2 �57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='74* �19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='86** (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='635) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='935) Sample S HA S HA S HA Individual fixed effects Yes Yes Yes Yes Yes Yes Week fixed effects Yes Yes Yes Yes Yes Yes Individual linear trends Yes Yes Yes Yes Yes Yes Individual quadratic trends Yes Yes Yes Yes Yes Yes Observations 5,500 7,707 5,629 7,836 5,629 7,836 Individuals 124 129 129 129 129 129 Adjusted R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='34 VCE Robust Robust Robust Robust Robust Robust Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' S, sedentary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' HA, highly active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 15 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' users, the benefits of an additional active leaderboard member diminish much faster (after 5 users) than they do for highly active users (after 11 users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Implications of Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The smaller optimal size for sedentary individuals suggest that they accrue benefits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', positive impacts on their exercise refer- ence points) even when leaderboards are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' How- ever, benefits diminish as leaderboards become larger, and leaderboards can even be harmful if they become excessively large (leaderboard sizes that become harm- ful to sedentary users were uncommon in our data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, highly active individuals seem to be de- motivated by leaderboards with too few individuals and only start to benefit on larger, more competitive leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These dynamics for highly active individ- uals reinforce the notion that these users require large leaderboards to derive benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Summary of Findings from Heterogeneous Effect Analysis Ex ante, we theorized that competition and social influ- ence are key mechanisms underlying leaderboard effects but that these mechanisms may introduce both motiva- tional and de-motivational effects of leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The heterogeneous effects we identify in this section point to the importance of these mechanisms as well as the poten- tial for nuance in their effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, we find robust posi- tive impacts of ranking first on a leaderboard, suggesting that successfully competing on leaderboards improves motivation for most users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Interestingly, and contrary to our theoretical conjecture, the benefits of competition hold even for sedentary users: they accrue positive effects from ranking first and are not harmed from leaderboards when they do not rank first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We attribute the robust benefit of leaderboards for sedentary users to the positive impacts of leaderboards on their exercise reference points and the likelihood of being held accountable by other users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', social influ- ence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, our results also suggest that social influence enabled by leaderboards can have negative impacts on motivation for some users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', negative impacts on exercise reference points for highly active users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Interestingly, these harms for highly active indi- viduals are attenuated when leaderboards are highly competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The nuanced impact of social influence is further demonstrated by the impacts of leaderboard size on physical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' While larger leaderboards gen- erally increase physical activity, sedentary users see diminishing value from larger leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This result is in line with our conjecture that social influence effects may diminish for sedentary users if they get “lost in the crowd” of larger leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In contrast, highly active individuals thrive in large leaderboards, substantiating our conjecture that highly active individuals become more likely to show positive impacts of competition and social influence as leaderboard size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Conclusions and Discussion The rapid and increasingly broad adoption of health wearables coupled with the gamification services built on top of them provides a potentially powerful vehicle for improving health behaviors at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results lend credence to this potential value, particularly when con- sidered over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' In our data, the average user partici- pated in a leaderboard for 237 days (conditional on participating in a leaderboard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, sedentary users who participated in a leaderboard for at least 237 days took more than 300,000 additional steps (with a conserva- tive estimate of 1,300 additional daily steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' To put this in context, the aggregated benefit of leaderboards (for these participants) amounts to 150 miles of distance (at 2,000 steps a mile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Importantly, the benefit is not homo- geneous and there is a decrease in daily steps of nearly equal measure for those who, prior to adoption, were highly active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, this health harm to the highly active subgroup may not be symmetrical to the gain for those who were previously sedentary, as this group remains very active in absolute terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also identify additional heterogeneity in benefit based on the number of other active participants and leaderboard rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This research has some important limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, we use secondary data in which individuals organically choose to adopt leaderboards, rather than being ran- domly assigned to the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although we put in significant effort to address potential confounding fac- tors for our analysis, only a large-scale randomized con- trol trial can provide a theoretical guarantee that the treatment is unconfounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Additionally, we were missing variables in our data set which can be particu- larly informative of leaderboard mechanisms and the heterogeneous value they generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifically, we do Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (Color online) Heterogeneous Effect by Active Users and Prior Activity 9,000 6,000 3,000 0 3,000 Effect on Step Counts 0 5 10 15 20 Number of Active Users Sedentary Highly-Active Hydari, Adjerid, and Striegel: Gamification and Healthful Activity 16 Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' not have deterministic measures of whether the focal user initiated the leaderboard or whether it was initiated by another user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A more comprehensive measure of leaderboard initiation could have provided important insights into how different types of users are initiating leaderboards, whether initiating a leaderboard impacts competitive dynamics, and whether individuals are selecting into leaderboards of value to them or if others are prompting them to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We hope to address these questions in subsequent data collection efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We also do not observe granular data on Fitbit app and device usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, this lack of usage data could attenuate our results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', lack of use could drive our treatment effects closer to zero) as we are currently assuming all adopters to be using leaderboards in all periods after adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This makes our results more conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Another potential limitation is that the student popu- lation in the sample may not be representative of the general population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', they may be healthier, more physically active, or have more free time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although broad generalization of results to the average popula- tion may be somewhat uncertain, it is useful to note that the largest leaderboard benefits accrue to the least active participants in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These individuals are more comparable to the average population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, a younger population may be more amenable to gamification approaches and be more likely to be moti- vated by such interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, we observe in our data the impact on physical activity and do not observe other downstream health outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', weight loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, given the documented relationship between physical exercise and other health outcomes and the magnitude of our effects, it is likely that individuals are, on average, healthier after leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' These limitations notwithstanding, our results have sig- nificant implications for research and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' (2013) argue for the potential of wearable technolo- gies and mobile health more generally to be “leveraged to more promptly assess and reward behaviors on a popula- tion scale, further reducing the need for prohibitively costly incentives” (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 666);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' however, there has been lim- ited research validating this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We help substan- tiate this notion by demonstrating the significant potential for gamification-based interventions to meaningfully im- pact physical activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Moreover, we find that the effects of leaderboards persist over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results also have implications for the general liter- ature on motivating changes in health behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Specifi- cally, gamification interventions coupled with health wearables provide notable advantages over other ap- proaches studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' First, and unlike most other behavioral interventions, the widespread adoption of health wearables allows for much larger scale inter- ventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Second, because they are based in digital plat- forms, the design of these interventions can be tailored at the individual or group level to maximize benefit to the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' This customization is particularly valua- ble given the heterogeneity in benefit between subpopu- lations in our study and recognition by scholars of the limits of “one size fits all” approaches typically taken by most of the extant literature (Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2014, Rogers and Feller 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our research also highlights potential areas for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' The variation in leaderboard effects points to the complex interplay between competition and social influence that underlies leaderboard effects and im- pacts motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Given that gamification approaches are highly varied, even for the same class of interven- tions (leaderboards can vary in terms of who can partici- pate, who on the leaderboard is made salient to users, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' ), more work needs to be done to rigorously evaluate the potential benefits of these varied gamification app- roaches, the key mechanisms which drive their effects, and how they may have differential impacts across health contexts and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Finally, there is a need to explore whether digital gamification interventions can be used in conjunction with classic interventions (decision aids, economic incentives, wellness programs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=') to unlock further health benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results also have important implications for firms and policy makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' From the perspective of firms that sell health wearables and their related platforms, a key ele- ment of their value proposition is that their products pos- itively impact health and well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results help substantiate the notion that physical activity can be posi- tively impacted for large swaths of their adopters but also that some of their adopters (perhaps their earliest and most enthusiastic ones) can be harmed by some of the interventions they offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results also suggests that these interventions need to be designed with careful con- sideration for how mechanisms intended to be motiva- tional may not be so for all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' More so, this highlights the value of customizing gamification interventions to individuals and the value of nudging individuals toward a set of features that are likely to benefit them most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Thus, these firms may need to better incorporate insights from behavioral research in the design of gamification inter- ventions available alongside health wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Furthermore, policy makers, employers, and insurers are experimenting with health wearables and gamifica- tion as they have significant incentives to encourage more active lifestyles, which may improve health, im- prove worker performance, and lower healthcare costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Scholars similarly suggest that gamification coupled with health wearables can be a cost-effective way to encourage increases in physical activity and to do so at scale (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' However, a salient concern with these approaches is that less healthy individuals, who are often of primary interest to policy makers and employers, can be demotivated when competing with their more active counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Our results highlight that Hydari, Adjerid, and Striegel: Gamification and Healthful Activity Management Science, Articles in Advance, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 1–19, © 2022 The Author(s) 17 Downloaded from informs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='org by [104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' this concern may be unfounded or at least less salient, as sedentary individuals may benefit substantially from these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Although the harm to highly active individuals is not ideal, some of these harms can be alle- viated by tailoring leaderboards for these groups, and in net, these approaches are still likely to be valuable to employers and policy makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' A lingering challenge with reaping this value may be encouraging less healthy individuals to take up these interventions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', join lead- erboards), and employers and policy makers may need to invest in incentives to increase uptake for this subset of individuals to maximize potential value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Endnotes 1 The World Health Organization (WHO) Constitution defines health as “a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity.” 2 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='int/news-room/fact-sheets/detail/physical-activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 3 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='gov/chronicdisease/resources/publications/factsh eets/physical-activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 4 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='lexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='com/en/definition/activity_tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 5 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='wired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='com/story/science-says-fitness-trackers-dont-work- wear-one-anyway and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='fastcompany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='com/3031324/why-your- company-should-think-twice-about-gamification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 6 Descriptive results substantiate this conjecture and suggest that highly active individuals are much more likely to dominate smaller leaderboards compared with larger ones, that is, the smaller leader- boards do not seem to challenge these highly active individuals as they easily dominate them even with their reduced activity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 7 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='merriam-webster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='com/dictionary/leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 8 Although we chose a particular form factor and a particular ven- dor, which is arguably the market leader at the time of the study, the similarity of leaderboards across platforms means our results may be relevant to other platforms as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 9 Fitbit previously offered 30-day fixed-time leaderboards that, to our knowledge, have been discontinued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 10 Mechanisms of individual accountability and changes to exercise reference points are distinct from competition mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For instance, I may have a friend or family member on my leaderboard who is not credibly competing with me (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', because their perform- ance exceeds my own by a huge margin) but this individual can still reach out and hold me accountable for my exercise goals or impact my perception of what is achievable for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 11 Approximately 600 students were recruited but we are left with 516 students after excluding early dropouts and always adopters of leaderboards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 12 Due to the coarseness of the survey data relative to the Fitbit data and some nonresponse in the sample, we use these controls for robustness checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 13 The adoption of Fitbit is not a central concern in our study as every user in our sample is a Fitbit user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 14 Another method to increase the plausibility of the common trends assumptions is to use propensity score to achieve covariate balance across the treated and untreated individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' We perform this propen- sity score–based adjustment as a robustness check in a later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 15 In Online Appendix C, we discuss GBM, IPTW, and IPTW use with DID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Furthermore, Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3 shows pre- and post-IPTW covariate bal- ance, and Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='4 and Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='1 explain the influence of covariates on leaderboard adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 16 Please see Abadie and Cattaneo (2018) for a recent discussion on such examinations of the common trend assumption and Abadie and Dermisi (2008) for an example in a two-period setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 17 Therefore, Li(t+0),:::, Li(t+9) denote the weekly breakdown of the actual adoption, whereas Li(t�9),:::, Li(t�1) denote the placebo treat- ments in the weeks prior to actual adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Furthermore, Li(t+9) denotes period 9 and beyond, whereas Li(t�10) denotes period �10 and before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 18 Please see apnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='com/article/2700956044de4517a471a47c3243078b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 19 Because the average number of participants on leaderboards was relatively small (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='3), a binary indicator for being first was sufficient to capture the impact of rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Results are consistent when we use a continuous measure of the prior week’s rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 20 We focused on active users because the Fitbit leaderboard hides inactive users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=', with zero steps) from the view of the focal user and does not use them in the ranking presented to individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 21 The labels “sedentary” and “highly active” are specific to our population and may not reflect average steps for sedentary or highly active individuals in the general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' 22 Please see Online Appendix Table H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='11 for similar analysis using full data and interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' References Abadie A, Cattaneo MD (2018) Econometric methods for program evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' Econom.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content='139] on 06 January 2023, at 06:00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} +page_content=' For personal use only, all rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E0T4oBgHgl3EQf6wJO/content/2301.02767v1.pdf'} diff --git a/jNFAT4oBgHgl3EQfaR1d/vector_store/index.pkl b/jNFAT4oBgHgl3EQfaR1d/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d978c8feefdf279f418f9d670cb9630c02b6a18a --- /dev/null +++ b/jNFAT4oBgHgl3EQfaR1d/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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+E-mail: vasiliy.ulitko@urfu.ru +Abstract. +A classical Monte Carlo algorithm based on the quasi-classical approximation is +applied to the pseudospin Hamiltonian of the model cuprate. The model takes into account +both local and non-local correlations, Heisenberg spin-exchange interaction, single-particle and +correlated two-particle transfer. We define the state selection rule that gives both the uniform +distribution of states in the phase space and the doped charge conservation. The simulation +results show a qualitative agreement of a phase diagrams with the experimental ones. +1. Introduction +The phase diagram of doped HTSC cuprates is the subject of active experimental [1, 2] and +theoretical research, despite the huge amount of work on this topic to date. A striking feature of +the phase diagram of HTSC cuprates is the competition and coexistence of antiferromagnetic, +superconducting, and charge orderings [3], manifested in pseudogap phase, strange metal phase, +a variety of static and dynamic fluctuations. The studies are complicated by the presence of +heterogeneity due to dopants or non-isovalent substitution, as well as to the internal electronic +tendency to heterogeneity [4]. Phase separation may be the cause of simultaneous detection +of the preformed pairs and BEC superconductivity in cuprates [2], a number of experimental +observations of the typical Fermi liquid behavior, at least in overdoped cuprates. For models +describing such complex multiphase states, the calculation of phase diagrams within the exact +schemes is obstructed due to the absence of one leading parameter, and therefore, to obtain +physically reliable results it is natural to use straightforward techniques, such as the mean field +approximation and the classical Monte Carlo method. +Previously, we developed a minimal model of the HTSC cuprates [5, 6], where the CuO2 planes +are considered as lattices of centers, which are the main element of the crystal and electronic +structure of cuprates. In this model, on-site Hilbert space is formed by three effective valence +states of the CuO4 cluster: [CuO4]7−, [CuO4]6−, and [CuO4]5−. +The necessity to consider +these valence states of CuO4 center on an equal basis is related to the strong relaxation effects +of the electron lattice in cuprates [7, 8]. +The valence states of CuO4 center have different +spin states: s = 1/2 for the [CuO4]6− center and s = 0 for the [CuO4]7− and [CuO4]5−, +respectively, and different symmetry of the orbital states: B1g for the ground states of the +[CuO4]6− center, A1g for the [CuO4]7− center, and the Zhang-Rice A1g or more complicated +low-lying non-Zhang-Rice states for the [CuO4]5− center. For these many-electron states with +strong p−d covalence and strong intra-center correlations, electrons cannot be described within +arXiv:2301.11708v1 [physics.comp-ph] 27 Jan 2023 + +conventional (quasi)particle approach that addresses the [CuO4]7−,6−,5− centers within the on- +site hole representation |n⟩, n = 0, 1, 2, respectively. We make use of a real space on-site S = 1 +pseudospin formalism to describe the charge triplets instead of conventional quasiparticle k- +momentum description. The pseudospin approach is used for the strongly correlated electron +systems [9, 10] and for the superconductivity [11] of cuprates for a long time. In our model, +the effective pseudospin Hamiltonian takes into account both local and nonlocal correlations, +single and two-particle transport, as well as Heisenberg spin-exchange interaction. Earlier, we +investigated a simplified static version of the spin-pseudospin model, for which phase diagrams +of the ground state and at a finite temperature were constructed, both analytically, in the +mean field approximation [12], and as a result of Monte Carlo simulations [13]. The use of +pseudospin formalism provides opportunities for numerical modeling using the well-developed +classical Monte Carlo (MC) method, the construction of phase diagrams and the study of the +features of the thermodynamic properties of the system. A similar effective S = 1 spin-charge +model for cuprates and its MC implementation were considered in papers [14, 15]. +We organize the article as follows. In Section 2, we present the pseudospin formalism and the +effective spin-pseudospin Hamiltonian of the model and introduce quasi-classical approximation. +In Section 3, we formulate the state selection algorithm and explore the features of the probability +distribution. +The results of classical MC simulations of our model and their discussion are +presented in Section 4. +2. Model +We develop a pseudospin model of cuprates [5, 6] where the CuO2 planes are considered as +lattices of CuO4 clusters, which are the main element of the crystal and electronic structure +of cuprates. The on-site Hilbert space is formed by 4 states. The effective valence states of +the cluster, [CuO4]7−, [CuO4]6−, and [CuO4]5−, have different spin states: formally one-hole +[CuO4]6− center is the s = 1/2 doublet, while the [CuO4]7− and [CuO4]5− centers are the spin +singlets. As a result, the basis |SM; sµ⟩ on a given site is the quartet of states +� +|11; 00⟩, +��10; 1 +2 +1 +2 +� +, +��10; 1 +2, − 1 +2 +� +, |1, −1; 00⟩ +� +. +The effective pseudospin Hamiltonian of the model cuprate +H = Hpot + H(1) +kin + H(2) +kin + Hex +(1) +takes into account both local and nonlocal charge correlations +Hpot = +� +i +� +∆S2 +zi − µSzi +� ++ V +� +⟨ij⟩ +SziSzj, +(2) +the three types of the correlated single-particle transport +H(1) +kin = − +� +⟨ij⟩ν +� +tpP ν +i+P ν +j− + tnNν +i+Nν +j− + tpn +2 +� +P ν +i+Nν +j− + Nν +i+P ν +j− +� ++ h.c. +� +, +(3) +the two-particle transport +H(2) +kin = −tb +� +⟨ij⟩ +� +S2 +i+S2 +j− + S2 +j+S2 +i− +� +, +(4) +and finally, the antiferromagnetic Heisenberg spin-exchange interaction for the CuO6− +4 +centers, +Hex = Js2 � +⟨ij⟩ +σiσj, +(5) + +where σ = P0 s/s operators take into account the on-site spin density P0 = 1 − S2 +z, and s +is the spin s = 1/2 operator. +The pseudospin operator Sz in (2) gives the value of charge +counted from ”parent” [CuO4]6− state on a given site, so the term with chemical potential µ +allows to account for the charge density constraint, nN = ⟨� +i Szi⟩ = const. Operators P ν ++ +in (3) create holes with the spin projection ν and change the states +��00; 1 +2, −ν +� +into the states +|11; 00⟩. Likewise, operators Nν ++ also create holes with the spin projection ν, but they transform +the states |1, −1; 00⟩ into +��00; 1 +2ν +� +. Operators S2 ++ in (4) creates the singlet hole pairs on the +[CuO4]7− centers, and, obviously, the following relations for the one-hole and two-hole creation +operators are fulfilled: S2 ++ = P ν ++N−ν ++ . The explicit form of matrices for operators in equations +(2–5) in the basis of states |SM; sµ⟩ is given in Appendix. +Using the quasi-classical approximation, we write the on-site wave function as follows +|Ψ⟩ = c1 |11; 00⟩ + c↑ +��10; 1 +2 +1 +2 +� ++ c↓ +��10; 1 +2, − 1 +2 +� ++ c−1 |1, −1; 00⟩ , +(6) +where the complex coefficients can be written in the following form: +ck = rk eiφk, +� +k +r2 +k = 1, +(7) +with phases φk ∈ [0, 2π], and we parametrize magnitudes rk by angles θ, ϕ, ψ ∈ [0, π +2 ]: +r1 += +cos θ cos ϕ, +(8) +r↑ += +sin θ cos ψ, +(9) +r↓ += +sin θ sin ψ, +(10) +r−1 += +cos θ sin ϕ. +(11) +The average values for all operators in the Hamiltonian (1) are given in Appendix. +The energy for a model (1) in the quasi-classical approximation +E = +� � +i +Ψi +��� H +��� +� +i +Ψi +� +(12) + +have the following form: +E = +� +i +(∆ − µ cos 2ϕi) cos2 θi + V +� +⟨ij⟩ +cos2 θi cos 2ϕi cos2 θj cos 2ϕj − +− tp +2 +� +⟨ij⟩ +sin 2θi cos ϕi sin 2θj cos ϕj +� +cos ψi cos ψj cos (φ1i − φ↑i − φ1j + φ↑j) + ++ sin ψi sin ψj cos (φ1i − φ↓i − φ1j + φ↓j) +� +− +− tn +2 +� +⟨ij⟩ +sin 2θi sin ϕi sin 2θj sin ϕj +� +cos ψi cos ψj cos (φ−1i − φ↑i − φ−1j + φ↑j) + ++ sin ψi sin ψj cos (φ−1i − φ↓i − φ−1j + φ↓j) +� +− +− tpn +4 +� +⟨ij⟩ +sin 2θi sin 2θj +� +cos ϕi sin ϕj +� +sin ψi cos ψj cos (φ1i − φ↓i + φ−1j − φ↑j) + ++ cos ψi sin ψj cos (φ1i − φ↑i + φ−1j − φ↓j) +� ++ ++ sin ϕi cos ϕj +� +sin ψi cos ψj cos (φ−1i − φ↓i + φ1j − φ↑j) + ++ cos ψi sin ψj cos (φ−1i − φ↑i + φ1j − φ↓j) +�� +− +− tb +2 +� +⟨ij⟩ +cos2 θi sin 2ϕi cos2 θj sin 2ϕj cos (φ−1i − φ1i − φ−1j + φ1j) + ++ Js2 � +⟨ij⟩ +sin2 θi sin2 θj +� +sin 2ψi sin 2ψj cos (φ↑i − φ↓i − φ↑j + φ↓j) + cos 2ψi cos 2ψj +� +. +(13) +3. State selection algorithm +The magnitudes of coefficients rk in Eq. (7) correspond to points in the octant of the 4- +dimensional unit sphere. In the Metropolis algorithm, randomly generated states should form a +uniform distribution in the phase space. For the parametrization (8–11), the solid angle element +is dΩ = cos θ sin θ dθ dϕ dψ, thus, the state selection algorithm should consist of generation of +uniformly distributed phases φk ∈ [0, 2π], uniformly distributed angle variables ϕ, ψ ∈ [0, π/2], +and uniformly distributed value m = cos2 θ ∈ [0, 1], where θ ∈ [0, π/2]. In this case, the MC +simulation of model (13) involves using the chemical potential µ as external fixed parameter and +the subsequent recalculation of the results in the variables charge density, n, and temperature, +T. +To study the features of the parametrization (7–11) we can find the on-site charge density +distribution which is generated by the state selection algorithm formulated above. For the on- +site charge density, we obtain the following expression in terms of uniformly distributed variables +ϕ and m: +n = r2 +1 − r2 +−1 = m cos 2ϕ. +(14) +The domains D(n) where m cos 2ϕ < n are shown in Fig. 1(a). Integrating over domain D(n), +we find the on-site charge distribution function F(n) +F(n) = 2 +π +� +D(n) +dm dϕ = − 1 +π arccos n + n +π ln 1 + +√ +1 − n2 +|n| +, +(15) + +Figure 1. (a) The constant value lines for the on-site charge density n defined by Eq. (14); (b) +the probability density function f(n); (c) the probability distribution function F(n). +and the corresponding probability density function f(n) +f(n) = 1 +π ln 1 + +√ +1 − n2 +|n| +. +(16) +These functions are shown in Fig 1(b,c). As a specific feature of the parametrization (7–11), the +probability density f(n) has a logarithmic singularity at n = 0. +One of the phase states in model (1) is the charge ordering. In this case, the function n(µ) +has a typical step-like feature, when a small change in µ causes a large jump in n, from n1 to n2, +so, taking into account the statistical nature of the Monte Carlo method, it is difficult to obtain +trustworthy simulation results for the range (n1, n2). Further, we will consider an algorithm +where the lattice state changes simultaneously on a pair of sites, but the total charge of the +pair is conserved. This ensures the conservation of the total charge of the system during the +simulation and allows us to study in detail the phase states of the system for all n. +If the states of a pair of sites 1 and 2 generated independently, the probability density to +have the charge of the pair 2n = n1 + n2 for a given charge n1 at the site 1 is +f1(n1; 2n) = f(n1)f(2n − n1) +Φ(2n) +(17) +where +Φ(2n) = +� n1,max +n1,min +f(x)f(2n − x) dx, +(18) +and the function f(n) is defined by Eq. (16). The minimal and maximal values of n1 at given +2n are +n1,min(2n) = −1 + n + |n|, +n1,max(2n) = 1 + n − |n|. +(19) +The cumulative distribution function F1(n1; 2n) of the charge n1 at the site 1 for the fixed pair +charge 2n has the following form: +F1(n1; 2n) = +� n1 +n1,min +f1(x; 2n) dx. +(20) +The normalized probability density function f1(t; 2n) = ∆n1f1 (∆n1t + n1,min; 2n), where +∆n1 = n1,max − n1,min, and cumulative distribution function F1(t; 2n) are shown in Fig. 2. The + +Figure 2. The normalized probability density function f1(t; 2n) for values of the pair charge +(a) 2|n| = 0.0, 0.3, 0.6, 0.9; (b) 2|n| = 0.9, 1.0, 1.1, 1.9; (c) the cumulative distribution function +F1(t; 2n) for 2|n| = 0.0, 0.3, 0.6, 1.0, 1.9. +probability density function f1 has logarithmic singularities if 2|n| < 1 as shown in Fig. 2(a), and +the corresponding distribution function F1 has vertical tangents at these points. If 1 ≤ 2|n| < 2, +the probability density function has no singularities, so the distribution function only slightly +deviates from the case of uniform distribution. +The uniform distribution in a phase space entails the constant probability density function +f(ϕ, m) = 2/π in the domain 0 ≤ m ≤ 1, 0 ≤ ϕ ≤ π +2 shown in Fig. 1(a). Since one of the new +variables must be n1, we choose them as (n1, m). The domain in variables (ϕ, m) is mapped +onto the domain in variables (n1, m) shown in Fig.3(a). The new density function p(n1, m) is +defined from equations +p(n1, m) dn1 dm = 2 +π +���� +∂ϕ +∂n1 +���� dn1 dm = +dn1 dm +π +� +m2 − n2 +1 +. +(21) +This allows us to find the conditional density function, +p2(m|n1) = +1 +πf(n1) +� +m2 − n2 +1 +, +(22) +and the conditional distribution function: +F2(m|n1) = +ln +� +m + +� +m2 − n2 +1 +� +− ln |n1| +ln +� +1 + +� +1 − n2 +1 +� +− ln |n1| +, +|n1| ≤ m ≤ 1. +(23) +Fig.3(b,c) show the normalized conditional density function p2(t|n1) = a p2(at + |n1||n1), +a = (1 − |n1|), and corresponding conditional distribution function F2(t|n1) for some values +of n1. The most significant variations of these functions take place in the region of small values +of the parameter n1, therefore, values decreasing on a logarithmic scale are considered. For +the state selection algorithm, it is necessary so solve an equation F2(m|n1) = γ at given n1, so +small values of n1 can potentially lead to large inaccuracies. Fortunately, the explicit solution +of equation F2(m|n1) = γ can be written: +m = 1 +2 +� +|n1|1−γ +� +1 + +� +1 − n2 +1 +�γ ++ |n1|1+γ +� +1 + +� +1 − n2 +1 +�−γ� +. +(24) +The state selection algorithm for the quasi-classical Monte Carlo simulation of the model (1) +that conserves the total charge consists of the following steps: + +Figure 3. +(a) The shaded area is the domain of functions in variables (n1, m); (b) the +conditional density function p2 for given values of n1; (c) the conditional distribution function +F2 for given values of n1. +(i) calculation of the total charge 2n = n1,0 + n2,0 for the randomly selected pair of sites 1 and +2; +(ii) calculation of the value n1 from equation F1(n1; 2n) = γ, where γ ∈ [0, 1] is the uniformly +distributed random value, and the function F1(n1; 2n) is defined by Eq. (20); +(iii) calculation of the value n2 = 2n − n1; +(iv) calculation of values mi, i = 1, 2, from equations F2(mi|ni) = γi, where γ ∈ [0, 1] is the +uniformly distributed random value, the function F2(m|n) is defined by Eq. (23), and the +explicit solution is given by Eq.(24); +(v) calculation of ϕi i = 1, 2, from equations cos(2ϕi) = ni/mi; +(vi) calculation of θi, i = 1, 2, from equations cos2 θi = mi; +(vii) generation of uniformly distributed random values φ(i) +k +∈ [0, 2π], i = 1, 2, k = +1, −1, ↑, ↓, +and ψi ∈ [0, π +2 ], i = 1, 2. +This allows us to find new states on the selected pair of sites using Eq. (6). +4. Results +In MC simulation, we calculated the structure factors +Fq(A, B) = +1 +N2 +� +lm +eiq (rl−rm) ⟨AlBm⟩ , +(25) +where Al and Bm are the on-site operators and the summation is performed over all sites of the +square lattice. To determine the type of ordering, we monitored the following structure factors: +• F(π,π)(σ, σ) for antiferromagnetic (AFM) order, +• F(π,π)(Sz, Sz) for the charge order (CO), +• F(0,0)(S2 ++, S2 +−) for the superconducting order (SC), +• F(0,0)(P +, P) for the “metal” phase (M). +Acknowledgments +The research was supported by UrFU Development Program ”Priority-2030”. + +Appendix +The matrices of pseudospin operators on a given site in the basis +� +|11; 00⟩, +��10; 1 +2 +1 +2 +� +, +��10; 1 +2, − 1 +2 +� +, +|1, −1; 00⟩ +� +have the following form: +Sz = +� +� +� +1 0 0 +0 +0 0 0 +0 +0 0 0 +0 +0 0 0 −1 +� +� +� , S2 +z = +� +� +� +1 0 0 0 +0 0 0 0 +0 0 0 0 +0 0 0 1 +� +� +� , S2 ++ = +� +� +� +0 0 0 1 +0 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� , S2 +− = +� +� +� +0 0 0 0 +0 0 0 0 +0 0 0 0 +1 0 0 0 +� +� +� , +(26) +P ↓ ++ = +� +� +� +0 1 0 0 +0 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� , P ↓ +− = +� +� +� +0 0 0 0 +1 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� , P ↑ ++ = +� +� +� +0 0 1 0 +0 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� , P ↑ +− = +� +� +� +0 0 0 0 +0 0 0 0 +1 0 0 0 +0 0 0 0 +� +� +� , +(27) +N↑ ++ = +� +� +� +0 0 0 0 +0 0 0 1 +0 0 0 0 +0 0 0 0 +� +� +� , N ↑ +− = +� +� +� +0 0 0 0 +0 0 0 0 +0 0 0 0 +0 1 0 0 +� +� +� , N ↓ ++ = +� +� +� +0 0 0 0 +0 0 0 0 +0 0 0 1 +0 0 0 0 +� +� +� , N ↓ +− = +� +� +� +0 0 0 0 +0 0 0 0 +0 0 0 0 +0 0 1 0 +� +� +� , +(28) +σz = +� +� +� +0 0 +0 +0 +0 1 +0 +0 +0 0 −1 0 +0 0 +0 +0 +� +� +� , +σx = +� +� +� +0 0 0 0 +0 0 1 0 +0 1 0 0 +0 0 0 0 +� +� +� , +σy = +� +� +� +0 0 +0 +0 +0 0 −i 0 +0 i +0 +0 +0 0 +0 +0 +� +� +� . +(29) +Using equations (6–11), we can write average values ⟨A⟩ = ⟨Ψ| A |Ψ⟩ for all operators in the +Hamiltonian (1) on a given site: +⟨Sz⟩ += +cos2 θ cos 2ϕ, +(30) +� +S2 +z +� += +cos2 θ, +(31) +� +S2 ++ +� += +1 +2 e−i(φ1−φ−1) cos2 θ sin 2ϕ, +� +S2 +− +� += +� +S2 ++ +�∗ , +(32) +� +P ↑ ++ +� += +1 +2 ei(φ↓−φ1) sin 2θ sin ψ cos ϕ, +� +P ↑ +− +� += +� +P ↑ ++ +�∗, +(33) +� +P ↓ ++ +� += +1 +2 ei(φ↑−φ1) sin 2θ cos ψ cos ϕ, +� +P ↓ +− +� += +� +P ↓ ++ +�∗, +(34) +� +N↑ ++ +� += +1 +2 e−i(φ↑−φ−1) sin 2θ cos ψ sin ϕ, +� +N↑ +− +� += +� +N↑ ++ +�∗, +(35) +� +N↓ ++ +� += +1 +2 e−i(φ↓−φ−1) sin 2θ sin ψ sin ϕ, +� +N↓ +− +� += +� +N↓ ++ +�∗, +(36) +⟨σx⟩ += +sin2 θ sin 2ψ cos (φ↓ − φ↑) , +(37) +⟨σy⟩ += +sin2 θ sin 2ψ sin (φ↓ − φ↑) , +(38) +⟨σz⟩ += +sin2 θ cos 2ψ. +(39) +References +[1] Boˇzovi´c I, He X, Wu J and Bollinger A T 2016 Nature 536 309–311 + +[2] Boˇzovi´c I, Wu J, He X and Bollinger A 2019 Physica C: Superconductivity and its +Applications 558 30–37 +[3] Fradkin E and Kivelson S A 2012 Nature Physics 8 864–866 +[4] Moskvin A S and Panov Y D 2019 Journal of Superconductivity and Novel Magnetism 32 +61–84 +[5] Moskvin A S 2011 Physical Review B 84 075116 +[6] Moskvin A S 2013 Journal of Physics: Condensed Matter 25 085601 +[7] Mallett B P P, Wolf T, Gilioli E, Licci F, Williams G V M, Kaiser A B, Ashcroft N W, +Suresh N and Tallon J L 2013 Physical Review Letters 111 237001 +[8] Moskvin A S and Panov Y D 2019 Physics of the Solid State 61 1553–1558 +[9] Castellani C, Castro C D, Feinberg D and Ranninger J 1979 Physical Review Letters 43 +1957–1960 +[10] Rice T M and Sneddon L 1981 Physical Review Letters 47 689–692 +[11] L¨ow U, Emery V J, Fabricius K and Kivelson S A 1994 Physical Review Letters 72 1918– +1921 +[12] Panov Y, Ulitko V, Budrin K, Chikov A and Moskvin A 2019 Journal of Magnetism and +Magnetic Materials 477 162–166 +[13] Yasinskaya D N, Ulitko V A and Panov Y D 2020 Physics of the Solid State 62 1713–1718 +[14] Cannas S A and Stariolo D A 2019 Physical Review E 99 042137 +[15] Frantz G L K, Schmidt M and Zimmer F M 2021 Physical Review E 103 032125 + diff --git a/ktFKT4oBgHgl3EQfCi2g/content/tmp_files/load_file.txt b/ktFKT4oBgHgl3EQfCi2g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f9af792e881803d5a5af36f118596b4041abd3b --- /dev/null +++ b/ktFKT4oBgHgl3EQfCi2g/content/tmp_files/load_file.txt @@ -0,0 +1,279 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf,len=278 +page_content='Classical Monte Carlo algorithm for simulation of a pseudospin model for cuprates V A Ulitko1, Yu D Panov1, A S Moskvin1 1Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=', Ekaterinburg , Russia E-mail: vasiliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='ulitko@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='ru Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' A classical Monte Carlo algorithm based on the quasi-classical approximation is applied to the pseudospin Hamiltonian of the model cuprate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The model takes into account both local and non-local correlations, Heisenberg spin-exchange interaction, single-particle and correlated two-particle transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' We define the state selection rule that gives both the uniform distribution of states in the phase space and the doped charge conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The simulation results show a qualitative agreement of a phase diagrams with the experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Introduction The phase diagram of doped HTSC cuprates is the subject of active experimental [1, 2] and theoretical research, despite the huge amount of work on this topic to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' A striking feature of the phase diagram of HTSC cuprates is the competition and coexistence of antiferromagnetic, superconducting, and charge orderings [3], manifested in pseudogap phase, strange metal phase, a variety of static and dynamic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The studies are complicated by the presence of heterogeneity due to dopants or non-isovalent substitution, as well as to the internal electronic tendency to heterogeneity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Phase separation may be the cause of simultaneous detection of the preformed pairs and BEC superconductivity in cuprates [2], a number of experimental observations of the typical Fermi liquid behavior, at least in overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' For models describing such complex multiphase states, the calculation of phase diagrams within the exact schemes is obstructed due to the absence of one leading parameter, and therefore, to obtain physically reliable results it is natural to use straightforward techniques, such as the mean field approximation and the classical Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Previously, we developed a minimal model of the HTSC cuprates [5, 6], where the CuO2 planes are considered as lattices of centers, which are the main element of the crystal and electronic structure of cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In this model, on-site Hilbert space is formed by three effective valence states of the CuO4 cluster: [CuO4]7−, [CuO4]6−, and [CuO4]5−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The necessity to consider these valence states of CuO4 center on an equal basis is related to the strong relaxation effects of the electron lattice in cuprates [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The valence states of CuO4 center have different spin states: s = 1/2 for the [CuO4]6− center and s = 0 for the [CuO4]7− and [CuO4]5−, respectively, and different symmetry of the orbital states: B1g for the ground states of the [CuO4]6− center, A1g for the [CuO4]7− center, and the Zhang-Rice A1g or more complicated low-lying non-Zhang-Rice states for the [CuO4]5− center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' For these many-electron states with strong p−d covalence and strong intra-center correlations, electrons cannot be described within arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='11708v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='comp-ph] 27 Jan 2023 conventional (quasi)particle approach that addresses the [CuO4]7−,6−,5− centers within the on- site hole representation |n⟩, n = 0, 1, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' We make use of a real space on-site S = 1 pseudospin formalism to describe the charge triplets instead of conventional quasiparticle k- momentum description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The pseudospin approach is used for the strongly correlated electron systems [9, 10] and for the superconductivity [11] of cuprates for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In our model, the effective pseudospin Hamiltonian takes into account both local and nonlocal correlations, single and two-particle transport, as well as Heisenberg spin-exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Earlier, we investigated a simplified static version of the spin-pseudospin model, for which phase diagrams of the ground state and at a finite temperature were constructed, both analytically, in the mean field approximation [12], and as a result of Monte Carlo simulations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The use of pseudospin formalism provides opportunities for numerical modeling using the well-developed classical Monte Carlo (MC) method, the construction of phase diagrams and the study of the features of the thermodynamic properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' A similar effective S = 1 spin-charge model for cuprates and its MC implementation were considered in papers [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' We organize the article as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In Section 2, we present the pseudospin formalism and the effective spin-pseudospin Hamiltonian of the model and introduce quasi-classical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In Section 3, we formulate the state selection algorithm and explore the features of the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The results of classical MC simulations of our model and their discussion are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Model We develop a pseudospin model of cuprates [5, 6] where the CuO2 planes are considered as lattices of CuO4 clusters, which are the main element of the crystal and electronic structure of cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The on-site Hilbert space is formed by 4 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The effective valence states of the cluster, [CuO4]7−, [CuO4]6−, and [CuO4]5−, have different spin states: formally one-hole [CuO4]6− center is the s = 1/2 doublet, while the [CuO4]7− and [CuO4]5− centers are the spin singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' As a result, the basis |SM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' sµ⟩ on a given site is the quartet of states � |11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩, ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2 1 2 � , ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2, − 1 2 � , |1, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The effective pseudospin Hamiltonian of the model cuprate H = Hpot + H(1) kin + H(2) kin + Hex (1) takes into account both local and nonlocal charge correlations Hpot = � i � ∆S2 zi − µSzi � + V � ⟨ij⟩ SziSzj, (2) the three types of the correlated single-particle transport H(1) kin = − � ⟨ij⟩ν � tpP ν i+P ν j− + tnNν i+Nν j− + tpn 2 � P ν i+Nν j− + Nν i+P ν j− � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � , (3) the two-particle transport H(2) kin = −tb � ⟨ij⟩ � S2 i+S2 j− + S2 j+S2 i− � , (4) and finally, the antiferromagnetic Heisenberg spin-exchange interaction for the CuO6− 4 centers, Hex = Js2 � ⟨ij⟩ σiσj, (5) where σ = P0 s/s operators take into account the on-site spin density P0 = 1 − S2 z, and s is the spin s = 1/2 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The pseudospin operator Sz in (2) gives the value of charge counted from ”parent” [CuO4]6− state on a given site, so the term with chemical potential µ allows to account for the charge density constraint, nN = ⟨� i Szi⟩ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Operators P ν + in (3) create holes with the spin projection ν and change the states ��00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2, −ν � into the states |11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Likewise, operators Nν + also create holes with the spin projection ν, but they transform the states |1, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩ into ��00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Operators S2 + in (4) creates the singlet hole pairs on the [CuO4]7− centers, and, obviously, the following relations for the one-hole and two-hole creation operators are fulfilled: S2 + = P ν +N−ν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The explicit form of matrices for operators in equations (2–5) in the basis of states |SM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' sµ⟩ is given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Using the quasi-classical approximation, we write the on-site wave function as follows |Ψ⟩ = c1 |11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩ + c↑ ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2 1 2 � + c↓ ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2, − 1 2 � + c−1 |1, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩ , (6) where the complex coefficients can be written in the following form: ck = rk eiφk, � k r2 k = 1, (7) with phases φk ∈ [0, 2π], and we parametrize magnitudes rk by angles θ, ϕ, ψ ∈ [0, π 2 ]: r1 = cos θ cos ϕ, (8) r↑ = sin θ cos ψ, (9) r↓ = sin θ sin ψ, (10) r−1 = cos θ sin ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (11) The average values for all operators in the Hamiltonian (1) are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='The energy for a model (1) in the quasi-classical approximation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='E = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='Ψi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='��� H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='sin ψi cos ψj cos (φ1i − φ↓i + φ−1j − φ↑j) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='+ cos ψi sin ψj cos (φ1i − φ↑i + φ−1j − φ↓j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='+ sin ϕi cos ϕj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='sin ψi cos ψj cos (φ−1i − φ↓i + φ1j − φ↑j) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='+ cos ψi sin ψj cos (φ−1i − φ↑i + φ1j − φ↓j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='− tb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='⟨ij⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='cos2 θi sin 2ϕi cos2 θj sin 2ϕj cos (φ−1i − φ1i − φ−1j + φ1j) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='+ Js2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='⟨ij⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='sin2 θi sin2 θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='sin 2ψi sin 2ψj cos (φ↑i − φ↓i − φ↑j + φ↓j) + cos 2ψi cos 2ψj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' State selection algorithm The magnitudes of coefficients rk in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (7) correspond to points in the octant of the 4- dimensional unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In the Metropolis algorithm, randomly generated states should form a uniform distribution in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' For the parametrization (8–11), the solid angle element is dΩ = cos θ sin θ dθ dϕ dψ, thus, the state selection algorithm should consist of generation of uniformly distributed phases φk ∈ [0, 2π], uniformly distributed angle variables ϕ, ψ ∈ [0, π/2], and uniformly distributed value m = cos2 θ ∈ [0, 1], where θ ∈ [0, π/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In this case, the MC simulation of model (13) involves using the chemical potential µ as external fixed parameter and the subsequent recalculation of the results in the variables charge density, n, and temperature, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' To study the features of the parametrization (7–11) we can find the on-site charge density distribution which is generated by the state selection algorithm formulated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' For the on- site charge density, we obtain the following expression in terms of uniformly distributed variables ϕ and m: n = r2 1 − r2 −1 = m cos 2ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (14) The domains D(n) where m cos 2ϕ < n are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Integrating over domain D(n), we find the on-site charge distribution function F(n) F(n) = 2 π � D(n) dm dϕ = − 1 π arccos n + n π ln 1 + √ 1 − n2 |n| , (15) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (a) The constant value lines for the on-site charge density n defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (b) the probability density function f(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (c) the probability distribution function F(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' and the corresponding probability density function f(n) f(n) = 1 π ln 1 + √ 1 − n2 |n| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (16) These functions are shown in Fig 1(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' As a specific feature of the parametrization (7–11), the probability density f(n) has a logarithmic singularity at n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' One of the phase states in model (1) is the charge ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' In this case, the function n(µ) has a typical step-like feature, when a small change in µ causes a large jump in n, from n1 to n2, so, taking into account the statistical nature of the Monte Carlo method, it is difficult to obtain trustworthy simulation results for the range (n1, n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Further, we will consider an algorithm where the lattice state changes simultaneously on a pair of sites, but the total charge of the pair is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' This ensures the conservation of the total charge of the system during the simulation and allows us to study in detail the phase states of the system for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' If the states of a pair of sites 1 and 2 generated independently, the probability density to have the charge of the pair 2n = n1 + n2 for a given charge n1 at the site 1 is f1(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) = f(n1)f(2n − n1) Φ(2n) (17) where Φ(2n) = � n1,max n1,min f(x)f(2n − x) dx, (18) and the function f(n) is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The minimal and maximal values of n1 at given 2n are n1,min(2n) = −1 + n + |n|, n1,max(2n) = 1 + n − |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (19) The cumulative distribution function F1(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) of the charge n1 at the site 1 for the fixed pair charge 2n has the following form: F1(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) = � n1 n1,min f1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (20) The normalized probability density function f1(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) = ∆n1f1 (∆n1t + n1,min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n), where ∆n1 = n1,max − n1,min, and cumulative distribution function F1(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The normalized probability density function f1(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) for values of the pair charge (a) 2|n| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (b) 2|n| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (c) the cumulative distribution function F1(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) for 2|n| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' probability density function f1 has logarithmic singularities if 2|n| < 1 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2(a), and the corresponding distribution function F1 has vertical tangents at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' If 1 ≤ 2|n| < 2, the probability density function has no singularities, so the distribution function only slightly deviates from the case of uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The uniform distribution in a phase space entails the constant probability density function f(ϕ, m) = 2/π in the domain 0 ≤ m ≤ 1, 0 ≤ ϕ ≤ π 2 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Since one of the new variables must be n1, we choose them as (n1, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The domain in variables (ϕ, m) is mapped onto the domain in variables (n1, m) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The new density function p(n1, m) is defined from equations p(n1, m) dn1 dm = 2 π ���� ∂ϕ ∂n1 ���� dn1 dm = dn1 dm π � m2 − n2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (21) This allows us to find the conditional density function, p2(m|n1) = 1 πf(n1) � m2 − n2 1 , (22) and the conditional distribution function: F2(m|n1) = ln � m + � m2 − n2 1 � − ln |n1| ln � 1 + � 1 − n2 1 � − ln |n1| , |n1| ≤ m ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (23) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content='3(b,c) show the normalized conditional density function p2(t|n1) = a p2(at + |n1||n1), a = (1 − |n1|), and corresponding conditional distribution function F2(t|n1) for some values of n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' The most significant variations of these functions take place in the region of small values of the parameter n1, therefore, values decreasing on a logarithmic scale are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' For the state selection algorithm, it is necessary so solve an equation F2(m|n1) = γ at given n1, so small values of n1 can potentially lead to large inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Fortunately, the explicit solution of equation F2(m|n1) = γ can be written: m = 1 2 � |n1|1−γ � 1 + � 1 − n2 1 �γ + |n1|1+γ � 1 + � 1 − n2 1 �−γ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (24) The state selection algorithm for the quasi-classical Monte Carlo simulation of the model (1) that conserves the total charge consists of the following steps: Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (a) The shaded area is the domain of functions in variables (n1, m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (b) the conditional density function p2 for given values of n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (c) the conditional distribution function F2 for given values of n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (i) calculation of the total charge 2n = n1,0 + n2,0 for the randomly selected pair of sites 1 and 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (ii) calculation of the value n1 from equation F1(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) = γ, where γ ∈ [0, 1] is the uniformly distributed random value, and the function F1(n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 2n) is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (iii) calculation of the value n2 = 2n − n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (iv) calculation of values mi, i = 1, 2, from equations F2(mi|ni) = γi, where γ ∈ [0, 1] is the uniformly distributed random value, the function F2(m|n) is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (23), and the explicit solution is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (24);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (v) calculation of ϕi i = 1, 2, from equations cos(2ϕi) = ni/mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (vi) calculation of θi, i = 1, 2, from equations cos2 θi = mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (vii) generation of uniformly distributed random values φ(i) k ∈ [0, 2π], i = 1, 2, k = +1, −1, ↑, ↓, and ψi ∈ [0, π 2 ], i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' This allows us to find new states on the selected pair of sites using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Results In MC simulation, we calculated the structure factors Fq(A, B) = 1 N2 � lm eiq (rl−rm) ⟨AlBm⟩ , (25) where Al and Bm are the on-site operators and the summation is performed over all sites of the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' To determine the type of ordering, we monitored the following structure factors: F(π,π)(σ, σ) for antiferromagnetic (AFM) order, F(π,π)(Sz, Sz) for the charge order (CO), F(0,0)(S2 +, S2 −) for the superconducting order (SC), F(0,0)(P +, P) for the “metal” phase (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Acknowledgments The research was supported by UrFU Development Program ”Priority-2030”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' Appendix The matrices of pseudospin operators on a given site in the basis � |11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩, ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2 1 2 � , ��10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 1 2, − 1 2 � , |1, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' 00⟩ � have the following form: Sz = � � � 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' S2 z = � � � 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' S2 + = � � � 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' S2 − = � � � 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (26) P ↓ + = � � � 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' P ↓ − = � � � 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' P ↑ + = � � � 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' P ↑ − = � � � 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (27) N↑ + = � � � 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' N ↑ − = � � � 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' N ↓ + = � � � 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' N ↓ − = � � � 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (28) σz = � � � 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' σx = � � � 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' σy = � � � 0 0 0 0 0 0 −i 0 0 i 0 0 0 0 0 0 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (29) Using equations (6–11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' we can write average values ⟨A⟩ = ⟨Ψ| A |Ψ⟩ for all operators in the Hamiltonian (1) on a given site: ⟨Sz⟩ = cos2 θ cos 2ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (30) � S2 z � = cos2 θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (31) � S2 + � = 1 2 e−i(φ1−φ−1) cos2 θ sin 2ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � S2 − � = � S2 + �∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (32) � P ↑ + � = 1 2 ei(φ↓−φ1) sin 2θ sin ψ cos ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � P ↑ − � = � P ↑ + �∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (33) � P ↓ + � = 1 2 ei(φ↑−φ1) sin 2θ cos ψ cos ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � P ↓ − � = � P ↓ + �∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (34) � N↑ + � = 1 2 e−i(φ↑−φ−1) sin 2θ cos ψ sin ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � N↑ − � = � N↑ + �∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (35) � N↓ + � = 1 2 e−i(φ↓−φ−1) sin 2θ sin ψ sin ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' � N↓ − � = � N↓ + �∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (36) ⟨σx⟩ = sin2 θ sin 2ψ cos (φ↓ − φ↑) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (37) ⟨σy⟩ = sin2 θ sin 2ψ sin (φ↓ − φ↑) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (38) ⟨σz⟩ = sin2 θ cos 2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' (39) References [1] Boˇzovi´c I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfCi2g/content/2301.11708v1.pdf'} +page_content=' He X,' 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a/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/2301.13372v1.pdf.txt b/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/2301.13372v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78a06527a6222f986c167100e3cb5b1ee7f4a4c5 --- /dev/null +++ b/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/2301.13372v1.pdf.txt @@ -0,0 +1,966 @@ +Improving Open-Domain Dialogue Evaluation +with a Causal Inference Model +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +Abstract Effective evaluation methods remain a significant challenge for research +on open-domain conversational dialogue systems. Explicit satisfaction ratings can +be elicited from users, but users often do not provide ratings when asked, and those +they give can be highly subjective. Post-hoc ratings by experts are an alternative, but +these can be both expensive and complex to collect. Here, we explore the creation of +automated methods for predicting both expert and user ratings of open-domain dia- +logues. We compare four different approaches. First, we train a baseline model using +an end-to-end transformer to predict ratings directly from the raw dialogue text. The +other three methods are variants of a two-stage approach in which we first extract in- +terpretable features at the turn level that capture, among other aspects, user dialogue +behaviors indicating contradiction, repetition, disinterest, compliments, or criticism. +We project these features to the dialogue level and train a dialogue-level MLP re- +gression model, a dialogue-level LSTM, and a novel causal inference model called +counterfactual-LSTM (CF-LSTM) to predict ratings. The proposed CF-LSTM is a +sequential model over turn-level features which predicts ratings using multiple re- +gressors depending on hypotheses derived from the turn-level features. As a causal +inference model, CF-LSTM aims to learn the underlying causes of a specific event, +such as a low rating. We also bin the user ratings and perform classification experi- +ments with all four models. In evaluation experiments on conversational data from +the Alexa Prize SocialBot, we show that the CF-LSTM achieves the best perfor- +mance for predicting dialogue ratings and classification. +Key words: open-domain dialogue, user ratings, dialogue evaluation, causal infer- +ence, user satisfaction +Cat P. Le +Duke University, e-mail: cat.le@duke.edu. +This work was done while Cat P. Le was a research intern at Amazon Alexa AI. +Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +Amazon Alexa AI, e-mail: \{lukedai,mjohnstn,yangliud,wamarilk,ghanadan\ +}@amazon.com +1 +arXiv:2301.13372v1 [cs.CL] 31 Jan 2023 + +2 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +1 Introduction +Evaluation has always been a complex challenge for interactive dialogue systems. +For task-oriented dialogues, frameworks such as Paradise [51] model the relation- +ship between user satisfaction, task completion, and cost factors such as dialogue +length, word error rate, and dialogue behaviors. However, open-domain dialogue +systems such as those built for the Alexa Prize SocialBot Grand Challenge [36, 9], +where there is no clearly defined task, require new metrics and methods for evalua- +tion that better reflect their affordances [49, 21, 13, 12, 16, 47]. +In Alexa Prize SocialBot, user conversation ratings are elicited primarily for +comparative evaluation of systems competing in the Grand Challenge. Numerous +other use cases for dialogue ratings include selecting training data and running A/B +tests to evaluate new capabilities, features, and models. Manually collected user rat- +ings are limited in that only a fraction of users of “Alexa Let’s Chat” leave ratings, +and these ratings can be highly subjective. An alternative is the post-hoc annota- +tion of dialogues by trained annotators, which is complex, expensive, and thus not +scalable. Consequently, a significant amount of dialogue data is unavailable for ex- +perimentation and modeling tasks. +To overcome this problem and increase the number of rated dialogues, we eval- +uate four methods for automatically predicting both user (128K) and expert (1.2K) +ratings of de-identified conversations collected from interactions with an Alexa So- +cialBot. First, we train a baseline method using a transformer-based model [5] to +predict dialogue ratings based on the raw input dialogue sequence. The other three +methods are variants of a two-stage approach in which we use pre-trained models +to extract interpretable features at the turn level. These models include classifiers +we call ODES (Open Domain Evaluation Signals) that detect user inputs such as +criticism, insults, compliments, requests to stop, and statements that indicate mis- +understanding, repetition, or contradiction among other classes [49]. DialoGPT [57] +and DialogRPT [10] are applied to the dialogue sequence to provide measures of re- +sponse relevance and specificity, and FED [30] is applied to each turn-pair to provide +metrics for the interestingness, engagingness, specificity, relevance, correctness, se- +mantic appropriateness, understandability and fluency of responses. In addition to +these derived measures, we also include ASR confidence and sentiment scores based +on processing the acoustics of user utterances. These features are projected to the +dialogue level as averages, counts, initial, final, and penultimate values and used as +input to our three models. +The first model trains an MLP regressor to predict dialogue ratings. The second +model, called dialogue-level LSTM, uses an LSTM model on the sequence of turn- +level features. The third approach models the rating using a shared sequential model +over the turn-pair level vectors and introduces a causal structure into the predic- +tion model. This proposed Counterfactual-LSTM (CF-LSTM) is a causal inference +model that uses different models to predict the overall rating under different hy- +potheses derived from the ODES classifiers. In particular, a treatment (e.g., 0 or 1) +is applied to each dialog based on the text classification. For instance, dialog with +negative texts, such as critique, disinterest, and insult, will be assigned a treatment + +Improving Open-Domain Dialogue Evaluation with a Causal Inference Model +3 +of 1. On the other hand, dialogs without negative texts will be assigned a treatment +of 0. Next, a causal inference neural network is trained using the dialog features and +their corresponding treatments to predict the customer’s rating. This approach intro- +duces the domain knowledge into the prediction model and helps further improve +the performance. Multiple different regression functions are used based on the hy- +pothesis. Lastly, we apply all four models to classify dialogues into bins according +to their ratings (e.g., binary and 5-class classification). Our contributions are: +1. We identify and train classifiers to extract various types of dialogue features (e.g., +sentiment, response relevance, and specificity) and use these features to predict +dialogue ratings. +2. We observe the special conditions of the open-domain dialogue evaluation prob- +lem and make three assumptions to apply causal inference to this problem. +3. We introduce three approaches and propose a novel causal inference model, CF- +LSTM, that is used to evaluate open-domain dialogues via mapping hidden dia- +logue features to the dialogue ratings. +2 Related Work +Much recent work on dialogue evaluation has focused on the utterance level, mea- +suring the goodness of a particular system response in the dialogue context. This +type of evaluation can be either reference-based or referenceless. Reference-based +approaches re-apply metrics from tasks such as machine translation or summariza- +tion and compare a proposed system utterance to a reference response, which may +be from a pre-existing dialogue [15], or collected via crowd-sourcing [20]. For in- +stance, BLEU [32] computes the n-gram precision of the system’s response string +when compared to a reference response, while METEOR [3] considers the stems +and synonyms of the reference and ROUGE [28] uses n-gram recall instead of n- +gram precision. More recent neural versions of these metrics, such as BERTScore +and BLEURT, use word embeddings rather than the raw strings [55, 41, 43]. How- +ever, it is well-known that these metrics have limited value for task-oriented dia- +logue and even less utility when evaluating the highly varied responses typical of +open-domain dialogues [38, 11, 29]. +Work on referenceless metrics aims to predict the quality of system response +in the dialogue context without using reference utterances. These metrics evaluate +the proposed system utterance in terms of coherence, interestingness, engagingness, +fluency, specificity, relevance, empathy, or other measures. For example, GRADE +measures coherence in terms of a graph-based representation of topic relatedness +[19]. DialogRPT can be used to provide metrics that estimate the likelihood that an +utterance generates many responses, directly or indirectly [10], based on training +with 133M utterance pairs from Reddit, consisting of utterances and their upvotes, +downvotes, and number of responses. +Another idea is to score a proposed system utterance by the probability that it +elicits a particular user response type, such as disinterest or criticism [46, 31, 49, 12]. + +4 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +For example, the FED framework scores proposed utterances by utilizing the Di- +aloGPT LM probabilities for the subsequent user utterances such as “That is in- +teresting” [56, 31]. Predictive models can also be generated from large dialogue +corpora by using the following user utterance as weak supervision [12, 46]. +Other research aims to predict the overall quality of the dialogue by developing +models for predicting a quality measure in terms of particular types of dialogue +events. The most straightforward framework is end-to-end (E2E) approaches using +the raw dialogue text as input to a transformer model [13, 5, 14]. This approach +performs well with expert-rated data but does not predict user-assigned ratings well. +We use this approach as our E2E baseline below. +Other work first derives several intermediate metrics at the turn or utterance level +and then uses those to predict the dialogue quality, as we do here. The first frame- +work like this was PARADISE, which included task completion as an input to the +model, along with other metrics such as ASR error rates, system dialogue acts, or +user requests to start over [51, 50]. Recent work on PARADISE has used other +features, such as user complaints and compliments, to apply it to open-domain di- +alogue [49]. Many recent approaches work with embedding features, such as senti- +ment scores and utterance classification, before predicting the user rating [53, 14]. +Sentiment analysis [21, 12] and dialogue features analysis [30, 42, 49] methods, for +instance, show solid results for predicting user-assigned ratings. +Furthermore, many approaches to open-domain dialogue evaluation use large- +scale corpora [4, 7, 6, 18, 44]. Since the Alexa Prize challenge was introduced in +recent years, several new controlled dialogue datasets have been created [54, 37, +27, 15]. However, evaluating the Alexa Prize SocialBot remains a challenging prob- +lem due to the nature of the collected dialogues. These dialogues are de-identified +conversations between thousands of users and the Alexa SocialBot covering numer- +ous topics and genres, with widely varying lengths (e.g., the SocialBot dialogues +vary in length from 3 to 230 turn-pairs). Additionally, the SocialBot is expected to +have up-to-date knowledge about news, sports, movies, music, books, etc. Hence, +the evaluation of the Alexa SocialBot requires a dynamic structure that takes into +account domain knowledge. +In this paper, we introduce a novel causal inference structure to open-domain di- +alogue evaluation to improve the performance of the prediction model for the Alexa +SocialBot. Causal inference [40, 39] has been extensively investigated in the neural +network literature [8, 45, 1, 52, 2] because of the black box nature of neural mod- +els. Notably, the TARNet model [45], a causal inference neural network, has been +applied to causal inference for various regression and classification problems. It can +be applied to many machine learning models, such as image generative models [52] +and Gaussian processes [1]. However, we have yet to be aware of previous work +that uses causal inference for open-domain dialogue evaluation. Here we propose a +novel LSTM causal structure for predicting the quality of open-domain dialogue to +make the evaluation model more robust. + +Improving Open-Domain Dialogue Evaluation with a Causal Inference Model +5 +3 Proposed Methods +We propose three approaches to tackle open-domain dialogue evaluation. These ap- +proaches convert each turn pair in the dialogue into a vector representation, then +process the vectors to predict the dialogue level rating. Our proposed methods con- +sist of two main components. +1. Dialogue Turn-level Feature Extractors. These extractors are applied to extract +meaningful hidden features from the dialogue as in Figure 1. Subsequently, these +turn-level features are converted to the dialogue-level features that are fed into +the prediction models. +2. Prediction Models. We utilize an end-to-end transformer on the raw dialogue +text to create a baseline. We then compare this baseline to three proposed mod- +els for predicting both user and expert ratings from the hidden dialogue-level +features: +• +Dialogue-level MLP (Section 3.2) +• +Dialogue-level LSTM (Section 3.2) +• +Counterfactual LSTM (Section 3.3) +The dialogue-level MLP uses an MLP regressor to project the turn features to the +user ratings, while the dialogue-level LSTM utilizes a standard LSTM to predict +ratings from the sequence of turn-level features. The counterfactual LSTM (CF- +LSTM) is a novel causal inference model. It consists of the LSTM layers augmented +with the causal analysis model. This causal model consists of multiple MLP regres- +sors, each trained to predict the ratings based on a specific hypothesis. +3.1 Dialogue Turn-level Feature Extractors +To extract features for each utterance-response pair, we apply 6 models (i.e., 2 mod- +els for the utterances and 4 models for the responses), as shown in Figure 1. We ap- +ply the Open-Dialogue Evaluation Signals (ODES) Classifier and an Audio-based +Sentiment Classifier for the utterance data. The ODES classifier, which is based on +the T5 architecture [35], was trained on 8K annotated utterances to categorize them +into the classes shown in Table 1: (i) user disinterest, (ii) user critique, (iii) user +not understand, (iv) user requests topic switch, (v) user obscenity, (vi) user rejects +topic switch, (vii) user requests repeat, (viii) user requests to stop, (ix) user insult, +(x) user compliment, (xi) user calls out repetition, (xii) user calls out contradiction, +(xiii) system not understand, and (xiv) others. This T5 classifier achieves above 95% +accuracy on the testing corpora. Next, the ODES categories assign the treatment T +for the CF-LSTM. +We also extract 3 different types of utterance sentiment features: sentiment va- +lence, satisfaction, and activation, which are highly correlated with user ratings +[21, 12]. + +6 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +Fig. 1 The dialogue turn-level feature extractors consist of 2 utterance models (i.e., ODES clas- +sifier, sentiment classifier) and 4 response models (i.e., FED model, Mean Norm IDF, DialoGPT +Relevance, and DialogRPT Specificity). The treatment assignments for our causal model are ex- +tracted using the ODES classifiers. +To generate scores for the responses, we apply the FED model [30], the Di- +aloGPT model [56] for response relevance, the DialogRPT model[10] for response +specificity width and depth, and Mean Norm IDF as another measure of response +specificity [30, 56]. The concatenation of these extracted features forms the repre- +sentation of the utterance-response pair, called turn-level features. These features +are normalized to their Z-scores before being converted into dialogue-level features +to train the prediction models. +3.2 Dialogue-level MLP and LSTM +The dialogue-level MLP is the two-level approach that uses the MLP [33] regres- +sion to project the turn pair features to the ratings. Here, we consider the dialogue +features that are an aggregation of the turn-level features by choosing a few fun- +damental values, such as average, initial, final, and penultimate values of the se- +quence of turn-level features. In this case, the dialogue features (i.e., input of the + +Dialogue i +Utterance Models +Turn-level Features +Utterance +ODES +ODES +Treatment +Classifier +Categories +Response +Sentiment +Sentiment +Sentiment +Utterance +Categories +Classifier +Valence +Response +ASR +Sentiment +Confidence +Satisfaction +User +Response Models +Sentiment +Utterance +Activation +FED +System +FED Metrics +Response +Model +Utterance +Specificity Mean +Mean Norm IDF +Norm IDF +Response +DialoGPT +Response +Relevance +Relevance +Utterance +Response +DialogRPT +Response +Specificity +Specificity (W) +.. : +Utterance +Response +Response +Specificity (D)Improving Open-Domain Dialogue Evaluation with a Causal Inference Model +7 +Table 1 Examples of dialogues for each ODES category in the Alexa SocialBot dataset. +Class ODES Name +Counts +Example +i +User +10,938 +SocialBot: Did you know that otters sleep holding hands? +disinterest +User: I really couldn’t care less. +ii +User +12,290 +SocialBot: Who do you think will win Superbowl 2019? +critique +User: You are really a very stupid bot. +iii +User not +12,399 +SocialBot: What is your favorite genre of video game? +understand +User: I don’t know what genre means. +iv +User requests +21,278 +SocialBot: Who is your favorite Batman actor? +topic switch +User: Can we talk about something else? +v +User +66,532 +SocialBot: Do you have a favorite movie? +obscenity +User: B*** me. +vi +User rejects +4,389 +SocialBot: I love country music too. But I was wondering, +topic switch +do you have a favorite sport? +User: No, I want to keep talking about music. +vii +User requests +26,779 +SocialBot: Would you rather climb the Eiffel Tower or the +to repeat +Empire State Building? +User: Could you say that again? +viii +User requests +78,504 +SocialBot: I just love talking about music. What is your +to stop +favorite kind of music? +User: Please stop I need to go to bed. +ix +User insult +12,173 +SocialBot: Wow, Taylor Swift has 97 albums. That’s a lot! +User: You are so full of sh**. +x +User +57,539 +SocialBot: Did you know LeBron James Jr has a college +compliment +football scholarship, and he’s only 12 years old? +User: That’s really interesting. +xi +User calls out +15,052 +SocialBot: What’s your favourite football team? +repetition +User: You already asked me that question twice. +xii +User calls out +3,147 +SocialBot: I don’t have any pets. +contradiction +User: You just said you had a cat. +xiii +System not +12,534 +User: Can we talk about Elle King? +understand +SocialBot: I like BB King too. +User: That’s not what I said. +xiv +Others +2,320,515 SocialBot: Do you like K-Pop music? +User: Yes, I often listen to Blackpink and BTS. +Dialogue-level MLP) have the same length and can be fed into the MLP regressor +for prediction. +The dialogue-level LSTM is a method that uses a standard LSTM [17] on the +dialogue-level features (i.e., a sequence of turn-level features) as in Table 3.1. This +model is similar to the one used for speech sentiment analysis [21], in which the +LSTM model is used to map the sequence of turn-level sentiment features (i.e., +sentiment valence, sentiment satisfaction, sentiment activation sentiment scores) to +the ratings. Here, the input of the model is the concatenated turn-level features. +Since this model is based on LSTM layers, it can handle the varied-length input +(e.g., dialogue-level feature vectors). + +8 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +3.3 Counterfactual LSTM +In recent natural language processing (NLP) tasks, the assumption that training and +test data are identically distributed is often not satisfied. Bias and variance from the +data are often introduced into the standard NLP approaches (e.g., non-causal mod- +els), which makes the training procedure difficult, and leads to poor results in real- +world test data. We propose using a causal structure containing domain knowledge +to overcome this issue. This structure helps introduce inductive biases and leads to +more robust predictors [8]. To apply causal inference to our problem, three common +conditions must be satisfied: Ignorability, Positivity, and Consistency [8]. +• +Ignorability. In causal inference and statistics, the ignorability requires that the +data collection method be independent of the treatment assignment. In other +words, the outcomes (i.e., factual and counterfactual) do not depend on the treat- +ment assignment. +T ⊥⊥ Y(T = a), +∀a ∈ {0,1} +(1) +• +Positivity. The probability of a sample being assigned the treatment T = a for +a ∈ {0,1} is bounded between 0 and 1. +0 < IP(T = a +��X = x) < 1, +∀a ∈ {0,1} +(2) +• +Consistency. A sample under a treatment T = a, for a ∈ {0,1} has an unique +outcome. The outcome of a sample under treatment T = a is identical to the +outcome if that sample is again assigned to T = a. +T = a ⇐⇒ Y(a) = Y, +∀a ∈ {0,1} +(3) +Here, the positivity and consistency conditions are satisfied. To satisfy the ignor- +ability condition, we must assume that the extracted dialogue features, described in +Section 3.1, capture everything relevant about the dialogue to predict the ratings. In +other words, we assume that all confounding variables X are observed and that there +is no hidden confounder. This property, called conditional ignorability, is described +as follows: +� +T ⊥⊥ (Y(a) +���X, +∀a ∈ {0,1} +(4) +We attempt to satisfy the ignorability condition by using many features in our +models that were proposed in previous work, as well as introducing our own novel +ODES classifiers, but we acknowledge that future work may derive additional fea- +Fig. 2 The causal relation- +ship between outcomes, treat- +ments, and input features. The +outcome Y (e.g., customer +ratings) is a function of the +dialogue features X, and the +treatments T. + +Dialog Features +X +Rating +T +TreatmentImproving Open-Domain Dialogue Evaluation with a Causal Inference Model +9 +Fig. 3 The diagram of the proposed counterfactual LSTM model. The model consists of the LSTM +layers and the causal structure h0,h1. It learns to map the dialogue-level features to the user ratings +based on the treatment T. +tures from the dialogue events, such as dialogue acts or sub-dialogue level features +[50, 34]. +Next, we propose a novel causal inference model called the Counterfactual +LSTM, which maps the dialogue-level features to the ratings based on a specific +hypothesis. This paper considers a simple case with only 2 hypotheses. First, the +turn-level features are stacked into a dialogue-level feature vector for each dialogue. +Since each dialogue has a different number of turn pairs, the dialogue-level features +are varied-length vectors. These dialogue-level features represent the dialogue tex- +tual data fed into the CF-LSTM model to predict the dialogue ratings. As shown +in Figure 3, the proposed CF-LSTM model consists of the LSTM layers and 2 in- +dividual MLP regressors, each used to train on a specific hypothesis or treatment. +The inputs of the LSTM layers are the dialogue-level feature vectors, which are se- +quences of turn-level features. Since the LSTM can take varied-length inputs, this +property also holds for CF-LSTM and can take varied-length dialogue-level feature +vectors. Next, the output features Φ of the LSTM are fed to the specific MLP (h0 or +h1) based on the hypothesis and trained to predict dialogue ratings. +3.3.1 Definition of Treatments +We utilize the extracted ODES features from Section 3.1 as the treatment groups. +Let T denote the treatment assignment. The treatment T = 0 is the hypothesis group +whose dialogues consist of only the ODES category (xiv), i.e., other. The treatment +T = 1 is the hypothesis group whose dialogues consist of one or more ODES cat- +egories (i) to (xii). Since the CF-LSTM has a dedicated MLP for each dialogue +hypothesis, it can predict dialogue ratings for similar dialogue with a different hy- +pothesis. For instance, two dialogues with different dialogue ratings have the same +negative sentiment features. One dialogue is categorized into ODES group (i), i.e., +user disinterest, and is assigned T = 1. On the other hand, the sentiment of the other +dialogue is classified as unfavorable, and the user appears to be satisfied with the + +ho +Turn 1 +T=O +Dialogue-level +yo +features +Turn 2 +LMSE (ho(Φ), Yo) +Φ +Turn 3 +LSTM +treatment T +Dialogue i +Turn 4 +Turn n +LMSE (h(Φ), Y) +Turn-level +treatments +IPM +features +W(p(Φ, T=0), p(Φ, T=1)10 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +conversation, e.g., T = 0. Non-causal regressors can quickly group these dialogues +since they contain the same sentiment features. Still, the CF-LSTM with a causal +structure can identify the difference between these two dialogues and better predict +the rating. +3.3.2 Loss Function of CF-LSTM +Here, we define the loss function for our proposed CF-LSTM model. First, most +dialogues with the treatment T = 1 have relatively low ratings. For example, the +dialogues in group (viii), i.e., user insult, will often receive ratings of 2 or less. +Additionally, the ratings are given non-uniformly among different groups of users. +These are the source of bias and variance in the dataset, and a large proportion of +users choose not to leave a rating. In order to avoid introducing variance and bias +into the model, the Integral probability metric (IPM) is established in [45]. This +metric is used as the upper bound for this source of variance. The IPM measures +the distance between two distributions: p(Φ,T = 0), p(Φ,T = 1). In this paper, we +use Wasserstein distance [48] as the IPM measurement in our CF-LSTM model. +Let Y0 and Y1 denote the ground truth ratings for the treatment T = 0 and T = 1, +respectively. The loss function for the CF-LSTM model is defined as follows: +L = LMSE +� +h0(Φ),Y0 +��T = 0 +� ++LMSE +� +h1(Φ),Y1 +��T = 1 +� ++W +� +p(Φ,T = 0), p(Φ,T = 1) +� +(5) +where L denotes the loss function of the CF-LSTM model, LMSE is the mean +square error (MSE) loss, W is the Wasserstein distance, h0() and h1() are the outputs +of the MLP regressors h0 and h1, respectively. This loss function consists of two +main parts, the MSE loss, and the IPM loss. The MSE loss is the error between +the predicted and ground truth ratings under every hypothesis (e.g., T = 0 and T = +1). The IPM loss is defined as the Wasserstein distance between 2 distributions of +ratings described above. +3.3.3 Data Generation for Classification Tasks +We consider the classification task of detecting low-quality dialogues at run-time. +The model needs to be trained with sufficient data samples to dynamically identify +the low-rated dialogues with high accuracy. It is often not satisfied in practice since +most collected data samples have good ratings. As a result, we want to generate +more dialogues with low ratings so that the model can be trained to identify these +types of data quickly. It is also essential since it determines whether the model can +be applied to evaluate dialogues at run-time. This online-evaluation mechanism can +also improve the overall user experience since the system could navigate to a new +topic if it realizes that the current conversation is rated poorly. Here, we first select +the poorly-rated dialogue i and augment these data by masking utterances. We can + +Improving Open-Domain Dialogue Evaluation with a Causal Inference Model +11 +Fig. 4 The procedure of generating new dialogues by masking utterances. Let dialogue i be the +low-rated dialogue with n+1 utterances. Each utterance is given the treatment of 1 and 0. At turn +m, if the treatment for utterance m is 1, we define a new poorly-rated dialogue with the utterances +0 to m and masking the remaining utterances, i.e., utterances (m+1) to n. +assign a treatment to each utterance in the dialogue from the extracted turn-level +features. At utterance m, if the treatment is 1, we define a new dialogue, labeled as +0 (i.e., poorly rated dialogue), that consists of the utterance 0 to the utterance m. In +other words, we mask all of the utterances after m from the original dialogue, as +illustrated in Figure 4. Lastly, this newly generated data is used to fine-tune the clas- +sification model. The resulting model can be applied online to dynamically improve +the user experience at run-time or used as a data selection mechanism that identifies +high-quality dialogues from low-quality dialogues. +3.3.4 Transfer Learning with CF-LSTM +As described above, the CF-LSTM is designed to predict the rating based on a +fixed number of hypotheses (e.g., 2) in the dataset. However, CF-LSTM can han- +dle the increase or decrease in the number of hypotheses using transfer learn- +ing [23, 25, 22, 24, 26]. In other words, we can transfer the prior knowledge from +the pre-trained CF-LSTM to a new causal NLP task using transfer learning tech- +niques [23, 22]. For example, the new data samples arrive with added treatment +types. Here, we modify the trained model by adding new MLP regressors (e.g., rep- +resenting different hypotheses) to the existing structure. We initialize the weights +randomly for the added regressors while freezing the other weights of the pre-trained + +Dialogue i +T +Utterance 0 +0 +The poorly-rated generated dialogues +Response +Utterance 0 +Utterance 0 +Utterance O +Response +Response +Response +Utterance 1 +1 +Response +Utterance 1 +Utterance 1 +Utterance 1 +Response +Response +Response +1 +Utterance 2 +Response +Utterance 2 +Utterance 2 +Response +Response +Utterance n-1 +1 +Response +Utterance n-1 +Response +0 +Utterance n +Response12 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +model. Lastly, we update the entire model by fine-tuning it using the new data sam- +ples. In the next section, we discuss the experimental results of our proposed meth- +ods when applied to our real-world dataset for both regression and classification +problems. +4 Experimental Results +In the following experiments, we apply our proposed methods to a dataset of di- +alogues with user ratings collected from users interacting with the Alexa Social- +Bot [36]. This dataset consists of 128K de-identified conversational dialogues. On +average, each dialogue has 20 utterances, and the data is limited to dialogue consist- +ing of at least three turn-pairs. For each utterance in a dialogue, we assign treatments +based on the ODES categories (Please see Section 3.3.1). Table 1 indicates the num- +ber of utterances of each ODES category in the Alexa SocialBot dataset. When con- +sidering this dataset, 70% of dialogues are assigned T = 0, and 30% of dialogues +are assigned T = 1. Next, we compare our methods to the baseline approach, the +End-to-end (E2E) Transformer. It directly uses a transformer-based model [5] to +map the dialogue textual data into the user ratings. +First, we consider the regression problem of predicting the user ratings for the +given dialogue. Let L1d denote the daily average prediction and L7d denote the 7- +day rolling daily average prediction. Table 2 shows the three methods’ performances +when compared with the E2E baseline in terms of the Pearson correlation between +the predicted ratings and the ground truth. As shown in Table 2, CF-LSTM achieves +higher correlations in the individual prediction, the daily average prediction, and the +7-day rolling daily average prediction. It also shows a performance boost compared +to the Dialogue-level MLP approach, which suffers from information loss when +converting turn-level to dialogue-level features. +Table 2 The comparisons between open-dialogue evaluation methods for the regression problem +in terms of Pearson correlation (e.g., individual, L1d, L7d predictions) and the classification prob- +lems (e.g., binary, 5-class) in terms of prediction accuracy, on SocialBot conversations +INDIVIDUAL +L1D⋆ +L7D† +BINARY 5-CLASS +METHODS +PREDICTION PREDICTION PREDICTION CLASS. CLASS. +E2E TRANSFORMER +0.22 +0.30 +0.47 +54.1% +32.8% +DIALOGUE-LEVEL LSTM +0.30 +0.41 +0.59 +64.6% +43.5% +DIALOGUE-LEVEL MLP +0.31 +0.40 +0.66 +62.5% +46.1% +CF-LSTM +0.34 +0.46 +0.68 +67.8% +48.2% +⋆ DAILY AVERAGE PREDICTION, † 7-DAY ROLLING AVERAGE PREDICTION +In addition, to understand the impact of assigning the treatment (i.e., ODES cat- +egories) to the dialogues, we consider the situation where the treatments of the col- + +Improving Open-Domain Dialogue Evaluation with a Causal Inference Model +13 +Table 3 The correlations of +CF-LSTM when the treatment +assignments for all dialogues +are inverted. +PEARSON +ORIGINAL +INVERTED +CORRELATION +TREATMENTS TREATMENTS +INDIVIDUAL PREDICTION +0.34 +0.26 +L1D PREDICTION +0.46 +0.35 +L7D PREDICTION +0.68 +0.52 +⋆ DAILY AVERAGE, † 7-DAY ROLLING AVERAGE +lected dialogues are inverted (e.g., treatment T = 0 becomes T = 1, and vice versa). +For the same dialogue feature vector, its treatment is inverted. This setting allows +us to investigate the impact of the ODES Classifier when assigning the incorrect +treatments and whether the models can overcome this misinformation. We feed the +modified data samples to the well-trained CF-LSTM capable of predicting the coun- +terfactual outcomes for these dialogues. Table 3 shows the correlations between the +predicted counterfactual outcomes and the original ground truth. Here, we want to +investigate whether the model can predict correctly even if the treatment assign- +ment is inverted (e.g., the ODES classifier malfunctions). As shown in Table 3, the +individual prediction and daily average prediction (L1d) of the inverted treatments +scenario still have a relatively high correlation with the original ground truth (i.e., +when the treatments are not inverted). The mean square error between the factual +outcomes Y1 and the counterfactual outcomes Y0 is 0.3034. The average treatment +effect (ATE) is defined as follows for our problem: +ATE = E[Y1i −Y0i] = −0.7809 +The ATE shows that, on average, a dialogue being assigned treatment T = 1 (e.g., +consists of disinterest, insult) will have a lower user rating than the dialogue with +T = 1 by 0.7809. +We randomly select a set of 1213 dialogues and ask an expert to rate the con- +versational quality. Let Rexpert be the rating given by the expert, and Ruser be the +ratings given by users. The expert ratings have a low Pearson correlation with the +user ratings, demonstrating the challenge in predicting user ratings in open-domain +dialogue since they are only loosely correlated with an expert evaluation of dialogue +quality. +ρ = corr(Rexpert,Ruser) = 0.2056 +Next, we use the CF-LSTM and the Dialogue-level MLP to predict the expert’s +rating in the given dialogues. Table 4 indicates the correlations obtained from CF- +LSTM and Dialogue-level MLP on expert-rated dialogues from Socialbot conver- +sations. Due to the simplicity of the network architecture, the Dialogue-level MLP +achieves competitive results in individual and daily average predictions. However, it +performs poorly in 7-day rolling average predictions because of the sparsity across +time of the small dataset of 1213 dialogues. On the other hand, CF-LSTM performs +well on all types of predictions and shows flexibility in training for both large and +small datasets. + +14 +Cat P. Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan +Table 4 The correlations of +prediction models on 1213 +expert-rated dialogues from +Socialbot conversations. +PEARSON +DIALOGUE- CF-LSTM +CORRELATION +LEVEL MLP +(OURS) +INDIVIDUAL PREDICTION +0.35 +0.26 +L1D PREDICTION +0.42 +0.36 +L7D PREDICTION +0.12 +0.51 +⋆ DAILY AVERAGE, † 7-DAY ROLLING AVERAGE +Lastly, we consider the problem of classifying the dialogues into 2-class (i.e., bi- +nary) and 5-class classification. For binary classification, we label any dialogue with +a rating less than 3 as 0 (e.g., low-rated dialogues). Hence, the remaining dialogues +with ratings greater or equal to 3 are labeled as 1 (i.e., highly-rated dialogues). For +the 5-class classification, we round up or down (i.e., round half-up) the ratings into +five groups. For example, a dialogue with a rating of 4.5 is classified into group 5, +and a dialogue with a rating of 3.4 is classified into group 3. In this dataset, we ob- +serve that most dialogues are often rated above 3 out of 5. As a result, we apply the +data generation method described in Section 3.3.3 to generate more poorly-rated +dialogues. This newly generated data is used to fine-tune the binary classification +model. The resulting model can be applied online to improve the user experience +at run-time. Table 2 indicates the performance of our model and other methods, in +terms of classification accuracy, in binary and 5-class classification problems. Our +proposed method CF-LSTM outperforms other approaches in both binary classifi- +cation and 5-class classification. +5 Conclusion and Future Work +We propose a novel causal inference structure for open-domain dialogue evalua- +tion. This structure utilizes turn-level features, such as sentiment analysis, textual +categories, response relevance, and response specificity, and uses them to predict +user ratings. Our method performs competitively in an evaluation with rated con- +versations with the Alexa SocialBot, in regression and classification problems. As +a causal inference model, this CF-LSTM model is more robust in learning com- +plex representations and capable of predicting the ratings for the dialogues under +different hypotheses. In addition to offline uses, this model can also be applied at +run-time to dynamically identify low-quality dialogues and trigger the introduction +of different topics to help improve the user experience. This paper considers only 2 +treatments, which are all the ODES categories, versus others. In future works, we +can further boost the flexibility of our model by increasing the number of treatments +corresponding to the different ODES groups. 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Dolan. +Dialogpt: Large-scale generative pre-training for conversational response generation, 2020. + diff --git a/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/load_file.txt b/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..941ebacf9ed14cb7ee70fff328cf80a818ddda4f --- /dev/null +++ b/l9FQT4oBgHgl3EQfoTY7/content/tmp_files/load_file.txt @@ -0,0 +1,1099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf,len=1098 +page_content='Improving Open-Domain Dialogue Evaluation with a Causal Inference Model Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan Abstract Effective evaluation methods remain a significant challenge for research on open-domain conversational dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Explicit satisfaction ratings can be elicited from users, but users often do not provide ratings when asked, and those they give can be highly subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Post-hoc ratings by experts are an alternative, but these can be both expensive and complex to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, we explore the creation of automated methods for predicting both expert and user ratings of open-domain dia- logues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We compare four different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' First, we train a baseline model using an end-to-end transformer to predict ratings directly from the raw dialogue text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The other three methods are variants of a two-stage approach in which we first extract in- terpretable features at the turn level that capture, among other aspects, user dialogue behaviors indicating contradiction, repetition, disinterest, compliments, or criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We project these features to the dialogue level and train a dialogue-level MLP re- gression model, a dialogue-level LSTM, and a novel causal inference model called counterfactual-LSTM (CF-LSTM) to predict ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The proposed CF-LSTM is a sequential model over turn-level features which predicts ratings using multiple re- gressors depending on hypotheses derived from the turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As a causal inference model, CF-LSTM aims to learn the underlying causes of a specific event, such as a low rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We also bin the user ratings and perform classification experi- ments with all four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In evaluation experiments on conversational data from the Alexa Prize SocialBot, we show that the CF-LSTM achieves the best perfor- mance for predicting dialogue ratings and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Key words: open-domain dialogue, user ratings, dialogue evaluation, causal infer- ence, user satisfaction Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le Duke University, e-mail: cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='le@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This work was done while Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le was a research intern at Amazon Alexa AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan Amazon Alexa AI, e-mail: \\{lukedai,mjohnstn,yangliud,wamarilk,ghanadan\\ }@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='13372v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='CL] 31 Jan 2023 2 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan 1 Introduction Evaluation has always been a complex challenge for interactive dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For task-oriented dialogues, frameworks such as Paradise [51] model the relation- ship between user satisfaction, task completion, and cost factors such as dialogue length, word error rate, and dialogue behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' However, open-domain dialogue systems such as those built for the Alexa Prize SocialBot Grand Challenge [36, 9], where there is no clearly defined task, require new metrics and methods for evalua- tion that better reflect their affordances [49, 21, 13, 12, 16, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In Alexa Prize SocialBot, user conversation ratings are elicited primarily for comparative evaluation of systems competing in the Grand Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Numerous other use cases for dialogue ratings include selecting training data and running A/B tests to evaluate new capabilities, features, and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Manually collected user rat- ings are limited in that only a fraction of users of “Alexa Let’s Chat” leave ratings, and these ratings can be highly subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' An alternative is the post-hoc annota- tion of dialogues by trained annotators, which is complex, expensive, and thus not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Consequently, a significant amount of dialogue data is unavailable for ex- perimentation and modeling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' To overcome this problem and increase the number of rated dialogues, we eval- uate four methods for automatically predicting both user (128K) and expert (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2K) ratings of de-identified conversations collected from interactions with an Alexa So- cialBot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' First, we train a baseline method using a transformer-based model [5] to predict dialogue ratings based on the raw input dialogue sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The other three methods are variants of a two-stage approach in which we use pre-trained models to extract interpretable features at the turn level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These models include classifiers we call ODES (Open Domain Evaluation Signals) that detect user inputs such as criticism, insults, compliments, requests to stop, and statements that indicate mis- understanding, repetition, or contradiction among other classes [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' DialoGPT [57] and DialogRPT [10] are applied to the dialogue sequence to provide measures of re- sponse relevance and specificity, and FED [30] is applied to each turn-pair to provide metrics for the interestingness, engagingness, specificity, relevance, correctness, se- mantic appropriateness, understandability and fluency of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In addition to these derived measures, we also include ASR confidence and sentiment scores based on processing the acoustics of user utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These features are projected to the dialogue level as averages, counts, initial, final, and penultimate values and used as input to our three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The first model trains an MLP regressor to predict dialogue ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The second model, called dialogue-level LSTM, uses an LSTM model on the sequence of turn- level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The third approach models the rating using a shared sequential model over the turn-pair level vectors and introduces a causal structure into the predic- tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This proposed Counterfactual-LSTM (CF-LSTM) is a causal inference model that uses different models to predict the overall rating under different hy- potheses derived from the ODES classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In particular, a treatment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', 0 or 1) is applied to each dialog based on the text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For instance, dialog with negative texts, such as critique, disinterest, and insult, will be assigned a treatment Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 3 of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' On the other hand, dialogs without negative texts will be assigned a treatment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Next, a causal inference neural network is trained using the dialog features and their corresponding treatments to predict the customer’s rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This approach intro- duces the domain knowledge into the prediction model and helps further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Multiple different regression functions are used based on the hy- pothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Lastly, we apply all four models to classify dialogues into bins according to their ratings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', binary and 5-class classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Our contributions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We identify and train classifiers to extract various types of dialogue features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', sentiment, response relevance, and specificity) and use these features to predict dialogue ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We observe the special conditions of the open-domain dialogue evaluation prob- lem and make three assumptions to apply causal inference to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We introduce three approaches and propose a novel causal inference model, CF- LSTM, that is used to evaluate open-domain dialogues via mapping hidden dia- logue features to the dialogue ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 2 Related Work Much recent work on dialogue evaluation has focused on the utterance level, mea- suring the goodness of a particular system response in the dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This type of evaluation can be either reference-based or referenceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Reference-based approaches re-apply metrics from tasks such as machine translation or summariza- tion and compare a proposed system utterance to a reference response, which may be from a pre-existing dialogue [15], or collected via crowd-sourcing [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For in- stance, BLEU [32] computes the n-gram precision of the system’s response string when compared to a reference response, while METEOR [3] considers the stems and synonyms of the reference and ROUGE [28] uses n-gram recall instead of n- gram precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' More recent neural versions of these metrics, such as BERTScore and BLEURT, use word embeddings rather than the raw strings [55, 41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' How- ever, it is well-known that these metrics have limited value for task-oriented dia- logue and even less utility when evaluating the highly varied responses typical of open-domain dialogues [38, 11, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Work on referenceless metrics aims to predict the quality of system response in the dialogue context without using reference utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These metrics evaluate the proposed system utterance in terms of coherence, interestingness, engagingness, fluency, specificity, relevance, empathy, or other measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For example, GRADE measures coherence in terms of a graph-based representation of topic relatedness [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' DialogRPT can be used to provide metrics that estimate the likelihood that an utterance generates many responses, directly or indirectly [10], based on training with 133M utterance pairs from Reddit, consisting of utterances and their upvotes, downvotes, and number of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Another idea is to score a proposed system utterance by the probability that it elicits a particular user response type, such as disinterest or criticism [46, 31, 49, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 4 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan For example, the FED framework scores proposed utterances by utilizing the Di- aloGPT LM probabilities for the subsequent user utterances such as “That is in- teresting” [56, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Predictive models can also be generated from large dialogue corpora by using the following user utterance as weak supervision [12, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Other research aims to predict the overall quality of the dialogue by developing models for predicting a quality measure in terms of particular types of dialogue events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The most straightforward framework is end-to-end (E2E) approaches using the raw dialogue text as input to a transformer model [13, 5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This approach performs well with expert-rated data but does not predict user-assigned ratings well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We use this approach as our E2E baseline below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Other work first derives several intermediate metrics at the turn or utterance level and then uses those to predict the dialogue quality, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The first frame- work like this was PARADISE, which included task completion as an input to the model, along with other metrics such as ASR error rates, system dialogue acts, or user requests to start over [51, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Recent work on PARADISE has used other features, such as user complaints and compliments, to apply it to open-domain di- alogue [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Many recent approaches work with embedding features, such as senti- ment scores and utterance classification, before predicting the user rating [53, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Sentiment analysis [21, 12] and dialogue features analysis [30, 42, 49] methods, for instance, show solid results for predicting user-assigned ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Furthermore, many approaches to open-domain dialogue evaluation use large- scale corpora [4, 7, 6, 18, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Since the Alexa Prize challenge was introduced in recent years, several new controlled dialogue datasets have been created [54, 37, 27, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' However, evaluating the Alexa Prize SocialBot remains a challenging prob- lem due to the nature of the collected dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These dialogues are de-identified conversations between thousands of users and the Alexa SocialBot covering numer- ous topics and genres, with widely varying lengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', the SocialBot dialogues vary in length from 3 to 230 turn-pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Additionally, the SocialBot is expected to have up-to-date knowledge about news, sports, movies, music, books, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Hence, the evaluation of the Alexa SocialBot requires a dynamic structure that takes into account domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In this paper, we introduce a novel causal inference structure to open-domain di- alogue evaluation to improve the performance of the prediction model for the Alexa SocialBot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Causal inference [40, 39] has been extensively investigated in the neural network literature [8, 45, 1, 52, 2] because of the black box nature of neural mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Notably, the TARNet model [45], a causal inference neural network, has been applied to causal inference for various regression and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It can be applied to many machine learning models, such as image generative models [52] and Gaussian processes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' However, we have yet to be aware of previous work that uses causal inference for open-domain dialogue evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here we propose a novel LSTM causal structure for predicting the quality of open-domain dialogue to make the evaluation model more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 5 3 Proposed Methods We propose three approaches to tackle open-domain dialogue evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These ap- proaches convert each turn pair in the dialogue into a vector representation, then process the vectors to predict the dialogue level rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Our proposed methods con- sist of two main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Dialogue Turn-level Feature Extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These extractors are applied to extract meaningful hidden features from the dialogue as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Subsequently, these turn-level features are converted to the dialogue-level features that are fed into the prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Prediction Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We utilize an end-to-end transformer on the raw dialogue text to create a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We then compare this baseline to three proposed mod- els for predicting both user and expert ratings from the hidden dialogue-level features: Dialogue-level MLP (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2) Dialogue-level LSTM (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2) Counterfactual LSTM (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3) The dialogue-level MLP uses an MLP regressor to project the turn features to the user ratings, while the dialogue-level LSTM utilizes a standard LSTM to predict ratings from the sequence of turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The counterfactual LSTM (CF- LSTM) is a novel causal inference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It consists of the LSTM layers augmented with the causal analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This causal model consists of multiple MLP regres- sors, each trained to predict the ratings based on a specific hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 Dialogue Turn-level Feature Extractors To extract features for each utterance-response pair, we apply 6 models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', 2 mod- els for the utterances and 4 models for the responses), as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We ap- ply the Open-Dialogue Evaluation Signals (ODES) Classifier and an Audio-based Sentiment Classifier for the utterance data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The ODES classifier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' which is based on the T5 architecture [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' was trained on 8K annotated utterances to categorize them into the classes shown in Table 1: (i) user disinterest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (ii) user critique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (iii) user not understand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (iv) user requests topic switch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (v) user obscenity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (vi) user rejects topic switch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (vii) user requests repeat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (viii) user requests to stop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (ix) user insult,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (x) user compliment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (xi) user calls out repetition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (xii) user calls out contradiction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' (xiii) system not understand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' and (xiv) others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This T5 classifier achieves above 95% accuracy on the testing corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Next, the ODES categories assign the treatment T for the CF-LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We also extract 3 different types of utterance sentiment features: sentiment va- lence, satisfaction, and activation, which are highly correlated with user ratings [21, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 6 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 1 The dialogue turn-level feature extractors consist of 2 utterance models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', ODES clas- sifier, sentiment classifier) and 4 response models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', FED model, Mean Norm IDF, DialoGPT Relevance, and DialogRPT Specificity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The treatment assignments for our causal model are ex- tracted using the ODES classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' To generate scores for the responses, we apply the FED model [30], the Di- aloGPT model [56] for response relevance, the DialogRPT model[10] for response specificity width and depth, and Mean Norm IDF as another measure of response specificity [30, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The concatenation of these extracted features forms the repre- sentation of the utterance-response pair, called turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These features are normalized to their Z-scores before being converted into dialogue-level features to train the prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2 Dialogue-level MLP and LSTM The dialogue-level MLP is the two-level approach that uses the MLP [33] regres- sion to project the turn pair features to the ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, we consider the dialogue features that are an aggregation of the turn-level features by choosing a few fun- damental values, such as average, initial, final, and penultimate values of the se- quence of turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In this case, the dialogue features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' input of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Dialogue i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Turn-level Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='ODES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='ODES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Treatment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Categories ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Categories ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Valence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='ASR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Confidence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Satisfaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Sentiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='FED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='FED Metrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Specificity Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Mean Norm IDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Norm IDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='DialoGPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Relevance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Relevance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='DialogRPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Specificity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Specificity (W) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='. : Utterance Response Response Specificity (D)Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 7 Table 1 Examples of dialogues for each ODES category in the Alexa SocialBot dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Class ODES Name Counts Example i User 10,938 SocialBot: Did you know that otters sleep holding hands?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' disinterest User: I really couldn’t care less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' ii User 12,290 SocialBot: Who do you think will win Superbowl 2019?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' critique User: You are really a very stupid bot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' iii User not 12,399 SocialBot: What is your favorite genre of video game?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' understand User: I don’t know what genre means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' iv User requests 21,278 SocialBot: Who is your favorite Batman actor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' topic switch User: Can we talk about something else?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' v User 66,532 SocialBot: Do you have a favorite movie?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' obscenity User: B*** me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' vi User rejects 4,389 SocialBot: I love country music too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' But I was wondering, topic switch do you have a favorite sport?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: No, I want to keep talking about music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' vii User requests 26,779 SocialBot: Would you rather climb the Eiffel Tower or the to repeat Empire State Building?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: Could you say that again?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' viii User requests 78,504 SocialBot: I just love talking about music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' What is your to stop favorite kind of music?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: Please stop I need to go to bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' ix User insult 12,173 SocialBot: Wow, Taylor Swift has 97 albums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' That’s a lot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: You are so full of sh**.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' x User 57,539 SocialBot: Did you know LeBron James Jr has a college compliment football scholarship, and he’s only 12 years old?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: That’s really interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' xi User calls out 15,052 SocialBot: What’s your favourite football team?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' repetition User: You already asked me that question twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' xii User calls out 3,147 SocialBot: I don’t have any pets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' contradiction User: You just said you had a cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' xiii System not 12,534 User: Can we talk about Elle King?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' understand SocialBot: I like BB King too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: That’s not what I said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' xiv Others 2,320,515 SocialBot: Do you like K-Pop music?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' User: Yes, I often listen to Blackpink and BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Dialogue-level MLP) have the same length and can be fed into the MLP regressor for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The dialogue-level LSTM is a method that uses a standard LSTM [17] on the dialogue-level features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', a sequence of turn-level features) as in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This model is similar to the one used for speech sentiment analysis [21], in which the LSTM model is used to map the sequence of turn-level sentiment features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', sentiment valence, sentiment satisfaction, sentiment activation sentiment scores) to the ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, the input of the model is the concatenated turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Since this model is based on LSTM layers, it can handle the varied-length input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', dialogue-level feature vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 8 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3 Counterfactual LSTM In recent natural language processing (NLP) tasks, the assumption that training and test data are identically distributed is often not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Bias and variance from the data are often introduced into the standard NLP approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', non-causal mod- els), which makes the training procedure difficult, and leads to poor results in real- world test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We propose using a causal structure containing domain knowledge to overcome this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This structure helps introduce inductive biases and leads to more robust predictors [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' To apply causal inference to our problem, three common conditions must be satisfied: Ignorability, Positivity, and Consistency [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Ignorability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In causal inference and statistics, the ignorability requires that the data collection method be independent of the treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In other words, the outcomes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', factual and counterfactual) do not depend on the treat- ment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' T ⊥⊥ Y(T = a), ∀a ∈ {0,1} (1) Positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The probability of a sample being assigned the treatment T = a for a ∈ {0,1} is bounded between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 0 < IP(T = a ��X = x) < 1, ∀a ∈ {0,1} (2) Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' A sample under a treatment T = a, for a ∈ {0,1} has an unique outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The outcome of a sample under treatment T = a is identical to the outcome if that sample is again assigned to T = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' T = a ⇐⇒ Y(a) = Y, ∀a ∈ {0,1} (3) Here, the positivity and consistency conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' To satisfy the ignor- ability condition, we must assume that the extracted dialogue features, described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1, capture everything relevant about the dialogue to predict the ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In other words, we assume that all confounding variables X are observed and that there is no hidden confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This property, called conditional ignorability, is described as follows: � T ⊥⊥ (Y(a) ���X, ∀a ∈ {0,1} (4) We attempt to satisfy the ignorability condition by using many features in our models that were proposed in previous work, as well as introducing our own novel ODES classifiers, but we acknowledge that future work may derive additional fea- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 2 The causal relation- ship between outcomes, treat- ments, and input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The outcome Y (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', customer ratings) is a function of the dialogue features X, and the treatments T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Dialog Features X Rating T TreatmentImproving Open-Domain Dialogue Evaluation with a Causal Inference Model 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3 The diagram of the proposed counterfactual LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The model consists of the LSTM layers and the causal structure h0,h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It learns to map the dialogue-level features to the user ratings based on the treatment T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' tures from the dialogue events, such as dialogue acts or sub-dialogue level features [50, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Next, we propose a novel causal inference model called the Counterfactual LSTM, which maps the dialogue-level features to the ratings based on a specific hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This paper considers a simple case with only 2 hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' First, the turn-level features are stacked into a dialogue-level feature vector for each dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Since each dialogue has a different number of turn pairs, the dialogue-level features are varied-length vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These dialogue-level features represent the dialogue tex- tual data fed into the CF-LSTM model to predict the dialogue ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As shown in Figure 3, the proposed CF-LSTM model consists of the LSTM layers and 2 in- dividual MLP regressors, each used to train on a specific hypothesis or treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The inputs of the LSTM layers are the dialogue-level feature vectors, which are se- quences of turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Since the LSTM can take varied-length inputs, this property also holds for CF-LSTM and can take varied-length dialogue-level feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Next, the output features Φ of the LSTM are fed to the specific MLP (h0 or h1) based on the hypothesis and trained to predict dialogue ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 Definition of Treatments We utilize the extracted ODES features from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 as the treatment groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Let T denote the treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The treatment T = 0 is the hypothesis group whose dialogues consist of only the ODES category (xiv), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The treatment T = 1 is the hypothesis group whose dialogues consist of one or more ODES cat- egories (i) to (xii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Since the CF-LSTM has a dedicated MLP for each dialogue hypothesis, it can predict dialogue ratings for similar dialogue with a different hy- pothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For instance, two dialogues with different dialogue ratings have the same negative sentiment features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' One dialogue is categorized into ODES group (i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', user disinterest, and is assigned T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' On the other hand, the sentiment of the other dialogue is classified as unfavorable, and the user appears to be satisfied with the ho Turn 1 T=O Dialogue-level yo features Turn 2 LMSE (ho(Φ), Yo) Φ Turn 3 LSTM treatment T Dialogue i Turn 4 Turn n LMSE (h(Φ), Y) Turn-level treatments IPM features W(p(Φ, T=0), p(Φ, T=1)10 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan conversation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Non-causal regressors can quickly group these dialogues since they contain the same sentiment features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Still, the CF-LSTM with a causal structure can identify the difference between these two dialogues and better predict the rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2 Loss Function of CF-LSTM Here, we define the loss function for our proposed CF-LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' First, most dialogues with the treatment T = 1 have relatively low ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For example, the dialogues in group (viii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', user insult, will often receive ratings of 2 or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Additionally, the ratings are given non-uniformly among different groups of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' These are the source of bias and variance in the dataset, and a large proportion of users choose not to leave a rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In order to avoid introducing variance and bias into the model, the Integral probability metric (IPM) is established in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This metric is used as the upper bound for this source of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The IPM measures the distance between two distributions: p(Φ,T = 0), p(Φ,T = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In this paper, we use Wasserstein distance [48] as the IPM measurement in our CF-LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Let Y0 and Y1 denote the ground truth ratings for the treatment T = 0 and T = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The loss function for the CF-LSTM model is defined as follows: L = LMSE � h0(Φ),Y0 ��T = 0 � +LMSE � h1(Φ),Y1 ��T = 1 � +W � p(Φ,T = 0), p(Φ,T = 1) � (5) where L denotes the loss function of the CF-LSTM model, LMSE is the mean square error (MSE) loss, W is the Wasserstein distance, h0() and h1() are the outputs of the MLP regressors h0 and h1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This loss function consists of two main parts, the MSE loss, and the IPM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The MSE loss is the error between the predicted and ground truth ratings under every hypothesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', T = 0 and T = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The IPM loss is defined as the Wasserstein distance between 2 distributions of ratings described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3 Data Generation for Classification Tasks We consider the classification task of detecting low-quality dialogues at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The model needs to be trained with sufficient data samples to dynamically identify the low-rated dialogues with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It is often not satisfied in practice since most collected data samples have good ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As a result, we want to generate more dialogues with low ratings so that the model can be trained to identify these types of data quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It is also essential since it determines whether the model can be applied to evaluate dialogues at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This online-evaluation mechanism can also improve the overall user experience since the system could navigate to a new topic if it realizes that the current conversation is rated poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, we first select the poorly-rated dialogue i and augment these data by masking utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We can Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 4 The procedure of generating new dialogues by masking utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Let dialogue i be the low-rated dialogue with n+1 utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Each utterance is given the treatment of 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' At turn m, if the treatment for utterance m is 1, we define a new poorly-rated dialogue with the utterances 0 to m and masking the remaining utterances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', utterances (m+1) to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' assign a treatment to each utterance in the dialogue from the extracted turn-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' At utterance m, if the treatment is 1, we define a new dialogue, labeled as 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', poorly rated dialogue), that consists of the utterance 0 to the utterance m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In other words, we mask all of the utterances after m from the original dialogue, as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Lastly, this newly generated data is used to fine-tune the clas- sification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The resulting model can be applied online to dynamically improve the user experience at run-time or used as a data selection mechanism that identifies high-quality dialogues from low-quality dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='4 Transfer Learning with CF-LSTM As described above, the CF-LSTM is designed to predict the rating based on a fixed number of hypotheses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', 2) in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' However, CF-LSTM can han- dle the increase or decrease in the number of hypotheses using transfer learn- ing [23, 25, 22, 24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In other words, we can transfer the prior knowledge from the pre-trained CF-LSTM to a new causal NLP task using transfer learning tech- niques [23, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For example, the new data samples arrive with added treatment types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, we modify the trained model by adding new MLP regressors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', rep- resenting different hypotheses) to the existing structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We initialize the weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='randomly for the added regressors while freezing the other weights of the pre-trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Dialogue i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='The poorly-rated generated dialogues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance n-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance n-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Utterance n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Response12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Lastly, we update the entire model by fine-tuning it using the new data sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In the next section, we discuss the experimental results of our proposed meth- ods when applied to our real-world dataset for both regression and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 4 Experimental Results In the following experiments, we apply our proposed methods to a dataset of di- alogues with user ratings collected from users interacting with the Alexa Social- Bot [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This dataset consists of 128K de-identified conversational dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' On average, each dialogue has 20 utterances, and the data is limited to dialogue consist- ing of at least three turn-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For each utterance in a dialogue, we assign treatments based on the ODES categories (Please see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 1 indicates the num- ber of utterances of each ODES category in the Alexa SocialBot dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' When con- sidering this dataset, 70% of dialogues are assigned T = 0, and 30% of dialogues are assigned T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Next, we compare our methods to the baseline approach, the End-to-end (E2E) Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It directly uses a transformer-based model [5] to map the dialogue textual data into the user ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' First, we consider the regression problem of predicting the user ratings for the given dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Let L1d denote the daily average prediction and L7d denote the 7- day rolling daily average prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 2 shows the three methods’ performances when compared with the E2E baseline in terms of the Pearson correlation between the predicted ratings and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As shown in Table 2, CF-LSTM achieves higher correlations in the individual prediction, the daily average prediction, and the 7-day rolling daily average prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' It also shows a performance boost compared to the Dialogue-level MLP approach, which suffers from information loss when converting turn-level to dialogue-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 2 The comparisons between open-dialogue evaluation methods for the regression problem in terms of Pearson correlation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', individual, L1d, L7d predictions) and the classification prob- lems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', binary, 5-class) in terms of prediction accuracy, on SocialBot conversations INDIVIDUAL L1D⋆ L7D† BINARY 5-CLASS METHODS PREDICTION PREDICTION PREDICTION CLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' CLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' E2E TRANSFORMER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='47 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='8% DIALOGUE-LEVEL LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='59 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='6% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='5% DIALOGUE-LEVEL MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='66 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='5% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='1% CF-LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='68 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='8% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2% ⋆ DAILY AVERAGE PREDICTION, † 7-DAY ROLLING AVERAGE PREDICTION In addition, to understand the impact of assigning the treatment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', ODES cat- egories) to the dialogues, we consider the situation where the treatments of the col- Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 13 Table 3 The correlations of CF-LSTM when the treatment assignments for all dialogues are inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' PEARSON ORIGINAL INVERTED CORRELATION TREATMENTS TREATMENTS INDIVIDUAL PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='26 L1D PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='35 L7D PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='52 ⋆ DAILY AVERAGE, † 7-DAY ROLLING AVERAGE lected dialogues are inverted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', treatment T = 0 becomes T = 1, and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For the same dialogue feature vector, its treatment is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This setting allows us to investigate the impact of the ODES Classifier when assigning the incorrect treatments and whether the models can overcome this misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We feed the modified data samples to the well-trained CF-LSTM capable of predicting the coun- terfactual outcomes for these dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 3 shows the correlations between the predicted counterfactual outcomes and the original ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Here, we want to investigate whether the model can predict correctly even if the treatment assign- ment is inverted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', the ODES classifier malfunctions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As shown in Table 3, the individual prediction and daily average prediction (L1d) of the inverted treatments scenario still have a relatively high correlation with the original ground truth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', when the treatments are not inverted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The mean square error between the factual outcomes Y1 and the counterfactual outcomes Y0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The average treatment effect (ATE) is defined as follows for our problem: ATE = E[Y1i −Y0i] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='7809 The ATE shows that, on average, a dialogue being assigned treatment T = 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', consists of disinterest, insult) will have a lower user rating than the dialogue with T = 1 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='7809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' We randomly select a set of 1213 dialogues and ask an expert to rate the con- versational quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Let Rexpert be the rating given by the expert, and Ruser be the ratings given by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The expert ratings have a low Pearson correlation with the user ratings, demonstrating the challenge in predicting user ratings in open-domain dialogue since they are only loosely correlated with an expert evaluation of dialogue quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' ρ = corr(Rexpert,Ruser) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='2056 Next, we use the CF-LSTM and the Dialogue-level MLP to predict the expert’s rating in the given dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 4 indicates the correlations obtained from CF- LSTM and Dialogue-level MLP on expert-rated dialogues from Socialbot conver- sations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Due to the simplicity of the network architecture, the Dialogue-level MLP achieves competitive results in individual and daily average predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' However, it performs poorly in 7-day rolling average predictions because of the sparsity across time of the small dataset of 1213 dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' On the other hand, CF-LSTM performs well on all types of predictions and shows flexibility in training for both large and small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 14 Cat P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Le, Luke Dai, Michael Johnston, Yang Liu, Marilyn Walker, Reza Ghanadan Table 4 The correlations of prediction models on 1213 expert-rated dialogues from Socialbot conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' PEARSON DIALOGUE- CF-LSTM CORRELATION LEVEL MLP (OURS) INDIVIDUAL PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='26 L1D PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='36 L7D PREDICTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='51 ⋆ DAILY AVERAGE, † 7-DAY ROLLING AVERAGE Lastly, we consider the problem of classifying the dialogues into 2-class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', bi- nary) and 5-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For binary classification, we label any dialogue with a rating less than 3 as 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', low-rated dialogues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Hence, the remaining dialogues with ratings greater or equal to 3 are labeled as 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', highly-rated dialogues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For the 5-class classification, we round up or down (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', round half-up) the ratings into five groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' For example, a dialogue with a rating of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='5 is classified into group 5, and a dialogue with a rating of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='4 is classified into group 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In this dataset, we ob- serve that most dialogues are often rated above 3 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As a result, we apply the data generation method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='3 to generate more poorly-rated dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This newly generated data is used to fine-tune the binary classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' The resulting model can be applied online to improve the user experience at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Table 2 indicates the performance of our model and other methods, in terms of classification accuracy, in binary and 5-class classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Our proposed method CF-LSTM outperforms other approaches in both binary classifi- cation and 5-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' 5 Conclusion and Future Work We propose a novel causal inference structure for open-domain dialogue evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This structure utilizes turn-level features, such as sentiment analysis, textual categories, response relevance, and response specificity, and uses them to predict user ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Our method performs competitively in an evaluation with rated con- versations with the Alexa SocialBot, in regression and classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' As a causal inference model, this CF-LSTM model is more robust in learning com- plex representations and capable of predicting the ratings for the dialogues under different hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In addition to offline uses, this model can also be applied at run-time to dynamically identify low-quality dialogues and trigger the introduction of different topics to help improve the user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' This paper considers only 2 treatments, which are all the ODES categories, versus others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' In future works, we can further boost the flexibility of our model by increasing the number of treatments corresponding to the different ODES groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Additionally, other essential dialogue features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=', text and sentiment embeddings) can be included in the input features to help improve the overall performance of the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Improving Open-Domain Dialogue Evaluation with a Causal Inference Model 15 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Alaa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Van Der Schaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Bayesian inference of individualized treatment effects using multi-task gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Brockett, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Liu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Dolan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} +page_content=' Dialogpt: Large-scale generative pre-training for conversational response generation, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FQT4oBgHgl3EQfoTY7/content/2301.13372v1.pdf'} diff --git a/m9E2T4oBgHgl3EQfegeM/content/tmp_files/2301.03917v1.pdf.txt b/m9E2T4oBgHgl3EQfegeM/content/tmp_files/2301.03917v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c4ef2c46786ced04fd9ce31f96982a219d73974 --- /dev/null +++ b/m9E2T4oBgHgl3EQfegeM/content/tmp_files/2301.03917v1.pdf.txt @@ -0,0 +1,325 @@ +arXiv:2301.03917v1 [math.GR] 10 Jan 2023 +p-GROUPS AND ZEROS OF CHARACTERS +ALEXANDER MORET´O, GABRIEL NAVARRO +Abstract. Fix a prime p and an integer n ≥ 0. Among the non-linear irreducible +characters of the p-groups of order pn, what is the minimum number of elements +that take the value 0? +1. Introduction +Dihedral, semi-dihedral and generalized quaternion groups are ubiquitous in finite +group theory. +They have been characterized along the years in several ways: as +the non-cyclic 2-groups whose number of involutions is 1 modulo 4 (Alperin-Feit- +Thompson); as the non-abelian 2-groups whose commutator subgroup has index 4 +(O. Taussky-Todd), as the 2-groups of maximal class; or, using fields of values of +characters, as the 2-groups with exactly five rational-valued irreducible characters +([4]), for instance. +Using zeros of characters, the following is yet another one. It is somewhat re- +markable that a finite group can be characterized by the number of zeros of a single +irreducible character. +Theorem A. Suppose that G is a 2-group of order 2n. Let χ be an irreducible non- +linear complex character of G. Then χ(g) = 0 for at least 2n−1 + 2 elements g ∈ G. +Furthermore, there exists χ ∈ Irr(G) that vanishes at exactly 2n−1 +2 elements if and +only if G is dihedral, semidihedral or generalized quaternion. +The situation for general p-groups is more mysterious, and difficult. The following +also includes the harder implication in Theorem A (when p is even). +Theorem B. Let G be a p-group of order pn. Let χ be an irreducible non-linear +character of G. Then χ(g) = 0 for at least pn − pn−1 + p2 − p elements g ∈ G. If +equality holds, then G is a p-group of maximal class with a maximal abelian subgroup. +The minimum number of elements taking the value zero among all the non-linear +characters of the groups of order 55 is 2600 > 55 − 54 + 52 − 5 = 2520. On the other +Date: January 11, 2023. +2010 Mathematics Subject Classification. Primary 20C15. +We thank Eamonn O’Brien for helpful conversations and computer calculations supporting Con- +jectures 2.6 and 2.9. The research of both authors is supported by Ministerio de Ciencia e Innovaci´on +(Grant PID2019-103854GB-I00 funded by MCIN/AEI/ 10.13039/501100011033). The first author +is also supported by Generalitat Valenciana CIAICO/2021/163. +1 + +2 +ALEXANDER MORET´O, GABRIEL NAVARRO +hand, among groups of order 75, this number is exactly 1448 = 75−74+72−7. This is +related to some results in [6], and makes us suspect that an explicit minimum bound +for the number of zeros among p-groups of order pn might not be easy to discover. +(See Corollary 2.8 below and the paragraph that follows it.) +The converse of Theorem B is not true, as shown for instance by SmallGroup(55, 30), +a 5-group with maximal class and an abelian maximal normal subgroup, although it +is likely to be true if p = 3 (as we shall explain). +Our renewed interest on zeros of characters comes from a recent intriguing conjec- +ture by A. Miller [5] that deserves attention. Using a non-trivial number theoretic +result by Siegel, J. G. Thompson proved many years ago that at least 1/3 of the +elements of a finite group take a zero or a root of unity value on every irreducible +character of G (see Problem 2.15 of [3]). Now A. Miller [5] has conjectured that +it should be at least 1/2 of the elements. Using number theory, Miller gives in [5] +lower bounds for the number of zeros of characters for nilpotent groups, which are +improved by our Theorem B. +At the time of this writing, unfortunately, we cannot contribute much to Miller’s +conjecture. The data seems to endorse it but a proof –even for solvable groups– seems +elusive. (As a matter of fact, the same data suggest a much stronger statement: that +outside any given normal subgroup, the proportion of elements that take zero or root +unity values is again 1/2.) +As pointed out by Miller, the proportion of zero and root of unity values is exactly +1/2 in certain dihedral groups. +Since these groups are supersolvable, it may be +of interest to consider that case. +We conclude this note with a proof of Miller’s +conjecture for a family of groups that includes supersolvable groups. +Theorem C. Suppose that χ is an irreducible character of a finite group G. If G +has a Sylow tower, then χ(g) is zero or a root of unity for at least |G|/2 elements of +G. +We notice that, unlike in the case of nilpotent groups, roots of unity are definitely +necessary here. For instance, the non-linear characters of degree 2 of SL2(3) vanish +at exactly 6 elements. +2. p-groups +Our notation follows [2, 3]. In this section we prove Theorem B and, as a conse- +quence, deduce Theorem A. We start with an elementary lemma. +Lemma 2.1. Let G be a finite group and χ ∈ Irr(G) faithful. Suppose that there +exists U ≤ G and λ ∈ Irr(U) linear such that χ = λG. Then U is abelian. +Proof. Since χ is faithful, Lemma 5.11 of [3] implies that 1 = � +g∈G ker λg. Hence, U +embeds into the direct product of the abelian groups G/ ker λg. The result follows. +□ + +p-GROUPS AND ZEROS OF CHARACTERS +3 +We need the following technical lemma. +Lemma 2.2. Suppose that G = UV is a p-group, where U is abelian, V is non- +abelian of maximal class, U, V are maximal subgroups of G and |Z(G)| = p. Then G +has maximal class. +Proof. Recall that a non-abelian p-group P has maximal class if and only if there +is x ∈ P such that |CP(x)| = p2 ([2, Satz III.14.23]). Put W = U ∩ V ⊳ G. Since +|Z(G)| = p, we have that Z(G) ⊆ W. Note that the hypotheses imply that W is +abelian, |G : W| = p2, and |G| ≥ p4 (since V is non-abelian). If |G| = p4 and G +does not have maximal class, then we have that G/Z(G) is abelian and G′ = Z(G) +has order p. Then 1 = [x, y]p = [xp, y] for x, y ∈ G, and thus Φ(G) ⊆ Z(G). But +there are no extra-special groups of order p4, so we may assume that |G| ≥ p5. Since +V is non-abelian and has maximal class, there exists v ∈ V such that |CV (v)| = p2. +Since |W| ≥ p3, this implies that v ∈ V − W. Let u ∈ U − W. Since Z(G) ≤ W, +in particular u ̸∈ Z(G). Since V = ⟨v⟩W and u centralizes W, we deduce that u +and v do not commute. +Thus |CU(v)| = p. +Since G = U⟨v⟩, we conclude that +CG(v) = CU(v)⟨v⟩ has order p2. We conclude that G has maximal class. +□ +The case of groups of class 2 of Theorem B follows easily from well-known results. +Lemma 2.3. Let G be a p-group of order pn class 2. Then for any χ ∈ Irr(G), +χ(g) = 0 for at least pn − pn−2 elements g ∈ G. In particular, χ vanishes at at least +pn − pn−1 + p2 − p elements and if equality holds then n = 3. +Proof. Let Z/ ker χ = Z(χ)/ ker χ. By Theorem 2.1 of [3], p2 ≤ χ(1)2 = |G : Z|. +Using Problem 6.3 of [3], we deduce that χ vanishes on G − Z. Since |G − Z| = +pn − pn−2, the result follows. The second part is straightforward. +□ +The following is a more detailed version of Theorem B. +Theorem 2.4. Let G be a p-group of order pn. If χ ∈ Irr(G) is non-linear, then G +vanishes on at least pn − pn−1 + p2 − p elements of G. If equality holds then +(i) χ is faithful and χ(1) = p. +(ii) G is a p-group of maximal class with a maximal abelian subgroup U. +(iii) If n > 3, then U is the unique maximal subgroup of G with a character that +induces χ and the set of zeros of χ is (G − U) ∪ (Z2(G) − Z(G)). +Proof. If n = 3 then G is an extraspecial p-group and the result is well-known. We +assume in the remaining that n > 3. +We prove the first part by induction on n. By Lemma 2.3, we may assume that G +does not has class 2. Since χ is monomial, there exists U maximal in G such that χ +is induced from U. Suppose first that there exists V ̸= U maximal in G such that χ +is also induced from V . Then χ vanishes on (G−U)∪(G−V ) = G−(U ∩V ). There +are pn −pn−2 elements in this set, and this number exceeds pn −pn−1 +p2 −p. Hence, +we will assume in the remaining that U is the unique maximal subgroup of G with a + +4 +ALEXANDER MORET´O, GABRIEL NAVARRO +character that induces χ. Let θ ∈ Irr(U) such that θG = χ. Since G is not cyclic, let +V be another maximal subgroup of G. Set W = U ∩ V . Then, using Corollary 6.19 +of [3], we have that χV ∈ Irr(V ) and by Mackey (Problem 5.2 of [3]) χV = (θW)V . +By the inductive hypothesis, χV vanishes on at least pn−1 − pn−2 + p2 − p elements. +Since χV is induced from θW, then χV vanishes on the pn−1−pn−2 elements of V −W. +Therefore, χV vanishes at least on p2−p elements that belong to W. Since χ vanishes +on G − U and at these p2 − p elements in W, the first part of the result follows. +Assume now and for the rest of the proof that equality holds. First, we prove that +χ is faithful. Let K = ker χ. Put |K| = pm. Let χ be the character χ viewed as a +character of G/K. For any element xK that is a zero of χ, χ vanishes on the coset +xK. By the first part, χ vanishes on at least pn−m+pn−m+1 +p2 −p elements. Hence, +the number of zeros of χ is at least pm(pn−m + pn−m+1 + p2 − p). Since the number +of zeros of χ is pn − pn−1 + p2 − p, this forces m = 0. This proves that χ is faithful. +Next, we see that χ vanishes on Z2(G) − Z(G). Let x ∈ Z2(G) and g ∈ G such +that [x, g] ̸= 1. Let λ ∈ Irr(Z(G)) lying under χ. Note that λ is faithful. Hence +χ(x) = χ(xg) = χ(x[x, g]) = χ(x)λ([x, g]), +which implies that χ(x) = 0, as wanted. +Now, we claim that Z2(G) ≤ U. By Theorem 6.22 of [3] χ is an M-character +over Z2(G). This means that there exists Z2(G) ⊆ H ⊆ G and ψ ∈ Irr(H) such +that ψG = χ and ψZ2(G) is irreducible. +If H < G, by uniqueness of U, we have +that H ⊆ U, and the claim is proven. +Thus we may assume that H = G and +that τ = χZ2(G) ∈ Irr(Z2(G)). Since χ(1) > 1, we have that Z2(G) is not abelian. +Assume by contradiction that Z2(G) ̸≤ U, so that G = Z2(G)U. Suppose first that +|Z2(G)| = pt > p3. Since Z2(G) has class 2, we deduce that τ has at least pt − pt−2 +zeros by Lemma 2.3. Since by Mackey (θU∩Z2(G))Z2(G) = τ, then τ is zero on the +pt−pt−1 elements of Z2(G)−(U ∩Z2(G)). Hence, there are at least pt−1−pt−2 > p2−p +zeros of τ in U ∩ Z2(G). Since these are zeros of χ, we conclude that χ has at least +pn − pn−1 + pt−1 − pt−2 zeros, which is a contradiction. Now, we may assume that +|Z2(G)| = p3. Therefore, χ(1) = τ(1) = p. Since χ is faithful and induced from +U, we conclude from Lemma 2.1 that U is abelian. Now, [G′, Z2(G)] = 1 (see [2, +Hauptsatz III.2.11]) and since G′ is contained in the abelian group U, it follows that +G′ is central in G, so G has class 2. This contradicts Lemma 2.3, proving the claim. +We have thus seen that the set of zeros of χ is (G−U)∪(Z2(G)−Z(G)), where the +union is disjoint. Therefore |Z2(G)−Z(G)| = p2−p, and we deduce that |Z2(G)| = p2 +and |Z(G)| = p. +Next, we claim that χ(1) = p. Suppose that χ(1) > p. Since, again, χ is an +M-character over Z2(G), there exists Z2(G) ≤ H < U such that χ is induced from +H. In particular, χ is zero on G − � +g∈G Hg. Since � +g∈G Hg ⊊ U (by Lemma 3.1 of +[6], for instance), this implies that χ has zeros in U − Z2(G), a contradiction. This +proves the claim. + +p-GROUPS AND ZEROS OF CHARACTERS +5 +As a consequence, we obtain that θ ∈ Irr(U), the character that induces χ, is +linear. Since χ is faithful, Lemma 2.1 implies that U is abelian. +It remains to see that G has maximal class. We prove this by induction on |G|. +Let Z2(G) ≤ X ≤ U be such that X ⊴ G and G/X is elementary abelian of order +p2 (it exists because since G/Z(G) is not abelian, G/Z2(G) cannot be cyclic). Let +V ̸= U such that X < V < G. By the uniqueness of U, τ = χV ∈ Irr(V ) and +by Mackey, τ = (θX)V vanishes on V − X. Since all the zeros of χ in X are in +Z2(G), τ does not have zeros in X − Z2(G). We conclude that the set of zeros of τ +is (V − X) ∪ (Z2(G) − Z(G)), which has cardinality pn−1 − pn−2 + p2 − p. By the +inductive hypothesis, V has maximal class. It follows from Lemma 2.2 that G has +maximal class. +□ +The proof of Theorem A now follows easily. +Theorem 2.5. Suppose that G is a 2-group of order 2n. Let χ be an irreducible non- +linear complex character of G. Then χ(g) = 0 for at least 2n−1 + 2 elements g ∈ G. +Furthermore, there exists χ ∈ Irr(G) that vanishes at exactly 2n−1 +2 elements if and +only if G is dihedral, semidihedral or generalized quaternion. +Proof. By Theorem B, we only have to prove that if G is dihedral, semidihedral or +generalized quaternion and χ ∈ Irr(G) is faithful, then χ vanishes on exactly 2n−1 +2 +elements of G. But this is easy. Let U be the maximal abelian subgroup of G, and let +g ∈ G such that G = ⟨g, U⟩ with xg = xi, where i = −1 is G is dihedral or quaternion +and i = 2n−2 − 1 if G is semidihedral. We have that χ = λG where λ ∈ Irr(U) is +faithful and |G : U| = 2. Now, for any y ∈ U, λ(y) = ε is a primitive o(y)-th root of +unity, and λ(x) + λg(x) = ε + ε−i = 0 if and only if o(x) = 4. +□ +We expect the following to hold for p = 3. +Conjecture 2.6. Let G be a 3-group of order 3n. Then G has an irreducible character +that vanishes at exactly 3n−3n−1+6 elements if and only if G is a 3-group of maximal +class with a maximal abelian subgroup. +Note that the “only if” part follows from Theorem B. We recall that the 3-groups +of maximal class (as well as the p-groups of maximal class with a maximal abelian +subgroup for any prime p) were classified by Blackburn [1]. However, it does not +seem easy to prove that they possess an irreducible character that vanishes at exactly +3n − 3n−1 + 6 elements. Eamonn O’Brien, has checked that this is true for groups of +order at most 310. +As we have mentioned, the converse of Theorem B does not hold for p > 3. This +situation is related to [6]. In [6] it was proved that the number of conjugacy classes +of zeros of any non-linear irreducible character of a p-group is at least p2 − 1 (see +Theorem C of [6]). Furthermore, if equality holds and the character is faithful then +G is a p-group of maximal class with a maximal abelian subgroup U and the set of +zeros of the character is (G −U) ∪(Z2(G) −Z(G)) (see the proof of Theorem C of [6] + +6 +ALEXANDER MORET´O, GABRIEL NAVARRO +and the paragraph that follows it). Now, we make clear the relation between both +problems. (Note that this relation is only transparent after proving Theorem 2.4.) +Theorem 2.7. Let G be a non-abelian p-group of order pn and χ ∈ Irr(G) faithful. +Then χ vanishes at exactly pn − pn−1 + p2 − p elements if and only if χ vanishes at +exactly p2 − 1 conjugacy classes. +Proof. This is clear if n = 3 so we may assume that n > 3. +Suppose first that χ vanishes at exactly p2 − 1 conjugacy classes. As we have just +mentioned, then G is a p-group of maximal class with a maximal abelian subgroup +U and the set of zeros of the χ is (G − U) ∪ (Z2(G) − Z(G). Since the cardinality of +this set is pn − pn−1 + p2 − p the result follows. +Conversely, assume that χ vanishes at exactly pn − pn−1 + p2 − p elements. By +Theorem 2.4, G is a p-group of maximal class with a maximal abelian subgroup U +and the set of zeros of the character is (G − U) ∪ (Z2(G) − Z(G). Let g ∈ G − U, so +that G = ⟨g⟩U. Since |Z(G)| = p, CU(g) = Z(G), so |CG(g)| = p2. In other words, +the conjugacy classes in G − U have size pn−2. Therefore, the number of conjugacy +classes of G contained in G − U is (pn − pn−1)/pn−2 = p2 − p. Since |Z2(G)| = p2, +the conjugacy classes in Z2(G) − Z(G) have size p. Hence, the number of conjugacy +classes of G contained in this subset is p − 1. It follows that χ vanishes at exactly +p2 − 1 conjugacy classes, as wanted. +□ +Now, we can use Theorem D of [6] to see that if p > 3 and equality holds in +Theorem B then |G| is bounded in terms of p. +Corollary 2.8. Let G be a p-group of order pn, where p > 3. If G has an irreducible +character χ that vanishes at exactly pn − pn−1 + p2 − p elements, then |G| ≤ pr+1, +where r is the smallest prime that does not divide p − 1. +Proof. By Theorem 2.4, we know that χ is faithful. Now, by Theorem 2.7, χ vanishes +at exactly p2 − 1 conjugacy classes and the result follows from Theorem D of [6]. +□ +Let us summarize. If G is a non-abelian group, and mz(G) is the minimum number +of elements of G taking the zero value among the non-linear irreducible characters of +G, we let +mz(pn) = min{mz(G) | |G| = pn} . +We have shown in Theorem B that mz(pn) ≤ pn − pn−1 + p2 − p, and in Theorem A +that equality holds if p = 2. (We suspect that the same holds if p = 3.) Also the +proof oif Theorem B and computer calculations performed by O’Brien suggest that +the following could be true. +Conjecture 2.9. Let G be a p-group of order pn. Then mz(pn) = mz(G) if and only +if G has maximal class with an abelian maximal normal subgroup. + +p-GROUPS AND ZEROS OF CHARACTERS +7 +3. Groups with a Sylow tower +We conclude with the proof of Theorem C, which we restate. Our interest now +also includes roots of unity values of characters. +Theorem 3.1. Let G be a group with a Sylow tower and let χ ∈ Irr(G). Then the +proportion of elements x ∈ G such that χ(x) = 0 or χ(x) is a root of unity is at least +1/2. +Proof. We argue by induction on |G|. There exists a prime p that divides |G| and G +has a normal Hall p′-subgroup N. Let P ∈ Sylp(G), so that G = PN. Since G/N is +a p-group, it follows from Theorem 6.22 of [3] that χ is a relative M-character with +respect to N. Thus there exists N ≤ H ≤ G and ψ ∈ Irr(H) such that χ = ψG and +ψN ∈ Irr(N). Suppose first that H < G. Since G/N is a p-group, every maximal +subgroup U of G that contains N is normal in G. Since χ is induced U, it follows +that χ vanishes on G − U. There are at least |G|/2 elements in this set. Thus the +theorem holds in this case. +Now, we may assume that H = G. In other words, θ = χN ∈ Irr(N). Let ˆθ be +the canonical extension of θ to G. We claim that the proportion of zeros and root of +unity values of ˆθ exceeds 1/2. Let Gp be the set of p-elements of G. Therefore +G = +� +x∈Gp +CN(x)x +is a disjoint union by Lemma 8.18 of [3]. Now, if 1 ̸= x ∈ Gp, c ∈ CN(x), and +θ∗ ∈ Irr(CN(x)) is the x-Glauberman correspondent of θ, we have by Theorem 13.32 +of [3] that +ˆθ(cx) = ǫθ∗(c) , +where ǫ is a sign. Since G has a Sylow tower, we have that CN(x) has a Sylow tower. +Let Ax be the set of elements of CN(x) where θ∗ has the value zero or a root of unity. +value 0 or root of unity. By induction, we have that +|Ax| ≥ |CN(x)|/2 +for every x ∈ Gp. If y ∈ Ax, then ˆθ(yx) is a zero or a root of unity, and therefore, ˆθ +has at least +� +x∈Gp +|Ax| ≥ |G|/2 +roots of unity or zero values. +Now, by Gallagher’s Corollary 6.17 of [3], we have that χ = µˆθ, where µ ∈ +Irr(G/N) = Irr(P). +If µ is not linear, then the result follows from the p-group +case. If µ is linear, then |χ(x)| = |ˆθ(x)| and the result follows from Problem 3.2 of +[3] and the previous paragraph +□ + +8 +ALEXANDER MORET´O, GABRIEL NAVARRO +It is easy to build examples of nonsolvable groups with irreducible characters that +either vanish or take root of unity values at exactly one-half of its elements. Consider +for instance G = S ≀D10, where S is any simple group. However, if Miller’s conjecture +is true, then it seems reasonable to expect that if equality holds and χ ∈ Irr(G) +is a character that either vanishes or takes a root of unity value at one-half of the +elements of G, then χ is monomial of degree 2 and G/ ker χ is supersolvable. +References +[1] N. Blackburn, On a special class of p-groups. Acta Math. 100 (1958), 45–92. +[2] B. Huppert, Endliche Gruppen I. Springer-Verlag, 1967. +[3] I. M. Isaacs, Character Theory of Finite Groups. AMS-Chelsea, Providence, 2006. +[4] I. M. Isaacs, G. Navarro, J. Sangroniz, p-groups having few almost-rational irreducible +characters, Israel J. Math. 189 (2012), 65–96. +[5] A. Miller, Zeros and roots of unity in character tables, Enseign. Math., to appear, +arXiv:2003.13238. +[6] A. Moret´o, J. Sangroniz, On the number of conjugacy classes of zeros of characters. Israel +J. Math. 142 (2004), 163–187. +Departament de Matem`atiques, Universitat de Val`encia, 46100 Burjassot, Val`encia, +Spain +Email address: alexander.moreto@uv.es +Email address: gabriel@uv.es + diff --git a/m9E2T4oBgHgl3EQfegeM/content/tmp_files/load_file.txt b/m9E2T4oBgHgl3EQfegeM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83a26f8040f7f1fd61c989acf5913c4bce4e9728 --- /dev/null +++ b/m9E2T4oBgHgl3EQfegeM/content/tmp_files/load_file.txt @@ -0,0 +1,341 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf,len=340 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='03917v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='GR] 10 Jan 2023 p-GROUPS AND ZEROS OF CHARACTERS ALEXANDER MORET´O, GABRIEL NAVARRO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Fix a prime p and an integer n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Among the non-linear irreducible characters of the p-groups of order pn, what is the minimum number of elements that take the value 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Introduction Dihedral, semi-dihedral and generalized quaternion groups are ubiquitous in finite group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' They have been characterized along the years in several ways: as the non-cyclic 2-groups whose number of involutions is 1 modulo 4 (Alperin-Feit- Thompson);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' as the non-abelian 2-groups whose commutator subgroup has index 4 (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Taussky-Todd), as the 2-groups of maximal class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' or, using fields of values of characters, as the 2-groups with exactly five rational-valued irreducible characters ([4]), for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Using zeros of characters, the following is yet another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' It is somewhat re- markable that a finite group can be characterized by the number of zeros of a single irreducible character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that G is a 2-group of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let χ be an irreducible non- linear complex character of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then χ(g) = 0 for at least 2n−1 + 2 elements g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Furthermore, there exists χ ∈ Irr(G) that vanishes at exactly 2n−1 +2 elements if and only if G is dihedral, semidihedral or generalized quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The situation for general p-groups is more mysterious, and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The following also includes the harder implication in Theorem A (when p is even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a p-group of order pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let χ be an irreducible non-linear character of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then χ(g) = 0 for at least pn − pn−1 + p2 − p elements g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If equality holds, then G is a p-group of maximal class with a maximal abelian subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The minimum number of elements taking the value zero among all the non-linear characters of the groups of order 55 is 2600 > 55 − 54 + 52 − 5 = 2520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' On the other Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Primary 20C15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We thank Eamonn O’Brien for helpful conversations and computer calculations supporting Con- jectures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The research of both authors is supported by Ministerio de Ciencia e Innovaci´on (Grant PID2019-103854GB-I00 funded by MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='13039/501100011033).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The first author is also supported by Generalitat Valenciana CIAICO/2021/163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 1 2 ALEXANDER MORET´O, GABRIEL NAVARRO hand, among groups of order 75, this number is exactly 1448 = 75−74+72−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This is related to some results in [6], and makes us suspect that an explicit minimum bound for the number of zeros among p-groups of order pn might not be easy to discover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (See Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='8 below and the paragraph that follows it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=') The converse of Theorem B is not true, as shown for instance by SmallGroup(55, 30), a 5-group with maximal class and an abelian maximal normal subgroup, although it is likely to be true if p = 3 (as we shall explain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Our renewed interest on zeros of characters comes from a recent intriguing conjec- ture by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Miller [5] that deserves attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Using a non-trivial number theoretic result by Siegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Thompson proved many years ago that at least 1/3 of the elements of a finite group take a zero or a root of unity value on every irreducible character of G (see Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='15 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Miller [5] has conjectured that it should be at least 1/2 of the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Using number theory, Miller gives in [5] lower bounds for the number of zeros of characters for nilpotent groups, which are improved by our Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' At the time of this writing, unfortunately, we cannot contribute much to Miller’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The data seems to endorse it but a proof –even for solvable groups– seems elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (As a matter of fact, the same data suggest a much stronger statement: that outside any given normal subgroup, the proportion of elements that take zero or root unity values is again 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=') As pointed out by Miller, the proportion of zero and root of unity values is exactly 1/2 in certain dihedral groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since these groups are supersolvable, it may be of interest to consider that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We conclude this note with a proof of Miller’s conjecture for a family of groups that includes supersolvable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that χ is an irreducible character of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If G has a Sylow tower, then χ(g) is zero or a root of unity for at least |G|/2 elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We notice that, unlike in the case of nilpotent groups, roots of unity are definitely necessary here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' For instance, the non-linear characters of degree 2 of SL2(3) vanish at exactly 6 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' p-groups Our notation follows [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In this section we prove Theorem B and, as a conse- quence, deduce Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We start with an elementary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a finite group and χ ∈ Irr(G) faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that there exists U ≤ G and λ ∈ Irr(U) linear such that χ = λG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then U is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ is faithful, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='11 of [3] implies that 1 = � g∈G ker λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence, U embeds into the direct product of the abelian groups G/ ker λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ p-GROUPS AND ZEROS OF CHARACTERS 3 We need the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that G = UV is a p-group, where U is abelian, V is non- abelian of maximal class, U, V are maximal subgroups of G and |Z(G)| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then G has maximal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Recall that a non-abelian p-group P has maximal class if and only if there is x ∈ P such that |CP(x)| = p2 ([2, Satz III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Put W = U ∩ V ⊳ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since |Z(G)| = p, we have that Z(G) ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Note that the hypotheses imply that W is abelian, |G : W| = p2, and |G| ≥ p4 (since V is non-abelian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If |G| = p4 and G does not have maximal class, then we have that G/Z(G) is abelian and G′ = Z(G) has order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then 1 = [x, y]p = [xp, y] for x, y ∈ G, and thus Φ(G) ⊆ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' But there are no extra-special groups of order p4, so we may assume that |G| ≥ p5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since V is non-abelian and has maximal class, there exists v ∈ V such that |CV (v)| = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since |W| ≥ p3, this implies that v ∈ V − W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let u ∈ U − W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since Z(G) ≤ W, in particular u ̸∈ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since V = ⟨v⟩W and u centralizes W, we deduce that u and v do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Thus |CU(v)| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since G = U⟨v⟩, we conclude that CG(v) = CU(v)⟨v⟩ has order p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We conclude that G has maximal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ The case of groups of class 2 of Theorem B follows easily from well-known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a p-group of order pn class 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then for any χ ∈ Irr(G), χ(g) = 0 for at least pn − pn−2 elements g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In particular, χ vanishes at at least pn − pn−1 + p2 − p elements and if equality holds then n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let Z/ ker χ = Z(χ)/ ker χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1 of [3], p2 ≤ χ(1)2 = |G : Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Using Problem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='3 of [3], we deduce that χ vanishes on G − Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since |G − Z| = pn − pn−2, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' The second part is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ The following is a more detailed version of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a p-group of order pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If χ ∈ Irr(G) is non-linear, then G vanishes on at least pn − pn−1 + p2 − p elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If equality holds then (i) χ is faithful and χ(1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (ii) G is a p-group of maximal class with a maximal abelian subgroup U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (iii) If n > 3, then U is the unique maximal subgroup of G with a character that induces χ and the set of zeros of χ is (G − U) ∪ (Z2(G) − Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If n = 3 then G is an extraspecial p-group and the result is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We assume in the remaining that n > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We prove the first part by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='3, we may assume that G does not has class 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ is monomial, there exists U maximal in G such that χ is induced from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose first that there exists V ̸= U maximal in G such that χ is also induced from V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then χ vanishes on (G−U)∪(G−V ) = G−(U ∩V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' There are pn −pn−2 elements in this set, and this number exceeds pn −pn−1 +p2 −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence, we will assume in the remaining that U is the unique maximal subgroup of G with a 4 ALEXANDER MORET´O, GABRIEL NAVARRO character that induces χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let θ ∈ Irr(U) such that θG = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since G is not cyclic, let V be another maximal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Set W = U ∩ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then, using Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='19 of [3], we have that χV ∈ Irr(V ) and by Mackey (Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='2 of [3]) χV = (θW)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By the inductive hypothesis, χV vanishes on at least pn−1 − pn−2 + p2 − p elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χV is induced from θW, then χV vanishes on the pn−1−pn−2 elements of V −W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Therefore, χV vanishes at least on p2−p elements that belong to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ vanishes on G − U and at these p2 − p elements in W, the first part of the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Assume now and for the rest of the proof that equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' First, we prove that χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let K = ker χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Put |K| = pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let χ be the character χ viewed as a character of G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' For any element xK that is a zero of χ, χ vanishes on the coset xK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By the first part, χ vanishes on at least pn−m+pn−m+1 +p2 −p elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence, the number of zeros of χ is at least pm(pn−m + pn−m+1 + p2 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since the number of zeros of χ is pn − pn−1 + p2 − p, this forces m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This proves that χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Next, we see that χ vanishes on Z2(G) − Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let x ∈ Z2(G) and g ∈ G such that [x, g] ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let λ ∈ Irr(Z(G)) lying under χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Note that λ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence χ(x) = χ(xg) = χ(x[x, g]) = χ(x)λ([x, g]), which implies that χ(x) = 0, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, we claim that Z2(G) ≤ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='22 of [3] χ is an M-character over Z2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This means that there exists Z2(G) ⊆ H ⊆ G and ψ ∈ Irr(H) such that ψG = χ and ψZ2(G) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If H < G, by uniqueness of U, we have that H ⊆ U, and the claim is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Thus we may assume that H = G and that τ = χZ2(G) ∈ Irr(Z2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ(1) > 1, we have that Z2(G) is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Assume by contradiction that Z2(G) ̸≤ U, so that G = Z2(G)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose first that |Z2(G)| = pt > p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since Z2(G) has class 2, we deduce that τ has at least pt − pt−2 zeros by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since by Mackey (θU∩Z2(G))Z2(G) = τ, then τ is zero on the pt−pt−1 elements of Z2(G)−(U ∩Z2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence, there are at least pt−1−pt−2 > p2−p zeros of τ in U ∩ Z2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since these are zeros of χ, we conclude that χ has at least pn − pn−1 + pt−1 − pt−2 zeros, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, we may assume that |Z2(G)| = p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Therefore, χ(1) = τ(1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ is faithful and induced from U, we conclude from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1 that U is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, [G′, Z2(G)] = 1 (see [2, Hauptsatz III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='11]) and since G′ is contained in the abelian group U, it follows that G′ is central in G, so G has class 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This contradicts Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='3, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We have thus seen that the set of zeros of χ is (G−U)∪(Z2(G)−Z(G)), where the union is disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Therefore |Z2(G)−Z(G)| = p2−p, and we deduce that |Z2(G)| = p2 and |Z(G)| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Next, we claim that χ(1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that χ(1) > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since, again, χ is an M-character over Z2(G), there exists Z2(G) ≤ H < U such that χ is induced from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In particular, χ is zero on G − � g∈G Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since � g∈G Hg ⊊ U (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1 of [6], for instance), this implies that χ has zeros in U − Z2(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' p-GROUPS AND ZEROS OF CHARACTERS 5 As a consequence, we obtain that θ ∈ Irr(U), the character that induces χ, is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ is faithful, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1 implies that U is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' It remains to see that G has maximal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We prove this by induction on |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let Z2(G) ≤ X ≤ U be such that X ⊴ G and G/X is elementary abelian of order p2 (it exists because since G/Z(G) is not abelian, G/Z2(G) cannot be cyclic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let V ̸= U such that X < V < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By the uniqueness of U, τ = χV ∈ Irr(V ) and by Mackey, τ = (θX)V vanishes on V − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since all the zeros of χ in X are in Z2(G), τ does not have zeros in X − Z2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We conclude that the set of zeros of τ is (V − X) ∪ (Z2(G) − Z(G)), which has cardinality pn−1 − pn−2 + p2 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By the inductive hypothesis, V has maximal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='2 that G has maximal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ The proof of Theorem A now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose that G is a 2-group of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let χ be an irreducible non- linear complex character of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then χ(g) = 0 for at least 2n−1 + 2 elements g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Furthermore, there exists χ ∈ Irr(G) that vanishes at exactly 2n−1 +2 elements if and only if G is dihedral, semidihedral or generalized quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Theorem B, we only have to prove that if G is dihedral, semidihedral or generalized quaternion and χ ∈ Irr(G) is faithful, then χ vanishes on exactly 2n−1 +2 elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' But this is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let U be the maximal abelian subgroup of G, and let g ∈ G such that G = ⟨g, U⟩ with xg = xi, where i = −1 is G is dihedral or quaternion and i = 2n−2 − 1 if G is semidihedral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We have that χ = λG where λ ∈ Irr(U) is faithful and |G : U| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, for any y ∈ U, λ(y) = ε is a primitive o(y)-th root of unity, and λ(x) + λg(x) = ε + ε−i = 0 if and only if o(x) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ We expect the following to hold for p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a 3-group of order 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then G has an irreducible character that vanishes at exactly 3n−3n−1+6 elements if and only if G is a 3-group of maximal class with a maximal abelian subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Note that the “only if” part follows from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We recall that the 3-groups of maximal class (as well as the p-groups of maximal class with a maximal abelian subgroup for any prime p) were classified by Blackburn [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' However, it does not seem easy to prove that they possess an irreducible character that vanishes at exactly 3n − 3n−1 + 6 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Eamonn O’Brien, has checked that this is true for groups of order at most 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' As we have mentioned, the converse of Theorem B does not hold for p > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This situation is related to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In [6] it was proved that the number of conjugacy classes of zeros of any non-linear irreducible character of a p-group is at least p2 − 1 (see Theorem C of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Furthermore, if equality holds and the character is faithful then G is a p-group of maximal class with a maximal abelian subgroup U and the set of zeros of the character is (G −U) ∪(Z2(G) −Z(G)) (see the proof of Theorem C of [6] 6 ALEXANDER MORET´O, GABRIEL NAVARRO and the paragraph that follows it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, we make clear the relation between both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (Note that this relation is only transparent after proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=') Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a non-abelian p-group of order pn and χ ∈ Irr(G) faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then χ vanishes at exactly pn − pn−1 + p2 − p elements if and only if χ vanishes at exactly p2 − 1 conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' This is clear if n = 3 so we may assume that n > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose first that χ vanishes at exactly p2 − 1 conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' As we have just mentioned, then G is a p-group of maximal class with a maximal abelian subgroup U and the set of zeros of the χ is (G − U) ∪ (Z2(G) − Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since the cardinality of this set is pn − pn−1 + p2 − p the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Conversely, assume that χ vanishes at exactly pn − pn−1 + p2 − p elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='4, G is a p-group of maximal class with a maximal abelian subgroup U and the set of zeros of the character is (G − U) ∪ (Z2(G) − Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let g ∈ G − U, so that G = ⟨g⟩U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since |Z(G)| = p, CU(g) = Z(G), so |CG(g)| = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In other words, the conjugacy classes in G − U have size pn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Therefore, the number of conjugacy classes of G contained in G − U is (pn − pn−1)/pn−2 = p2 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since |Z2(G)| = p2, the conjugacy classes in Z2(G) − Z(G) have size p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Hence, the number of conjugacy classes of G contained in this subset is p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' It follows that χ vanishes at exactly p2 − 1 conjugacy classes, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ Now, we can use Theorem D of [6] to see that if p > 3 and equality holds in Theorem B then |G| is bounded in terms of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a p-group of order pn, where p > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If G has an irreducible character χ that vanishes at exactly pn − pn−1 + p2 − p elements, then |G| ≤ pr+1, where r is the smallest prime that does not divide p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='4, we know that χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='7, χ vanishes at exactly p2 − 1 conjugacy classes and the result follows from Theorem D of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' □ Let us summarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If G is a non-abelian group, and mz(G) is the minimum number of elements of G taking the zero value among the non-linear irreducible characters of G, we let mz(pn) = min{mz(G) | |G| = pn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We have shown in Theorem B that mz(pn) ≤ pn − pn−1 + p2 − p, and in Theorem A that equality holds if p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' (We suspect that the same holds if p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=') Also the proof oif Theorem B and computer calculations performed by O’Brien suggest that the following could be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a p-group of order pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then mz(pn) = mz(G) if and only if G has maximal class with an abelian maximal normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' p-GROUPS AND ZEROS OF CHARACTERS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Groups with a Sylow tower We conclude with the proof of Theorem C, which we restate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Our interest now also includes roots of unity values of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let G be a group with a Sylow tower and let χ ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Then the proportion of elements x ∈ G such that χ(x) = 0 or χ(x) is a root of unity is at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We argue by induction on |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' There exists a prime p that divides |G| and G has a normal Hall p′-subgroup N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let P ∈ Sylp(G), so that G = PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since G/N is a p-group, it follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='22 of [3] that χ is a relative M-character with respect to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Thus there exists N ≤ H ≤ G and ψ ∈ Irr(H) such that χ = ψG and ψN ∈ Irr(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Suppose first that H < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since G/N is a p-group, every maximal subgroup U of G that contains N is normal in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since χ is induced U, it follows that χ vanishes on G − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' There are at least |G|/2 elements in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Thus the theorem holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, we may assume that H = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' In other words, θ = χN ∈ Irr(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let ˆθ be the canonical extension of θ to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' We claim that the proportion of zeros and root of unity values of ˆθ exceeds 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let Gp be the set of p-elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Therefore G = � x∈Gp CN(x)x is a disjoint union by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='18 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, if 1 ̸= x ∈ Gp, c ∈ CN(x), and θ∗ ∈ Irr(CN(x)) is the x-Glauberman correspondent of θ, we have by Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='32 of [3] that ˆθ(cx) = ǫθ∗(c) , where ǫ is a sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Since G has a Sylow tower, we have that CN(x) has a Sylow tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Let Ax be the set of elements of CN(x) where θ∗ has the value zero or a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' value 0 or root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' By induction, we have that |Ax| ≥ |CN(x)|/2 for every x ∈ Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If y ∈ Ax, then ˆθ(yx) is a zero or a root of unity, and therefore, ˆθ has at least � x∈Gp |Ax| ≥ |G|/2 roots of unity or zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Now, by Gallagher’s Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='17 of [3], we have that χ = µˆθ, where µ ∈ Irr(G/N) = Irr(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If µ is not linear, then the result follows from the p-group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' If µ is linear, then |χ(x)| = |ˆθ(x)| and the result follows from Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='2 of [3] and the previous paragraph □ 8 ALEXANDER MORET´O, GABRIEL NAVARRO It is easy to build examples of nonsolvable groups with irreducible characters that either vanish or take root of unity values at exactly one-half of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Consider for instance G = S ≀D10, where S is any simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' However, if Miller’s conjecture is true, then it seems reasonable to expect that if equality holds and χ ∈ Irr(G) is a character that either vanishes or takes a root of unity value at one-half of the elements of G, then χ is monomial of degree 2 and G/ ker χ is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Blackburn, On a special class of p-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 100 (1958), 45–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Huppert, Endliche Gruppen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Springer-Verlag, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Isaacs, Character Theory of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' AMS-Chelsea, Providence, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Isaacs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Sangroniz, p-groups having few almost-rational irreducible characters, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 189 (2012), 65–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Miller, Zeros and roots of unity in character tables, Enseign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=', to appear, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='13238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Moret´o, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Sangroniz, On the number of conjugacy classes of zeros of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' 142 (2004), 163–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content=' Departament de Matem`atiques, Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain Email address: alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='moreto@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='es Email address: gabriel@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} +page_content='es' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfegeM/content/2301.03917v1.pdf'} diff --git 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b/mtAyT4oBgHgl3EQf__o3/content/tmp_files/2301.00917v1.pdf.txt @@ -0,0 +1,787 @@ +Anomalous circular phonon dichroism in transition metal dichalcogenides +Wen-Yu Shan1 +1Department of Physics, School of Physics and Materials Science, +Guangzhou University, Guangzhou 510006, China +(Dated: January 4, 2023) +Magnetic field can generally induce circular phonon dichroism based on the formation of Landau +levels of electrons. Here we study the magnetization-induced circular phonon dichroism in transition +metal dichalcogenides, without forming the Landau levels. We find that, instead of the conventional +deformation potential coupling, the pseudogauge-type electron-phonon coupling plays an essential +role in the emergence of the phenomenon. +As a concrete example, a large dichroism signal is +obtained in monolayer MoTe2 on a EuO substrate, even without considering the Rashba spin-orbit +coupling. Due to the two-dimensional spin-valley-coupled band structure, MoTe2 shows a reciprocal +and nonreciprocal absorption of circularly polarized acoustic phonons upon reversing the direction +of phonon propagation and magnetization, respectively. +By varying the gate voltage, a tunable +circular phonon dichroism can be realized, which paves a way toward new physics and applications +of two-dimensional acoustoelectronics. +Introduction.—Recent years have seen a surge of inter- +est in investigating topological properties in the nonelec- +tronic systems, e.g., photonic, magnonic and phononic +materials. For phonons, the concepts of band topology +and geometry have brought into new ingredients: chiral +phonons [1–4], angular momentum [5–8], orbital mag- +netic moments of phonons [9–12], phonon angular mo- +mentum Hall effect [13], phonon rotoelectric effect [14] +and so on. +In metals, the interplay between phonons +and electrons with nontrivial band topology or geometry +may further induce distinctive features, such as phonon +helicity [15] and phonon magnetochiral effect [16, 17]. +Circular dichroism, the differential absorption between +left- and right-handed circularly polarized light, has been +widely used in examining topological phases of mat- +ter [18–24]. A phononic analog, namely, circular phonon +dichroism (CPD), is later proposed in three-dimensional +Weyl semimetals [25]. However, a direct analogy between +phonons and photons is not that obvious. The reasons +are twofold. +First, the photon wave vector is usually +much smaller than the Fermi wave vector of electrons, +thus only inducing the interband transition of electrons; +whereas the phonon wave vector may be comparable to +that of electrons, giving rise to either interband or intra- +band transition (see Fig. 1 (b)). Second, light waves con- +sist of only transverse modes, whereas acoustic waves in +solids have both longitudinal and transverse modes. Par- +ticularly, when dealing with two-dimensional (2D) mate- +rials, one has to mix longitudinal and transverse in-plane +modes to create circular phonons [26], in marked con- +trast to the case of light. This indicates that 2D circular +phonon dichroism is intrinsically different from the circu- +lar dichroism of light, where the former has received far +less attention. +Experimentally, several works have unveiled the effect +of Landau levels of electrons on the phonon dispersion +or circular dichroism in graphene [26–28], such as the +magnetophonon resonance. Nevertheless, the treatment +of Landau levels inevitably induces topology, even into +an originally trivial system. In this sense, the CPD can +not resolve the real band topology or geometry of the un- +derlying system. Another way of breaking time-reversal +symmetry is to introduce the magnetic exchange inter- +action, which does not require the formation of Lan- +dau levels and could retain the basic topology or ge- +ometry of the band structure. Up to now, the intrinsic +magnetization-induced CPD in 2D materials like mono- +layer transition metal dichalcogenides, remains unknown. +This generalization of magnetization bears similarities +to the case of anomalous Hall effect, hence the name +anomalous circular phonon dichroism. +The distinct +spin-valley-coupled band structure of transition metal +dichalcogenides may further contribute to the anoma- +lous behaviors of CPD and their nonreciprocal relations. +Therefore studying this new type of CPD would be de- +sirable for a better understanding and manipulation of +band geometry or topology in 2D materials. +In this paper, we explore the magnetization-induced +CPD in monolayer transition metal dichalcogenides. To +allow this effect, the pseudogauge-type electron-phonon +coupling is necessary instead of the conventional defor- +mation potential coupling. We obtain a large dichroism +signal in monolayer MoTe2 on a EuO substrate, even in +the absence of Rashba spin-orbit coupling. Due to the +unique spin-valley coupling, we find that MoTe2 shows +a reciprocal (nonreciprocal) absorption of circularly po- +larized acoustic phonons upon reversing the direction +of phonon propagation (magnetization). Our study re- +freshes our knowledge on the effect of electron-phonon +coupling on phonon dynamics, and paves the way toward +acoustoelectronics for 2D materials. +Model Hamiltonian.—We take the pristine 2H-phase +transition metal dichalcogenides MoTe2 on a EuO sub- +strate as a prototype (see Fig. +1 (a)). +The effective +electronic Hamiltonian is given by He = � +k ψ+(k)[H0 + +arXiv:2301.00917v1 [cond-mat.mes-hall] 3 Jan 2023 + +2 +FIG. 1. +Schematics of (a) the setup and (b) electronic band +structure of monolayer MoTe2. In (a), 2H-phase monolayer +MoTe2 is deposited on the EuO substrate. +In (b), yellow +(green) region corresponds to spin-up (-down) bands. Tran- +sition process of electrons due to acoustic phonons (photons) +is indicated by the blue solid (red dashed) line. +Hsoc + Hex + HR]ψ(k), where [29, 30] +H0 = ¯hv(τσxkx + σyky) + ∆ +2 σz, +Hsoc = τsz(λcσ+ + λvσ−), +Hex = −s · n(Bcσ+ + Bvσ−), +HR = λR(τsyσx − sxσy). +(1) +ψ+(k) and ψ(k) are the creation and annihilation op- +erator of electrons. +Hsoc, Hex and HR correspond to +the Ising-type spin-orbit coupling, proximity-induced ex- +change and Rashba interaction, respectively. s and σ are +Pauli matrices acting on spin {↑, ↓} and orbit subspace +{|dz2⟩, +1 +√ +2(|dx2−y2⟩ + iτ|dxy⟩)}, and σ± = +1 +2(σ0 ± σz). +τ = ±1 labels valley K±. λc/v describes the spin splitting +of the conduction and valence bands, respectively. Bc/v +is the effective Zeeman field experienced by the conduc- +tion and valence bands, arising from the exchange cou- +pling with the magnetic substrate. The out-of-plane z- +direction magnetization n = ez is considered (see Fig. 1 +(a)). For the moment, we set λR = 0 in order to have an- +alytical expressions and intuitive physical picture. The +role of λR will be clarified later. +The electronic band +structure upon magnetization is schematically shown in +Fig. +1 (b), where the signature of spin-valley coupling +can be seen explicitly. The Fermi level is pinned at the +valence bands, where the effect of spin-valley coupling is +manifest. The magnetization Bc/v shifts the opposite- +spin states from different valleys in opposite directions, +and thus breaks the time-reversal symmetry. +For the phononic part, we consider two branches of +in-plane acoustic phonon modes. Due to the low sound +velocity cl/t (l/t for longitudinal/transverse phonon po- +larization), the acoustic phonon energy ωl/t = ¯hcl/t|q| is +much smaller than the valence band splitting of electrons, +i.e., 2(Bv ± λv), where q is the phonon wave vector. As +a result, only intraband transitions of electrons are trig- +gered by acoustic phonons (see Fig. 1 (b)). By contrast, +optical phonon modes with larger energy, may enable +either intraband or interband transitions. Nonetheless, +the basic physical picture should be similar. For simplic- +ity, we further study the long-wavelength limit of phonon +modes, which allows us to neglect the intervalley scatter- +ing process of electrons. +Based on the theory of elasticity [31–33], the electron- +acoustic-phonon coupling in MoTe2 contains two terms: +He−ph = Hd +e−ph + Hp +e−ph, where Hd +e−ph (Hp +e−ph) refers +to the deformation (pseudogauge) potential coupling. +Hd/p +e−ph has a general form [34, 35] +Hd/p +e−ph = +� +k,q +ψ+(k + q)[u(q) · ˆTd/p(q)]ψ(k), +(2) +where u(q) is the Fourier transform of the in-plane collec- +tive displacement u(r) for acoustic modes [32] and ˆT (q) +is the Fourier transform of the “effective” force operator +ˆT (r) acting on atoms by electrons. +For the deforma- +tion potential Hd +e−ph, ˆTd(q) = igdq, which is indepen- +dent of the valley index τ. For the pseudogauge poten- +tial Hp +e−ph, the force operator becomes valley-dependent, +that is, ˆT τ=−1 +p +(q) = igp[q · σ, (q × σ)z] and ˆT τ=1 +p +(q) = +K[ ˆT τ=−1 +p +(−q)]. K is the complex conjugation operator. +The relation between ˆT τ=1 +p +(q) and ˆT τ=−1 +p +(−q) preserves +the time-reversal symmetry of electron-phonon coupling +in the absence of magnetization. +Phonon equation of motion.—For the phonon dynam- +ics, we consider the phonon equation of motion in the +frequency-momentum (ω, q) domain [25] +ω2uα(q) = +� +β +[Φαβ(q) + ¯hχαβ(q, ω)]uβ(q), +(3) +where α, β = x, y and Φ(q) is the dynamical matrix. +χαβ(q, ω) is a retarded response function arising from the +electron-phonon coupling and follows at each valley [35] +χτ +αβ(q, ω + iδ) = +� +n,m +� +¯hd2k +ρ(2π)2 +fτ,m,k − fτ,n,k−q +ω + iδ + Eτ,m,k − Eτ,n,k−q +× ⟨τ, m, k| ˆT τ +α(q)|τ, n, k − q⟩⟨τ, n, k − q| ˆT τ +β (−q)|τ, m, k⟩. +(4) +Eτ,m,k and |τ, m, k⟩ are the dispersion and electronic +wave function of Hamiltonian (1), respectively. +fτ,m,k +(fτ,n,k−q) is the Fermi distribution function, ρ is the 2D + +(a) +MoTe2 +(a) +Te +Mo +Euo +Te +EuO +(b) +acoustic +photon +phonon +>SS +EF +-K +K +-K3 +mass density, δ is a positive infinitesimal. Since only in- +traband transitions (band indices m = n) of electrons are +allowed by acoustic modes in the low-temperature limit, +m, n reduce to the ones intersected by the Fermi level, +i.e., spin-split valence bands at valley K± (see Fig. 1 +(b)). +Circular phonon dichroism.—Our main interest lies in +the anti-Hermitian part of χ(q, ω), that is, −2iωγ(q, ω), +where γ(q, ω) is a Hermitian matrix satisfying γ+(q, ω) = +γ(q, ω). This matrix corresponds to the non-Hermitian +part of the phonon self-energy, which physically origi- +nates from the phonon absorption by electrons. In the +basis of {ˆx, ˆy}T , γ matrix has the form +γ(q, ω) = +� +� +D(q, ω) + ¯D(q, ω) +¯A(q, ω) + iA(q, ω) +¯A(q, ω) − iA(q, ω) D(q, ω) − ¯D(q, ω) +� +� . +(5) +Different from the Weyl semimetals [25], new terms +¯D(q, ω) and ¯A(q, ω) occur in monolayer MoTe2 as a re- +sult of D3h point-group symmetry. +For the left- and +right-handed circularly polarized phonons, |uL/R⟩ = +1 +√ +2[1 ±i]T , the damping (absorption) coefficients read +γL/R = D(q, ω) ∓ A(q, ω). The relative difference be- +tween γL and γR defines the circular phonon dichroism +(CPD). One can see that the behavior of CPD is totally +determined by A(q, ω)/D(q, ω). +For longitudinal or transverse phonons, the polariza- +tion is linear as |ul⟩ = [cos φq sin φq]T and |ut⟩ = [− sin φq +cos φq]T , where the angular variable φq = tan−1(qy/qx). +The damping coefficients are given by γl/t = D(q, ω) +± cos 2φq ¯D(q, ω) ± sin 2φq ¯A(q, ω), which explicitly de- +pends on the phonon propagation direction q. Here, dif- +ferent from the circular phonons, the damping coefficients +γl/t for the linear phonons depend on the parameters +¯D(q, ω) and ¯A(q, ω). +Specifically for the deformation potential Hd +e−ph, γ ma- +trix is proportional to [35] +γ(q, ω) ∝ +� +� +q2 +x +qxqy +qxqy +q2 +y +� +� . +(6) +This immediately leads to A(q, ω) = 0, meaning that +the CPD vanishes when only the deformation potential +coupling is taken in account. Meanwhile, γt = 0, sug- +gesting that there is no absorption for the transverse +phonon modes. This agrees with the fact that the de- +formation potential only couples electrons to the longi- +tudinal phonon modes [34]. +For the pseudogauge potential Hp +e−ph, the situation +is more complex. When only focusing on the acoustic +modes, analytical expressions for all elements of γ(q, ω) +matrix can be obtained [35]. For example, when a sin- +gle valence band at valley Kτ is intersected by the Fermi +γ l +γ t +EF +EF +valley + K+/− +valley K- +(b) +A/D +q / kτ=−1 +F +(a) +EF = −0.48eV +t mode +l mode +q=0.108 +q=0.5*0.108 +(c) +A/D +EF [eV] +K+ +K− +K− +(d) +A/D +∆ [eV] +FIG. 2. +(a) Angular φq-dependence of the damping coeffi- +cients γl (γt) for the longitudinal (transverse) acoustic phonon +modes. +(b)-(d) Relations of the circular phonon dichroism +A/D versus the phonon wave vector q/kτ=−1 +F +, Fermi energy +EF and gap function ∆, respectively. In (b), the Fermi energy +is fixed: EF = −0.48 eV. The inset shows the details of the +cyan region, and the two peaks are due to the longitudinal +and transverse phonon modes, respectively. The peaks in the +cyan region are given by a summation of valleys K±, whereas +the peaks near q/kτ=−1 +F += 2 are only determined by valley +K−. kτ=−1 +F +is the Fermi wave vector at valley K−. The red +dot refers to the case of (a). In (c), different values of q are +adopted. The locations of the Fermi level at the peaks are +shown in the inset: both valley K± are intersected at smaller +EF ; only valley K− is intersected at larger EF . In (d), the +black dot indicates the value of ∆ in (a)-(c): ∆ = 1.05 eV. +Parameters: λv = 0.11 eV, λc = 0.029 eV, ¯hv = 2.33 eV·˚A, +Bc = 0.206 eV, Bv = 0.17 eV [29], longitudinal and trans- +verse sound velocity cl = 3.64 × 103 m/s and ct = 2.21 × 103 +m/s [36], mass density ρ = 9.40 × 10−6 kg/m2 [37] and the +electron-phonon coupling constant gp = 0.32 eV [38]. +level, A(q, ω)/D(q, ω) reduces to +Aτ(q, ω) +Dτ(q, ω) = τω ∆ − τλc + τλv + Bc − Bv +2[ωxτ +F − (¯hvkτ +F )2] +Θ(kτ +F − q +2 − kτ +0), +(7) +where the Heaviside step function Θ(· · · ) constrains +the +magnitude +of +phonon +wave +vector +q += +|q|. +kτ +F is a valley-dependent Fermi wave vector of elec- +trons. +kτ +0 = +ω +2¯hv +� +1 + (∆−τλc+τλv+Bc−Bv)2 +(¯hvq)2−ω2 +and xτ +F = +� +( ∆−τλc+τλv+Bc−Bv +2 +)2 + (¯hvkτ +F )2. +However, such a +simple +relation +fails +when +both +valleys +are +inter- +sected by the Fermi level, +given that A(q, ω) += +� +τ Aτ(q, ω) and D(q, ω) += +� +τ Dτ(q, ω). +On the +other hand, both +¯A(q, ω) and +¯D(q, ω) become φq- +dependent +[35]: +¯A(q, ω) += +−F(q, ω) sin 4φq +and + +4 +¯D(q, ω) = F(q, ω) cos 4φq, with a φq-independent fac- +tor F(q, ω). By substituting these into γl/t, we find for +linearly polarized phonons, +γl/t = D(q, ω) ± F(q, ω) cos 6φq. +(8) +One can see that γl/t has a six-fold (C6) rotational sym- +metry on φq (see Fig. +2 (a)), which is different from +the three-fold (C3) rotational symmetry of the underly- +ing crystals. The reason for the symmetry mismatch is +due to the reciprocal behaviors of γl/t upon reversing the +direction of phonon propagation, i.e., q → −q, as shown +in Table I. For phonons, cl/t ≪ v, giving rise to [35] +D(q, ω) ≈ −F(q, ω). As a result, γl ≈ 2D(q, ωl) sin2 3φq +and γt ≈ 2D(q, ωt) sin2 3(φq − π +6 ). This means that there +is an angular shift π +6 in φq between γl and γt, as shown +in Fig. 2 (a). +For circularly polarized phonons, numerical results of +A(q, ω)/D(q, ω) as functions of the rescaled phonon wave +vector q/kτ=−1 +F +, Fermi energy EF and ∆ are shown in +Fig. +2 (b)-(d), respectively. +The Fermi wave vector +kτ=−1 +F +rather than kτ=1 +F +is selected since the valence +band edge of valley K− is higher than K+, as shown +in Fig. 1. In Fig. 2 (b), a non-monotonic behavior of +A/D as q increases can be seen explicitly. The jumps at +q/kτ=−1 +F +≈ 0.47 and 1.98 originate from the sudden van- +ishing of valley K+ and K−, respectively, as required by +the factor Θ(kτ +F − q +2 −kτ +0) in Eq. (7). Such a factor can be +understood as a result of the energy and momentum con- +servation for the electron-phonon scattering process. For +acoustic phonons, the electron scattering approximately +occurs on the Fermi surface. In this sense, the phonon +wave vector q must be smaller than the maximum value of +momentum transfer of electrons, that is, q < 2kτ +F . kτ +0 is a +small offset wave vector arising from the acoustic phonon +dispersion ω. As seen in the inset of Fig. 2 (b), there are +actually two adjacent peaks (jumps) in the highlighted +region corresponding to the l and t mode, respectively, +since kτ +0 is different for ω = ωl/t. As the sound velocity +cl > ct, kτ +0 is larger for the longitudinal mode, leading +to a smaller transition value of q. In Fig. 2 (c), the lo- +cations of the Fermi level for the peaks are indicated in +the inset. The peaks at the lower (higher) Fermi level +are dominated by valley K+ (K−), which exhibit oppo- +site signs of A/D. For each valley, the magnitude |A/D| +increases when the Fermi level is tuned toward the band +edge. Different values of q are also compared. We find +TABLE I. Transformation properties of parameters D, ¯D, A +and ¯A. +Transformation +D(q, ω) +¯D(q, ω) +A(q, ω) +¯A(q, ω) +q → −q ++ ++ ++ ++ +n → −n ++ ++ +− ++ +that by adopting a smaller q, the peaks are shifted to a +higher Fermi level, as kτ +F becomes smaller. The peaks +also show a larger magnitude and become sharper, par- +ticularly for the second peaks. Therefore this provides +a means of tuning the sign and magnitude of the CPD. +In Fig. +2 (d), the value of ∆ adopted in Fig. +2 (a)- +(c) is indicated. We can see that the magnitude |A/D| +is basically enhanced when ∆ increases, expect for the +discontinuous points. +That is the reason why we pro- +pose monolayer transition metal dichalcogenides as can- +didate materials, which have large band gap and thus +large CPD signals. For an order-of-magnitude estimate, +we consider the parameters corresponding to the red dot +in Fig. +2 (b), which also refer to the case of Fig. +2 +(a). We find D = 1.90 × 107/s and A = −7.81 × 105/s. +This yields a difference of the attenuation between the +left- and right-handed circularly polarized waves, that is, +(γL−γR)/¯c ∼ 534/m, where ¯c = (cl+ct)/2 is the average +sound velocity. Such difference is much larger than that +of the Weyl semimetals [25], and should be observable in +ultrasonic experiments. +Nonreciprocal absorption.—Given that both the space- +inversion and time-reversal symmetry are broken in our +system, the absorption of circularly polarized phonons is +expected to be nonreciprocal. To see this, we consider in +Table I the transformation properties of parameters D, +¯D, A and ¯A upon reversing the direction of phonon prop- +agation q or magnetization n. We find that D, ¯D and ¯A +are even functions of q and n, whereas A is an even (odd) +function of q (n). +Accordingly, the absorption coeffi- +cients of circular phonons γL/R remain unchanged under +the transformation q → −q, whereas γL/R interchange +with each other under the transformation n → −n. This +represents a reciprocal and nonreciprocal CPD upon re- +versing the direction of phonon propagation and magne- +tization, respectively. Such result is similar to that of +the Faraday rotation of light polarization [39], where the +rotation angle only depends on the magnetic field direc- +tion. However, the origin is different. The absorption +of circularly polarized photons is actually nonreciprocal +when q → −q, but the chirality of circular photons also +depends on the light propagation direction q. As a re- +sult, the reciprocal q-dependence of the rotation angle +is recovered. On the other hand, due to the 2D nature, +the chirality of in-plane circular phonons is independent +of the phonon propagation direction q, giving rise to the +reciprocal absorption. This also indicates that there is +no directional dichroism [40] or phonon magnetochiral +effect [16, 17] in our system. +Roles of Rashba spin-orbit coupling.—Now +we +take +into account the Rashba term HR and treat it as a per- +turbation. Analytical expressions are calculated [35] and +numerical results of electronic band structure and CPD +are shown in Fig. +3. +In Fig. +3 (a), we find that the +Rashba term shifts the conduction (valence) band edge +to higher (lower) energy at valley K−, whereas it hardly + +5 +λR +λR +eV +λR +eV +(b) +K- +E [eV] +k +λR +λR +eV +λR +eV +K+ +(a) +A/D +EF [eV] +FIG. 3. +(a) Electronic band structure of monolayer MoTe2 +and (b) circular phonon dichroism A/D versus the Fermi en- +ergy EF for different strength of Rashba spin-orbit coupling. +changes the band structure at valley K+. This explains +the phenomenon of peak shift in Fig. +3 (b) since the +peaks are always close to the band edge. By increasing +the strength of Rashba spin-orbit coupling, the magni- +tude of CPD can be enhanced, which provides us a knob +to tune the CPD. For a realistic strength λR = 0.072 +eV [29], the behavior of A/D is similar to the case with- +out Rashba spin-orbit coupling, thus validating our above +treatment. +Particularly, we find that the introduction +of HR does not change the reciprocal behaviors of ab- +sorption coefficients γL/R under q → −q. Therefore, to +obtain the nonreciprocity, additional ingredients should +be taken into account, such as the cyclotron motion of +electrons [26–28, 41, 42] or phonon-magnon coupling [43]. +Discussion and conclusion.—We have studied the cir- +cular phonon dichroism in magnetic two-dimensional ma- +terials, i.e., monolayer MoTe2 in proximity to the EuO +substrate. +Large dichroism signal is obtained for the +pseudogauge-type electron-phonon coupling, even with- +out introducing the Landau levels or Rashba spin-orbit +coupling. Such a signal is reciprocal (nonreciprocal) upon +reversing the direction of phonon propagation (magneti- +zation). +By varying the gate voltage, the CPD signal +can be tuned through the role of Fermi level and Rashba +spin-orbit coupling. +The proposed CPD effect can also be applied to +other transition-metal dichalcogenides with spin-valley- +coupled band structure, or their van der waals het- +erostructures. The effect can be detected by the pulse- +echo technique [44, 45] based on the different absorption +coefficients between left- and right-handed circularly po- +larized phonons. An alternative detection is the Raman +spectroscopy analysis [26, 27] of phonon polarization by +injecting a linearly polarized acoustic waves. +Acknowledgments. +This work is supported by the +National Natural Science Foundation of China (NSFC, +Grant No. 11904062). We also acknowledge the support +of a startup grant from Guangzhou University. +[1] L. Zhang and Q. Niu, Phys. Rev. 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L¨uthi, Physical Acoustics in the Solid State (Springer- +Verlag, Berlin, 2004). + diff --git a/mtAyT4oBgHgl3EQf__o3/content/tmp_files/load_file.txt b/mtAyT4oBgHgl3EQf__o3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c192ba3738ce00981e3ed5673b4617c5459aeefe --- /dev/null +++ b/mtAyT4oBgHgl3EQf__o3/content/tmp_files/load_file.txt @@ -0,0 +1,628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf,len=627 +page_content='Anomalous circular phonon dichroism in transition metal dichalcogenides Wen-Yu Shan1 1Department of Physics, School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China (Dated: January 4, 2023) Magnetic field can generally induce circular phonon dichroism based on the formation of Landau levels of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Here we study the magnetization-induced circular phonon dichroism in transition metal dichalcogenides, without forming the Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We find that, instead of the conventional deformation potential coupling, the pseudogauge-type electron-phonon coupling plays an essential role in the emergence of the phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As a concrete example, a large dichroism signal is obtained in monolayer MoTe2 on a EuO substrate, even without considering the Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Due to the two-dimensional spin-valley-coupled band structure, MoTe2 shows a reciprocal and nonreciprocal absorption of circularly polarized acoustic phonons upon reversing the direction of phonon propagation and magnetization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' By varying the gate voltage, a tunable circular phonon dichroism can be realized, which paves a way toward new physics and applications of two-dimensional acoustoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—Recent years have seen a surge of inter- est in investigating topological properties in the nonelec- tronic systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=', photonic, magnonic and phononic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For phonons, the concepts of band topology and geometry have brought into new ingredients: chiral phonons [1–4], angular momentum [5–8], orbital mag- netic moments of phonons [9–12], phonon angular mo- mentum Hall effect [13], phonon rotoelectric effect [14] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In metals, the interplay between phonons and electrons with nontrivial band topology or geometry may further induce distinctive features, such as phonon helicity [15] and phonon magnetochiral effect [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Circular dichroism, the differential absorption between left- and right-handed circularly polarized light, has been widely used in examining topological phases of mat- ter [18–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' A phononic analog, namely, circular phonon dichroism (CPD), is later proposed in three-dimensional Weyl semimetals [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' However, a direct analogy between phonons and photons is not that obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The reasons are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' First, the photon wave vector is usually much smaller than the Fermi wave vector of electrons, thus only inducing the interband transition of electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' whereas the phonon wave vector may be comparable to that of electrons, giving rise to either interband or intra- band transition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Second, light waves con- sist of only transverse modes, whereas acoustic waves in solids have both longitudinal and transverse modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Par- ticularly, when dealing with two-dimensional (2D) mate- rials, one has to mix longitudinal and transverse in-plane modes to create circular phonons [26], in marked con- trast to the case of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This indicates that 2D circular phonon dichroism is intrinsically different from the circu- lar dichroism of light, where the former has received far less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Experimentally, several works have unveiled the effect of Landau levels of electrons on the phonon dispersion or circular dichroism in graphene [26–28], such as the magnetophonon resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Nevertheless, the treatment of Landau levels inevitably induces topology, even into an originally trivial system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In this sense, the CPD can not resolve the real band topology or geometry of the un- derlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Another way of breaking time-reversal symmetry is to introduce the magnetic exchange inter- action, which does not require the formation of Lan- dau levels and could retain the basic topology or ge- ometry of the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Up to now, the intrinsic magnetization-induced CPD in 2D materials like mono- layer transition metal dichalcogenides, remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This generalization of magnetization bears similarities to the case of anomalous Hall effect, hence the name anomalous circular phonon dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The distinct spin-valley-coupled band structure of transition metal dichalcogenides may further contribute to the anoma- lous behaviors of CPD and their nonreciprocal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Therefore studying this new type of CPD would be de- sirable for a better understanding and manipulation of band geometry or topology in 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In this paper, we explore the magnetization-induced CPD in monolayer transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' To allow this effect, the pseudogauge-type electron-phonon coupling is necessary instead of the conventional defor- mation potential coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We obtain a large dichroism signal in monolayer MoTe2 on a EuO substrate, even in the absence of Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Due to the unique spin-valley coupling, we find that MoTe2 shows a reciprocal (nonreciprocal) absorption of circularly po- larized acoustic phonons upon reversing the direction of phonon propagation (magnetization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Our study re- freshes our knowledge on the effect of electron-phonon coupling on phonon dynamics, and paves the way toward acoustoelectronics for 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—We take the pristine 2H-phase transition metal dichalcogenides MoTe2 on a EuO sub- strate as a prototype (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The effective electronic Hamiltonian is given by He = � k ψ+(k)[H0 + arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='00917v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='mes-hall] 3 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Schematics of (a) the setup and (b) electronic band structure of monolayer MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In (a), 2H-phase monolayer MoTe2 is deposited on the EuO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In (b), yellow (green) region corresponds to spin-up (-down) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Tran- sition process of electrons due to acoustic phonons (photons) is indicated by the blue solid (red dashed) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Hsoc + Hex + HR]ψ(k), where [29, 30] H0 = ¯hv(τσxkx + σyky) + ∆ 2 σz, Hsoc = τsz(λcσ+ + λvσ−), Hex = −s · n(Bcσ+ + Bvσ−), HR = λR(τsyσx − sxσy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (1) ψ+(k) and ψ(k) are the creation and annihilation op- erator of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Hsoc, Hex and HR correspond to the Ising-type spin-orbit coupling, proximity-induced ex- change and Rashba interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' s and σ are Pauli matrices acting on spin {↑, ↓} and orbit subspace {|dz2⟩, 1 √ 2(|dx2−y2⟩ + iτ|dxy⟩)}, and σ± = 1 2(σ0 ± σz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' τ = ±1 labels valley K±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' λc/v describes the spin splitting of the conduction and valence bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Bc/v is the effective Zeeman field experienced by the conduc- tion and valence bands, arising from the exchange cou- pling with the magnetic substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The out-of-plane z- direction magnetization n = ez is considered (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the moment, we set λR = 0 in order to have an- alytical expressions and intuitive physical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The role of λR will be clarified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The electronic band structure upon magnetization is schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (b), where the signature of spin-valley coupling can be seen explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The Fermi level is pinned at the valence bands, where the effect of spin-valley coupling is manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The magnetization Bc/v shifts the opposite- spin states from different valleys in opposite directions, and thus breaks the time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the phononic part, we consider two branches of in-plane acoustic phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Due to the low sound velocity cl/t (l/t for longitudinal/transverse phonon po- larization), the acoustic phonon energy ωl/t = ¯hcl/t|q| is much smaller than the valence band splitting of electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=', 2(Bv ± λv), where q is the phonon wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As a result, only intraband transitions of electrons are trig- gered by acoustic phonons (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' By contrast, optical phonon modes with larger energy, may enable either intraband or interband transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Nonetheless, the basic physical picture should be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For simplic- ity, we further study the long-wavelength limit of phonon modes, which allows us to neglect the intervalley scatter- ing process of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Based on the theory of elasticity [31–33], the electron- acoustic-phonon coupling in MoTe2 contains two terms: He−ph = Hd e−ph + Hp e−ph, where Hd e−ph (Hp e−ph) refers to the deformation (pseudogauge) potential coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Hd/p e−ph has a general form [34, 35] Hd/p e−ph = � k,q ψ+(k + q)[u(q) · ˆTd/p(q)]ψ(k), (2) where u(q) is the Fourier transform of the in-plane collec- tive displacement u(r) for acoustic modes [32] and ˆT (q) is the Fourier transform of the “effective” force operator ˆT (r) acting on atoms by electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the deforma- tion potential Hd e−ph, ˆTd(q) = igdq, which is indepen- dent of the valley index τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the pseudogauge poten- tial Hp e−ph, the force operator becomes valley-dependent, that is, ˆT τ=−1 p (q) = igp[q · σ, (q × σ)z] and ˆT τ=1 p (q) = K[ ˆT τ=−1 p (−q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' K is the complex conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The relation between ˆT τ=1 p (q) and ˆT τ=−1 p (−q) preserves the time-reversal symmetry of electron-phonon coupling in the absence of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Phonon equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—For the phonon dynam- ics, we consider the phonon equation of motion in the frequency-momentum (ω, q) domain [25] ω2uα(q) = � β [Φαβ(q) + ¯hχαβ(q, ω)]uβ(q), (3) where α, β = x, y and Φ(q) is the dynamical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' χαβ(q, ω) is a retarded response function arising from the electron-phonon coupling and follows at each valley [35] χτ αβ(q, ω + iδ) = � n,m � ¯hd2k ρ(2π)2 fτ,m,k − fτ,n,k−q ω + iδ + Eτ,m,k − Eτ,n,k−q × ⟨τ, m, k| ˆT τ α(q)|τ, n, k − q⟩⟨τ, n, k − q| ˆT τ β (−q)|τ, m, k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (4) Eτ,m,k and |τ, m, k⟩ are the dispersion and electronic wave function of Hamiltonian (1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' fτ,m,k (fτ,n,k−q) is the Fermi distribution function, ρ is the 2D (a) MoTe2 (a) Te Mo Euo Te EuO (b) acoustic photon phonon >SS EF K K K3 mass density, δ is a positive infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Since only in- traband transitions (band indices m = n) of electrons are allowed by acoustic modes in the low-temperature limit, m, n reduce to the ones intersected by the Fermi level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=', spin-split valence bands at valley K± (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Circular phonon dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—Our main interest lies in the anti-Hermitian part of χ(q, ω), that is, −2iωγ(q, ω), where γ(q, ω) is a Hermitian matrix satisfying γ+(q, ω) = γ(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This matrix corresponds to the non-Hermitian part of the phonon self-energy, which physically origi- nates from the phonon absorption by electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In the basis of {ˆx, ˆy}T , γ matrix has the form γ(q, ω) = � � D(q, ω) + ¯D(q, ω) ¯A(q, ω) + iA(q, ω) ¯A(q, ω) − iA(q, ω) D(q, ω) − ¯D(q, ω) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (5) Different from the Weyl semimetals [25], new terms ¯D(q, ω) and ¯A(q, ω) occur in monolayer MoTe2 as a re- sult of D3h point-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the left- and right-handed circularly polarized phonons, |uL/R⟩ = 1 √ 2[1 ±i]T , the damping (absorption) coefficients read γL/R = D(q, ω) ∓ A(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The relative difference be- tween γL and γR defines the circular phonon dichroism (CPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' One can see that the behavior of CPD is totally determined by A(q, ω)/D(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For longitudinal or transverse phonons, the polariza- tion is linear as |ul⟩ = [cos φq sin φq]T and |ut⟩ = [− sin φq cos φq]T , where the angular variable φq = tan−1(qy/qx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The damping coefficients are given by γl/t = D(q, ω) ± cos 2φq ¯D(q, ω) ± sin 2φq ¯A(q, ω), which explicitly de- pends on the phonon propagation direction q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Here, dif- ferent from the circular phonons, the damping coefficients γl/t for the linear phonons depend on the parameters ¯D(q, ω) and ¯A(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Specifically for the deformation potential Hd e−ph, γ ma- trix is proportional to [35] γ(q, ω) ∝ � � q2 x qxqy qxqy q2 y � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (6) This immediately leads to A(q, ω) = 0, meaning that the CPD vanishes when only the deformation potential coupling is taken in account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Meanwhile, γt = 0, sug- gesting that there is no absorption for the transverse phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This agrees with the fact that the de- formation potential only couples electrons to the longi- tudinal phonon modes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For the pseudogauge potential Hp e−ph, the situation is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' When only focusing on the acoustic modes, analytical expressions for all elements of γ(q, ω) matrix can be obtained [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For example, when a sin- gle valence band at valley Kτ is intersected by the Fermi γ l γ t EF EF valley K+/− valley K- (b) A/D q / kτ=−1 F (a) EF = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='48eV t mode l mode q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='108 q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='5*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='108 (c) A/D EF [eV] K+ K− K− (d) A/D ∆ [eV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (a) Angular φq-dependence of the damping coeffi- cients γl (γt) for the longitudinal (transverse) acoustic phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (b)-(d) Relations of the circular phonon dichroism A/D versus the phonon wave vector q/kτ=−1 F , Fermi energy EF and gap function ∆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In (b), the Fermi energy is fixed: EF = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='48 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The inset shows the details of the cyan region, and the two peaks are due to the longitudinal and transverse phonon modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The peaks in the cyan region are given by a summation of valleys K±, whereas the peaks near q/kτ=−1 F = 2 are only determined by valley K−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' kτ=−1 F is the Fermi wave vector at valley K−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The red dot refers to the case of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In (c), different values of q are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The locations of the Fermi level at the peaks are shown in the inset: both valley K± are intersected at smaller EF ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' only valley K− is intersected at larger EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In (d), the black dot indicates the value of ∆ in (a)-(c): ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='05 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Parameters: λv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='11 eV, λc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='029 eV, ¯hv = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='33 eV·˚A, Bc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='206 eV, Bv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='17 eV [29], longitudinal and trans- verse sound velocity cl = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='64 × 103 m/s and ct = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='21 × 103 m/s [36], mass density ρ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='40 × 10−6 kg/m2 [37] and the electron-phonon coupling constant gp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='32 eV [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' level, A(q, ω)/D(q, ω) reduces to Aτ(q, ω) Dτ(q, ω) = τω ∆ − τλc + τλv + Bc − Bv 2[ωxτ F − (¯hvkτ F )2] Θ(kτ F − q 2 − kτ 0), (7) where the Heaviside step function Θ(· · · ) constrains the magnitude of phonon wave vector q = |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' kτ F is a valley-dependent Fermi wave vector of elec- trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' kτ 0 = ω 2¯hv � 1 + (∆−τλc+τλv+Bc−Bv)2 (¯hvq)2−ω2 and xτ F = � ( ∆−τλc+τλv+Bc−Bv 2 )2 + (¯hvkτ F )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' However, such a simple relation fails when both valleys are inter- sected by the Fermi level, given that A(q, ω) = � τ Aτ(q, ω) and D(q, ω) = � τ Dτ(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' On the other hand, both ¯A(q, ω) and ¯D(q, ω) become φq- dependent [35]: ¯A(q, ω) = −F(q, ω) sin 4φq and 4 ¯D(q, ω) = F(q, ω) cos 4φq, with a φq-independent fac- tor F(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' By substituting these into γl/t, we find for linearly polarized phonons, γl/t = D(q, ω) ± F(q, ω) cos 6φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (8) One can see that γl/t has a six-fold (C6) rotational sym- metry on φq (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (a)), which is different from the three-fold (C3) rotational symmetry of the underly- ing crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The reason for the symmetry mismatch is due to the reciprocal behaviors of γl/t upon reversing the direction of phonon propagation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=', q → −q, as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For phonons, cl/t ≪ v, giving rise to [35] D(q, ω) ≈ −F(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As a result, γl ≈ 2D(q, ωl) sin2 3φq and γt ≈ 2D(q, ωt) sin2 3(φq − π 6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This means that there is an angular shift π 6 in φq between γl and γt, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For circularly polarized phonons, numerical results of A(q, ω)/D(q, ω) as functions of the rescaled phonon wave vector q/kτ=−1 F , Fermi energy EF and ∆ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (b)-(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The Fermi wave vector kτ=−1 F rather than kτ=1 F is selected since the valence band edge of valley K− is higher than K+, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (b), a non-monotonic behavior of A/D as q increases can be seen explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The jumps at q/kτ=−1 F ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='47 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='98 originate from the sudden van- ishing of valley K+ and K−, respectively, as required by the factor Θ(kτ F − q 2 −kτ 0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Such a factor can be understood as a result of the energy and momentum con- servation for the electron-phonon scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For acoustic phonons, the electron scattering approximately occurs on the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In this sense, the phonon wave vector q must be smaller than the maximum value of momentum transfer of electrons, that is, q < 2kτ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' kτ 0 is a small offset wave vector arising from the acoustic phonon dispersion ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (b), there are actually two adjacent peaks (jumps) in the highlighted region corresponding to the l and t mode, respectively, since kτ 0 is different for ω = ωl/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As the sound velocity cl > ct, kτ 0 is larger for the longitudinal mode, leading to a smaller transition value of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (c), the lo- cations of the Fermi level for the peaks are indicated in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The peaks at the lower (higher) Fermi level are dominated by valley K+ (K−), which exhibit oppo- site signs of A/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For each valley, the magnitude |A/D| increases when the Fermi level is tuned toward the band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Different values of q are also compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We find TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Transformation properties of parameters D, ¯D, A and ¯A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Transformation D(q, ω) ¯D(q, ω) A(q, ω) ¯A(q, ω) q → −q + + + + n → −n + + − + that by adopting a smaller q, the peaks are shifted to a higher Fermi level, as kτ F becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The peaks also show a larger magnitude and become sharper, par- ticularly for the second peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Therefore this provides a means of tuning the sign and magnitude of the CPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (d), the value of ∆ adopted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (a)- (c) is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We can see that the magnitude |A/D| is basically enhanced when ∆ increases, expect for the discontinuous points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' That is the reason why we pro- pose monolayer transition metal dichalcogenides as can- didate materials, which have large band gap and thus large CPD signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For an order-of-magnitude estimate, we consider the parameters corresponding to the red dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (b), which also refer to the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We find D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='90 × 107/s and A = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='81 × 105/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This yields a difference of the attenuation between the left- and right-handed circularly polarized waves, that is, (γL−γR)/¯c ∼ 534/m, where ¯c = (cl+ct)/2 is the average sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Such difference is much larger than that of the Weyl semimetals [25], and should be observable in ultrasonic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Nonreciprocal absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—Given that both the space- inversion and time-reversal symmetry are broken in our system, the absorption of circularly polarized phonons is expected to be nonreciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' To see this, we consider in Table I the transformation properties of parameters D, ¯D, A and ¯A upon reversing the direction of phonon prop- agation q or magnetization n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We find that D, ¯D and ¯A are even functions of q and n, whereas A is an even (odd) function of q (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Accordingly, the absorption coeffi- cients of circular phonons γL/R remain unchanged under the transformation q → −q, whereas γL/R interchange with each other under the transformation n → −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This represents a reciprocal and nonreciprocal CPD upon re- versing the direction of phonon propagation and magne- tization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Such result is similar to that of the Faraday rotation of light polarization [39], where the rotation angle only depends on the magnetic field direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' However, the origin is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The absorption of circularly polarized photons is actually nonreciprocal when q → −q, but the chirality of circular photons also depends on the light propagation direction q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' As a re- sult, the reciprocal q-dependence of the rotation angle is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' On the other hand, due to the 2D nature, the chirality of in-plane circular phonons is independent of the phonon propagation direction q, giving rise to the reciprocal absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This also indicates that there is no directional dichroism [40] or phonon magnetochiral effect [16, 17] in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Roles of Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—Now we take into account the Rashba term HR and treat it as a per- turbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Analytical expressions are calculated [35] and numerical results of electronic band structure and CPD are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 3 (a), we find that the Rashba term shifts the conduction (valence) band edge to higher (lower) energy at valley K−, whereas it hardly 5 λR λR eV λR eV (b) K- E [eV] k λR λR eV λR eV K+ (a) A/D EF [eV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' (a) Electronic band structure of monolayer MoTe2 and (b) circular phonon dichroism A/D versus the Fermi en- ergy EF for different strength of Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' changes the band structure at valley K+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This explains the phenomenon of peak shift in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 3 (b) since the peaks are always close to the band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' By increasing the strength of Rashba spin-orbit coupling, the magni- tude of CPD can be enhanced, which provides us a knob to tune the CPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' For a realistic strength λR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='072 eV [29], the behavior of A/D is similar to the case with- out Rashba spin-orbit coupling, thus validating our above treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Particularly, we find that the introduction of HR does not change the reciprocal behaviors of ab- sorption coefficients γL/R under q → −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Therefore, to obtain the nonreciprocity, additional ingredients should be taken into account, such as the cyclotron motion of electrons [26–28, 41, 42] or phonon-magnon coupling [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Discussion and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='—We have studied the cir- cular phonon dichroism in magnetic two-dimensional ma- terials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=', monolayer MoTe2 in proximity to the EuO substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Large dichroism signal is obtained for the pseudogauge-type electron-phonon coupling, even with- out introducing the Landau levels or Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Such a signal is reciprocal (nonreciprocal) upon reversing the direction of phonon propagation (magneti- zation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' By varying the gate voltage, the CPD signal can be tuned through the role of Fermi level and Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The proposed CPD effect can also be applied to other transition-metal dichalcogenides with spin-valley- coupled band structure, or their van der waals het- erostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' The effect can be detected by the pulse- echo technique [44, 45] based on the different absorption coefficients between left- and right-handed circularly po- larized phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' An alternative detection is the Raman spectroscopy analysis [26, 27] of phonon polarization by injecting a linearly polarized acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' This work is supported by the National Natural Science Foundation of China (NSFC, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' 11904062).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' We also acknowledge the support of a startup grant from Guangzhou University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Zhang and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Niu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' Lett.' metadata={'source': 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+page_content=' Chick, Ultrasonic Meth- ods in Solid State Physics (Academic Press, New York, 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} +page_content=' L¨uthi, Physical Acoustics in the Solid State (Springer- Verlag, Berlin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAyT4oBgHgl3EQf__o3/content/2301.00917v1.pdf'} diff --git a/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/2301.03469v1.pdf.txt b/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/2301.03469v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b764b22c91c99e415211de09b166fdf1c3dfc99 --- /dev/null +++ b/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/2301.03469v1.pdf.txt @@ -0,0 +1,1793 @@ +KIDS: kinematics-based (in)activity detection and +segmentation in a sleep case study +Omar Elnaggar1, Roselina Arelhi2, Frans Coenen3, Andrew Hopkinson4, Lyndon +Mason5,6, and Paolo Paoletti1,* +1School of Engineering, University of Liverpool, Liverpool L69 3GH, United Kingdom +2Faculty of Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom +3School of Electrical Engineering, Electronics and Computer Science, University of Liverpool, Liverpool L69 3BX, +United Kingdom +4School of Psychology,University of Liverpool, Liverpool L69 7ZA, United Kingdom +5School of Medicine, University of Liverpool, Liverpool L69 3GE, United Kingdom +6Department of Trauma and Orthopaedics, Liverpool University Hospitals NHS Foundation Trust, Liverpool L9 7AL, +United Kingdom +*P.Paoletti@liverpool.ac.uk +ABSTRACT +Sleep behaviour and in-bed movements contain rich information on the neurophysiological health of people, and have a direct +link to the general well-being and quality of life. Standard clinical practices rely on polysomnography for sleep assessment; +however, it is intrusive, performed in unfamiliar environments and requires trained personnel. Progress has been made +on less invasive sensor technologies, such as actigraphy, but clinical validation raises concerns over their reliability and +precision. Additionally, the field lacks a widely acceptable algorithm, with proposed approaches ranging from raw signal +or feature thresholding to data-hungry classification models, many of which are unfamiliar to medical staff. This paper +proposes an online Bayesian probabilistic framework for objective (in)activity detection and segmentation based on clinically +meaningful joint kinematics, measured by a custom-made wearable sensor. Intuitive three-dimensional visualisations of +kinematic timeseries were accomplished through dimension reduction based preprocessing, offering out-of-the-box framework +explainability potentially useful for clinical monitoring and diagnosis. The proposed framework attained up to 99.2% F1-score +and 0.96 Pearson’s correlation coefficient in, respectively, the posture change detection and inactivity segmentation tasks. +The work paves the way for a reliable home-based analysis of movements during sleep which would serve patient-centred +longitudinal care plans. +Introduction +The study of a human sleep behaviour reveals their state of health and well-being. Habitual in-bed behaviour can reveal +physiological and neurological disorders that are otherwise latent during wakefulness1 such as restless leg syndrome and +periodic leg movements. Sleep deprivation and intermittent sleep were found to be linked to multiple health risks2–5. In-bed +sleep behaviour (movements and postures) can sometimes also cause health complications, such as pressure sores6, apnoea7 +and painful spasms8,9. +In the light of the clinical context outlined above, there has been a growing interest within the research community to +study human sleep behaviour. Different aspects have been investigated including sleep posture classification10,11, detection +of in-bed movements and posture transitions12,13, sleep staging14, sleep physiology and vital sign monitoring15,16. Various +technologies have been employed for at-home and in-clinic sleep monitoring. The clinical gold standard for the assessment of +sleep-related disorders is polysomnography (PSG) which measures multiple physiological parameters. There are, however, +disadvantages to using PSG such as sensor and electrode intrusiveness, unfamiliar sleep environment, and cost of personnel +training and technology. Therefore, alternatives to PSG have been proposed to make less sophisticated sleep assessments. +Popular options included the less intrusive accelerometer-based sensing (actigraphy17) which involves an actigraphic device, +such as a smartwatch worn around the wrist or ankle, to record motor activity during sleep and measure parameters like sleep +quality and duration. Other solutions adopted bed-embodied sensors, such as load cells12, and in-bedroom sensors such as +app-empowered smartphones18 which incorporates multiple sensors like accelerometers and microphones. +Within the large field of in-bed movement analysis, there are commonly three research directions reported in the literature: +active/idle state detection, wake/sleep state detection and sleep stage estimation. From the literature, these directions +broadly rely on similar methodologies, namely threshold-based, classification-based and hybrid approaches. Threshold-based +arXiv:2301.03469v1 [eess.SP] 4 Jan 2023 + +Stage 1 +Stage 2 (First Approach) +Stage 2 (Second Approach) +Stage 3A +Stage 3B +Figure 1. Graphical illustration of the proposed kinematics-based (in)activity detection and segmentation (KIDS) framework. +2/18 + +Participant Study Pipeline +LW +180° +5 +Bluetooth +Dual-IMU +Server +Wearable Sensor +Evenly-spaced +Evenly-spaced +Axes of Rotation +Angles of Rotation +Synthetic Axis-Angle +OMATLAB +Orientation Dataset +Segment-to-Segment Orientation +Uniform Manifold +Optional Input +Approximation and Projection +(for Visualisation) +parent +Z. +child +0.5 +Ychild +parent +0 +-0.5 +45° +-1 +Analytical Dimension +Reduction +Unconstrained 3D +Embedding Space +i +Real Unseen +Orientation Embedding +Constrained 3D +Embedding Space +个 +Non- +Downsampling +informative +Informative +Real Unseen +Prior +Prior +Orientation Embedding +Online Bayesian Inference +Current Segment +Changepoint Detection +Run Length Estimate +Inactivity Segmentationapproaches12,17,19–25 are the most popular and they apply a predefined threshold hyperparameter to a predictor variable (raw +data or processed features) to segment sensor timeseries. Classification-based approaches18,20,21,26 employ classifiers to +identify which states, e.g. active or idle, sensor measurements belong to. The less popular hybrid approaches17,27 use a mixture +of threshold- and classification-based approaches; a thresholding algorithm typically produces preliminary labels which are +then refined by a classifier to improve performance. The previous approaches have shortcomings, such as the detection of +short-lasting wake state surrounded by long-lasting sleep state28. Therefore, it is common in the literature to employ handcrafted +“re-scoring rules” to correct for such systematic errors29. Nevertheless, this set of rules need to be applied with caution as they +may favour accuracy over F1 score or even degrade both of them28. Additional insights can be found in comparative studies28,30 +which analysed the performance of previous approaches with focus on actigraphy. +The limitations of existing work on body movement analysis during sleep can therefore be summarised as follows. On +one hand, threshold-based approaches have tricky-to-tweak hyperparameter(s), as different subjects exhibit varying in-bed +behaviour and movement intensities. These approaches also rely on raw sensor data or manually extracted features which are not +necessarily the best representation of information for movement analysis. On the other hand, classification-based approaches +require large-size datasets for classifier training, and these datasets are typically imbalanced in nature (disproportionate +class-wise sample size). Another issue is the performance dependency of classifiers on the dataset population which lack +diversity of ethnicity, age, etc28. +This paper presents a novel kinematics-based (in)activity detection and segmentation (KIDS) framework (depicted in Fig. 1) +based on dimension reduction (DR) and Bayesian inference, without the need for naïve hard rules or longitudinal collection of +unbalanced training data. KIDS leverages Bayesian inference to: (i) perform probabilistic modelling of preprocessed joint +kinematic timeseries, (ii) objectively detect the temporal locations of posture changepoint events and (iii) segment preprocessed +timeseries into segments of inactivity according to the estimated Bayesian statistics, i.e. mean and variance. The decision +making of the KIDS framework is primarily based on the current segment run length, +, probabilistically determined at each +time instant. Inactivity segments contain kinematic observations of consistent statistics which allow +to (linearly) increase +in value, whereas activity periods typically come with abrupt and permanent change in the estimated statistics that result in +repetitive resetting of +to zero. On the user end, a semi-automatic reset detection logic with an adjustable duration threshold +parameter is available to detect changepoints and segment the timeseries data into sleep postures according to specific needs, +without altering the underlying Bayesian inference of (in)activity. This paper demonstrates a possible use of this parameter +to select sufficiently long segments while ruling out short-lasting ones associated with periods of activity. However, clinical +users may opt for different uses as per their needs, for example, to extract short-lasting segments instead. The disentanglement +of the threshold parameter from the Bayesian perception of (in)activity highlights a fundamental difference from existing +threshold-based approaches where thresholding is the backbone of (in)activity perception. +KIDS is a generalised framework that exploits joint kinematics. In this work, the left wrist (LW) joint is considered for +implementation, but other body joints could be substituted. The LW joint kinematics describe the hand-to-forearm orientation +and were captured by a custom-made miniature wearable sensor module with two embedded inertial measurement units +(IMUs). The measured orientations is natively represented in the four-dimensional (4D) axis-angle space, which is not +readily visualisable and likely unfamiliar to non-technical medical experts. Therefore, two DR methods were employed to +preprocess the axis-angle space, mapping the wrist kinematics to a three-dimensional (3D) space, ready for direct visualisation +and subsequent Bayesian inference at lower computational complexity. The first dimension reduction method produced an +unconstrained 3D embedding space using Uniform Manifold Approximation and Projection (UMAP)31. The second method +was a proposed analytical approximation of the UMAP embedding space which produces a fully constrained 3D embedding +space. We compare both 3D embedding spaces, and discuss the implications these have on the Bayesian inference and the +overall performance of the (in)activity detection and segmentation. +The choice of segment-to-segment kinematics was motivated by the authors’ recent work which showed the effectiveness +of using similar kinematic cues from four extremity joints (wrists and ankles) in recognising twelve sleep postures11. This +previous work required manual segmentation of each posture to showcase the posture classification performance. The KIDS +framework proposed in this paper provides autonomous (in)activity detection and segmentation, which have great potential to +empower several clinical applications, including in-bed posture analysis and sleep behaviour disorder screening. Even though +kinematic measurements from the wrists and ankles were necessary to discriminate between postures, this paper shows evidence +that the kinematic profile of the LW alone is sufficient for the (in)activity detection and segmentation task. +Results +The proposed KIDS framework, shown in Fig. 1, involves three stages: (i) wearable inertial sensing for the measurement of +segment-to-segment orientation across the LW joint, (ii) DR-based joint kinematics preprocessing and visualisation, and (iii) +kinematics-based Bayesian inference for (in)activity detection and segmentation. Presented in this section are highlights from +each stage. +3/18 + +Figure 2. A timeseries describing the hand-to-forearm orientation in the axis-angle space. +Simulated Sleep Protocol and Wrist Kinematics Measurement +A simulated sleep experimental protocol (discussed in the Methods Section) was devised to validate the proposed KIDS +framework. The protocol emulates real sleep by guiding participants through a sequential replication of twelve common +sleep postures in a shuffled order. The collected inertial measurements from wearable sensors were subsequently fused to +estimate the hand-to-forearm orientation in the form of a quaternion, which was subsequently converted to the 4D axis-angle +representation (shown in Fig. 2) for subsequent preprocessing and joint kinematics visualisation. In this paper, xxx ∈ R4 +(= x1 · ˆi+x2 · ˆj +x3 · ˆk +x4 · ˆw) represents the sensor-measured segment-to-segment orientation in the axis(ˆi, ˆj, ˆk)-angle( ˆw) +space. The relative orientation timeseries is indexed using a timestamp vector ttt = t1,t2,...,tT. +Joint Kinematics Preprocessing and Visualisation +Enabled by two different DR methods, UMAP-based and ADR-based, the preprocessing stage produces a reduced dimensional +representation of sensor-measured joint orientation, xxx, allowing for intuitive 3D visualisation potentially beneficial to non- +technical medical experts, while lowering the computational complexity of subsequent Bayesian inference. The output +preprocessed orientation embedding is mathematically denoted by ooo ∈ R3 (= o1 · ˆi+o2 · ˆj +o3 · ˆk). The complete embeddings +dataset O encloses all preprocessed ooo over the timestamp vector, ttt. +A computer graphics (CG) pipeline (outlined in the Methods Section) was adopted to produce a synthetic dataset of nearly +50,000 axis-angle orientations, enabling UMAP to learn the 4D-to-3D mapping task without longitudinal collection of sensor +data. In the 3D embedding space, UMAP represented the synthetic orientations dataset as a thick-crust, egg-shaped point +cloud shown in Fig. 1. Manual investigation showed that the latitudinal and longitudinal navigation of the 3D point cloud +corresponded to different axes of rotation, whereas the radial distance from the cloud center was found proportional to the +angle of rotation. Afterwards, the UMAP was presented with over 60 minutes of sensor-measured LW joint orientations from a +random participant as depicted in Fig. 3a. From the figure it can be seen that the wrist orientations evidently occupy finite +regions of the 3D embedding space while leaving some blank due to the anatomical joint constraints. Surprisingly, the twelve +sleep postures are discriminable by the LW joint orientation alone (see Fig. 3c) except for minor overlaps which are typical. +The success of UMAP in the visualisation of joint kinematics does not necessarily lead to effective Bayesian inference +(more on this will be discussed later). UMAP is intrinsically a stochastic method, meaning that different runs are not guaranteed +to produce the same embedding space. This limitation can be partly addressed by saving the pre-trained UMAP model for +later use. However, the unconstrained nature of the embedding space (non-origin centered, unevenly scaled and geometrically +deformed point cloud) remain inevitable. Therefore, a second DR method was proposed to analytically approximate the +nonlinear UMAP mapping function whilst giving full control over the embedding space. From here onwards, the latter method +is regarded as Analytical Dimension Reduction (ADR). +Unlike the UMAP-based method, the proposed ADR does not require pre-training, and is designed to mathematically +produce an origin-centered, thick-crust sphere with radius ranging from 1 to 2 in the Cartesian space. The latitudinal, +longitudinal and radial displacements within the ADR embedding space have the same kinematic interpretations of that of the +UMAP embedding space. For illustration purpose only, the synthetic dataset (designated for UMAP) was visualised using the +ADR method as shown in Fig. 1 with no geometrical artefacts. Fig. 3b shows the ADR-preprocessed LW joint orientations +measured by the miniature wearable sensor from a random participant with similar observations in regards to anatomical +constraints and discriminability of sleep postures. +4/18 + +x1 +x2 +0 +x3 +0 +-1 +x4 +(radians +2182 +2242 +2303 +2364 +2424 +2485 +2545 +2606 +2667 +2727 +tk(a) +(b) +(c) +Figure 3. Visualisation of a participant’s wrist kinematics using UMAP and ADR techniques: (a) a 3D point cloud formed by +all LW joint orientations observed in the datasets (postures and transitions are colour-coded), and (b) the blue- and red-coloured +3D UMAP embedding clusters produced from the dual replication of each sleep posture. +(In)activity Detection and Segmentation +The perception of physical (in)activity during sleep is probabilistically handled by Bayesian inference. The Bayesian inference +operates on O after downsampling (decimation factor = 100), indexed by a timestamp vector ⇓ttt = ⇓t1, ⇓t2,...,⇓tT. The KIDS +framework capitalises on a Bayesian inference algorithm32 which evaluates weighted hypotheses on +k at each arbitrary time +step, ⇓tk and sequentially estimates the posterior predictive parameters (e.g. mean and precision) of an input timeseries. The +method presented in this paper reduces these weighted hypotheses to produce a single number which is the posterior mean +estimate, ˆ k, per time step tk. A significant drop in this estimate implies that there is an increased probability of a short run +length, i.e that the recent data likely belongs to a different distribution which, in turns, implies human subject activity. Further +reset detection logic is applied to ˆ k to detect changepoint events and segment the periods of inactivity in the timeseries data. It +was found that ˆ k, though rarely, can exhibit a gradual reset with consecutive magnitude drops over two or three time steps; +therefore, a postprocessing algorithm was applied to the run length estimate ( ˆ k → ˆ p +k) to better detect reset events at both +sudden and gradual falls. Complete details on (in)activity detection and segmentation are provided in the Methods Section. +To realise the added value of each stage in the proposed methodology, four variations (two major and two minor) of the +KIDS framework were evaluated. The two major variations come from the choice of the kinematic preprocessing method +(either UMAP or ADR). As it will be shown later, the choice of the DR method does not only affect the topology of O, but also +5/18 + +UMAP 3D Embeddings +UMAP +2 +Transitions +· Pose 10 +: Pose 11 +0 +Pose12 +Pose 1 +-2 +Pose 2 +Pose 3 +Pose 4 +-4- +Pose 5 +N +Pose 6 +-6- +Pose 7 +Pose 8 +Pose 9 +-8 +-10 - +-12 +5 +-6 +-4 +-2 +0 +0 +2 +4 +-5 +6 +X +YADR +ADR 3D Embeddings +Transitions +2 +Pose 10 +Pose 11 +1.5 +Pose 12 +Pose 1 +1 +Pose 2 +Pose 3 +0.5 +Pose 4 +Pose 5 +N +0 +Pose 6 +Pose 7 +-0.5 +Pose 8 +Pose 9 +-1 - +-1.5 +-1 +0 +-2 +2 +1 +1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +2 +Y +-2 +XPosture-wise UMAP Embeddings +Pose 1 +Pose 2 +Pose 3 +Pose 4 +Pose 5 +Pose 6 +0 +0 +0 +0 +0 +0 +-5 +N +-5 +N +N +-5 +N +-5 +N +-5 +-10 +-10 +-10 +-10 +-10 +-10 +-5 +-5 +-5 +-5 +-5 +-5 +0 +0 +-5 +-5 +-5 +-5 +5 +0 +0 +0 +0 +0 +0 +0 +0 +0 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +X +X +X +X +Y +Y +X +Pose 7 +Pose 8 +Pose 9 +Pose 10 +Pose 11 +Pose 12 +0 +0 +0 +0 +0 +0 +-5 +N +-5 +N +-5 +N -5 +V +-10 +-10 +-10 +-10 +-10 +-10 +5 +5 +5 +-5 +-5 +-5 +5 +-5 +0 +5 +0 +-5 +0 +0 +-5 +0 +0 +0 +0 +0 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +Y +Y +Y +x +Y +Y +X +Y +Xsubstantially governs the level of information on O encoded in the initial prior presented to the Bayesian inference algorithm. +The other two minor variations correspond to whether the postprocessing algorithm was used. Presented below are the detailed +results for the best- and least-performing variations, respectively, of KIDS: (i) ADR with (w/) postprocessing and (ii) UMAP +without (w/o) postprocessing. Nonetheless, the overall performance evaluation metrics are available for all four variations. +Analysis of ADR-based Bayesian Inference with Postprocessing (Best-performing Algorithm) +As discussed earlier, ADR produces a constrained 3D embedding space. Such controlled space advantageously facilitates +crafting of the so-called “informative prior” (see Fig. 1) to be presented to the Bayesian inference. The prior is typically +encoded in the form of a parameterised probability distribution, describing the state of knowledge on ADRO before evidence +measurements are obtained. More details are provided in the Methods section on the encoding of the informative prior belief. +A random participant dataset was selected to comment on the (in)activity detection and segmentation results of KIDS shown +in Fig. 4. For this dataset, KIDS achieves a 100% F1-score in the changepoint detection task and a satisfactory correlation +coefficient (R = 0.94) between predicted and ground-truth (GT) inactivity durations. Fig. 4a shows the 3D ooo embeddings along +with the predicted current segment mean and standard deviation (by-products of Bayesian inference). The estimated statistics +appear to effectively model the underlying (hidden) data sampling process unique to each segment of (in)activity. Upon the +onset of each inactive segment, the 1-Sigma confidence interval (brown strip) gradually converges to the true underlying spread +of the timeseries segment as shown in the exploded view given in Fig. 4a. The effectiveness of the segment-aware statistical +modelling owes to the capability of the Bayesian inference to correctly assign probabilities (weights) to different hypotheses on +k as shown in the grayscale matrix representation of P( +k | ooo(⇓t1:k)) in Fig. 4b. +The KIDS framework employs multi-stage processing of P( +k | ooo(⇓t1:k)) to accomplish (in)activity detection and segmen- +tation. First, a Least Mean Square (LMS) Bayesian estimator is used to elect one mean estimate, ˆ k from each column of +P( +k | ooo(⇓t1:k)). Second, the point estimates of the run length are postprocessed, producing ˆ p +k shown in Fig. 4c. Third, a reset +detection logic (further explained in the Methods Section) then operates on log10 ˆ p +k to detect the temporal locations of the +changepoint events (end of inactivity). Fourth, combining the temporal locations of changepoint events with the point estimates +of run length ˆ p +k, the postural inactivity was segmented as shown in Fig. 4d. A lower limit of 20 samples was imposed on +the duration of segmented inactivity to eliminate incidental resets (false positives), which typicaly occur as the sleeper takes +few transitions before settling on a new posture. The value of the lower limit was informed by the multimodal histogram (see +Fig. 4e) of magnitude drops in ˆ p +k at the temporal locations of changepoint events, where the rightmost peak corresponds to the +aforementioned incidental resets. +In rare cases where ooo embeddings are highly similar across two consecutive sleep postures (e.g. 1st and 2nd inactive +segments in Fig. 4d), it was found that ˆ p +k may exhibit a transient drop (partial reset) during the posture transition. If this +occurs, the duration of the second inactive segment would be slightly overestimated. Therefore, an upper limit (= elapsed +time in samples since the previous changepoint event) is imposed on the estimated inactivity duration. Portrayed in Fig. 4f +is the correlation plot between predicted and GT inactivity segment durations, indicating a satisfactory performance despite +systematic errors due to signal discretisation and downsampling. +Analysis of UMAP-based Bayesian Inference without Postprocessing (Least-performing Algorithm) +Unlike ADR, UMAP produces an unconstrained 3D embedding space. Consequently, a “non-informative prior” is presented to +the Bayesian inference (see Fig. 1) to expand its ability of modelling UMAPO for which the data topology is ambiguous. As +it will be further elaborated in the Methods section, the multivariate non-informative prior spreads widely (almost flat) over +the mean-covariance space such that no particular combination is favoured in any way. This allows the Bayesian inference to +objectively construct the posterior distribution prominently based on observed UMAPO. +The same participant dataset was again used to comment on the (in)activity detection and segmentation results of this +variant of the KIDS framework, depicted in Fig. 5. Overall, this variant showed poor performance. The changepoint detection +F1-score dropped to 76.9% and the correlation coefficient (R) between predicted and GT inactivity durations dipped drastically +to 7e−2. +Due to the unconstrained pointcloud topology of UMAP, Fig. 5a demonstrates that o1, o2 and o3 had varying scales and +offsets from zero. Despite it being the same participant dataset, UMAPO nonetheless exhibits more aggressive signal dynamics +(large spikes) than ADRO, specifically during posture transition times. This behaviour could simply be due to the stretched +topology of the UMAP pointcloud (see Fig. 3a), or alternatively, it may suggest non-linearities (or discontinuities) in the UMAP +mapping function though this would require further investigation to confirm. In regard to the current segment statistics, the +Bayesian predicted mean and standard deviation demonstrate an underdamped dynamic response (see Fig. 5a) which contrasts +with the damped statistical predictions from the earlier KIDS variant in Fig. 4a. This is due to the absence of an informative +prior, forcing the Bayesian inference to largely rely on observed ooo to estimate the timeseries statistics. This observation is +supported by the 1-Sigma confidence interval that fits tightly onto ooo embeddings (refer to the exploded view of Fig. 5a), unlike +6/18 + +a +b +c +d +e +f +Figure 4. Results from a participant dataset using ADR w/ postprocessing of ˆ k. +7/18 + +Preprocessed Joint Orientation Embeddings, +2 +Embeddings +0 +Predicted +Statistics +02 + Mean +2 +1-Sigma +0 +03 +Confidence +1 +Interval +-2 +Posterior Distribution over Rk, P(Rk | o(t1:k)) +100 +50 +P +三0 +0 +Least Mean Square Estimate of Rk w/ Postprocessing +100 +50 +Predicted (blue) and Ground-truth (green) Inactivity Segments +10 +12 +4 +18 +20 +22 +24 +15 +17 +19 +21 +23 +0 +200 +400 +600 +800 +1000 +1200 +40 +Histogram of ? +20 +0 +-120 +-100 +-80 +09- +-40 +-20 +0 +Correlation between Predicted and Ground-truth Inactivity Durations +55 +R = 0.94 +50 +16 + Durations +45 +40 +35 +1:1 Reference Line +5. +30 +20 +Note: the red number next to +each point corresponds to +22 +the chronological order of +25 +ground-truth inactivity +10 +segments, i.e. points assigned +20 +the same number imply +multiple changepoint events +15 +associated with one +ground-truth segment. +10 +5 +0 L +0 +10 +20 +30 +40 +50 +Ground-truth Inactivity Durationsa +b +c +d +e +f +Figure 5. Results from a participant dataset using UMAP w/o postprocessing of ˆ k. +8/18 + +UMAP +Preprocessed Joint Orientation Embeddings, +5 +Embeddings +01 +-5 +Predicted +4 +Statistics +02 +-2 + Mean +10 +5 +1-Sigma +03 +Confidence +o0 + Interval +Posterior Distribution over Rk, P(Rk | o(t1:k) +50 +P=0 +Least Mean Square Estimate of Rk w/o Postprocessing +60 +40 +Aomm +20 +h +0 +Predicted (blue) and Ground-truth (green) Inactivity Segments +10 +12 +20 +8 +16 +22 +24 +3 +7 +9 +11 +13 +15 +17 +23 +■ +0 +200 +400 +600 +800 +1000 +1200 +Vtk +200 +Histogram of Rk + k-1 +100 +0 +-60 +-50 +-40 +-30 +-20 +-10 +0 +Correlation between Predicted and Ground-truth Inactivity Durations +60 +R = 0.07 +2 +50 +Predicted Inactivity Durations +18 +40 +13 +16 +9 +1:1 Reference Line +5 +30 +Note: the red number next to +0:21 +3 24 +each point corresponds to +17 +the chronological order of +22 +ground-truth inactivity +20 +14 +segments, i.e. points assigned +8 +the same number imply +multiple changepoint events +associated with one +10 + ground-truth segment. +12 +18 +8 +12 +. +0 +10 +20 +30 +40 +50 +Ground-truth Inactivity Durationsthe posterior statistics of the previous KIDS variant which takes only a few time steps to gradually converge from the prior +covariance to the true underlying covariance of the timeseries. +The heavy dependency of Bayesian inference on ooo embeddings led to a higher uncertainty in the Bayesian multi-hypothesis +evaluation of +k. The uncertainty can be visually observed in Fig. 5b in the form of “salt and pepper noise” contaminating the +grayscale matrix representation of P( +k | ooo(⇓t1:k)). As a result, the LMS Bayesian estimator outputted a less reliable point +estimate ˆ k (refer to Fig. 5c) that was too sensitive to minor variations in UMAPO. These minor variations are typically due to +involuntary micro-body movements, such as hand twitches and breathing-related perturbations to the wrist pose. +A multi-stage processing of P( +k | ooo(⇓t1:k)), similar to that of the previous variant of KIDS, was performed except that no +postprocessing was applied to the LMS point estimate ˆ k. Three main observations can be concluded from the (in)activity +detection and segmentation results shown in Fig. 5d. First, numerous repetitive resets were detected (see Fig. 5e), some of +which were unfiltered incidental resets, explaining the presence of short-lasting inactive segments (see the 12th and 18th GT +inactive segments). Second, few GT inactive segments (6th, 10th and 11th) were neither detected nor segmented. Third, the +correlation plot in Fig. 5f shows strong evidence of underestimation of inactivity durations, i.e. most segmentations are located +below the 1:1 reference line. All three observations were primarily caused by the poor reliability of the run length estimation +performed by this KIDS variant. +Performance Metrics: A Comparative Analysis on Variants of the KIDS Framework +The performance evaluation metrics are essential not only to make a fair comparison between the four variants of the proposed +KIDS framework, but also to recognise the added value of each of the adopted methods, such as UMAP, ADR and postprocessing +of ˆ k. In this work, the F1-score, Sensitivity (Se) and Positive Predictive Value (PPV) are the metrics used to evaluate the +changepoint detection performance, while the Pearson’s correlation coefficient (R) is reserved for assessing the quality of +inactivity segmentation. These are standard metrics commonly reported in relevant works28 and would establish a good ground +for benchmarking. +Eq. (1), where TP, FP and FN denote true positives, false positives and false negatives respectively, shows how each of the +changepoint detection metrics is determined. The predictive positive value (or precision) is the ratio of correct KIDS-derived +changepoint events (KIDSnc) to the total number of changepoint events (KIDSn); a detected changepoint event is regarded as +“correct” if it is no more than 3 samples away from the end time of its closest GT inactive segment. The sensitivity (or recall) is +the ratio of KIDSnc to the total number of GT inactive segments (GTn). Lastly, the F1-score is the harmonic mean of PPV and +Se. The Pearson’s R coefficient is a test statistic that measures the statistical relationship between two continuous variables. +According to Eq. (2), R is mathematically defined as the ratio of the covariance for two arbitrary random variates, X and Y, to +the product of their standard deviations, σX and σY. This ratio ranges from +1 (complete positive correlation) to -1 (complete +negative correlation), with 0 indicating no correlation. In this work, the sample populations of X and Y are the predicted and +GT inactive segment durations respectively. +PPV = +KIDSnc +KIDSn +� += +TP +TP + FP +� +Se = +KIDSnc +GTn +� += +TP +TP + FN +� +F1-score = 2×PPV×Se +PPV + Se +(1) +R = Cov(X,Y) +σX ·σY +(2) +For the changepoint detection performance (see Fig. 6a), ADR-based KIDS variants attained +96% F1 scores while +UMAP-based variants scored below 91% for the same metric. The >5% gap in performance underscores the positive impact +brought by both the ADR 3D embedding space and the informative Bayesian prior to the detection of changepoint events. The +postprocessing of ˆ k evidently improved the changepoint detection performance in the ADR case by up to 6.7%, 12.5% and +4.3% gains in the F1-score, Se and PPV metrics respectively. Importantly, the employed postprocessing algorithm stands out +from the conventional re-scoring rules in the literature which are often criticised for compromising the F1-score and Se in +favour of the overall accuracy28. This is because existing re-scoring rules aim to improve the performance of an algorithm +using assumptions on the sleeper, unlike the case in this work where the postprocessing algorithm was crafted based on the +studied behaviour of the KIDS framework itself. Nevertheless, in the case of UMAP-based KIDS, postprocessing was found to +have no to minor adverse effect on their changepoint detection performance, with an average 1% drop in the F1-score after +9/18 + +(a) +(b) +Figure 6. Performance evaluation metrics of all four variants of the proposed KIDS framework. +the incorporation of the postprocessing algorithm. The unviability of ˆ k postprocessing for UMAP-based KIDS comes as no +surprise, since the main problem was the intrinsic Bayesian uncertainty (due to missing an informative prior). +In regard to the inactivity segmentation performance (refer to Fig. 6b), the average difference in the Pearson’s R coefficient +between respective ADR- and UMAP-based KIDS variants is 0.18. The wide gap again signifies the importance of the ADR +3D embedding space and the informative prior in Bayesian inference. Besides, the postprocessing algorithm brought a surge +in segmentation performance, with an average 23.5% and 18.2% enhancement to the Pearson’s R coefficient of UMAP- and +ADR-based KIDS respectively. Notably, the best-performing KIDS variant (ADR w/ postprocessing) accomplished an average +R of 0.96, which is far more superior than the least-performing variant (UMAP w/o postprocessing) at around 0.63 only. +The key takeaways from the comparative analysis given in Fig. 6 are as follows. First, ADR-based KIDS variants generally +perform better than UMAP-based variants at both changepoint detection and inactivity segmentation tasks, regardless of +postprocessing ˆ k. Second, when the postprocessing algorithm was factored into the performance evaluation, its effect was +different for ADR- and UMAP-based KIDS. In case of ADR, the postprocessing algorithm was found to always enhance +the framework performance in both tasks of changepoint detection and inactivity segmentation. However, for UMAP-based +variants, postprocessing was only viable at the inactivity segmentation task for reasons covered earlier. +Discussion +This paper has presented KIDS, which according to the authors’ best knowledge, is the first in-bed movement analysis +framework to fulfil two intertwined clinical needs: (in)activity detection and segmentation. Unlike the previous work reported +in the literature that typically addresses either one of the two needs, the KIDS framework reformulates the problem of in-bed +movement analysis by providing a joint answer to the two interconnected research questions. Moreover, while the prevailing +literature used hard-coded thresholding of raw/processed sensor data, data-hungry classifiers or a mixture of the two approaches; +KIDS leverages Bayesian probability at its core to provide an objective assessment of body (in)activity. The input to KIDS +is wearable sensor measurements of the underlying kinematic profile of the left wrist joint. Such information is clinically +meaningful and comprehensible to medical staff, and its use cases can be expanded beyond (in)activity analysis. +The KIDS framework complements a previous study11 on sleep posture recognition where four wearable sensor modules +mounted on the wrists and ankles were used to classify twelve postures based on quadruple joint kinematic cues. For that +purpose, four joints were monitored to minimise the likelihood of overlap between sleep postures sharing similar extremity +limb positions. However, for the objective of (in)activity detection and segmentation during sleep, monitoring quadruple joints +is not well justified. The hand is probably one of the most moved parts of the human body, and being lightweight, it potentially +carries much of the information on body mobility during sleep. Therefore, as a starting point, this paper tested the hypothesis +that monitoring the left wrist alone would be sufficient to obtain reliable performance. +Low power consumption and real-time performance are desired criteria for portable sleep monitoring devices. The choice +of the sensor data rate directly determines the update rate of the in-device algorithm, and power consumption consequently +10/18 + +Changepoint Detection Performance +100 +90 +80 +70 +60 +50 +40 +30 +20 +10 +0 +F1 +Se +ADRpp +UMAPwpp +ADR w/o pp +UMAP w/o pp +PPVInactivity Segmentation Performance +R +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +ADRwpp +UMAPWPP +ADRw/opp +UMAPw/oppvaries depending on the computational cost of the algorithm. As covered in the Introduction, it is a predominant practice in +the literature to make use of actigraphy signals for wearable-based sleep analysis. While actigraphy captures higher-order +kinematics of the human body, it typically necessitates a relatively fast-updating movement analysis algorithm in order not +to miss posture transitions. Alternatively, the KIDS framework employs a computationally efficient inertial sensor fusion +algorithm to fuse high-frequency inertial signals and produce a filtered estimate of the LW joint orientation that is robust to +sensor artefacts and environmental disturbances. KIDS lays down the assumption that upon every posture transition, the LW +joint orientation is permanently changed, meaning that the requirement of a fast-updating algorithm can be dropped. Therefore, +downsampling of the joint orientation profile was safely incorporated into KIDS without concerns over the (in)activity detection +and segmentation performance. As a result, the tight time constraints on the computation cycle of the in-device algorithm were +substantially relaxed, allowing for the adoption of more advanced body movement analysis approaches such as the Bayesian +probabilistic framework adopted in KIDS without compromising real-time performance. +For further algorithm explainability and more optimal real-time performance, KIDS incorporates a novel joint kinematics +preprocessing stage based on dimension reduction to learn a low-dimensional representation of joint orientations. The LW joint +kinematics recordings obtained from in-vivo experiments were embedded into intuitive 3D visualisations showing a decent +intra-posture clustering and inter-posture separability. Visualisations produced by two DR methods (UMAP and ADR) were +compared, and fully constrained 3D embedding spaces were only guaranteed by ADR. A side benefit of the constrained ADR +3D embedding space was that it facilitated the design of informative priors useful for Bayesian inference. +The general working principle of KIDS relies on statistical modelling of the 3D preprocessed joint kinematic timeseries to +evaluate multiple hypotheses on the current segment run length. The output weighted hypotheses are then fused and processed +by a reset detection logic to perform (in)activity detection and segmentation. The performance of four variants of the KIDS +framework was quantitatively and qualitatively studied and comparisons were made based on standard metrics. The primary +variants of KIDS come from the choice of the DR method used for preprocessing joint kinematics, while secondary variants +correspond to whether the current segment run length estimate was refined by a postprocessing algorithm. It was found +that ADR-based KIDS variants generally outperformed UMAP-based variants. Furthermore, even though the postprocessing +algorithm enhanced the segmentation performance of UMAP-based KIDS, its added value was more evident in case of +ADR-based KIDS improving both its changepoint detection and segmentation. Lastly, performance evaluation showed that the +assumption of KIDS on permanent change in joint orientation upon each posture transition is valid except for only one case. +For this case, the LW joint orientation of a participant barely changed after one posture transition; nonetheless, the timeseries +modelling of KIDS was still able to partially reset ˆ k and the transition was successfully detected by the reset detection logic. +Methods +This section presents the methodology and experimental protocol behind the KIDS framework. First, we discuss the participant +study pipeline and the simulated sleep protocol devised for the validation of the framework. Second, an overview on the +custom-made wearable sensors and the algorithm for wrist kinematics measurement is provided. Third, we outline the proposed +DR-based preprocessing of measured joint kinematics using two methods; UMAP and Analytical dimension reduction. Lastly, +the Bayesian probabilistic framework enabling the (in)activity detection and segmentation is outlined. +Participant Study +Five healthy adult participants (age: 36 ± 15.8 years, height: 169 ± 11 cm, body weight: 72.8 ± 23.2 kg) took part in the +study upon signing an informed consent. The methods were carried out in accordance with relevant guidelines and regulations +and all experimental protocols were approved by The University of Liverpool Research Ethics Committee (review reference: +9850). A leaflet containing pictures of twelve sleep postures (refer to previous work11 for posture definitions) was handed +to each participant to assist them in replicating the postures, with each sleep posture replicated twice (two trials). To ensure +postural data resembles that of a realistic sleep scenario, a random pose shuffling technique was used to ensure statistical +independence of samples across the dataset. Posture and transition durations may vary as per the participant’s comfort and need +for guidance by on-site researchers. For all datasets, a bespoke wearable sensor module was used to capture body segment and +joint kinematics, and to transmit these data to a localhost server at a data rate of 30 Hz. +Bespoke Wearable Sensor Module +The participant study employed four wearable sensor modules to monitor the joint orientations of the two wrists and two ankles +simultaneously. In a previous work11, the multi-sensor data were suitable for a robust sleep posture classification. This work +however exploits the LW joint kinematics alone, as a starting point, for body (in)activity analysis. The custom-made sensor +module provides dual-segment orientation tracking across the LW joint, empowered by two embedded IMU sensors mounted +on the forearm and hand. The IMU model is the BNO055 from Bosch Sensortec© (Bosch Sensortec GmbH, Reutlingen, DE). +Both IMU sensors are managed by a single ESP32-WROOM-32D microcontroller from Espressif Systems© (Espressif Systems +11/18 + +Shanghai Co Ltd, Shanghai, CN) featuring Bluetooth connectivity for wireless data transmission. At about 6 cubic centimeters +in volume for each IMU case, the sensor module is sufficiently slim and small for wearability during sleep. +Inertial Sensor Fusion for Wrist Kinematics Measurement +Intra- and inter-sensor fusion were employed in the work. An inertial sensor fusion algorithm was needed to estimate the +attitude of each IMU sensor (intra-sensor fusion), that is, a function of the body segment it is mounted on. Given the two +segment orientations derived from both IMU sensors, an inter-sensor fusion step is subsequently applied to determine the relative +joint orientation. Prior to each in-vivo experiment, all IMU sensors were calibrated according to standard procedures33,34 to +estimate and reduce errors owing to constant bias, scale factors, cross-axis sensitivity and response nonlinearity. +For intra-sensor fusion, the Madgwick filter35 is employed to fuse the IMU geo-inertial measurements and provide a filtered +estimate of the absolute segment quaternion with respect to the Earth reference frame. The filter has low computational cost +and operates in the quaternion space, allowing for efficient and singularity-free IMU attitude estimation. For inter-sensor fusion, +the relative segment-to-segment quaternion was found through kinematic transformation which maps the orientation of the +child segment (hand) to that of the parent segment (forearm). For further details on the mathematical formulations surrounding +sensor fusion, it is advised to refer to previous work11. Lastly, the segment-to-segment orientation was then converted from +the quaternion space to the axis-angle space, where a unique xxx(tk) ∈ R4 exists at each arbitrary time step tk. The disentangled +axis-angle representation of joint orientations is more intuitive compared to quaternions and allowed for the extraction of +meaningful postural analytics as shown in previous work11. +Joint Kinematics Preprocessing +The preprocessing of joint kinematics was enabled by dimension reduction, allowing for intuitive 3D visualisations of sensor- +measured joint orientations and computationally efficient Bayesian inference subsequently. Such 3D visualisation of joint +kinematic timeseries facilitated the design of the informative prior which was found to enhance the Bayesian inference, and +could potentially be useful for medical screening and/or diagnosis. Dimension reduction has been successfully applied to +visualise data across different domains, from wearable sensing11 through speech processing36 to knowledge exchange37. The +following text elaborates on the two DR methods employed in this paper. +Uniform Manifold Approximation and Projection +UMAP31 is a nonlinear DR method that iteratively constructs a lower dimensional force-directed graph representative of some +high-dimensional dataset. It is capable of handling nonlinear data manifolds and is known for preserving both the local and +global data structures in the low-dimensional embedding space. In principle, UMAP performs dimension reduction over two +stages: (i) identifying nearest neighbours and constructing a neighbour graph, and (ii) learning a low-dimensional representation +through iterative minimisation of a cost function. Further details on UMAP can be found in its original paper31. +The use of UMAP in this work is distinct from the common use of DR in human motion analysis works38–41. Herein UMAP +was run twice; a first run to learn the mapping from 4D joint orientation data to a 3D manifold using synthetic data and a second +run to embed sensor-measured joint orientations (unseen data) into the pretrained 3D embedding space. The synthetic dataset +was carefully designed to uniformly sample the axis-angle orientation space (given some resolution), hence enabling UMAP +to construct a reliable neighbour graph. Afterwards, the 4D sensor-measured joint orientations were passed to UMAP to be +projected into the pretrained 3D embedding space accordingly. The 3D visualisations of synthetic and real joint orientations are +presented in Figs. 1 and 3a respectively. +A computer graphics (CG) pipeline (shown in Fig. 7) was used to generate the synthetic axes of rotation. The pipeline +first employed a procedural mesh generation technique to construct a unit cube with face vertices, vvvf = {xf ,yf ,zf }. Then, the +ellipsoidal projection +vvve = +� +xe +ye +ze +� += +� +xf +� +1− 1 +2 y2 +f − 1 +2 z2 +f + 1 +3 y2 +f z2 +f +yf +� +1− 1 +2 x2 +f − 1 +2 z2 +f + 1 +3 x2 +f z2 +f +zf +� +1− 1 +2 x2 +f − 1 +2 y2 +f + 1 +3 x2 +f y2 +f +� +(3) +was used to project all vvvf onto the surface of a unit sphere, such that the projections are evenly distributed over the surface +of the sphere. The projected vertices, vvve, represent the synthetic axes of rotation, and were exported into a comma-separated +values (CSV) file. All procedural 3D modelling was implemented in the C# programming language and realised in Unity© +(Unity Technologies Inc., California, US). +The CSV file containing all vvve was subsequently imported into MATLAB© (The MathWorks, Massachusetts, US). Therein, +an orientation dataset generator script concatenated each synthetic axis of rotation with each angle of rotation from the closed +set { 36 +36π, 35 +36π, 34 +36π,..., 1 +36π}. The output synthetic axis-angle dataset had a total of 48,600 orientations. Further details on the +CG pipeline is available online in Supplementary Methods - CG Pipeline. +12/18 + +Figure 7. The computer graphics pipeline used for the generation of synthetic axes of rotation. +Analytical Dimension Reduction +The 3D point cloud from UMAP resembled the shape a thick-crust egg shell. Through visual inspection, this point cloud was +found not to be centred at the origin, and had geometrical artefacts such as uneven scaling. These are inevitable outcomes +inherent to the unconstrained embedding space of UMAP. To gain full control over the dimension reduction as a process, +analytical dimension reduction was proposed taking inspiration from the UMAP 3D visualisation. In a nutshell, ADR aims to +output a standardised point cloud taking the shape of a thick-crust sphere that is clear from any form of geometrical deformation +(see Fig. 1). +The ADR sphere have the following shape features. The sphere has an inner radius, irs = 1, and an outer radius, ors = 2. +The radial displacement from irs to ors corresponds to a change in the angle of rotation from 0 to π radians. Based on this +definition, it is then possible to interpolate the radial displacement, rs, given the sensor-measured angle of rotation x4: +rs = 1 +π x4 + irs +(4) +Additionally, the latitudinal and longitudinal displacements around the ADR sphere correspond to differently oriented axes +of rotation defined by x1, x2 and x3. Therefore, an arbitrary preprocessed orientation embedding, ADRooo, in the constrained +embedding space of ADR can be formulated as +ADRooo = +� +o1 +o2 +o3 +� += rs +� +x1 +x2 +x3 +� +(5) +Downsampling of Preprocessed Joint Kinematics +Sleep time is dominated by long durations of inactivity. Running Bayesian inference at every time step is a waste of +computational energy, and is not recommended for a wearable device that continuously streams sensor data at 30 Hz overnight. +In such case, downsampling is generally a good solution to reduce the computational and data storage requirements. Therefore, +the preprocessed joint kinematic timeseries was downsampled by a decimation factor of 100 before it was presented to the +Bayesian inference. +Bayesian Inference of the Current Segment Run Length +A prominent highlight of the KIDS framework is the use of Bayesian inference for (in)activity detection and segmentation. The +Bayesian inference was used to estimate the current segment run length, +k, at each time step ⇓tk. By definition, +k is the +length (in samples) of a data segment whose samples shares similar statistical characteristics. Essentially, +k is a key quantity +which carries information on the length of inactivity and the temporal locations of changepoints. +The Bayesian inference in this paper is largely inspired by a previously reported Bayesian run length estimation algorithm32, +which evaluates multiple probability-weighted hypotheses on +k at each time step. The original paper presented the general +Bayesian framework for the estimation of +k. Therefore, the following steps were incorporated to adapt the Bayesian framework +to the case study presented in this paper. First, the Bayesian inference problem was reformulated to simultaneously estimate +both the mean and precision of the preprocessed joint kinematic timeseries. The estimation of the mean was essential since the +13/18 + +Computer Graphics Pipeline +Procedural Construction of Unit +Ellipsoidal Projection of +Cube with Triangular Mesh +Cube FacesFigure 8. Graphical illustration of the Bayesian run length estimation algorithm. +14/18 + +Incremental Generation of Posterior Predictive Models +P(RT= T Io(t1:T) +(T + 1) × (T + 1) Probability Matrix +→ P(R2 = 2 / o(μt1:2) +3 +P(RT = 1 Io("t1:T) +2 +P(RT = 0 / o(t1:T) +1×1 2×1 +(T-1) ×1 +T×1 +Underlying Predictive Models +Compute +k×1 +OR +Predictive +Probabilities +个 +Informative +Non-Informative +Prior +Prior +Embedding +Updated Priors (i.e. Posteriors) +Timeseries +Bavesian Inference of Current Segment Run Lengthpreprocessed timeseries typical exhibit a mean shift upon a change in posture. The estimation of the precision was equally +important since the fluctuations in the joint orientation slightly varies from one posture to another due to factors, such as snoring +and breathing. Second, conjugate exponential distributions were adapted to the dual mean-precision estimation problem to +maintain real-time inference. Third, a Bayesian estimator was utilised to fuse the weighted hypotheses on +k and output a +single point estimate, ˆ k at each time step. Fourth, we complemented the original Bayesian framework with a reset detection +logic, which detected the temporal locations of changepoints based on the ˆ k estimate. +The following section explains the general working principle behind the Bayesian inference framework. Complete details +on the underlying mathematics are available online in Supplementary Methods - Bayesian Estimation of Current Segment Run +Length, including the construction of the prior probability distribution used for implicit encoding of information42 to tune the +Bayesian inference. +General Overview of the Bayesian Run Length Estimation +A graphical illustration of the Bayesian run length estimation algorithm is presented in Fig. 8. The algorithm requires two +pieces of information as inputs: (i) either informative or non-informative prior on O(⇓ttt), and (ii) the embedding timeseries +O(⇓ttt). Initially at ⇓t0 and before observing O, the algorithm assumes P( +0 = 0, ooo(⇓t0) = φ) = P( +0 = 0) = 1 which is the +probabilistic notation for +0 = 0. At ⇓t1 and upon observing ooo(⇓t1), the algorithm evaluates the predictive probability; the +probability that ooo(⇓t1) belong to the (non-)informative prior. Afterwards, the algorithm evaluates all possible hypotheses on +1. +Logically, +k can take one of two values: +k = +� +0, +if changepoint occurs at ⇓tk. +k−1 +1, +otherwise. +(6) +Nonetheless, the Bayesian framework does not directly set +k. Instead, it evaluates two types of transition probabilities: +(i) “growth probabilities” from any possible +k−1, and (ii) a “changepoint probability” from any possible +k−1. In Fig. 8, +growth transitions are represented by green diagonal lines whereas changepoint transitions are represented by red curves. At +⇓t1, the algorithm evaluates only two hypotheses on +1 in the form of joint probabilities: P( +1 = ζ, ooo(⇓t1)) for ζ = 0 and 1. +To evaluate these hypotheses, the algorithm forward passes P( +0 = 0, ooo(⇓t0) = φ) from the previous time step and fuses it +with the predictive probability. This step explains why the algorithm is often described as a “message-passing algorithm”. After +finding the joint probabilities at ⇓t1, the posterior distribution P( +1 = ζ | ooo(⇓t1)) was determined using Bayes rule. Since the +number of hypotheses on +k (possible values of ζ) grows linearly over time steps, a new predictive model (corresponding +to the new hypothesis) is concatenated to the set of underlying predictive models at the end of each Bayesian inference step. +In subsequent inference steps, a predictive probability is computed for ooo(⇓tk) given each underlying predictive model (or +hypothesis) , unlike ⇓t0 at which only the (non-)informative prior was available for prediction. +Point Estimation of the Current Segment Run Length +The aforementioned Bayesian estimation algorithm produced a posterior distribution over +k conditioned on the preprocessed +joint kinematic timeseries so far observed at each time step ⇓tk. However, +k is still a discrete random variable and each value +it takes under this distribution corresponds to one run length hypothesis. Therefore, the use of point estimation is proposed to +produce an estimate ˆ k that is close to the true value of +k in some probabilistic sense43. To this end, the least mean squares +(LMS) estimator is employed (see Eq. (7)) to produce ˆ k by minimising the mean squared error conditional on multiple +continuous observations, ooo(⇓t1:k). The mean squared error is a effective criterion that trades-off bias and variance for an +estimator. The output of the point estimation stage is a (T +1) vector containing the LMS estimate of the run length at each +time step. +ˆ k = argmin +ˆ +k +E +� +( +k − ˆ k)2 | ooo(⇓t1:k) +� += E +� +k | ooo(⇓t1:k) +� += ∑ +k +k P( +k | ooo(⇓t1:k)) +(7) +Reset Detection Logic for (In)activity Detection and Segmentation +Encoded in ˆ k is the information on the segments and changepoints present in O(⇓ttt). The preprocessed joint kinematic +timeseries is now regarded as quasistatic segments of data which end every time ˆ k is reset upon body movement or activity. +The detection of resets in ˆ k is necessary for the detection of changepoints which mark the start of active segments (or the end +of inactivity). +15/18 + +Scale-invariant Reset Detection (Activity Detection) +Reset detection is mainly about making a binary objective decision on whether the current segment of data samples is likely +to continue growing or terminate. Importantly, the decision making needs to be invariant to the scale of ˆ k, i.e. resets from +ˆ k = ζa and ˆ k = ζb should both be treated equally by the algorithm even if ζb ≫ ζa. Therefore, the application of threshold +on the magnitude drop, ˆ k − ˆ k−1, would not meet the scale-invariant criterion. Alternatively, it is proposed to perform +thresholding on the logarithmic scale of ˆ k since it effectively represents percent change or multiplicative factors. A reference +drop of log10 2 ≈ 0.3 on the logarithmic scale indicates halving of ˆ k in the linear scale, hence it constitutes a binary decision +boundary for run length termination. Therefore, a reset is detected if the consecutive difference term, log10 ˆ k −log10 ˆ k−1, +produced a drop larger than 0.3. +Postprocessing of Current Segment Run Length Point Estimates +Prior to the detection of resets, it is recommended to incorporate a postprocessing algorithm to refine ˆ k for a better performance +(refer to the Results Section). It was found that some gradual resets may occur, associated with few consecutive drops in ˆ k. +This resetting behaviour may be more challenging to be picked up by the reset detection logic. Therefore, a three-sample +moving filter (defined in Eq. (8)) was applied to ˆ k to merge these consecutive drops over one time step. The postprocessed +estimate of ˆ k is denoted by ˆ p +k: +ˆ p +k = +� ˆ k−1, +if ˆ k−1 > ˆ k > ˆ k+1. +ˆ k, +otherwise. +(8) +For clarity, if a two-consecutive-drop scenario is considered with ˆ k−1 = 30, ˆ k = 23 and ˆ k+1 = 0, then the corresponding +postprocessed LMS estimates will be ˆ p +k−1 = 30, ˆ p +k = 30 and ˆ p +k+1 = 0. In this example, the moving filter produced a total +drop of 30 between ˆ p +k+1 and ˆ p +k. Even though the changepoint would be detected one time step later than it actually was, the +postprocessing algorithm prevents the larger risk of not detecting the changepoint event. +Elimination of Repetitive Resets During Activity +During periods of activity, multiple resets may occur within a short time interval as the human participant makes few posture +adjustments until they settle at the new sleep posture. Such repetitive resets do not allow ˆ p +k to proliferate in value, and hence +they generally come after short run lengths. In this paper, the focus is more on sleep postures which are normally associated +with longer run lengths (periods of inactivity). Consequently, the run length before each detected reset was thresholded to +eliminate repetitive resets during periods of activity. Any arbitrary reset at ⇓tk with ˆ p +k−1 < 20 was eliminated. The minimum +run length of twenty samples was backed by histograms of ˆ p +k − ˆ p +k−1 from different participant datasets (example in Fig. 4e). +Inactivity Segmentation +After the filtration of repetitive resets, the remaining resets are those associated with sleep postures that persisted for sufficiently +long run lengths. The duration of each posture is equal to the postprocessed point estimate of the current segment run length +just before the detected reset. This was found to work better than taking the elapsed time between adjacent resets, simply +because the Bayesian estimation of +k is a recursive and self-correcting algorithm. 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Stability analysis of the t-sne algorithm for human activity pattern data. 1839–1845, +DOI: 10.1109/SMC.2018.00318 (2019). +42. Murphy, K. P. Conjugate bayesian analysis of the gaussian distribution. Def 1, 1–29 (2007). +43. Dekking, F. M., Kraaikamp, C., Lopuhaä, H. P. & Meester, L. E. A Modern Introduction to Probability and Statistics: +Understanding why and how (Springer, 2005). +Acknowledgements (not compulsory) +The authors would like to thank Daniel Potts and Shay Stanley for their assistance with the development of the wearable sensors +used in the participant study. +Author contributions statement +O. E. was responsible for the conceptualisation, research design, methodology and data acquisition. R. A. and A. H. contributed +to the methodology and interpretation of data. A.H. instructed on the participant experimental protocol and research ethics. L. +M. identified knowledge gaps from the literature, formulated the research problem and advised on the concept solution. F. C. +and P. P. provided input on conceptualisation and research supervision. All authors edited and reviewed the manuscript, and +approved the submitted version. +Competing interests +O.E. is supported by the University of Liverpool Doctoral Network in AI for Future Digital Health. The authors declare no +other competing interests. +18/18 + +KIDS: kinematics-based (in)activity detection and +segmentation in a sleep case study +Omar Elnaggar1,*, Roselina Arelhi2, Frans Coenen3, Andrew Hopkinson4, Lyndon +Mason5,6, and Paolo Paoletti1 +1School of Engineering, University of Liverpool, Liverpool L69 3GH, United Kingdom +2Faculty of Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom +3School of Electrical Engineering, Electronics and Computer Science, University of Liverpool, Liverpool L69 3BX, +United Kingdom +4School of Psychology,University of Liverpool, Liverpool L69 7ZA, United Kingdom +5School of Medicine, University of Liverpool, Liverpool L69 3GE, United Kingdom +6Department of Trauma and Orthopaedics, Liverpool University Hospitals NHS Foundation Trust, Liverpool L9 7AL, +United Kingdom +*Omar.Elnaggar@liverpool.ac.uk +Supplementary Information +Supplementary Methods - CG Pipeline +Even though the axis-angle representation is intuitive, its four dimensions does not facilitate easy visualisation of joint +orientation timeseries. Therefore, UMAP and analytical dimension reduction were used to reduce the dimensions to three. +With fewer dimensions, the computational cost of subsequent Bayesian inference was lowered. This section is focused on the +generation of synthetic orientation data essential for the UMAP-based dimension reduction. +Sensor-measured joint orientations are typically limited in size and does not cover the entire axis-angle orientation space +due to anatomical constraints. For these reasons, forcing the UMAP to learn the data manifold of xxx from sensor measurements +does not guarantee accurate manifold modelling and may suffer from mapping discontinuities. Therefore, this paper proposes +the generation of a synthetic axis-angle orientation dataset better suited for manifold learning by UMAP. Leveraging a CG +pipeline, synthetic axes of rotation evenly distributed over the surface of a unit sphere were obtained. Then, an equidistant +sequence of scalar angles of rotation was created. Finally, the synthetic orientation dataset was formed by exhausting all +possible combinations between the axes and angles of rotation. +The CG pipeline aims at the procedural 3D modelling of a unit sphere with evenly-distributed vertices (axes of rotation). +As depicted in Fig. S1, the pipeline generally consisted of two stages: (i) the construction of a unit cube with a structured +triangular mesh, and (ii) the projection of the cube vertices onto the surface of a unit sphere. All procedural 3D modelling was +implemented in the C# programming language and realised in Unity© (Unity Technologies Inc., California, US). +The first stage of the CG pipeline involves procedural construction of the six faces making the unit cube. Each quadrilateral +face is defined by a surface normal vector,⃗nface, and a predefined mesh resolution, ϒ, such that ϒ is the number of vertices +along any side of the quadrilateral face. Given these definitions, the vertex grid of the face was constructed with ϒ2 vertices, +each assigned a unique identifier index. The cube faces are all procedurally constructed using ϒ = 15 and⃗nface being the six +orthonormal directions of Unity’s world coordinate system, which yields a total of 1,350 vertices defining the cube. +Next, the cube vertices need to be projected onto the surface of a unit sphere to obtain the synthetic axes of rotation. To this +end, two approaches, based on Euclidean normalisation and ellipsoidal projection respectively, were employed and compared. +First Approach: Projection of Cube Vertices with Euclidean Normalisation +Let vvvc denote an arbitrary vertex on the unit cube, with individual coordinates (xc,yc,zc). This approach projects vvvc onto the +surface of a unit sphere through the normalisation of vvvc by its Euclidean norm as defined in Eq. (S1). +euclidvvvs = +1 +� +x2c +y2c +z2c +· +� +xc +yc +zc +� +(S1) +The result of this projection is displayed in Fig. S1a, which clearly shows projected vertices clumped together at the corners +of the faces than in the centres. Eventually, this projection would produce non-uniform samples over the data manifold in the +orientation space, and could potentially cause unstable dimension reduction. Therefore, a second approach was developed to +address this projection artefact. +arXiv:2301.03469v1 [eess.SP] 4 Jan 2023 + +a +b +Figure S1. The computer graphics pipeline used for the generation of synthetic axes of rotation. +Second Approach: Ellipsoidal Projection of Cube Faces +To obtain better distributed vertices, this approach projects constant-x, constant-y and constant-z square faces, which are the +basic building geometries of a cube. The principal idea is to project each face onto an ellipsoid, and the stitched face ellipsoidal +projections together make a unit sphere. Suppose a generic face vertex, vvvf , defined as +vvvf = {(xf ,yf ,zf ) | xf ,yf ,z f ∈ [−1,1]} +(S2) +where either xf , yf or zf is fixed at some constant value between -1 to 1, depending on the face orientation defined by its surface +normal vector. Mathematically, the ellipsoidal projection aims to map face vertices vvvf to vertices vvve on a 3D ellipsoid such that +the point of intersection with the coordinate axis remain unchanged, i.e. vvve = vvv f at the centre of the face, and the face vertices +are more normalised the closer they are to the corners of the face. For this requirement to hold true, the ellipsoid equation for +some constant-xf face, as an example, must be +1 = x2 +e +x2 +f ++ y2 +e +b2e ++ z2 +e +c2e +(S3) +where be and ce are constants, and be = ce due to symmetry about xe −ye and xe −ze planes. As xf is varied from -1 to 1, the +ellipsoid in Eq. (S3) make two intersection curves with the unit sphere, one curve for each of the upper and lower hemispheres. +Any known point along these intersection curves can be used to determine be and ce. The upper intersection curve meets the +2/5 + +Procedural Construction +Computer Graphics Pipeline +of Unit Cube +with Triangular Mesh +Euclidean +Ellipsoidal +Normalisation of +Projection of +Cube Vertices +Cube Faces +(First Approach) +(Second Approach)Figure S2. The effect of cube face displacement on the ellipsoidal projection. +xe −ye plane at +vvve = +� +xe +ye +ze +� += +� +1 +√ +2 xf +1 +√ +2 +� +1− 1 +2 x2 +f +1 +√ +2 +� +1− 1 +2 x2 +f +� +(S4) +Substituting Eq. (S4) into Eq. (S3) yields be = ce = 2−x2 +f . Hence, the ellipsoid for the constant-x face becomes +1 = x2 +e +x2 +f ++ +y2 +e +2−x2 +f ++ +z2 +e +2−x2 +f +(S5) +By symmetry, the constant-y and constant-z faces are similarly mapped to the ellipsoids defined by Eqs. (S6) and (S7) +respectively. +1 = +x2 +e +2−y2 +f ++ y2 +e +y2 +f ++ +z2 +e +2−y2 +f +(S6) +1 = +x2 +e +2−z2 +f ++ +y2 +e +2−z2 +f ++ z2 +e +z2 +f +(S7) +Then, the system of equations Eqs. (S5) to (S7) are simultaneously solved for xe, ye and ze to obtain the general expression +for the ellipsoidal projection, vvve +vvve = +� +xe +ye +ze +� += +� +xf +� +1− 1 +2 y2 +f − 1 +2 z2 +f + 1 +3 y2 +f z2 +f +y f +� +1− 1 +2 x2 +f − 1 +2 z2 +f + 1 +3 x2 +f z2 +f +zf +� +1− 1 +2 x2 +f − 1 +2 y2 +f + 1 +3 x2 +f y2 +f +� +(S8) +Fig. S2 shows the ellipsoids for constant-x, constant-y and constant-z faces at different displacements from the origin of the +Cartesian coordinate system. As xf , yf and z f converge to 1.0, their corresponding ellipsoids converge in 3D shape to a unit +sphere. This specific case of convergence applies to a cube with a side length of 1 unit, where the ellipsoidal projections of all +six faces form a complete sphere when stitched together. Therefore, a better distributed vertex projection ellipsvvvs (see Fig. S1b) +of the cube vertices onto the surface of a unit sphere was obtained using Eq. (S8). +Each projected vertex on the surface of the sphere represents a candidate Cartesian axis of rotation. All 1,350 axes of rotation +were written to a comma-separated values (CSV) file for subsequent import into MATLAB© (The MathWorks, Massachusetts, +US). Therein, an orientation dataset generator script concatenated each synthetic axis of rotation with each angle of rotation +from the closed set { 36 +36π, 35 +36π, 34 +36π,..., 1 +36π}. The output synthetic axis-angle dataset contained a total of 48,600 orientations. +3/5 + +Projected Faces +Ellipsoids +xf = 1.0 +yf = 1.0 +Xf = 0.8 +zf = 1.0 +yf = 0.8 +xf = 0.5 +8'0 = z +yf = 0.5 +zf = 0.5Supplementary Methods - Bayesian Estimation of Current Segment Run Length +Based on the original paper?, the general mathematical expression for the joint probability can be written as: +P( +k, ooo(⇓t1:k)) = ∑ +k−1 +P(ooo(⇓tk) | +k, ooo(⇓t1:k−1)) +� +�� +� +Predictive Term +P( +k | +k−1) +� +�� +� +Changepoint Prior +P( +k−1, ooo(⇓t1:k−1)) +� +�� +� +Recursive Term +(S9) +The Predictive Term. It is also referred to as the posterior predictive term because it predicts the next embedding given +the previously observed embeddings and +k. Thus, if a hypothesis states +k = ζ, only embeddings within the past ζ time steps +should contribute to the predictive term: +P(ooo(⇓tk) | +k = ζ, ooo(⇓t1:k−1)) = P(ooo(⇓tk) | +k = ζ, ooo(⇓tk−ζ:k−1)) +(S10) +In this paper, it is assumed that O(⇓ttt) is continuously distributed with unknown mean vector µµµO ∈ R3 and precision matrix +λλλ O ∈ R3×3. Additionally, a likelihood model, Pl(ooo(⇓tk) | ηηη), is defined with model parameters ηηη = {µµµO, λλλ O}. To compute +the predictive term, Eq. (S11) first finds the posterior distribution, P(ηηη | +k = ζ, ooo(⇓tk−ζ:k−1)), then marginalises ηηη, conditional +on +k. Hence, the predictive term becomes the posterior predictive distribution: +P(ooo(⇓tk) | +k = ζ, ooo(⇓tk−ζ:k−1)) = +� +ηηη Pl(ooo(⇓tk) | ηηη) P(ηηη | +k = ζ, ooo(⇓tk−ζ:k−1)) dηηη +(S11) +The posterior distribution can be re-written as P(ηηη | D, ααα), where D = ooo(⇓tk−ζ:k−1) and ααα denotes the parameters of the +prior distribution, P(ηηη | ααα). Using Bayes’ theorem, conditional probability and chain rule, P(ηηη | D, ααα) can be expanded as +follows: +P(ηηη | D, ααα) = Pl(D | ηηη, ααα) P(ηηη | ααα) +P(D | ααα) +(S12) +However, computing the integral in Eq. (S11) is intractable. Fortunately, conjugate-exponential models? provide an efficient +solution to this problem. The concept of conjugacy allows for the derivation of a closed form expression as a function of ααα +without the need for integration. Hence, given a likelihood model Pl(D | ηηη, ααα), its conjugate prior model can be leveraged to +simplify the posterior distribution to: +P(ηηη | D, ααα) = P(ηηη | ααα′) +(S13) +where both the prior and posterior distributions belong to the same distribution family with parameters ααα and ααα′ respectively. +Since both µµµO and λλλ O are inferred, the likelihood model follows a multivariate Gaussian distribution, for which the +natural conjugate prior is the Normal-Wishart distribution with quadruple parameterisation ααα = {µµµα, κα, να, ΣΣΣα}, +P(ηηη | ααα) = N W(ηηη | ααα) += N (µµµO | µµµα, (κα λλλ O)−1) W(λλλ O | ΣΣΣα, να) +(S14) +where µµµα ∈ R3 and κα ∈ R are the mean vector and the covariance scale coefficient, respectively, of the normal distribution, +and να ∈ R and ΣΣΣα ∈ R3×3 are the number of degrees of freedom and the scale matrix, respectively, of the Wishart distribution. +Since P(ηηη | ααα) is a conjugate prior, then P(ηηη | D, ααα) turns out to be a Normal-Wishart distribution too. Substituting +P(ηηη | D, ααα) into Eq. (S11), it can be shown that the posterior predictive distribution follows a Student’s t-distribution with +(ν′ +α −2) degrees of freedom: +P(ooo(⇓tk) | D, ααα) = +� +Pl(ooo(⇓tk) | ηηη) P(ηηη | D, ααα) dηηη += Tν′α−2(ooo(⇓tk) | µµµ′ +α, ΣΣΣ′ +α(κ′ +α +1)κ′ +α +−1(ν′ +α −2)−1) +(S15) +where µµµ′ +α and ΣΣΣ′ +α(κ′ +α +1)κ′ +α +−1(ν′ +α −2)−1 are the predicted µµµO and λλλ −1 +O , respectively, of the posterior predictive distribution. +Setting the prior on preprocessed joint kinematic timeseries. The set of parameters ααα(k,ζ) are updated at each time +step ⇓tk and for each hypothesis +k = ζ. The (non-)informative prior on ηηη was defined by setting the initial parameters ααα(0,0) +of the prior model to the appropriate values. In the case of the Normal-Wishart distribution, the non-informative prior, ααα(0,0), +was defined with µµµα(0,0) = +� +10−4 10−4 10−4� +, κα(0,0) = 10−4, να(0,0) = 4 and any matrix ΣΣΣα(0,0) provided that its +4/5 + +determinant |ΣΣΣα(0,0)| = 0. The non-informative prior is more suited for UMAPO due to the unconstrained embedding space of +UMAP. +In contrast to UMAP, ADR has a constrained embedding space and known 3D embedding topology which facilitated the +design of an informative prior. Visual inspection of the real preprocessed joint kinematics in the ADR embedding space informed +the following settings: (1) µµµO has an expected value E(µµµO) = 0 and an initial guess of III3×3 for its covariance matrix, and (2) +λλλ O has an expected value E[λλλ O] = να ΣΣΣα = 20 III3×3. Therefore, ααα(0,0) was set such that µµµα(0,0) = +� +10−4 10−4 10−4� +, +κα(0,0) = 1/20, να(0,0) = 4 and ΣΣΣα(0,0) = 5 III3×3. +Updating the prior model parameters. At the end of each Bayesian inference step, the prior model parameters ααα are +updated to represent the new set of hypotheses on +k. At ⇓tk and given an arbitrary hypothesis +k = ζ, the prior model +parameters denoted by ααα(k,ζ) are updated as follows: +µµµα(k,ζ) = κα(0,0)µµµα(0,0)+ζ oook−ζ+1:k +κα(0,0)+ζ +κα(k,ζ) = κα(0,0)+ζ +να(k,ζ) = να(0,0)+ζ +ΣΣΣα(k,ζ) = ΣΣΣα(0,0)+ κα(0,0) ζ +κα(0,0)+ζ (µµµα(0,0)−oook−ζ+1:k) (µµµα(0,0)−oook−ζ+1:k)T+ +ζ +∑ +i=1 +(ooo(⇓tk−i)−oook−ζ+1:k) (ooo(⇓tk−i)−oook−ζ+1:k)T +(S16) +where oook−ζ+1:k is the mean vector of the embeddings in ooo(⇓tk−ζ+1:k). +The Changepoint Prior. This term allows for encoding the prior knowledge on changepoints present in O(⇓ttt). The +current segment run length +k is is a discrete, nonnegative random variable this is subject to two possible outcomes at any +given ⇓tk; either reset to zero upon a changepoint or an incremental growth by one. Thus, it was sensible to model +k with +a geometric distribution of success probability, p, where success corresponds to the growth of +k. Based on this modelling +assumption, the changepoint prior term becomes +P( +k | +k−1) = +� +� +� +� +� +p, +if +k = 0. +1− p, +if +k = +k−1 +1. +0, +otherwise. +(S17) +Using the changepoint prior in Eq. (S17), the joint probability term in Eq. (S9) can be further simplified. For growth +transitions, it is redundant to sum over all possible +k−1 as the only valid run length is +k = +k−1 +1. Substituting Eqs. (S13) +and (S17) into Eq. (S9), the joint growth transition probability is defined as +P( +k, ooo(⇓t1:k)) = P(ooo(⇓tk) | ααα(k −1, +k−1)) (1− p) P( +k−1, ooo(⇓t1:k−1) +(S18) +For the changepoint transitions, the run length can be reset from any valid +k−1. Therefore, the summation over +k−1 is +necessary, +P( +k, ooo(⇓t1:k)) = ∑ +k−1 +P(ooo(⇓tk) | ααα(k −1, +k−1)) +p P( +k−1, ooo(⇓t1:k−1) +(S19) +After finding all joint probabilities P( +k, ooo(⇓t1:k)) at ⇓tk, the posterior distribution on +k was found using Bayes rule +P( +k | ooo(⇓t1:k)) = +P( +k, ooo(⇓t1:k)) +∑ +k P( +k, ooo(⇓t1:k)) +(S20) +where ∑ +k P( +k, ooo(⇓t1:k)) is the normalising evidence. +Once the posterior distribution over +k is determined ∀k ∈ [0,T], this concludes the Bayesian inference of the current +segment run length with P( +k | ooo(⇓t1:k)) being a (T + 1) × (T + 1) matrix indexed by k and different +k = ζ. The upper +triangular part of this matrix contains all-zero probabilities since +k ≤ k at any ⇓tk. +5/5 + diff --git a/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/load_file.txt b/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4244686052b940d95ec18190d9d88d050de1f2f --- /dev/null +++ b/n9E1T4oBgHgl3EQf1gWo/content/tmp_files/load_file.txt @@ -0,0 +1,1250 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf,len=1249 +page_content='KIDS: kinematics-based (in)activity detection and segmentation in a sleep case study Omar Elnaggar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Roselina Arelhi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Frans Coenen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Andrew Hopkinson4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lyndon Mason5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' and Paolo Paoletti1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='* 1School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3GH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 2Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Sheffield,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Sheffield S1 3JD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 3School of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Electronics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3BX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 4School of Psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 7ZA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 5School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3GE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 6Department of Trauma and Orthopaedics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool University Hospitals NHS Foundation Trust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L9 7AL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='Paoletti@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='uk ABSTRACT Sleep behaviour and in-bed movements contain rich information on the neurophysiological health of people, and have a direct link to the general well-being and quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Standard clinical practices rely on polysomnography for sleep assessment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' however, it is intrusive, performed in unfamiliar environments and requires trained personnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Progress has been made on less invasive sensor technologies, such as actigraphy, but clinical validation raises concerns over their reliability and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Additionally, the field lacks a widely acceptable algorithm, with proposed approaches ranging from raw signal or feature thresholding to data-hungry classification models, many of which are unfamiliar to medical staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This paper proposes an online Bayesian probabilistic framework for objective (in)activity detection and segmentation based on clinically meaningful joint kinematics, measured by a custom-made wearable sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Intuitive three-dimensional visualisations of kinematic timeseries were accomplished through dimension reduction based preprocessing, offering out-of-the-box framework explainability potentially useful for clinical monitoring and diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The proposed framework attained up to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='2% F1-score and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='96 Pearson’s correlation coefficient in, respectively, the posture change detection and inactivity segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The work paves the way for a reliable home-based analysis of movements during sleep which would serve patient-centred longitudinal care plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Introduction The study of a human sleep behaviour reveals their state of health and well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Habitual in-bed behaviour can reveal physiological and neurological disorders that are otherwise latent during wakefulness1 such as restless leg syndrome and periodic leg movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Sleep deprivation and intermittent sleep were found to be linked to multiple health risks2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In-bed sleep behaviour (movements and postures) can sometimes also cause health complications, such as pressure sores6, apnoea7 and painful spasms8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In the light of the clinical context outlined above, there has been a growing interest within the research community to study human sleep behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Different aspects have been investigated including sleep posture classification10,11, detection of in-bed movements and posture transitions12,13, sleep staging14, sleep physiology and vital sign monitoring15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Various technologies have been employed for at-home and in-clinic sleep monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The clinical gold standard for the assessment of sleep-related disorders is polysomnography (PSG) which measures multiple physiological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' There are, however, disadvantages to using PSG such as sensor and electrode intrusiveness, unfamiliar sleep environment, and cost of personnel training and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, alternatives to PSG have been proposed to make less sophisticated sleep assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Popular options included the less intrusive accelerometer-based sensing (actigraphy17) which involves an actigraphic device, such as a smartwatch worn around the wrist or ankle, to record motor activity during sleep and measure parameters like sleep quality and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Other solutions adopted bed-embodied sensors, such as load cells12, and in-bedroom sensors such as app-empowered smartphones18 which incorporates multiple sensors like accelerometers and microphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Within the large field of in-bed movement analysis, there are commonly three research directions reported in the literature: active/idle state detection, wake/sleep state detection and sleep stage estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' From the literature, these directions broadly rely on similar methodologies, namely threshold-based, classification-based and hybrid approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Threshold-based arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='03469v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='SP] 4 Jan 2023 Stage 1 Stage 2 (First Approach) Stage 2 (Second Approach) Stage 3A Stage 3B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Graphical illustration of the proposed kinematics-based (in)activity detection and segmentation (KIDS) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 2/18 Participant Study Pipeline LW 180° 5 Bluetooth Dual-IMU Server Wearable Sensor Evenly-spaced Evenly-spaced Axes of Rotation Angles of Rotation Synthetic Axis-Angle OMATLAB Orientation Dataset Segment-to-Segment Orientation Uniform Manifold Optional Input Approximation and Projection (for Visualisation) parent Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' child 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 Ychild parent 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 45° 1 Analytical Dimension Reduction Unconstrained 3D Embedding Space i Real Unseen Orientation Embedding Constrained 3D Embedding Space 个 Non- Downsampling informative Informative Real Unseen Prior Prior Orientation Embedding Online Bayesian Inference Current Segment Changepoint Detection Run Length Estimate Inactivity Segmentationapproaches12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='19–25 are the most popular and they apply a predefined threshold hyperparameter to a predictor variable (raw data or processed features) to segment sensor timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Classification-based approaches18,20,21,26 employ classifiers to identify which states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' active or idle, sensor measurements belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The less popular hybrid approaches17,27 use a mixture of threshold- and classification-based approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' a thresholding algorithm typically produces preliminary labels which are then refined by a classifier to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The previous approaches have shortcomings, such as the detection of short-lasting wake state surrounded by long-lasting sleep state28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, it is common in the literature to employ handcrafted “re-scoring rules” to correct for such systematic errors29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Nevertheless, this set of rules need to be applied with caution as they may favour accuracy over F1 score or even degrade both of them28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Additional insights can be found in comparative studies28,30 which analysed the performance of previous approaches with focus on actigraphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The limitations of existing work on body movement analysis during sleep can therefore be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' On one hand, threshold-based approaches have tricky-to-tweak hyperparameter(s), as different subjects exhibit varying in-bed behaviour and movement intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' These approaches also rely on raw sensor data or manually extracted features which are not necessarily the best representation of information for movement analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' On the other hand, classification-based approaches require large-size datasets for classifier training, and these datasets are typically imbalanced in nature (disproportionate class-wise sample size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Another issue is the performance dependency of classifiers on the dataset population which lack diversity of ethnicity, age, etc28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This paper presents a novel kinematics-based (in)activity detection and segmentation (KIDS) framework (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1) based on dimension reduction (DR) and Bayesian inference, without the need for naïve hard rules or longitudinal collection of unbalanced training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' KIDS leverages Bayesian inference to: (i) perform probabilistic modelling of preprocessed joint kinematic timeseries, (ii) objectively detect the temporal locations of posture changepoint events and (iii) segment preprocessed timeseries into segments of inactivity according to the estimated Bayesian statistics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The decision making of the KIDS framework is primarily based on the current segment run length, , probabilistically determined at each time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Inactivity segments contain kinematic observations of consistent statistics which allow to (linearly) increase in value, whereas activity periods typically come with abrupt and permanent change in the estimated statistics that result in repetitive resetting of to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' On the user end, a semi-automatic reset detection logic with an adjustable duration threshold parameter is available to detect changepoints and segment the timeseries data into sleep postures according to specific needs, without altering the underlying Bayesian inference of (in)activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This paper demonstrates a possible use of this parameter to select sufficiently long segments while ruling out short-lasting ones associated with periods of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' However, clinical users may opt for different uses as per their needs, for example, to extract short-lasting segments instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The disentanglement of the threshold parameter from the Bayesian perception of (in)activity highlights a fundamental difference from existing threshold-based approaches where thresholding is the backbone of (in)activity perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' KIDS is a generalised framework that exploits joint kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this work, the left wrist (LW) joint is considered for implementation, but other body joints could be substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The LW joint kinematics describe the hand-to-forearm orientation and were captured by a custom-made miniature wearable sensor module with two embedded inertial measurement units (IMUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The measured orientations is natively represented in the four-dimensional (4D) axis-angle space, which is not readily visualisable and likely unfamiliar to non-technical medical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, two DR methods were employed to preprocess the axis-angle space, mapping the wrist kinematics to a three-dimensional (3D) space, ready for direct visualisation and subsequent Bayesian inference at lower computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The first dimension reduction method produced an unconstrained 3D embedding space using Uniform Manifold Approximation and Projection (UMAP)31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The second method was a proposed analytical approximation of the UMAP embedding space which produces a fully constrained 3D embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' We compare both 3D embedding spaces, and discuss the implications these have on the Bayesian inference and the overall performance of the (in)activity detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The choice of segment-to-segment kinematics was motivated by the authors’ recent work which showed the effectiveness of using similar kinematic cues from four extremity joints (wrists and ankles) in recognising twelve sleep postures11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This previous work required manual segmentation of each posture to showcase the posture classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The KIDS framework proposed in this paper provides autonomous (in)activity detection and segmentation, which have great potential to empower several clinical applications, including in-bed posture analysis and sleep behaviour disorder screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Even though kinematic measurements from the wrists and ankles were necessary to discriminate between postures, this paper shows evidence that the kinematic profile of the LW alone is sufficient for the (in)activity detection and segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Results The proposed KIDS framework, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1, involves three stages: (i) wearable inertial sensing for the measurement of segment-to-segment orientation across the LW joint, (ii) DR-based joint kinematics preprocessing and visualisation, and (iii) kinematics-based Bayesian inference for (in)activity detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Presented in this section are highlights from each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3/18 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A timeseries describing the hand-to-forearm orientation in the axis-angle space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Simulated Sleep Protocol and Wrist Kinematics Measurement A simulated sleep experimental protocol (discussed in the Methods Section) was devised to validate the proposed KIDS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The protocol emulates real sleep by guiding participants through a sequential replication of twelve common sleep postures in a shuffled order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The collected inertial measurements from wearable sensors were subsequently fused to estimate the hand-to-forearm orientation in the form of a quaternion, which was subsequently converted to the 4D axis-angle representation (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 2) for subsequent preprocessing and joint kinematics visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this paper, xxx ∈ R4 (= x1 · ˆi+x2 · ˆj +x3 · ˆk +x4 · ˆw) represents the sensor-measured segment-to-segment orientation in the axis(ˆi, ˆj, ˆk)-angle( ˆw) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The relative orientation timeseries is indexed using a timestamp vector ttt = t1,t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=',tT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Joint Kinematics Preprocessing and Visualisation Enabled by two different DR methods, UMAP-based and ADR-based, the preprocessing stage produces a reduced dimensional representation of sensor-measured joint orientation, xxx, allowing for intuitive 3D visualisation potentially beneficial to non- technical medical experts, while lowering the computational complexity of subsequent Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The output preprocessed orientation embedding is mathematically denoted by ooo ∈ R3 (= o1 · ˆi+o2 · ˆj +o3 · ˆk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The complete embeddings dataset O encloses all preprocessed ooo over the timestamp vector, ttt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A computer graphics (CG) pipeline (outlined in the Methods Section) was adopted to produce a synthetic dataset of nearly 50,000 axis-angle orientations, enabling UMAP to learn the 4D-to-3D mapping task without longitudinal collection of sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In the 3D embedding space, UMAP represented the synthetic orientations dataset as a thick-crust, egg-shaped point cloud shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Manual investigation showed that the latitudinal and longitudinal navigation of the 3D point cloud corresponded to different axes of rotation, whereas the radial distance from the cloud center was found proportional to the angle of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Afterwards, the UMAP was presented with over 60 minutes of sensor-measured LW joint orientations from a random participant as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' From the figure it can be seen that the wrist orientations evidently occupy finite regions of the 3D embedding space while leaving some blank due to the anatomical joint constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Surprisingly, the twelve sleep postures are discriminable by the LW joint orientation alone (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3c) except for minor overlaps which are typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The success of UMAP in the visualisation of joint kinematics does not necessarily lead to effective Bayesian inference (more on this will be discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' UMAP is intrinsically a stochastic method, meaning that different runs are not guaranteed to produce the same embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This limitation can be partly addressed by saving the pre-trained UMAP model for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' However, the unconstrained nature of the embedding space (non-origin centered, unevenly scaled and geometrically deformed point cloud) remain inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, a second DR method was proposed to analytically approximate the nonlinear UMAP mapping function whilst giving full control over the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' From here onwards, the latter method is regarded as Analytical Dimension Reduction (ADR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Unlike the UMAP-based method, the proposed ADR does not require pre-training, and is designed to mathematically produce an origin-centered, thick-crust sphere with radius ranging from 1 to 2 in the Cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The latitudinal, longitudinal and radial displacements within the ADR embedding space have the same kinematic interpretations of that of the UMAP embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For illustration purpose only, the synthetic dataset (designated for UMAP) was visualised using the ADR method as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1 with no geometrical artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3b shows the ADR-preprocessed LW joint orientations measured by the miniature wearable sensor from a random participant with similar observations in regards to anatomical constraints and discriminability of sleep postures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4/18 x1 x2 0 x3 0 1 x4 (radians 2182 2242 2303 2364 2424 2485 2545 2606 2667 2727 tk(a) (b) (c) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Visualisation of a participant’s wrist kinematics using UMAP and ADR techniques: (a) a 3D point cloud formed by all LW joint orientations observed in the datasets (postures and transitions are colour-coded), and (b) the blue- and red-coloured 3D UMAP embedding clusters produced from the dual replication of each sleep posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (In)activity Detection and Segmentation The perception of physical (in)activity during sleep is probabilistically handled by Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The Bayesian inference operates on O after downsampling (decimation factor = 100), indexed by a timestamp vector ⇓ttt = ⇓t1, ⇓t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=',⇓tT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The KIDS framework capitalises on a Bayesian inference algorithm32 which evaluates weighted hypotheses on k at each arbitrary time step, ⇓tk and sequentially estimates the posterior predictive parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' mean and precision) of an input timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The method presented in this paper reduces these weighted hypotheses to produce a single number which is the posterior mean estimate, ˆ k, per time step tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A significant drop in this estimate implies that there is an increased probability of a short run length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e that the recent data likely belongs to a different distribution which, in turns, implies human subject activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Further reset detection logic is applied to ˆ k to detect changepoint events and segment the periods of inactivity in the timeseries data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' It was found that ˆ k, though rarely, can exhibit a gradual reset with consecutive magnitude drops over two or three time steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' therefore, a postprocessing algorithm was applied to the run length estimate ( ˆ k → ˆ p k) to better detect reset events at both sudden and gradual falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Complete details on (in)activity detection and segmentation are provided in the Methods Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To realise the added value of each stage in the proposed methodology, four variations (two major and two minor) of the KIDS framework were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The two major variations come from the choice of the kinematic preprocessing method (either UMAP or ADR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As it will be shown later, the choice of the DR method does not only affect the topology of O, but also 5/18 UMAP 3D Embeddings UMAP 2 Transitions Pose 10 : Pose 11 0 Pose12 Pose 1 2 Pose 2 Pose 3 Pose 4 4- Pose 5 N Pose 6 6- Pose 7 Pose 8 Pose 9 8 10 - 12 5 6 4 2 0 0 2 4 5 6 X YADR ADR 3D Embeddings Transitions 2 Pose 10 Pose 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 Pose 12 Pose 1 1 Pose 2 Pose 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 Pose 4 Pose 5 N 0 Pose 6 Pose 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 Pose 8 Pose 9 1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 1 0 2 2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='Xsubstantially governs the level of information on O encoded in the initial prior presented to the Bayesian inference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The other two minor variations correspond to whether the postprocessing algorithm was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Presented below are the detailed results for the best- and least-performing variations, respectively, of KIDS: (i) ADR with (w/) postprocessing and (ii) UMAP without (w/o) postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Nonetheless, the overall performance evaluation metrics are available for all four variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Analysis of ADR-based Bayesian Inference with Postprocessing (Best-performing Algorithm) As discussed earlier, ADR produces a constrained 3D embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Such controlled space advantageously facilitates crafting of the so-called “informative prior” (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1) to be presented to the Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The prior is typically encoded in the form of a parameterised probability distribution, describing the state of knowledge on ADRO before evidence measurements are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' More details are provided in the Methods section on the encoding of the informative prior belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A random participant dataset was selected to comment on the (in)activity detection and segmentation results of KIDS shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For this dataset, KIDS achieves a 100% F1-score in the changepoint detection task and a satisfactory correlation coefficient (R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='94) between predicted and ground-truth (GT) inactivity durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4a shows the 3D ooo embeddings along with the predicted current segment mean and standard deviation (by-products of Bayesian inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The estimated statistics appear to effectively model the underlying (hidden) data sampling process unique to each segment of (in)activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Upon the onset of each inactive segment, the 1-Sigma confidence interval (brown strip) gradually converges to the true underlying spread of the timeseries segment as shown in the exploded view given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The effectiveness of the segment-aware statistical modelling owes to the capability of the Bayesian inference to correctly assign probabilities (weights) to different hypotheses on k as shown in the grayscale matrix representation of P( k | ooo(⇓t1:k)) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The KIDS framework employs multi-stage processing of P( k | ooo(⇓t1:k)) to accomplish (in)activity detection and segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First, a Least Mean Square (LMS) Bayesian estimator is used to elect one mean estimate, ˆ k from each column of P( k | ooo(⇓t1:k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second, the point estimates of the run length are postprocessed, producing ˆ p k shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Third, a reset detection logic (further explained in the Methods Section) then operates on log10 ˆ p k to detect the temporal locations of the changepoint events (end of inactivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Fourth, combining the temporal locations of changepoint events with the point estimates of run length ˆ p k, the postural inactivity was segmented as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A lower limit of 20 samples was imposed on the duration of segmented inactivity to eliminate incidental resets (false positives), which typicaly occur as the sleeper takes few transitions before settling on a new posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The value of the lower limit was informed by the multimodal histogram (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4e) of magnitude drops in ˆ p k at the temporal locations of changepoint events, where the rightmost peak corresponds to the aforementioned incidental resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In rare cases where ooo embeddings are highly similar across two consecutive sleep postures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1st and 2nd inactive segments in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4d), it was found that ˆ p k may exhibit a transient drop (partial reset) during the posture transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' If this occurs, the duration of the second inactive segment would be slightly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, an upper limit (= elapsed time in samples since the previous changepoint event) is imposed on the estimated inactivity duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Portrayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4f is the correlation plot between predicted and GT inactivity segment durations, indicating a satisfactory performance despite systematic errors due to signal discretisation and downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Analysis of UMAP-based Bayesian Inference without Postprocessing (Least-performing Algorithm) Unlike ADR, UMAP produces an unconstrained 3D embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Consequently, a “non-informative prior” is presented to the Bayesian inference (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1) to expand its ability of modelling UMAPO for which the data topology is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As it will be further elaborated in the Methods section, the multivariate non-informative prior spreads widely (almost flat) over the mean-covariance space such that no particular combination is favoured in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This allows the Bayesian inference to objectively construct the posterior distribution prominently based on observed UMAPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The same participant dataset was again used to comment on the (in)activity detection and segmentation results of this variant of the KIDS framework, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Overall, this variant showed poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The changepoint detection F1-score dropped to 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='9% and the correlation coefficient (R) between predicted and GT inactivity durations dipped drastically to 7e−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Due to the unconstrained pointcloud topology of UMAP, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5a demonstrates that o1, o2 and o3 had varying scales and offsets from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Despite it being the same participant dataset, UMAPO nonetheless exhibits more aggressive signal dynamics (large spikes) than ADRO, specifically during posture transition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This behaviour could simply be due to the stretched topology of the UMAP pointcloud (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3a), or alternatively, it may suggest non-linearities (or discontinuities) in the UMAP mapping function though this would require further investigation to confirm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In regard to the current segment statistics, the Bayesian predicted mean and standard deviation demonstrate an underdamped dynamic response (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5a) which contrasts with the damped statistical predictions from the earlier KIDS variant in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This is due to the absence of an informative prior, forcing the Bayesian inference to largely rely on observed ooo to estimate the timeseries statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This observation is supported by the 1-Sigma confidence interval that fits tightly onto ooo embeddings (refer to the exploded view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5a), unlike 6/18 a b c d e f Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Results from a participant dataset using ADR w/ postprocessing of ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 7/18 Preprocessed Joint Orientation Embeddings, 2 Embeddings 0 Predicted Statistics 02 Mean 2 1-Sigma 0 03 Confidence 1 Interval 2 Posterior Distribution over Rk, P(Rk | o(t1:k)) 100 50 P 三0 0 Least Mean Square Estimate of Rk w/ Postprocessing 100 50 Predicted (blue) and Ground-truth (green) Inactivity Segments 10 12 4 18 20 22 24 15 17 19 21 23 0 200 400 600 800 1000 1200 40 Histogram of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 20 0 120 100 80 09- 40 20 0 Correlation between Predicted and Ground-truth Inactivity Durations 55 R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='94 50 16 Durations 45 40 35 1:1 Reference Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 30 20 Note: the red number next to each point corresponds to 22 the chronological order of 25 ground-truth inactivity 10 segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' points assigned 20 the same number imply multiple changepoint events 15 associated with one ground-truth segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 10 5 0 L 0 10 20 30 40 50 Ground-truth Inactivity Durationsa b c d e f Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Results from a participant dataset using UMAP w/o postprocessing of ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 8/18 UMAP Preprocessed Joint Orientation Embeddings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5 Embeddings 01 5 Predicted 4 Statistics 02 2 Mean 10 5 1-Sigma 03 Confidence o0 Interval Posterior Distribution over Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' P(Rk | o(t1:k) 50 P=0 Least Mean Square Estimate of Rk w/o Postprocessing 60 40 Aomm 20 h 0 Predicted (blue) and Ground-truth (green) Inactivity Segments 10 12 20 8 16 22 24 3 7 9 11 13 15 17 23 ■ 0 200 400 600 800 1000 1200 Vtk 200 Histogram of Rk k-1 100 0 60 50 40 30 20 10 0 Correlation between Predicted and Ground-truth Inactivity Durations 60 R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='07 2 50 Predicted Inactivity Durations 18 40 13 16 9 1:1 Reference Line 5 30 Note: the red number next to 0:21 3 24 each point corresponds to 17 the chronological order of 22 ground-truth inactivity 20 14 segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' points assigned 8 the same number imply multiple changepoint events associated with one 10 ground-truth segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 12 18 8 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 0 10 20 30 40 50 Ground-truth Inactivity Durationsthe posterior statistics of the previous KIDS variant which takes only a few time steps to gradually converge from the prior covariance to the true underlying covariance of the timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The heavy dependency of Bayesian inference on ooo embeddings led to a higher uncertainty in the Bayesian multi-hypothesis evaluation of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The uncertainty can be visually observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5b in the form of “salt and pepper noise” contaminating the grayscale matrix representation of P( k | ooo(⇓t1:k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As a result, the LMS Bayesian estimator outputted a less reliable point estimate ˆ k (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5c) that was too sensitive to minor variations in UMAPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' These minor variations are typically due to involuntary micro-body movements, such as hand twitches and breathing-related perturbations to the wrist pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A multi-stage processing of P( k | ooo(⇓t1:k)), similar to that of the previous variant of KIDS, was performed except that no postprocessing was applied to the LMS point estimate ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Three main observations can be concluded from the (in)activity detection and segmentation results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First, numerous repetitive resets were detected (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5e), some of which were unfiltered incidental resets, explaining the presence of short-lasting inactive segments (see the 12th and 18th GT inactive segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second, few GT inactive segments (6th, 10th and 11th) were neither detected nor segmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Third, the correlation plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5f shows strong evidence of underestimation of inactivity durations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' most segmentations are located below the 1:1 reference line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' All three observations were primarily caused by the poor reliability of the run length estimation performed by this KIDS variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Performance Metrics: A Comparative Analysis on Variants of the KIDS Framework The performance evaluation metrics are essential not only to make a fair comparison between the four variants of the proposed KIDS framework, but also to recognise the added value of each of the adopted methods, such as UMAP, ADR and postprocessing of ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this work, the F1-score, Sensitivity (Se) and Positive Predictive Value (PPV) are the metrics used to evaluate the changepoint detection performance, while the Pearson’s correlation coefficient (R) is reserved for assessing the quality of inactivity segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' These are standard metrics commonly reported in relevant works28 and would establish a good ground for benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (1), where TP, FP and FN denote true positives, false positives and false negatives respectively, shows how each of the changepoint detection metrics is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The predictive positive value (or precision) is the ratio of correct KIDS-derived changepoint events (KIDSnc) to the total number of changepoint events (KIDSn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' a detected changepoint event is regarded as “correct” if it is no more than 3 samples away from the end time of its closest GT inactive segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The sensitivity (or recall) is the ratio of KIDSnc to the total number of GT inactive segments (GTn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lastly, the F1-score is the harmonic mean of PPV and Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The Pearson’s R coefficient is a test statistic that measures the statistical relationship between two continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (2), R is mathematically defined as the ratio of the covariance for two arbitrary random variates, X and Y, to the product of their standard deviations, σX and σY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This ratio ranges from +1 (complete positive correlation) to -1 (complete negative correlation), with 0 indicating no correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this work, the sample populations of X and Y are the predicted and GT inactive segment durations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' PPV = KIDSnc KIDSn � = TP TP + FP � Se = KIDSnc GTn � = TP TP + FN � F1-score = 2×PPV×Se PPV + Se (1) R = Cov(X,Y) σX ·σY (2) For the changepoint detection performance (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 6a), ADR-based KIDS variants attained +96% F1 scores while UMAP-based variants scored below 91% for the same metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The >5% gap in performance underscores the positive impact brought by both the ADR 3D embedding space and the informative Bayesian prior to the detection of changepoint events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The postprocessing of ˆ k evidently improved the changepoint detection performance in the ADR case by up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='7%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='3% gains in the F1-score, Se and PPV metrics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Importantly, the employed postprocessing algorithm stands out from the conventional re-scoring rules in the literature which are often criticised for compromising the F1-score and Se in favour of the overall accuracy28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This is because existing re-scoring rules aim to improve the performance of an algorithm using assumptions on the sleeper, unlike the case in this work where the postprocessing algorithm was crafted based on the studied behaviour of the KIDS framework itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Nevertheless, in the case of UMAP-based KIDS, postprocessing was found to have no to minor adverse effect on their changepoint detection performance, with an average 1% drop in the F1-score after 9/18 (a) (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Performance evaluation metrics of all four variants of the proposed KIDS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' the incorporation of the postprocessing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The unviability of ˆ k postprocessing for UMAP-based KIDS comes as no surprise, since the main problem was the intrinsic Bayesian uncertainty (due to missing an informative prior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In regard to the inactivity segmentation performance (refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 6b), the average difference in the Pearson’s R coefficient between respective ADR- and UMAP-based KIDS variants is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The wide gap again signifies the importance of the ADR 3D embedding space and the informative prior in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Besides, the postprocessing algorithm brought a surge in segmentation performance, with an average 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5% and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='2% enhancement to the Pearson’s R coefficient of UMAP- and ADR-based KIDS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Notably, the best-performing KIDS variant (ADR w/ postprocessing) accomplished an average R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='96, which is far more superior than the least-performing variant (UMAP w/o postprocessing) at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='63 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The key takeaways from the comparative analysis given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 6 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First, ADR-based KIDS variants generally perform better than UMAP-based variants at both changepoint detection and inactivity segmentation tasks, regardless of postprocessing ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second, when the postprocessing algorithm was factored into the performance evaluation, its effect was different for ADR- and UMAP-based KIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In case of ADR, the postprocessing algorithm was found to always enhance the framework performance in both tasks of changepoint detection and inactivity segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' However, for UMAP-based variants, postprocessing was only viable at the inactivity segmentation task for reasons covered earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Discussion This paper has presented KIDS, which according to the authors’ best knowledge, is the first in-bed movement analysis framework to fulfil two intertwined clinical needs: (in)activity detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Unlike the previous work reported in the literature that typically addresses either one of the two needs, the KIDS framework reformulates the problem of in-bed movement analysis by providing a joint answer to the two interconnected research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Moreover, while the prevailing literature used hard-coded thresholding of raw/processed sensor data, data-hungry classifiers or a mixture of the two approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' KIDS leverages Bayesian probability at its core to provide an objective assessment of body (in)activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The input to KIDS is wearable sensor measurements of the underlying kinematic profile of the left wrist joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Such information is clinically meaningful and comprehensible to medical staff, and its use cases can be expanded beyond (in)activity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The KIDS framework complements a previous study11 on sleep posture recognition where four wearable sensor modules mounted on the wrists and ankles were used to classify twelve postures based on quadruple joint kinematic cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For that purpose, four joints were monitored to minimise the likelihood of overlap between sleep postures sharing similar extremity limb positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' However, for the objective of (in)activity detection and segmentation during sleep, monitoring quadruple joints is not well justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The hand is probably one of the most moved parts of the human body, and being lightweight, it potentially carries much of the information on body mobility during sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, as a starting point, this paper tested the hypothesis that monitoring the left wrist alone would be sufficient to obtain reliable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Low power consumption and real-time performance are desired criteria for portable sleep monitoring devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The choice of the sensor data rate directly determines the update rate of the in-device algorithm, and power consumption consequently 10/18 Changepoint Detection Performance 100 90 80 70 60 50 40 30 20 10 0 F1 Se ADRpp UMAPwpp ADR w/o pp UMAP w/o pp PPVInactivity Segmentation Performance R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='1 0 ADRwpp UMAPWPP ADRw/opp UMAPw/oppvaries depending on the computational cost of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As covered in the Introduction, it is a predominant practice in the literature to make use of actigraphy signals for wearable-based sleep analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' While actigraphy captures higher-order kinematics of the human body, it typically necessitates a relatively fast-updating movement analysis algorithm in order not to miss posture transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Alternatively, the KIDS framework employs a computationally efficient inertial sensor fusion algorithm to fuse high-frequency inertial signals and produce a filtered estimate of the LW joint orientation that is robust to sensor artefacts and environmental disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' KIDS lays down the assumption that upon every posture transition, the LW joint orientation is permanently changed, meaning that the requirement of a fast-updating algorithm can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, downsampling of the joint orientation profile was safely incorporated into KIDS without concerns over the (in)activity detection and segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As a result, the tight time constraints on the computation cycle of the in-device algorithm were substantially relaxed, allowing for the adoption of more advanced body movement analysis approaches such as the Bayesian probabilistic framework adopted in KIDS without compromising real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For further algorithm explainability and more optimal real-time performance, KIDS incorporates a novel joint kinematics preprocessing stage based on dimension reduction to learn a low-dimensional representation of joint orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The LW joint kinematics recordings obtained from in-vivo experiments were embedded into intuitive 3D visualisations showing a decent intra-posture clustering and inter-posture separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Visualisations produced by two DR methods (UMAP and ADR) were compared, and fully constrained 3D embedding spaces were only guaranteed by ADR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A side benefit of the constrained ADR 3D embedding space was that it facilitated the design of informative priors useful for Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The general working principle of KIDS relies on statistical modelling of the 3D preprocessed joint kinematic timeseries to evaluate multiple hypotheses on the current segment run length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The output weighted hypotheses are then fused and processed by a reset detection logic to perform (in)activity detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The performance of four variants of the KIDS framework was quantitatively and qualitatively studied and comparisons were made based on standard metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The primary variants of KIDS come from the choice of the DR method used for preprocessing joint kinematics, while secondary variants correspond to whether the current segment run length estimate was refined by a postprocessing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' It was found that ADR-based KIDS variants generally outperformed UMAP-based variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Furthermore, even though the postprocessing algorithm enhanced the segmentation performance of UMAP-based KIDS, its added value was more evident in case of ADR-based KIDS improving both its changepoint detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lastly, performance evaluation showed that the assumption of KIDS on permanent change in joint orientation upon each posture transition is valid except for only one case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For this case, the LW joint orientation of a participant barely changed after one posture transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' nonetheless, the timeseries modelling of KIDS was still able to partially reset ˆ k and the transition was successfully detected by the reset detection logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Methods This section presents the methodology and experimental protocol behind the KIDS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First, we discuss the participant study pipeline and the simulated sleep protocol devised for the validation of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second, an overview on the custom-made wearable sensors and the algorithm for wrist kinematics measurement is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Third, we outline the proposed DR-based preprocessing of measured joint kinematics using two methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' UMAP and Analytical dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lastly, the Bayesian probabilistic framework enabling the (in)activity detection and segmentation is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Participant Study Five healthy adult participants (age: 36 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='8 years, height: 169 ± 11 cm, body weight: 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='8 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='2 kg) took part in the study upon signing an informed consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The methods were carried out in accordance with relevant guidelines and regulations and all experimental protocols were approved by The University of Liverpool Research Ethics Committee (review reference: 9850).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A leaflet containing pictures of twelve sleep postures (refer to previous work11 for posture definitions) was handed to each participant to assist them in replicating the postures, with each sleep posture replicated twice (two trials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To ensure postural data resembles that of a realistic sleep scenario, a random pose shuffling technique was used to ensure statistical independence of samples across the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Posture and transition durations may vary as per the participant’s comfort and need for guidance by on-site researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For all datasets, a bespoke wearable sensor module was used to capture body segment and joint kinematics, and to transmit these data to a localhost server at a data rate of 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Bespoke Wearable Sensor Module The participant study employed four wearable sensor modules to monitor the joint orientations of the two wrists and two ankles simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In a previous work11, the multi-sensor data were suitable for a robust sleep posture classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This work however exploits the LW joint kinematics alone, as a starting point, for body (in)activity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The custom-made sensor module provides dual-segment orientation tracking across the LW joint, empowered by two embedded IMU sensors mounted on the forearm and hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The IMU model is the BNO055 from Bosch Sensortec© (Bosch Sensortec GmbH, Reutlingen, DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Both IMU sensors are managed by a single ESP32-WROOM-32D microcontroller from Espressif Systems© (Espressif Systems 11/18 Shanghai Co Ltd, Shanghai, CN) featuring Bluetooth connectivity for wireless data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' At about 6 cubic centimeters in volume for each IMU case, the sensor module is sufficiently slim and small for wearability during sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Inertial Sensor Fusion for Wrist Kinematics Measurement Intra- and inter-sensor fusion were employed in the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' An inertial sensor fusion algorithm was needed to estimate the attitude of each IMU sensor (intra-sensor fusion), that is, a function of the body segment it is mounted on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Given the two segment orientations derived from both IMU sensors, an inter-sensor fusion step is subsequently applied to determine the relative joint orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Prior to each in-vivo experiment, all IMU sensors were calibrated according to standard procedures33,34 to estimate and reduce errors owing to constant bias, scale factors, cross-axis sensitivity and response nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For intra-sensor fusion, the Madgwick filter35 is employed to fuse the IMU geo-inertial measurements and provide a filtered estimate of the absolute segment quaternion with respect to the Earth reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The filter has low computational cost and operates in the quaternion space, allowing for efficient and singularity-free IMU attitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For inter-sensor fusion, the relative segment-to-segment quaternion was found through kinematic transformation which maps the orientation of the child segment (hand) to that of the parent segment (forearm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For further details on the mathematical formulations surrounding sensor fusion, it is advised to refer to previous work11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lastly, the segment-to-segment orientation was then converted from the quaternion space to the axis-angle space, where a unique xxx(tk) ∈ R4 exists at each arbitrary time step tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The disentangled axis-angle representation of joint orientations is more intuitive compared to quaternions and allowed for the extraction of meaningful postural analytics as shown in previous work11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Joint Kinematics Preprocessing The preprocessing of joint kinematics was enabled by dimension reduction, allowing for intuitive 3D visualisations of sensor- measured joint orientations and computationally efficient Bayesian inference subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Such 3D visualisation of joint kinematic timeseries facilitated the design of the informative prior which was found to enhance the Bayesian inference, and could potentially be useful for medical screening and/or diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Dimension reduction has been successfully applied to visualise data across different domains, from wearable sensing11 through speech processing36 to knowledge exchange37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The following text elaborates on the two DR methods employed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Uniform Manifold Approximation and Projection UMAP31 is a nonlinear DR method that iteratively constructs a lower dimensional force-directed graph representative of some high-dimensional dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' It is capable of handling nonlinear data manifolds and is known for preserving both the local and global data structures in the low-dimensional embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In principle, UMAP performs dimension reduction over two stages: (i) identifying nearest neighbours and constructing a neighbour graph, and (ii) learning a low-dimensional representation through iterative minimisation of a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Further details on UMAP can be found in its original paper31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The use of UMAP in this work is distinct from the common use of DR in human motion analysis works38–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Herein UMAP was run twice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' a first run to learn the mapping from 4D joint orientation data to a 3D manifold using synthetic data and a second run to embed sensor-measured joint orientations (unseen data) into the pretrained 3D embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The synthetic dataset was carefully designed to uniformly sample the axis-angle orientation space (given some resolution), hence enabling UMAP to construct a reliable neighbour graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Afterwards, the 4D sensor-measured joint orientations were passed to UMAP to be projected into the pretrained 3D embedding space accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The 3D visualisations of synthetic and real joint orientations are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1 and 3a respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A computer graphics (CG) pipeline (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 7) was used to generate the synthetic axes of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The pipeline first employed a procedural mesh generation technique to construct a unit cube with face vertices, vvvf = {xf ,yf ,zf }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Then, the ellipsoidal projection vvve = � xe ye ze � = � xf � 1− 1 2 y2 f − 1 2 z2 f + 1 3 y2 f z2 f yf � 1− 1 2 x2 f − 1 2 z2 f + 1 3 x2 f z2 f zf � 1− 1 2 x2 f − 1 2 y2 f + 1 3 x2 f y2 f � (3) was used to project all vvvf onto the surface of a unit sphere, such that the projections are evenly distributed over the surface of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The projected vertices, vvve, represent the synthetic axes of rotation, and were exported into a comma-separated values (CSV) file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' All procedural 3D modelling was implemented in the C# programming language and realised in Unity© (Unity Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', California, US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The CSV file containing all vvve was subsequently imported into MATLAB© (The MathWorks, Massachusetts, US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therein, an orientation dataset generator script concatenated each synthetic axis of rotation with each angle of rotation from the closed set { 36 36π, 35 36π, 34 36π,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', 1 36π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The output synthetic axis-angle dataset had a total of 48,600 orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Further details on the CG pipeline is available online in Supplementary Methods - CG Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 12/18 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The computer graphics pipeline used for the generation of synthetic axes of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Analytical Dimension Reduction The 3D point cloud from UMAP resembled the shape a thick-crust egg shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Through visual inspection, this point cloud was found not to be centred at the origin, and had geometrical artefacts such as uneven scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' These are inevitable outcomes inherent to the unconstrained embedding space of UMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To gain full control over the dimension reduction as a process, analytical dimension reduction was proposed taking inspiration from the UMAP 3D visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In a nutshell, ADR aims to output a standardised point cloud taking the shape of a thick-crust sphere that is clear from any form of geometrical deformation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The ADR sphere have the following shape features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The sphere has an inner radius, irs = 1, and an outer radius, ors = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The radial displacement from irs to ors corresponds to a change in the angle of rotation from 0 to π radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Based on this definition, it is then possible to interpolate the radial displacement, rs, given the sensor-measured angle of rotation x4: rs = 1 π x4 + irs (4) Additionally, the latitudinal and longitudinal displacements around the ADR sphere correspond to differently oriented axes of rotation defined by x1, x2 and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, an arbitrary preprocessed orientation embedding, ADRooo, in the constrained embedding space of ADR can be formulated as ADRooo = � o1 o2 o3 � = rs � x1 x2 x3 � (5) Downsampling of Preprocessed Joint Kinematics Sleep time is dominated by long durations of inactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Running Bayesian inference at every time step is a waste of computational energy, and is not recommended for a wearable device that continuously streams sensor data at 30 Hz overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In such case, downsampling is generally a good solution to reduce the computational and data storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the preprocessed joint kinematic timeseries was downsampled by a decimation factor of 100 before it was presented to the Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Bayesian Inference of the Current Segment Run Length A prominent highlight of the KIDS framework is the use of Bayesian inference for (in)activity detection and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The Bayesian inference was used to estimate the current segment run length, k, at each time step ⇓tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' By definition, k is the length (in samples) of a data segment whose samples shares similar statistical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Essentially, k is a key quantity which carries information on the length of inactivity and the temporal locations of changepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The Bayesian inference in this paper is largely inspired by a previously reported Bayesian run length estimation algorithm32, which evaluates multiple probability-weighted hypotheses on k at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The original paper presented the general Bayesian framework for the estimation of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the following steps were incorporated to adapt the Bayesian framework to the case study presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First, the Bayesian inference problem was reformulated to simultaneously estimate both the mean and precision of the preprocessed joint kinematic timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The estimation of the mean was essential since the 13/18 Computer Graphics Pipeline Procedural Construction of Unit Ellipsoidal Projection of Cube with Triangular Mesh Cube FacesFigure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Graphical illustration of the Bayesian run length estimation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 14/18 Incremental Generation of Posterior Predictive Models P(RT= T Io(t1:T) (T + 1) × (T + 1) Probability Matrix → P(R2 = 2 / o(μt1:2) 3 P(RT = 1 Io("t1:T) 2 P(RT = 0 / o(t1:T) 1×1 2×1 (T-1) ×1 T×1 Underlying Predictive Models Compute k×1 OR Predictive Probabilities 个 Informative Non-Informative Prior Prior Embedding Updated Priors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Posteriors) Timeseries Bavesian Inference of Current Segment Run Lengthpreprocessed timeseries typical exhibit a mean shift upon a change in posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The estimation of the precision was equally important since the fluctuations in the joint orientation slightly varies from one posture to another due to factors, such as snoring and breathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second, conjugate exponential distributions were adapted to the dual mean-precision estimation problem to maintain real-time inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Third, a Bayesian estimator was utilised to fuse the weighted hypotheses on k and output a single point estimate, ˆ k at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Fourth, we complemented the original Bayesian framework with a reset detection logic, which detected the temporal locations of changepoints based on the ˆ k estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The following section explains the general working principle behind the Bayesian inference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Complete details on the underlying mathematics are available online in Supplementary Methods - Bayesian Estimation of Current Segment Run Length, including the construction of the prior probability distribution used for implicit encoding of information42 to tune the Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' General Overview of the Bayesian Run Length Estimation A graphical illustration of the Bayesian run length estimation algorithm is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The algorithm requires two pieces of information as inputs: (i) either informative or non-informative prior on O(⇓ttt), and (ii) the embedding timeseries O(⇓ttt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Initially at ⇓t0 and before observing O, the algorithm assumes P( 0 = 0, ooo(⇓t0) = φ) = P( 0 = 0) = 1 which is the probabilistic notation for 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' At ⇓t1 and upon observing ooo(⇓t1), the algorithm evaluates the predictive probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' the probability that ooo(⇓t1) belong to the (non-)informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Afterwards, the algorithm evaluates all possible hypotheses on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Logically, k can take one of two values: k = � 0, if changepoint occurs at ⇓tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' k−1 +1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (6) Nonetheless, the Bayesian framework does not directly set k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Instead, it evaluates two types of transition probabilities: (i) “growth probabilities” from any possible k−1, and (ii) a “changepoint probability” from any possible k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 8, growth transitions are represented by green diagonal lines whereas changepoint transitions are represented by red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' At ⇓t1, the algorithm evaluates only two hypotheses on 1 in the form of joint probabilities: P( 1 = ζ, ooo(⇓t1)) for ζ = 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To evaluate these hypotheses, the algorithm forward passes P( 0 = 0, ooo(⇓t0) = φ) from the previous time step and fuses it with the predictive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This step explains why the algorithm is often described as a “message-passing algorithm”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' After finding the joint probabilities at ⇓t1, the posterior distribution P( 1 = ζ | ooo(⇓t1)) was determined using Bayes rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Since the number of hypotheses on k (possible values of ζ) grows linearly over time steps, a new predictive model (corresponding to the new hypothesis) is concatenated to the set of underlying predictive models at the end of each Bayesian inference step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In subsequent inference steps, a predictive probability is computed for ooo(⇓tk) given each underlying predictive model (or hypothesis) , unlike ⇓t0 at which only the (non-)informative prior was available for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Point Estimation of the Current Segment Run Length The aforementioned Bayesian estimation algorithm produced a posterior distribution over k conditioned on the preprocessed joint kinematic timeseries so far observed at each time step ⇓tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' However, k is still a discrete random variable and each value it takes under this distribution corresponds to one run length hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the use of point estimation is proposed to produce an estimate ˆ k that is close to the true value of k in some probabilistic sense43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To this end, the least mean squares (LMS) estimator is employed (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (7)) to produce ˆ k by minimising the mean squared error conditional on multiple continuous observations, ooo(⇓t1:k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The mean squared error is a effective criterion that trades-off bias and variance for an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The output of the point estimation stage is a (T +1) vector containing the LMS estimate of the run length at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' ˆ k = argmin ˆ k E � ( k − ˆ k)2 | ooo(⇓t1:k) � = E � k | ooo(⇓t1:k) � = ∑ k k P( k | ooo(⇓t1:k)) (7) Reset Detection Logic for (In)activity Detection and Segmentation Encoded in ˆ k is the information on the segments and changepoints present in O(⇓ttt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The preprocessed joint kinematic timeseries is now regarded as quasistatic segments of data which end every time ˆ k is reset upon body movement or activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The detection of resets in ˆ k is necessary for the detection of changepoints which mark the start of active segments (or the end of inactivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 15/18 Scale-invariant Reset Detection (Activity Detection) Reset detection is mainly about making a binary objective decision on whether the current segment of data samples is likely to continue growing or terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Importantly, the decision making needs to be invariant to the scale of ˆ k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' resets from ˆ k = ζa and ˆ k = ζb should both be treated equally by the algorithm even if ζb ≫ ζa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the application of threshold on the magnitude drop, ˆ k − ˆ k−1, would not meet the scale-invariant criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Alternatively, it is proposed to perform thresholding on the logarithmic scale of ˆ k since it effectively represents percent change or multiplicative factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A reference drop of log10 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='3 on the logarithmic scale indicates halving of ˆ k in the linear scale, hence it constitutes a binary decision boundary for run length termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, a reset is detected if the consecutive difference term, log10 ˆ k −log10 ˆ k−1, produced a drop larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Postprocessing of Current Segment Run Length Point Estimates Prior to the detection of resets, it is recommended to incorporate a postprocessing algorithm to refine ˆ k for a better performance (refer to the Results Section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' It was found that some gradual resets may occur, associated with few consecutive drops in ˆ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This resetting behaviour may be more challenging to be picked up by the reset detection logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, a three-sample moving filter (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (8)) was applied to ˆ k to merge these consecutive drops over one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The postprocessed estimate of ˆ k is denoted by ˆ p k: ˆ p k = � ˆ k−1, if ˆ k−1 > ˆ k > ˆ k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' ˆ k, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (8) For clarity, if a two-consecutive-drop scenario is considered with ˆ k−1 = 30, ˆ k = 23 and ˆ k+1 = 0, then the corresponding postprocessed LMS estimates will be ˆ p k−1 = 30, ˆ p k = 30 and ˆ p k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this example, the moving filter produced a total drop of 30 between ˆ p k+1 and ˆ p k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Even though the changepoint would be detected one time step later than it actually was, the postprocessing algorithm prevents the larger risk of not detecting the changepoint event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Elimination of Repetitive Resets During Activity During periods of activity, multiple resets may occur within a short time interval as the human participant makes few posture adjustments until they settle at the new sleep posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Such repetitive resets do not allow ˆ p k to proliferate in value, and hence they generally come after short run lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In this paper, the focus is more on sleep postures which are normally associated with longer run lengths (periods of inactivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Consequently, the run length before each detected reset was thresholded to eliminate repetitive resets during periods of activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Any arbitrary reset at ⇓tk with ˆ p k−1 < 20 was eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The minimum run length of twenty samples was backed by histograms of ˆ p k − ˆ p k−1 from different participant datasets (example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Inactivity Segmentation After the filtration of repetitive resets, the remaining resets are those associated with sleep postures that persisted for sufficiently long run lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The duration of each posture is equal to the postprocessed point estimate of the current segment run length just before the detected reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This was found to work better than taking the elapsed time between adjacent resets, simply because the Bayesian estimation of k is a recursive and self-correcting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the last posterior distribution, P( k | ooo(⇓t1:k)), within a segment of inactivity is generally more reliable than previous posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Data availability The preprocessed joint kinematic timeseries belonging to participants can be accessed online on www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='abcxyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' References 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='00318 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Murphy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Conjugate bayesian analysis of the gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Def 1, 1–29 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Dekking, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', Kraaikamp, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', Lopuhaä, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' & Meester, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A Modern Introduction to Probability and Statistics: Understanding why and how (Springer, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Acknowledgements (not compulsory) The authors would like to thank Daniel Potts and Shay Stanley for their assistance with the development of the wearable sensors used in the participant study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Author contributions statement O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' was responsible for the conceptualisation, research design, methodology and data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' contributed to the methodology and interpretation of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' instructed on the participant experimental protocol and research ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' identified knowledge gaps from the literature, formulated the research problem and advised on the concept solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' provided input on conceptualisation and research supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' All authors edited and reviewed the manuscript, and approved the submitted version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Competing interests O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' is supported by the University of Liverpool Doctoral Network in AI for Future Digital Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The authors declare no other competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 18/18 KIDS: kinematics-based (in)activity detection and segmentation in a sleep case study Omar Elnaggar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Roselina Arelhi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Frans Coenen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Andrew Hopkinson4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Lyndon Mason5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' and Paolo Paoletti1 1School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3GH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 2Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Sheffield,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Sheffield S1 3JD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 3School of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Electronics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3BX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 4School of Psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 7ZA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 5School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L69 3GE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom 6Department of Trauma and Orthopaedics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool University Hospitals NHS Foundation Trust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Liverpool L9 7AL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' United Kingdom Omar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='Elnaggar@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='uk Supplementary Information Supplementary Methods - CG Pipeline Even though the axis-angle representation is intuitive, its four dimensions does not facilitate easy visualisation of joint orientation timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, UMAP and analytical dimension reduction were used to reduce the dimensions to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' With fewer dimensions, the computational cost of subsequent Bayesian inference was lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This section is focused on the generation of synthetic orientation data essential for the UMAP-based dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Sensor-measured joint orientations are typically limited in size and does not cover the entire axis-angle orientation space due to anatomical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For these reasons, forcing the UMAP to learn the data manifold of xxx from sensor measurements does not guarantee accurate manifold modelling and may suffer from mapping discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, this paper proposes the generation of a synthetic axis-angle orientation dataset better suited for manifold learning by UMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Leveraging a CG pipeline, synthetic axes of rotation evenly distributed over the surface of a unit sphere were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Then, an equidistant sequence of scalar angles of rotation was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Finally, the synthetic orientation dataset was formed by exhausting all possible combinations between the axes and angles of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The CG pipeline aims at the procedural 3D modelling of a unit sphere with evenly-distributed vertices (axes of rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' S1, the pipeline generally consisted of two stages: (i) the construction of a unit cube with a structured triangular mesh, and (ii) the projection of the cube vertices onto the surface of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' All procedural 3D modelling was implemented in the C# programming language and realised in Unity© (Unity Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', California, US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The first stage of the CG pipeline involves procedural construction of the six faces making the unit cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Each quadrilateral face is defined by a surface normal vector,⃗nface, and a predefined mesh resolution, ϒ, such that ϒ is the number of vertices along any side of the quadrilateral face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Given these definitions, the vertex grid of the face was constructed with ϒ2 vertices, each assigned a unique identifier index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The cube faces are all procedurally constructed using ϒ = 15 and⃗nface being the six orthonormal directions of Unity’s world coordinate system, which yields a total of 1,350 vertices defining the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Next, the cube vertices need to be projected onto the surface of a unit sphere to obtain the synthetic axes of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To this end, two approaches, based on Euclidean normalisation and ellipsoidal projection respectively, were employed and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' First Approach: Projection of Cube Vertices with Euclidean Normalisation Let vvvc denote an arbitrary vertex on the unit cube, with individual coordinates (xc,yc,zc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This approach projects vvvc onto the surface of a unit sphere through the normalisation of vvvc by its Euclidean norm as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' euclidvvvs = 1 � x2c +y2c +z2c � xc yc zc � (S1) The result of this projection is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' S1a, which clearly shows projected vertices clumped together at the corners of the faces than in the centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Eventually, this projection would produce non-uniform samples over the data manifold in the orientation space, and could potentially cause unstable dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, a second approach was developed to address this projection artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='03469v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='SP] 4 Jan 2023 a b Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The computer graphics pipeline used for the generation of synthetic axes of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Second Approach: Ellipsoidal Projection of Cube Faces To obtain better distributed vertices, this approach projects constant-x, constant-y and constant-z square faces, which are the basic building geometries of a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The principal idea is to project each face onto an ellipsoid, and the stitched face ellipsoidal projections together make a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Suppose a generic face vertex, vvvf , defined as vvvf = {(xf ,yf ,zf ) | xf ,yf ,z f ∈ [−1,1]} (S2) where either xf , yf or zf is fixed at some constant value between -1 to 1, depending on the face orientation defined by its surface normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Mathematically, the ellipsoidal projection aims to map face vertices vvvf to vertices vvve on a 3D ellipsoid such that the point of intersection with the coordinate axis remain unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' vvve = vvv f at the centre of the face, and the face vertices are more normalised the closer they are to the corners of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For this requirement to hold true, the ellipsoid equation for some constant-xf face, as an example, must be 1 = x2 e x2 f + y2 e b2e + z2 e c2e (S3) where be and ce are constants, and be = ce due to symmetry about xe −ye and xe −ze planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As xf is varied from -1 to 1, the ellipsoid in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S3) make two intersection curves with the unit sphere, one curve for each of the upper and lower hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Any known point along these intersection curves can be used to determine be and ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The upper intersection curve meets the 2/5 Procedural Construction Computer Graphics Pipeline of Unit Cube with Triangular Mesh Euclidean Ellipsoidal Normalisation of Projection of Cube Vertices Cube Faces (First Approach) (Second Approach)Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The effect of cube face displacement on the ellipsoidal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' xe −ye plane at vvve = � xe ye ze � = � 1 √ 2 xf 1 √ 2 � 1− 1 2 x2 f 1 √ 2 � 1− 1 2 x2 f � (S4) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S4) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S3) yields be = ce = 2−x2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Hence, the ellipsoid for the constant-x face becomes 1 = x2 e x2 f + y2 e 2−x2 f + z2 e 2−x2 f (S5) By symmetry, the constant-y and constant-z faces are similarly mapped to the ellipsoids defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S6) and (S7) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1 = x2 e 2−y2 f + y2 e y2 f + z2 e 2−y2 f (S6) 1 = x2 e 2−z2 f + y2 e 2−z2 f + z2 e z2 f (S7) Then, the system of equations Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S5) to (S7) are simultaneously solved for xe, ye and ze to obtain the general expression for the ellipsoidal projection, vvve vvve = � xe ye ze � = � xf � 1− 1 2 y2 f − 1 2 z2 f + 1 3 y2 f z2 f y f � 1− 1 2 x2 f − 1 2 z2 f + 1 3 x2 f z2 f zf � 1− 1 2 x2 f − 1 2 y2 f + 1 3 x2 f y2 f � (S8) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' S2 shows the ellipsoids for constant-x, constant-y and constant-z faces at different displacements from the origin of the Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' As xf , yf and z f converge to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='0, their corresponding ellipsoids converge in 3D shape to a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This specific case of convergence applies to a cube with a side length of 1 unit, where the ellipsoidal projections of all six faces form a complete sphere when stitched together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, a better distributed vertex projection ellipsvvvs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' S1b) of the cube vertices onto the surface of a unit sphere was obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Each projected vertex on the surface of the sphere represents a candidate Cartesian axis of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' All 1,350 axes of rotation were written to a comma-separated values (CSV) file for subsequent import into MATLAB© (The MathWorks, Massachusetts, US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therein, an orientation dataset generator script concatenated each synthetic axis of rotation with each angle of rotation from the closed set { 36 36π, 35 36π, 34 36π,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', 1 36π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The output synthetic axis-angle dataset contained a total of 48,600 orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 3/5 Projected Faces Ellipsoids xf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='0 yf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='0 Xf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='8 zf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='0 yf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='8 xf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content="5 8'0 = z yf = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5 zf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content='5Supplementary Methods - Bayesian Estimation of Current Segment Run Length Based on the original paper?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=', the general mathematical expression for the joint probability can be written as: P( k, ooo(⇓t1:k)) = ∑ k−1 P(ooo(⇓tk) | k, ooo(⇓t1:k−1)) � �� � Predictive Term P( k | k−1) � �� � Changepoint Prior P( k−1, ooo(⇓t1:k−1)) � �� � Recursive Term (S9) The Predictive Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' It is also referred to as the posterior predictive term because it predicts the next embedding given the previously observed embeddings and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Thus, if a hypothesis states k = ζ, only embeddings within the past ζ time steps should contribute to the predictive term: P(ooo(⇓tk) | k = ζ, ooo(⇓t1:k−1)) = P(ooo(⇓tk) | k = ζ, ooo(⇓tk−ζ:k−1)) (S10) In this paper, it is assumed that O(⇓ttt) is continuously distributed with unknown mean vector µµµO ∈ R3 and precision matrix λλλ O ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Additionally, a likelihood model, Pl(ooo(⇓tk) | ηηη), is defined with model parameters ηηη = {µµµO, λλλ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' To compute the predictive term, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S11) first finds the posterior distribution, P(ηηη | k = ζ, ooo(⇓tk−ζ:k−1)), then marginalises ηηη, conditional on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Hence, the predictive term becomes the posterior predictive distribution: P(ooo(⇓tk) | k = ζ, ooo(⇓tk−ζ:k−1)) = � ηηη Pl(ooo(⇓tk) | ηηη) P(ηηη | k = ζ, ooo(⇓tk−ζ:k−1)) dηηη (S11) The posterior distribution can be re-written as P(ηηη | D, ααα), where D = ooo(⇓tk−ζ:k−1) and ααα denotes the parameters of the prior distribution, P(ηηη | ααα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Using Bayes’ theorem, conditional probability and chain rule, P(ηηη | D, ααα) can be expanded as follows: P(ηηη | D, ααα) = Pl(D | ηηη, ααα) P(ηηη | ααα) P(D | ααα) (S12) However, computing the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S11) is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Fortunately, conjugate-exponential models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' provide an efficient solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The concept of conjugacy allows for the derivation of a closed form expression as a function of ααα without the need for integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Hence, given a likelihood model Pl(D | ηηη, ααα), its conjugate prior model can be leveraged to simplify the posterior distribution to: P(ηηη | D, ααα) = P(ηηη | ααα′) (S13) where both the prior and posterior distributions belong to the same distribution family with parameters ααα and ααα′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Since both µµµO and λλλ O are inferred,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' the likelihood model follows a multivariate Gaussian distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' for which the natural conjugate prior is the Normal-Wishart distribution with quadruple parameterisation ααα = {µµµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' κα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' να,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' ΣΣΣα},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' P(ηηη | ααα) = N W(ηηη | ααα) = N (µµµO | µµµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (κα λλλ O)−1) W(λλλ O | ΣΣΣα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' να) (S14) where µµµα ∈ R3 and κα ∈ R are the mean vector and the covariance scale coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' of the normal distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' and να ∈ R and ΣΣΣα ∈ R3×3 are the number of degrees of freedom and the scale matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' of the Wishart distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Since P(ηηη | ααα) is a conjugate prior, then P(ηηη | D, ααα) turns out to be a Normal-Wishart distribution too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Substituting P(ηηη | D, ααα) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S11), it can be shown that the posterior predictive distribution follows a Student’s t-distribution with (ν′ α −2) degrees of freedom: P(ooo(⇓tk) | D, ααα) = � Pl(ooo(⇓tk) | ηηη) P(ηηη | D, ααα) dηηη = Tν′α−2(ooo(⇓tk) | µµµ′ α, ΣΣΣ′ α(κ′ α +1)κ′ α −1(ν′ α −2)−1) (S15) where µµµ′ α and ΣΣΣ′ α(κ′ α +1)κ′ α −1(ν′ α −2)−1 are the predicted µµµO and λλλ −1 O , respectively, of the posterior predictive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Setting the prior on preprocessed joint kinematic timeseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The set of parameters ααα(k,ζ) are updated at each time step ⇓tk and for each hypothesis k = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The (non-)informative prior on ηηη was defined by setting the initial parameters ααα(0,0) of the prior model to the appropriate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In the case of the Normal-Wishart distribution, the non-informative prior, ααα(0,0), was defined with µµµα(0,0) = � 10−4 10−4 10−4� , κα(0,0) = 10−4, να(0,0) = 4 and any matrix ΣΣΣα(0,0) provided that its 4/5 determinant |ΣΣΣα(0,0)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The non-informative prior is more suited for UMAPO due to the unconstrained embedding space of UMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' In contrast to UMAP, ADR has a constrained embedding space and known 3D embedding topology which facilitated the design of an informative prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Visual inspection of the real preprocessed joint kinematics in the ADR embedding space informed the following settings: (1) µµµO has an expected value E(µµµO) = 0 and an initial guess of III3×3 for its covariance matrix, and (2) λλλ O has an expected value E[λλλ O] = να ΣΣΣα = 20 III3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, ααα(0,0) was set such that µµµα(0,0) = � 10−4 10−4 10−4� , κα(0,0) = 1/20, να(0,0) = 4 and ΣΣΣα(0,0) = 5 III3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Updating the prior model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' At the end of each Bayesian inference step, the prior model parameters ααα are updated to represent the new set of hypotheses on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' At ⇓tk and given an arbitrary hypothesis k = ζ, the prior model parameters denoted by ααα(k,ζ) are updated as follows: µµµα(k,ζ) = κα(0,0)µµµα(0,0)+ζ oook−ζ+1:k κα(0,0)+ζ κα(k,ζ) = κα(0,0)+ζ να(k,ζ) = να(0,0)+ζ ΣΣΣα(k,ζ) = ΣΣΣα(0,0)+ κα(0,0) ζ κα(0,0)+ζ (µµµα(0,0)−oook−ζ+1:k) (µµµα(0,0)−oook−ζ+1:k)T+ ζ ∑ i=1 (ooo(⇓tk−i)−oook−ζ+1:k) (ooo(⇓tk−i)−oook−ζ+1:k)T (S16) where oook−ζ+1:k is the mean vector of the embeddings in ooo(⇓tk−ζ+1:k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The Changepoint Prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' This term allows for encoding the prior knowledge on changepoints present in O(⇓ttt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The current segment run length k is is a discrete, nonnegative random variable this is subject to two possible outcomes at any given ⇓tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' either reset to zero upon a changepoint or an incremental growth by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Thus, it was sensible to model k with a geometric distribution of success probability, p, where success corresponds to the growth of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Based on this modelling assumption, the changepoint prior term becomes P( k | k−1) = � � � � � p, if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 1− p, if k = k−1 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S17) Using the changepoint prior in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S17), the joint probability term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S9) can be further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' For growth transitions, it is redundant to sum over all possible k−1 as the only valid run length is k = k−1 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S13) and (S17) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' (S9), the joint growth transition probability is defined as P( k, ooo(⇓t1:k)) = P(ooo(⇓tk) | ααα(k −1, k−1)) (1− p) P( k−1, ooo(⇓t1:k−1) (S18) For the changepoint transitions, the run length can be reset from any valid k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Therefore, the summation over k−1 is necessary, P( k, ooo(⇓t1:k)) = ∑ k−1 P(ooo(⇓tk) | ααα(k −1, k−1)) p P( k−1, ooo(⇓t1:k−1) (S19) After finding all joint probabilities P( k, ooo(⇓t1:k)) at ⇓tk, the posterior distribution on k was found using Bayes rule P( k | ooo(⇓t1:k)) = P( k, ooo(⇓t1:k)) ∑ k P( k, ooo(⇓t1:k)) (S20) where ∑ k P( k, ooo(⇓t1:k)) is the normalising evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' Once the posterior distribution over k is determined ∀k ∈ [0,T], this concludes the Bayesian inference of the current segment run length with P( k | ooo(⇓t1:k)) being a (T + 1) × (T + 1) matrix indexed by k and different k = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' The upper triangular part of this matrix contains all-zero probabilities since k ≤ k at any ⇓tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} +page_content=' 5/5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E1T4oBgHgl3EQf1gWo/content/2301.03469v1.pdf'} diff --git a/nNE1T4oBgHgl3EQfOQP4/vector_store/index.faiss b/nNE1T4oBgHgl3EQfOQP4/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..486e2b4e38f18a388beab3656a52fb34e9ead5f9 --- /dev/null +++ b/nNE1T4oBgHgl3EQfOQP4/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:da8e971f049f530df5ee7e7ac9dd75ada7cdaa36bdfa074b6eebc1f0e9c98772 +size 9699373 diff --git a/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/2301.00693v1.pdf.txt b/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/2301.00693v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1da2b51abc6e520b0160d38e776f79b70c6659a8 --- /dev/null +++ b/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/2301.00693v1.pdf.txt @@ -0,0 +1,486 @@ +Deep Recurrent Learning Through Long Short Term +Memory and TOPSIS +* +Rossi Kamal, Zuzana Kubincova +Comenius University Bratislava +kamal1,kubincova@uniba.sk +Mosaddek Hossain Kamal, Upama Kabir +University of Dhaka +mhktushar,upama2@gmail.com +Abstract—Enterprise resource planning (ERP) software brings +resources, data together to keep software-flow within business +processes in a company. However, cloud computing’s cheap, easy +and quick management promise pushes business-owners for a +transition from monolithic to a data-center/cloud based ERP. +Since cloud-ERP development involves a cyclic process, namely +planning, implementing, testing and upgrading, its adoption is +realized as a deep recurrent neural network problem. Eventually, +a classification algorithm based on long short term memory +(LSTM) and TOPSIS is proposed to identify and rank, respec- +tively, adoption features. Our theoretical model is validated over +a reference model by articulating key players, services, architec- +ture,functionalities. Qualitative survey is conducted among users +by considering technology, innovation and resistance issues,to +formulate hypotheses on key adoption factors. +Index Terms—Cloud ERP, Deep Neural Network, Recurrent +Neural Network, Long Short Term Memory, TOPIS, DOI, TOE, +MIR +I. INTRODUCTION +A. Research Motivation +ERP is a software suite maintained through a standard +database for establishing information exchange among differ- +ent business processes with standardized practices, however, +reflected through the enterprise cycle[1]. It is a software +suite which incorporates business processes into a software +module for fully featuring all enterprise activities within an +organization. It is functional as a center point for data and +information across different business processes and modules +because of its capability of managing, keeping and retrieving +information that exists within enterprise business. Thus, ERP +is characterized by information flow in accordance with man- +ufacturing and product flow which enables managers to have +full control over the entire business process. As functional +interventions from different internal modules mediate the +overall process of the business organization, business owners +pay full attention to the monolith conventional ERP from +the beginning of deployment. However, the penetration of +industrial informatics and the internet of things pushes us +to rethink monolithic software towards a data center or even +cloud based approach[2]-[4]. +Cloud-ERP emerges from companies’ growing interest with +ICT technology, for example, transition-mindset on data- +center/cloud computing to gain competitive advantage over +rivals with the advent of data analytics/visualization towards +Thanks to support provided by Professor Dr Zuzana Kubincova +better business decision-making[5][6][7] . Cloud ERP has +outperformed monolithic ERP in terms of cost, customization +and reduced time/effort among small to large enterprises. +Companies with less IT-infrastructure/human-resources merely +select subscription-based cloud services, rather than building +expensive IT infrastructure from scratch. Its on-demand cus- +tomization option allows it to easily update on-time with so- +simple pay-per pricing. Furthermore, cloud ERP service is +deployed and then personalized with minimal time and effort. +Thus, customers forget the network performance underneath, +and concentrate more about what services are served. +B. Problem Statement +A theoretical framework is necessary, that articulates re- +quirements, demands and feedback of various enterprise users +in a Cloud ERP software. In this context, the framework +should consider user-experience, benefit, privacy, customiza- +tion starting from interviewing and pre-planning phases as +well. Such theories generally address challenges and issues +faced by companies/people from planning to implementation +towards customization. Furthermore, such a framework should +accommodate comparative and historical data regarding adop- +tion across organizations. Lastly, the theoretical framework is +generally cross-matched with a reference architecture, having +major actors, components and functionalities required for +Cloud ERP adoption. +In this context, we are motivated to use the Deep Neural +Network (DNN) based analysis for Cloud ERP adoption. DNN +is coined by both ’multilayered concept’ and ‘joint inference +by connected neurons’.It acts just like the human exploits +consequent (i.e. multi-layered) observations (i.e neurons ) to +predict a near future incidence. Likewise human prediction- +likelihood increases over time, when previous observations +help upcoming prediction, DNN exploits simple layer to +layer inference and advances over time to detect and classify +instances, starting from initial belief, towards final prediction. +However, unlike conventional machine learning which uses +manual data augmentation and trial-error based training, DNN +extracts features in an automated and cheaper manner. +C. Contribution +Therefore, our contribution in ongoing research, is as fol- +lows. We have provided a reference architecture with major +arXiv:2301.00693v1 [cs.SE] 30 Dec 2022 + +players, components and functionalities for Cloud ERP im- +plementation.Then, we have proposed a deep recurrent neural +network based approach to infer the Cloud ERP adoption +factors by using long short term memory and TOPSIS. We +have provided a theoretical analysis to justify how Cloud +ERP adoption is amenable to time-series-problem, how deep +recurrent neural network fits to it and how LSTM ranking +provided by TOPSIS accurately predicts adoption factors. +Finally, we conduct a qualitative survey among software-users +by considering technology, innovation and resistance factors +towards Cloud ERP adoption. +II. REFERENCE MODEL +In this section, Cloud-ERP’s reference model is described +with architecture and functionality. Cloud-computing termi- +nologies are introduced first (Fig. 1a) to articulate Cloud-ERP +presence and its internal functionalities (Fig. 1b). +In cloud computing, users get access to services via browser +based apps. Data /system software is present in the data +center and service is accessed through the internet. Cloud +service is either software as a service (SAAS) or platform as a +service (PAAS) or infrastructure as a service(IAAS) (Fig. 1a). +Customers pay for only infrastructure or both infrastructure +and software platforms in IAAS, PAAS, respectively. However, +the SAAS model provides access to the vendor’s software mar- +ketplace. Cloud-ERP is generally hosted and delivered on the +internet. Likewise with the SAAS model, customers subscribe +to a paid service for accessing ERP services over the internet. +Cloud-ERP can be without SAAS, having infrastructure or +platform, or with SAAS , however, with or without cloud. +Cloud ERP is generally provided on a private, public or hybrid +cloud basis. ERP service is provided solely by thirty party +vendors in the public cloud concept. Private-cloud based ERP +services are provided by only internal-organization settings but +are less dependent on external internet. Whenever both internal +and external cloud functionalities are added, ERP comes with +a hybrid attire. +Typical +cloud-ERP +architecture +consists +of +two +players, +namely +customer/company/user +and +service +provider/vendor(Fig. 1b). In the first step, service-providers +plan software services according to the user demand . Users’ +hardware, software and cloud platforms are scrutinized to +check the compatibility issues. In the second step, service +providers publish new software services and users are notified +about the release. In the third step, software services are +configured in users’ private clouds. Users get complete access +to software services, such as manufacturing, supply chain, +inventory, sales, thanks to Cloud ERP(Fig. 1c) . User is kept +in the loop with the service provider for feedback on future +updates. +III. THEORETICAL ANALYSIS +We conduct theoretical analysis to justify first that too +simple or too complex LSTM results as under-fitting or over- +fitting problem. However, moderate number of layers and +neurons leads to accurate prediction of LSTM. Hence, the +selection of LSTM is amenable to TOPSIS ranking. TOPSIS +is able to yield accurate LSTM, classifier outcome with the +intervention of weight matrix. +Theorem 1: Cloud ERP adoption is amenable to deep +recurrent neural network problem. +Proof:Cloud-ERP adoption is conceptualized through four- +phase Enterprise System Experience Cycle which is how +business value is created and transmitted in a successful +enterprise.In the beginning, ERP project planning is conducted +by terms such as finance, timeline Then, development and in- +stallation of ERP Software is done in running business. Later, +system testing and bug-fixing are conducted after employee +feedback. Finally, upgrade of the system is performed with +the adjusted software changes. Each phase is dependent on +outcome of previous phases, where many latent phases/issues +are important and a cyclic process is maintained through out +the life-span of enterprise. In a typical artificial neural network, +layers are generally connected through neurons and especially, +in a deep recurrent neural network, each layer is a result of the +joint operations of neurons from input and hidden layers. Thus, +Cloud-ERP adoption resembling enterprise system experience +cycle is amenable to deep recurrent neural network problem. +Lemma 2: TOPSIS exploits the weight matrix for LSTM +ranking. +Proof: TOPIS with its multi criteria decision making abil- +ity, selects criteria among alternatives based on trade-offs. +As such, each LSTM architecture (i.e. criteria) has its own +objective toward optimal solution. Thus, each limitation is +compensated for for its relative benefit of competitive criteria. +A relative ranking mechanism among LSTM architectures is +performed with the intervention of a weight matrix . Thus, +weight of each LSTM architecture is managed through a +weight matrix, by which LSTM architectures are ranked. +Lemma 3: LSTM selection is amenable to TOPIS ranking. +Proof: LSTM is implemented with a single layer or multi- +ple layers.LSTM with multiple layers often boost prediction +performance by compromising the complexity, explainability. +On the contrary, LSTM with a single hidden layer is good +for time series data. However, the number of neurons in a +single layer is a critical decision to make.LSTM with complex +architecture and huge neurons causes overfitting in data. ‘Too +small number of neurons’ causes a simple model with less +accuracy. Thus,selection of hidden layers and neurons is a +complex decision..TOPSIS is a multi-criteria decision mecha- +nism for ranking classification algorithms such as LSTM.Thus, +selection of LSTM among alternatives is amenable to TOPSIS +ranking. +Lemma 4: LSTM with moderate number of neurons and +hidden layers results in better prediction. +Proof:Prediction accuracy is not desired level, when there +exists a limited number of neurons in the hidden layer. +However, as the neurons in the hidden layer increase, it is +easier to learn the data and then gradually achieve desired +prediction performance. Hence, a moderate number of neurons +in the hidden layer is capable of achieving desired prediction +accuracy. + +Private +cloud +Cloud ERP +Service + Management +Policy + User + + Iaas,Pass,Saas + Hardware OS ERP + Client Browser + ERP APP + + Iaas,Pass, + Hardware OS + Client Browser + ERP APP + + Platform OS + Iaas Virtual Hardware + Client Browser + Cloud Infrastructure + Cloud Platform + Cloud Application +Private +cloud + Management +Policy + Service Providers +Manufacturing +inventory +finance +purchase +sales +Supply chain + (a) IAAS, PAAS, SAAS Conceptualization +(b)Architecture and Functionality +(c) Services + Service Providers +User +Fig. 1. +Cloud ERP’s IAAS/PAAS/SAAS Conceptualization, Reference Architecture and Services +b. DNN with input,hidden,output layers and neurons + x1 + x2 + x3 + y +w2 +w3 +w1 +a. Layers, neurons in ANN +Chartering +Shake- +down +Con- +figuration +Onward +and +Upward +c. Enterprise Experience System Cycle +Fig. 2. ANN,DNN Representation and Cloud-ERP Adoption with conceptualizing Enterprise System Experience Cycle +Lemma 5: Excessive number of neurons in hidden layer +degrades LSTM performance. LSTM +Proof: In a regular classification, the similarity between input +and output sequence results in a good prediction. However, the +performance degrades due to excessive prior and consequent +inputs often with the presence of large numbers of neurons +in hidden layers.Thus, large data input results in a functional +LSTM , however, class-prediction performance degrades. +Lemma 6: Too complex LSTM suffers from over-fitting +problem: +Proof:Neural network architecture with too many hidden lay- +ers and/or neurons is time-consuming . Hence, LSTM suffers +from over-fitting. +Lemma 7: Too Simple LSTM with relatively small hidden +layers suffers from under-fitting. +Proof: Simple deep neural network architecture is unable to +process required information to detect any pattern. Thus, too +simple LSTM architecture suffers from error-prone prediction +and leads to under-fitting problems. +Lemma 8: No rules-of-thumb exists for the selection of ’ +hidden layer and/or neurons in LSTM’ +Proof:: It is hard to relate ‘hidden layer and/or neurons’ + +and ‘input/output sequence’. However, the degree of noise +in the dataset makes significant deviation in the performance +of hidden neurons/layers of LSTM. Thus, no rules of thumb +exists for hidden layer and/or neurons in LSTM +A. TOPSIS-based Cloud-ERP Adoption Factor Analysis +We now describe our TOPSIS technique, which is used to +rank/score classification outputs [i.e. adoption factors] pro- +vided by LSTM. We begin with the simplified explanation +of general TOPSIS and then continue internal operations of +TOPSIS-based Cloud ERP adoption factor analysis. +TOPSIS is an effective and simplified algorithm, which +belongs to the umbrella of MCDM mechanism. Given a +problem and multiple solutions, MCDM helps to find us the +highest ranked solution. TOPSIS finds geometric distances of +alternatives to find best-ranked alternatives, which is said to be +closest to best ideal solution, where the low-ranked alternative +indicates worst-solution. TOPSIS generally processes such all +information/rank on a decision matrix through the life-span. +Our mechanism involves the TOPSIS based MCDM tech- +nique to retrieve the final Cloud ERP adoption factors for +given inputs, where confidence scores are associated with +base classifiers. In this way,alternative solutions are available +for remaining classes, such that confidence scores are major +representatives of base classifiers. Thus an element is chosen +from the decision matrix, where confidence scores are selec- +tion criteria for alternative solutions.Eventually, we proceed +with all steps of TOPSIS on the decision matrix to finally +reach out highest ranked alternative. The process involves +column normalization of the matrix in accordance with the +TOPSIS. In this case, we calculate the sum of squares of +observed information and then the square root of sum. Then +the normalization is obtained by dividing the input score by +the square root of the sum. +IV. QUALITATIVE SURVEY +In this section, we have conducted a qualitative survey +among clients regarding CloudERP-adoption choices by con- +sidering technology, innovation and resistance. +We have given importance on the cloud based ERP adoption +by considering technology perspective, innovation charac- +teristics and resistance influence. In this context, we have +discovered the relationship and deviation among innovation, +resistance and technology and their combined impact on cloud +ERP adoption in industry scale. +We have adopted a mixed method based research model in +analyzing the most dominant factors in Cloud ERP adoption. +In this context, we begin with technology perspectives on +the human mind while adopting a new technology, followed +by experiments on welcoming behaviors toward creativity.We +begin with a holistic view on new technology use in terms +of authority/organization and supportive/regulatory towards +any new invention/technology. However, as cloud ERP is +a new technology domain, and provides citizens with new +practices/ideas, we are motivated to analyze human behavior +regarding innovation characteristics. However, innovation is +not always welcome by all societies, its adoption takes various +ways in different groups. In this context, we are motivated to +analyze Cloud ERP adoption by considering the reluctance of +the human mind towards any new creative technology. +With the intervention of the TOE framework, we assess +the influence of three factors, namely technology, organization +and environment in adopting Cloud ERP by organization +employees. We have found that ICT knowledge, network +infrastructure, high competition and support from management +are fundamental factors in cloud ERP adoption. Meanwhile, +business goals, technology integration skill and implementa- +tion challenges are also found factors in some contexts. +By MIR, we assess how innovation goes ahead over +some primary resistance prior to full adoption.Thus, the fu- +ture acceptability is determined from the duration of resis- +tance.However, the confusion on product usability or longevity +works against innovation adoption. Similarly, the greater the +complexity, the larger is the innovation resistance. We have +found among participants, that critical factors such as cyber +security, vendor recognition and personalization are resisting +cloud ERP adoption. +By DOI, we try to find factors which are responsible +for selection of cloud ERP solutions over existing technolo- +gies.Innovation is coined by a new idea or practice recognized +by group or nationality. Supremacy over existing technology +and solutions is a major concern which helps clients to choose +cloud ERP, an innovative approach.Plug and Play solution is +found to be one of major factors in adoption of a new technol- +ogy. However, if the system is complex, people don’t like it +unless a solution is presented alongside. Troubleshooting and +user-experience are found as important factors for the complete +adoption of Cloud ERP . +V. CONCLUSION +Cloud-ERP is trending with the promise of cheap, easy +and quick business-process management. +In this context, a +deep recurrent neural network based classifier is proposed +to infer its key adoption features.Classifier uses LSTM first +to classify features and then TOPSIS to rank identified fea- +tures. Moreover, a qualitative survey in terms of technology, +innovation and resistance is conducted to infer adoption factors +and make hypotheses. Theoretical justification is also provided +to show how cloud-ERP adoption is amenable to recurrent +neural network problem and how TOPSIS and LSTM based +classifier detects adoption features. +References +1.Cieciora, M., Bołkunow, W., Pietrzak, P., Gago, P. +and Rze´znik-Knotek, M., 2020. Critical success factors of +ERP/CRM implementation in SMEs in Poland: pilot study. +2. Vacek, J., Dvoˇr´akov´a, L. and Tauˇsl Proch´azkov´a, P., 2020. +Knowledge intensive services in society 4.0. +3.Nagy, J., Ol´ah, J., Erdei, E., M´at´e, D. and Popp, J., +2018. The role and impact of Industry 4.0 and the internet of +things on the business strategy of the value chain—the case +of Hungary. Sustainability, 10(10), p.3491. + + Dataset +Feature +Extraction +LSTM +Classified +Data +Data +Pre-processing +Ranked +Classified +Data + Weight + Matrix +TOPSIS +output +Normalize +Decision +Matrix +Find +Weighted +Matrix +Find +Ideal +Solution +Separate +Measure for +alternative + Close + solution tp +alternative +Rank +alternative +solution +input +(a). LSTM +TOPSIS (b). TOPSIS +Fig. 3. Classification Mechanism + Cloud + Infra[E] + Cloud +Skill[T] +TOE +Plug and +Play +Cyber +Security +User +Experience +MIR +Cloud ERP + Ease of +Use +Supremacy + Trouble +shooting +DOI + Top +Management + [O] +Fig. 4. TOE, DOI, MIR-based-Mixed Method Qualitative Study +4.Pfeifer, M.R., 2021. Development of a Smart Manufactur- +ing Execution System Architecture for SMEs: A Czech Case +Study. Sustainability, 13(18), p.10181. +5.Lisowska, R. and Pamula, A., 2020. Cloud comput- +ing adoption in small and medium-sized enterprises in +Poland–benefit analysis. Global Journal of Information Tech- +nology: Emerging Technologies, 10(2), pp.98-105. +6.Morawiec, P. and Sołtysik-Piorunkiewicz, A., 2022. Cloud +Computing, Big Data, and Blockchain Technology Adoption +in ERP Implementation Methodology. Sustainability, 14(7), +p.3714. +7.ˇSpatenka, J. and Koch, M., 2021. Sustainable Develop- +ment of Companies Using the ERP System as a Fundamental +Tool of Digital Transformation. Trends Economics and Man- +agement, 15(38), pp.61-70. + diff --git a/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/load_file.txt b/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..575a82dd85420d93202397a4475480a0ff009dda --- /dev/null +++ b/ndAyT4oBgHgl3EQfy_k5/content/tmp_files/load_file.txt @@ -0,0 +1,237 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf,len=236 +page_content='Deep Recurrent Learning Through Long Short Term Memory and TOPSIS Rossi Kamal, Zuzana Kubincova Comenius University Bratislava kamal1,kubincova@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='sk Mosaddek Hossain Kamal, Upama Kabir University of Dhaka mhktushar,upama2@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='com Abstract—Enterprise resource planning (ERP) software brings resources, data together to keep software-flow within business processes in a company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, cloud computing’s cheap, easy and quick management promise pushes business-owners for a transition from monolithic to a data-center/cloud based ERP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Since cloud-ERP development involves a cyclic process, namely planning, implementing, testing and upgrading, its adoption is realized as a deep recurrent neural network problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Eventually, a classification algorithm based on long short term memory (LSTM) and TOPSIS is proposed to identify and rank, respec- tively, adoption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Our theoretical model is validated over a reference model by articulating key players, services, architec- ture,functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Qualitative survey is conducted among users by considering technology, innovation and resistance issues,to formulate hypotheses on key adoption factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Index Terms—Cloud ERP, Deep Neural Network, Recurrent Neural Network, Long Short Term Memory, TOPIS, DOI, TOE, MIR I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Research Motivation ERP is a software suite maintained through a standard database for establishing information exchange among differ- ent business processes with standardized practices, however, reflected through the enterprise cycle[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' It is a software suite which incorporates business processes into a software module for fully featuring all enterprise activities within an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' It is functional as a center point for data and information across different business processes and modules because of its capability of managing, keeping and retrieving information that exists within enterprise business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, ERP is characterized by information flow in accordance with man- ufacturing and product flow which enables managers to have full control over the entire business process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' As functional interventions from different internal modules mediate the overall process of the business organization, business owners pay full attention to the monolith conventional ERP from the beginning of deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, the penetration of industrial informatics and the internet of things pushes us to rethink monolithic software towards a data center or even cloud based approach[2]-[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud-ERP emerges from companies’ growing interest with ICT technology, for example, transition-mindset on data- center/cloud computing to gain competitive advantage over rivals with the advent of data analytics/visualization towards Thanks to support provided by Professor Dr Zuzana Kubincova better business decision-making[5][6][7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud ERP has outperformed monolithic ERP in terms of cost, customization and reduced time/effort among small to large enterprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Companies with less IT-infrastructure/human-resources merely select subscription-based cloud services, rather than building expensive IT infrastructure from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Its on-demand cus- tomization option allows it to easily update on-time with so- simple pay-per pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Furthermore, cloud ERP service is deployed and then personalized with minimal time and effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, customers forget the network performance underneath, and concentrate more about what services are served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Problem Statement A theoretical framework is necessary, that articulates re- quirements, demands and feedback of various enterprise users in a Cloud ERP software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, the framework should consider user-experience, benefit, privacy, customiza- tion starting from interviewing and pre-planning phases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Such theories generally address challenges and issues faced by companies/people from planning to implementation towards customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Furthermore, such a framework should accommodate comparative and historical data regarding adop- tion across organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lastly, the theoretical framework is generally cross-matched with a reference architecture, having major actors, components and functionalities required for Cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, we are motivated to use the Deep Neural Network (DNN) based analysis for Cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' DNN is coined by both ’multilayered concept’ and ‘joint inference by connected neurons’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='It acts just like the human exploits consequent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' multi-layered) observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='e neurons ) to predict a near future incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Likewise human prediction- likelihood increases over time, when previous observations help upcoming prediction, DNN exploits simple layer to layer inference and advances over time to detect and classify instances, starting from initial belief, towards final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, unlike conventional machine learning which uses manual data augmentation and trial-error based training, DNN extracts features in an automated and cheaper manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Contribution Therefore, our contribution in ongoing research, is as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have provided a reference architecture with major arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='00693v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='SE] 30 Dec 2022 players, components and functionalities for Cloud ERP im- plementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Then, we have proposed a deep recurrent neural network based approach to infer the Cloud ERP adoption factors by using long short term memory and TOPSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have provided a theoretical analysis to justify how Cloud ERP adoption is amenable to time-series-problem, how deep recurrent neural network fits to it and how LSTM ranking provided by TOPSIS accurately predicts adoption factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Finally, we conduct a qualitative survey among software-users by considering technology, innovation and resistance factors towards Cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' REFERENCE MODEL In this section, Cloud-ERP’s reference model is described with architecture and functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud-computing termi- nologies are introduced first (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1a) to articulate Cloud-ERP presence and its internal functionalities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In cloud computing, users get access to services via browser based apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Data /system software is present in the data center and service is accessed through the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud service is either software as a service (SAAS) or platform as a service (PAAS) or infrastructure as a service(IAAS) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Customers pay for only infrastructure or both infrastructure and software platforms in IAAS, PAAS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, the SAAS model provides access to the vendor’s software mar- ketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud-ERP is generally hosted and delivered on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Likewise with the SAAS model, customers subscribe to a paid service for accessing ERP services over the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud-ERP can be without SAAS, having infrastructure or platform, or with SAAS , however, with or without cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud ERP is generally provided on a private, public or hybrid cloud basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' ERP service is provided solely by thirty party vendors in the public cloud concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Private-cloud based ERP services are provided by only internal-organization settings but are less dependent on external internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Whenever both internal and external cloud functionalities are added, ERP comes with a hybrid attire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Typical cloud-ERP architecture consists of two players, namely customer/company/user and service provider/vendor(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In the first step, service-providers plan software services according to the user demand .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Users’ hardware, software and cloud platforms are scrutinized to check the compatibility issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In the second step, service providers publish new software services and users are notified about the release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In the third step, software services are configured in users’ private clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Users get complete access to software services, such as manufacturing, supply chain, inventory, sales, thanks to Cloud ERP(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' User is kept in the loop with the service provider for feedback on future updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' THEORETICAL ANALYSIS We conduct theoretical analysis to justify first that too simple or too complex LSTM results as under-fitting or over- fitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, moderate number of layers and neurons leads to accurate prediction of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Hence, the selection of LSTM is amenable to TOPSIS ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS is able to yield accurate LSTM, classifier outcome with the intervention of weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Theorem 1: Cloud ERP adoption is amenable to deep recurrent neural network problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Proof:Cloud-ERP adoption is conceptualized through four- phase Enterprise System Experience Cycle which is how business value is created and transmitted in a successful enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='In the beginning, ERP project planning is conducted by terms such as finance, timeline Then, development and in- stallation of ERP Software is done in running business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Later, system testing and bug-fixing are conducted after employee feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Finally, upgrade of the system is performed with the adjusted software changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Each phase is dependent on outcome of previous phases, where many latent phases/issues are important and a cyclic process is maintained through out the life-span of enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In a typical artificial neural network, layers are generally connected through neurons and especially, in a deep recurrent neural network, each layer is a result of the joint operations of neurons from input and hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, Cloud-ERP adoption resembling enterprise system experience cycle is amenable to deep recurrent neural network problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 2: TOPSIS exploits the weight matrix for LSTM ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Proof: TOPIS with its multi criteria decision making abil- ity, selects criteria among alternatives based on trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' As such, each LSTM architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' criteria) has its own objective toward optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, each limitation is compensated for for its relative benefit of competitive criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' A relative ranking mechanism among LSTM architectures is performed with the intervention of a weight matrix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, weight of each LSTM architecture is managed through a weight matrix, by which LSTM architectures are ranked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 3: LSTM selection is amenable to TOPIS ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Proof: LSTM is implemented with a single layer or multi- ple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='LSTM with multiple layers often boost prediction performance by compromising the complexity, explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' On the contrary, LSTM with a single hidden layer is good for time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, the number of neurons in a single layer is a critical decision to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='LSTM with complex architecture and huge neurons causes overfitting in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' ‘Too small number of neurons’ causes a simple model with less accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus,selection of hidden layers and neurons is a complex decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='.TOPSIS is a multi-criteria decision mecha- nism for ranking classification algorithms such as LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Thus, selection of LSTM among alternatives is amenable to TOPSIS ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 4: LSTM with moderate number of neurons and hidden layers results in better prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Proof:Prediction accuracy is not desired level, when there exists a limited number of neurons in the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, as the neurons in the hidden layer increase, it is easier to learn the data and then gradually achieve desired prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Hence, a moderate number of neurons in the hidden layer is capable of achieving desired prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Private cloud Cloud ERP Service Management Policy User Iaas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Pass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Saas Hardware OS ERP Client Browser ERP APP Iaas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Pass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Hardware OS Client Browser ERP APP Platform OS Iaas Virtual Hardware Client Browser Cloud Infrastructure Cloud Platform Cloud Application Private cloud Management Policy Service Providers Manufacturing inventory finance purchase sales Supply chain (a) IAAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' PAAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' SAAS Conceptualization (b)Architecture and Functionality (c) Services Service Providers User Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud ERP’s IAAS/PAAS/SAAS Conceptualization, Reference Architecture and Services b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' DNN with input,hidden,output layers and neurons x1 x2 x3 y w2 w3 w1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Layers, neurons in ANN Chartering Shake- down Con- figuration Onward and Upward c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Enterprise Experience System Cycle Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' ANN,DNN Representation and Cloud-ERP Adoption with conceptualizing Enterprise System Experience Cycle Lemma 5: Excessive number of neurons in hidden layer degrades LSTM performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' LSTM Proof: In a regular classification, the similarity between input and output sequence results in a good prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, the performance degrades due to excessive prior and consequent inputs often with the presence of large numbers of neurons in hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Thus, large data input results in a functional LSTM , however, class-prediction performance degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 6: Too complex LSTM suffers from over-fitting problem: Proof:Neural network architecture with too many hidden lay- ers and/or neurons is time-consuming .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Hence, LSTM suffers from over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 7: Too Simple LSTM with relatively small hidden layers suffers from under-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Proof: Simple deep neural network architecture is unable to process required information to detect any pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, too simple LSTM architecture suffers from error-prone prediction and leads to under-fitting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Lemma 8: No rules-of-thumb exists for the selection of ’ hidden layer and/or neurons in LSTM’ Proof:: It is hard to relate ‘hidden layer and/or neurons’ and ‘input/output sequence’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, the degree of noise in the dataset makes significant deviation in the performance of hidden neurons/layers of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus, no rules of thumb exists for hidden layer and/or neurons in LSTM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS-based Cloud-ERP Adoption Factor Analysis We now describe our TOPSIS technique, which is used to rank/score classification outputs [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' adoption factors] pro- vided by LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We begin with the simplified explanation of general TOPSIS and then continue internal operations of TOPSIS-based Cloud ERP adoption factor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS is an effective and simplified algorithm, which belongs to the umbrella of MCDM mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Given a problem and multiple solutions, MCDM helps to find us the highest ranked solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS finds geometric distances of alternatives to find best-ranked alternatives, which is said to be closest to best ideal solution, where the low-ranked alternative indicates worst-solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS generally processes such all information/rank on a decision matrix through the life-span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Our mechanism involves the TOPSIS based MCDM tech- nique to retrieve the final Cloud ERP adoption factors for given inputs, where confidence scores are associated with base classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this way,alternative solutions are available for remaining classes, such that confidence scores are major representatives of base classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Thus an element is chosen from the decision matrix, where confidence scores are selec- tion criteria for alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Eventually, we proceed with all steps of TOPSIS on the decision matrix to finally reach out highest ranked alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' The process involves column normalization of the matrix in accordance with the TOPSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this case, we calculate the sum of squares of observed information and then the square root of sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Then the normalization is obtained by dividing the input score by the square root of the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' QUALITATIVE SURVEY In this section, we have conducted a qualitative survey among clients regarding CloudERP-adoption choices by con- sidering technology, innovation and resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have given importance on the cloud based ERP adoption by considering technology perspective, innovation charac- teristics and resistance influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, we have discovered the relationship and deviation among innovation, resistance and technology and their combined impact on cloud ERP adoption in industry scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have adopted a mixed method based research model in analyzing the most dominant factors in Cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, we begin with technology perspectives on the human mind while adopting a new technology, followed by experiments on welcoming behaviors toward creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='We begin with a holistic view on new technology use in terms of authority/organization and supportive/regulatory towards any new invention/technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, as cloud ERP is a new technology domain, and provides citizens with new practices/ideas, we are motivated to analyze human behavior regarding innovation characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, innovation is not always welcome by all societies, its adoption takes various ways in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, we are motivated to analyze Cloud ERP adoption by considering the reluctance of the human mind towards any new creative technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' With the intervention of the TOE framework, we assess the influence of three factors, namely technology, organization and environment in adopting Cloud ERP by organization employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have found that ICT knowledge, network infrastructure, high competition and support from management are fundamental factors in cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Meanwhile, business goals, technology integration skill and implementa- tion challenges are also found factors in some contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' By MIR, we assess how innovation goes ahead over some primary resistance prior to full adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Thus, the fu- ture acceptability is determined from the duration of resis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='However, the confusion on product usability or longevity works against innovation adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Similarly, the greater the complexity, the larger is the innovation resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' We have found among participants, that critical factors such as cyber security, vendor recognition and personalization are resisting cloud ERP adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' By DOI, we try to find factors which are responsible for selection of cloud ERP solutions over existing technolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Innovation is coined by a new idea or practice recognized by group or nationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Supremacy over existing technology and solutions is a major concern which helps clients to choose cloud ERP, an innovative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Plug and Play solution is found to be one of major factors in adoption of a new technol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' However, if the system is complex, people don’t like it unless a solution is presented alongside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Troubleshooting and user-experience are found as important factors for the complete adoption of Cloud ERP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' CONCLUSION Cloud-ERP is trending with the promise of cheap, easy and quick business-process management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' In this context, a deep recurrent neural network based classifier is proposed to infer its key adoption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Classifier uses LSTM first to classify features and then TOPSIS to rank identified fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Moreover, a qualitative survey in terms of technology, innovation and resistance is conducted to infer adoption factors and make hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Theoretical justification is also provided to show how cloud-ERP adoption is amenable to recurrent neural network problem and how TOPSIS and LSTM based classifier detects adoption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Cieciora, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Nagy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', Ol´ah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', Erdei, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', M´at´e, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' and Popp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' The role and impact of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='0 and the internet of things on the business strategy of the value chain—the case of Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Sustainability, 10(10), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='3491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Dataset Feature Extraction LSTM Classified Data Data Pre-processing Ranked Classified Data Weight Matrix TOPSIS output Normalize Decision Matrix Find Weighted Matrix Find Ideal Solution Separate Measure for alternative Close solution tp alternative Rank alternative solution input (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' LSTM +TOPSIS (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOPSIS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Classification Mechanism Cloud Infra[E] Cloud Skill[T] TOE Plug and Play Cyber Security User Experience MIR Cloud ERP Ease of Use Supremacy Trouble shooting DOI Top Management [O] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' TOE, DOI, MIR-based-Mixed Method Qualitative Study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Pfeifer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Development of a Smart Manufactur- ing Execution System Architecture for SMEs: A Czech Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Sustainability, 13(18), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='10181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Lisowska, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' and Pamula, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud comput- ing adoption in small and medium-sized enterprises in Poland–benefit analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Global Journal of Information Tech- nology: Emerging Technologies, 10(2), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='98-105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='Morawiec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' and Sołtysik-Piorunkiewicz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Cloud Computing, Big Data, and Blockchain Technology Adoption in ERP Implementation Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Sustainability, 14(7), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='3714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='ˇSpatenka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' and Koch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Sustainable Develop- ment of Companies Using the ERP System as a Fundamental Tool of Digital Transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content=' Trends Economics and Man- agement, 15(38), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} +page_content='61-70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQfy_k5/content/2301.00693v1.pdf'} diff --git a/otFPT4oBgHgl3EQf6zWU/vector_store/index.faiss b/otFPT4oBgHgl3EQf6zWU/vector_store/index.faiss new file mode 100644 index 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b/p9AzT4oBgHgl3EQfq_2l/content/tmp_files/2301.01638v1.pdf.txt @@ -0,0 +1,1059 @@ +Paper submitted to the Journal of Fluids Engineering – Special Issue in Honor of Prof. Kirti Ghia + +New Property Averaging Scheme for Volume of Fluid Method for Two-Phase Flows +with Large Viscosity Ratios + +Sucharitha Rajendran (ASME Member) +Thermal-Fluids and Thermal Processing Laboratory, +Department of Mechanical and Materials Engineering, +University of Cincinnati, Cincinnati, OH 45221 + +Raj M. Manglik (ASME Fellow) +Thermal-Fluids and Thermal Processing Laboratory, +Department of Mechanical and Materials Engineering, +University of Cincinnati, Cincinnati, OH 45221 + +Milind A. Jog1 (ASME Fellow) +Thermal-Fluids and Thermal Processing Laboratory, +Department of Mechanical and Materials Engineering, +University of Cincinnati, Cincinnati, OH 45221 + + + +1 Corresponding author. Email: Milind.Jog@uc.edu + +2 + +Abstract +To predict liquid-gas two-phase flow phenomena, accurate tracking and prediction of the evolving +liquid-gas interface is required. Volume-of-Fluid or VoF method has been used in the literature +for computationally modeling of such flows. In the VoF method, a single set of governing +equations are solved for both phases along with an advection equation for the volume fraction. The +properties in each computational cell are determined by a linear weighted average of the properties +of the two fluids based on the phase fraction. While the method predicts water-air flows well, the +predictions tend to deviate significantly from experimental data for liquids with high viscosity. A +new property averaging technique is proposed in this paper, which is shown to provide accurate +results for high viscosity liquids. Computational predictions using the open source VoF solver +interFoam (available as a part of the OpenFOAM computational tool), and those obtained using +the proposed method are compared with experimental data for multiple two-phase applications. +Four different problems, viz., suspended droplet in air, jet breakup, drop impact on thin films, and +air entrapment during drop interaction with liquid pool, are considered to extensively validate the +new method. Experimental data for water and aqueous solutions of propylene and ethylene glycol +are used to cover a range of surface tension (72 – 36 mN/m) and viscosities (1 – 40 mPa.s). For +all cases, the modified VoF solver is observed to perform significantly better than original VoF +method. It reduces any spurious currents in simulations of drop suspended in air. For the cases of +drop impacting on a pool and during drop generation from liquid jets, the time progression of the +surface tension governed dynamics is improved from the slower estimate of interFoam solver. In +the case of drop impacting on a thin liquid film, where it is critical to capture the intricate interplay +between the surface tension and viscous force, the former effect of surface tension is exaggerated +in the original method but correctly captured with the proposed new method. + +3 + +Keywords: Numerical modeling, VoF method, two-phase flow, interfacial phenomena +Introduction +Our colleague of many years, Professor Kirti “Karman” Ghia, was devoted to the +advancement of fundamentals and applications of computational fluid dynamics. An example is +his early work on the lid driven cavity flow [1] that made a seminal contribution to multigrid +methods and became a benchmark for work that followed (over 5000 citations to date!). We +enjoyed many technical discussions with him. This work is a tribute to Prof. Ghia, and it addresses +a limitation of the Volume-of-Fluid (VoF) method in accurately predicting two-phase flows when +the viscosity ratio of the two fluids is large. +Two-phase processes with jets, droplets, and bubbles such as droplet impact on dry surfaces, +or on thin films, or on liquid pools, jet breakup and drop formation, and bubble growth from an +orifice submerged in liquid, are encountered in a variety of industrial applications [2]. Numerical +modeling of such phenomena has steadily advanced, and along with experiments, it has provided +insights into the underlying physics of the processes. For example, Berberovic et al. [3] used +numerical techniques along with experimental observations to understand the dynamics of drop +impact on liquid surfaces and have characterized formation and propagation of the surface wave +during impact. Extensive investigation of dynamics of confined bubbles in laminar micro channel +flow by Khodaparast et al. [4] relies on experimental and numerical analysis of two-phase flow. +Similarly, research in jets and atomization characterization [5–7] has advanced with the help of +numerical techniques. In addition to flow characterization, coupled heat transfer solution has been +implemented by Trujillo et al. [8] to study drop impact heat transfer. Sanjivan et al. [9] have +incorporated interfacial surfactant adsorption/desorption kinetics in their computational + +4 + +simulations of bubble growth in aqueous surfactant solutions. These and many such other studies +have used numerical methods for predicting multiphase (or two-phase) flow behavior. +For accurate predictions of the two-phase phenomena encountered in applications associated +with liquid breakup, atomization, liquid collisions and entrainment, precise capture of the liquid +and gas interface is essential. Interfacial tracking methods in incompressible fluids include front +tracking [10–12] , boundary integral [13], volume of fluid (VoF) [14] and level set (LS) [15,16] +methods. In the past decade, the techniques that have received the most attention are the last two +– VoF and LS – due to their robustness and ease of implementation. As the interface evolves in a +fixed computational mesh, both these methods use a separate parameter to identify the different +phases in each cell in the domain. The primary difference in these methods is their treatment of +this phase parameter. In the VoF method, an advection equation governs the fraction of volume of +the primary phase as the interface moves in time and space. This makes the interface discontinuous +and therefore, reconstruction of the calculated interface is required to obtain a smooth curve for +curvature and normal estimations at each interface location. The interface is typically smeared over +a few cells owing to the change in volume fraction as it goes from one phase to the other. In +contrast, the LS method defines the interface by a smooth continuous function going from positive +in one fluid to negative in the other and having a value of zero at the interface. This eliminates the +requirement for reconstruction and creates an interface solution that is sharp unlike in the VoF +method. However, though accurate estimations of the interface topology are achievable, the LS +method does not inherently conserve mass by the advection step unlike the VoF method. The VoF +method has been implemented in many commercial solvers as it inherently conserves mass. This +is the method under consideration for the current study. As interface tracking is very sensitive to +the reconstruction, VoF technique was seen to yield acceptable results only for certain applications + +5 + +and working liquids and some efforts have been taken to improve the interface tracking for this +method. This is discussed later in the next section. +Prior numerical studies of jets and droplets that use VoF techniques have primarily looked at +fluids with viscosities similar to water. They have provided useful results with water, hydrocarbon +fuels, and refrigerants – liquids with viscosities close to or less than that of water. However, liquids +used in chemical, pharmaceutical and process industries can have significantly larger viscosities. +In such cases, with large differences in the viscosities of the two fluids, VoF method was found to +produce errors in numerical solution. The error in numerical prediction also increases with higher +viscosity working fluid. A few studies have coupled the LS and VoF techniques to improve +interface tracking while ensuring mass conservation in the domain [17,18]. The improved accuracy +of combined LS and VoF (CLSVOF) method can be observed in the work of Ray et al. (2015) [ ] +where the problem of liquid drop impact has been modeled. Some other studies have focused on +improving dynamic meshing methods along with the VoF implementation to refine the numerical +calculations at the interface [7,19]. The current study provides a modification to the VoF method +so that accurate predictions can be made for highly viscous liquids. Extensive validation of the +suggested modification to the VoF method is conducted with available experimental data for four +different physical processes. The VoF method used for comparison in this study is based on the +open source two-phase solver present in OpenFOAM®. This package is well suited to handle +complex geometries and is easily parallelizable without the limits of licensing. It is based on C++, +where equations can be in a form that has a close resemblance to its mathematical equivalent. This +VoF interface tracking solver was first implemented in 1999 by Ubbink & Issa [20]. A detailed +summary of the original solver has been presented by Deshpande et al. [21]. Modification to the +interface reconstruction is implemented to ensure better predictions for viscous working liquids. + +6 + + +Numerical Procedure +The numerical procedure to solve the two-phase incompressible flow problems in this +study, and as enabled in the open-source code OpenFOAM 2.2.1 is discussed here. The +implemented solver available for this is interFoam, which, as noted by Deshpande et al. [21], has +been reliably employed to solve a number of problems. However, the accuracy of this original +solver was noticed to drop when the viscosity of the working fluid is high. This VoF solver uses +finite volume discretization of the governing equations on a fixed grid in the domain. All the flow +variables are stored in the cell center in the associated finite volume technique. A modified method +is proposed herein and its efficacy and relatively more precise validity are demonstrated and +highlighted. + +Governing Equations +The numerical technique adopted in this study considers two immiscible fluids (a gas and +a liquid) that are separated by a sharp interface. However, the two fluids are not solved separately +with their different properties. Instead, a single set of equations are solved for the entire fluid +region with properties differing continuously from one fluid to the other. The two fluid media are +distinguished by a phase fraction property that has values between 0 and 1, where 1 is identified +as liquid and 0 as gas. The interface is defined by the region where the value of this phase fraction +transitions from 0 to 1. Thus, the interface is smeared in such an evaluation and is not sharp. The +definition for this phase fraction (α) is as follows: + +7 + +𝛼 = Volume of primary phase +Total control volume + +where, 𝛼 = { +1 +0 < 𝛼 < 1 +0 + +in primary fluid (liquid) +in transition region (interface) +in secondary fluid (air) + +(1) +Using this phase fraction identifier for the fluids the respective thermo-physical properties are +determined in each computation cell as follows: +𝜑 = 𝛼 𝜑1 + (1 − 𝛼)𝜑2 +(2) +where, 𝜑 is the property of the fluid and 𝛼 is the phase fraction in the cell and subscripts 1 and 2 +represent liquid and air, respectively. Equation (2) therefore allows the properties to be calculated +in each cell at each time instant in the computations of the evolving interface. The transport of this +newly defined fluid bulk property (phase fraction, α) is further defined by an advection equation, +𝜕𝛼 +𝜕𝑡 + 𝛻. (v̅𝛼) = 0 +(3) +Additionally, the finite volume solver uses the continuity and momentum equations to +computationally obtain the flow field. In this study, an incompressible flow solution is of interest +and hence the energy equation is not added. +Continuity: +∇. (v̅) = 0 +(4) +Momentum Conservation: + ∂(𝜌v̅) +∂t ++ ∇. (𝜌v̅ v̅) = −∇𝑝 + ∇. T̅ + 𝑓𝑠 + 𝜌g̅ + +(5) + +8 + +where 𝑇̅ is the viscous stress tensor and 𝑓𝑠 is the volumetric force due to surface tension. The +properties such as density and viscosity in these two equations are evaluated based on the phase +fraction (𝛼). +Equations (3) – (4) are solved numerically by adopting a control volume discretization. In +this process, the surface tension force term in the momentum equation calls for special +consideration as it is not inherently a volumetric term like the other components in the equation. +One way to resolve this would be to use this force as a boundary condition on the free surface and +the surface pressure is obtained by linear interpolation between the surface pressure required and +the fluid pressure inside the interface. This requires multiple iterations to obtain surface pressures +within some tolerance interval with respect to that at previous time step. After the correction of +the surface pressure, the momentum and continuity equations are re-solved before advancing to +the next time step. This was the original VoF method suggested by Hirt and Nichols in 1981 [14]. +In addition to higher computational time due to the iterative surface pressure correction, this +technique required approximate prior knowledge of the interface shape that was to be obtained +from upstream and downstream cells. +Brackbill et al. [22] in 1992, proposed a means to resolve the problems associated with this +surface treatment by converting the surface tension forces to an equivalent volume force that can +be added to the Navier-stokes equation as an additional body force term. Their method called +continuum surface force (CSF) calculates this equivalent volumetric surface tension force as: +𝑓𝑠 = 𝜎 𝜅 𝒏 𝛿𝑠 +(6) +where, σ is the surface tension coefficient, κ is the interface curvature, 𝐧 is the surface normal and +δs is a Dirac delta function that assists with concentrating this calculation on the interface. In this + +9 + +model, the interface curvature κ is determined as a function of local gradients of the surface +normal, 𝐧 which in turn is a function of the phase fraction (α) given by + 𝜅 = 𝛻. 𝒏̂ and 𝒏 = 𝛻. 𝛼 +(7) +Tang and Wrobel [23] have shown how this surface tension model can be written in terms of +pressure drop across the interface and expressed in terms of a volume force in the momentum +equation. They normalize the interface curvature by using the volume averaged density (ρ) to +obtain the following for the surface tension force: +𝑓𝑠 = 𝜎 𝜅 +𝜌 +0.5(𝜌1 + 𝜌2) 𝛻. 𝛼 +(8) +It must be noted that the numerical solution would depend on the estimation of the interface +normal, 𝐧̂ , which in turn, is a function of the phase fraction. Thus, correct estimation of these +factors determines the accuracy and performance of the numerical solution. + +Modifications to the VoF Solver +One significant cause for incorrect estimation of interface curvature and normal stems from +deviations in estimations of fluid properties across the smeared interface. Inaccurate interface +calculations, in turn, would result in unreliable surface tension force estimation and thus cause +failure in predicting correct interface pressure gradients. To resolve this difficulty in interface +tracking, some past studies have either looked at correcting the Courant number estimation, such +as the case in Beerners et al.[24], or have proposed alternative methods to estimate interface +curvature and therefore surface tension force [25], while still others have looked at better meshing +techniques to resolve the computational domain in a dynamic manner [26]. Other notable + +10 + +improvements to accurately predict surface tension-dominated flows are provided by Puckett et al. +[ ], Popinet & Zelenski [ ], and Gerlach et al. [ ]. In our preliminary analysis of the complication +in correct interface estimation, it was found that the relative viscosity of the two phases in question +was an underlying source of inaccuracy. Thus, an attempt to correct this is made by proposing a +modification to the interface property averaging technique. +Multiple experimental measurements have been used to test a generic exponential property +averaging equation that is defined as: +𝜑 = 𝜑2 + (𝜑1 − 𝜑2)𝛼𝐂 +(9) +The specific experimental case employed for this analysis is the dynamic development of the +crown on impact of a liquid drop on a thin film of the same liquid [33]. This case was +computationally modeled as well, and numerical solutions duplicating the experimental results +obtained with several different working liquid properties (Table 1) are used to identify the correct +expression for the constant C in the exponential property averaging of Eq. (9). The results obtained +are then correlated with their relative viscosity ratios and the consequent results are graphed in +Figure 1. Thus, a property estimation equation that depends on the viscosity of the two working +fluids (1 and 2) is determined and is given by + +𝜑 = 𝜑2 + (𝜑1 − 𝜑2)𝛼𝟏.𝟓(𝝂𝟏 𝝂𝟐 +⁄ +)+𝟎.𝟕𝟓 +where, 𝜈 is the kinematic viscosity of the fluid in consideration. + +(10) + + + + +11 + +Table 1: Properties of the Newtonian liquids used to determine the exponent C in Eqn.(9) +Liquids +Density +(kg/m3) +Viscosity +(Pa.s) +Surface Tension +(N/m) +Water +998.0 +0.0010 +0.0728 +25% by volume of +Propylene Glycol +(25% PG/ 75%Water) +1007.5 +0.00255 +0.0541 +50% by volume of +Propylene Glycol +(50% PG/ 50% Water) +1017.0 +0.0050 +0.0452 +75% by volume of +Propylene Glycol +(75% PG/ 25% Water) +1026.5 +0.0120 +0.0411 +Ethylene Glycol (EG) +1113.2 +0.0161 +0.0484 + +Though the proposed modification to the VoF method, expressed in Eqs. (9) and (10), was +derived based on an experimental study of drop impact on thin films, its veracity and general +applicability has been tested with other two-phase flow problems as discussed later. An example +of the significance of property estimation equation is shown in Figure 2. The plot shows the +variation in interface density in the x-axis, between air ( = 1.225 kg/m3) and a viscous liquid +(ethylene glycol,  = 1113.2 kg/m3). As the phase fraction (α) in the x-axis goes from 0 to 1, the +interface moves from air to that of the viscous liquid. As seen, from the two plots, the change in +density with the exponential averaging equation (red line) is not monotonically linear as is the case +with arithmetic averaging (green line). This ensures a sharper interface that is not smeared over + +12 + +multiple cells, thereby providing better estimates of the interface curvature and therefore, surface +tension force and interface pressure drop. + + +Figure 1: Equation for the determination of constant C in the property averaging expression I +Eq. (9), as a function of relative viscosity ratios and as obtained from the mesurements with +different air-liquid systems [note: EG – ethylene glycol, and x% PG – x% by volume propylene +glcycol and water solution]. + +𝜈1 𝜈2 +Τ += 𝜈𝐿𝑖𝑞𝑢𝑖𝑑 𝜈𝐴𝑖𝑟 +Τ + +where, +x % PG: x % by volume of +propylene glycol and water +mixtures +EG: ethylene glycol + +2.5 +EG +2 +75% PG +1.5x40.75 +1.5 +c +50% PG +1 +25%.PG +Water. +0.5 +0 +0 +0.5 +113 + + +Figure 2: Comparison of the variation of fluid property (density) at the interface given by the +arithmetic weighted averaging (green line) method of Eq. (2) and that by the new proposed +method (red line) given by Eq. (10). + +In addition to altering the property estimation, the curvature field is smoothened in +accordance with the Ubbink et al. [20] technique where the phase fraction smoothness is corrected +to influence the smoothness of the interface curvature, as the two are related by Eq. (7). This +involves averaging the phase fraction in each computational cell volume by area averaging with +respect to the values at each face as expressed in the following: +𝛼[𝑐𝑒𝑙𝑙] = ∑ 𝛼[𝑓𝑎𝑐𝑒] 𝐴𝑟𝑒𝑎[𝑓𝑎𝑐𝑒] +∑ 𝐴𝑟𝑒𝑎[𝑓𝑎𝑐𝑒] + +(11) +Thus, the final set of equations that are solved in the proposed new modified VoF method are the +continuity, momentum and phase transport equations along with the surface tension force +Phase fraction, 𝜶 +Density (kg/m3) + +1500 +........... +1000 +.... +..... +500 +....................... +...,.....,.....,.... +........................ +........... +......... +0 +0 +0.2 +0.4 +0.6 +0.814 + +determined by Eq. (8). The properties in this one fluid numerical approach (VoF) are estimated as +per Eq. (10), and the interface curvature is further smoothened by area averaged smoothing of the +phase fraction as given by Eq. (11). + +Solution Procedure +The equations described in the previous section are discretized by using upwind scheme +for the spatial terms and solved iteratively on the computational grid. From the computed variables +the average values are stored in the cell center. Reconstruction from these averaged values is +required to evaluate the cell interface values which need to be used for the subsequent iteration in +the solution process. Limited linear piecewise reconstruction is used to diminish large variations +during this reconstruction. The PIMPLE algorithm (combination of PISO and SIMPLE) is used as +the iterative procedure for coupling mass and momentum conservation. For this coupling, within +each time solution, this algorithm solves the pressure equation while invoking an explicit +correction to velocity. Optionally, each iteration step can begin with a solution to the momentum +equation. This is called the momentum predictor loop and is set to loop twice for each step. The +linear solvers used in the PIMPLE method are preconditioned Conjugate Gradient for pressure +and a preconditioned Bi-Conjugate Gradient for velocity components. +Because a transient solution is the goal of these numerical solutions, an implicit Euler time +discretization is used. This first order implicit scheme is preferred to an explicit scheme as this +guarantees boundedness and is unconditionally stable. For providing good numerical stability, +adaptive time stepping is used where at the beginning of each time loop, the time step Δt is +calculated dynamically using the following criteria: + +15 + +∆𝑡 = min {Comax +Coo ∆𝑡𝑜; (1 + 𝜆1 +Comax +Coo ) ∆𝑡𝑜; 𝜆2∆𝑡𝑜; ∆𝑡𝑚𝑎𝑥 } +(12) +In the above criteria, ∆𝑡𝑚𝑎𝑥and Comax are predefined values for maximum permissible time step +and maximum Courant number. The damping factors, λ1 and, λ2, are defined as 0.1 and 1.2 +respectively. The superscript o refers to values at the previous time-step. The maximum Courant +number is set to be 0.1 for the current numerical study. + +Numerical domain and mesh discretization +The numerical problems used in this study to test and establish the general applicability of +the modified VoF for two-phase flows are all considered to be axisymmetric. This assumption is +based on experimental evidence and helps reduce the computational domain and therefore makes +the numerical solution faster. The entire physical domain of the problem can be considered as +cylindrical with the axis representing the axis of symmetry. In OpenFOAM, the computational +domain used for these problems is a wedge-shaped thin sliver (with a small angle, < 5o) of this +cylinder. This wedge straddles the X-Y plane and runs along the central axis of symmetry as shown +in Figure 3 (a). The typical orthogonal structured non-uniform mesh for this domain is shown in +Figure 3 (b). The flow phenomena under examination are typically around the axis of symmetry +in all the problems considered. Therefore, the mesh is denser in this region than farther away. +Likewise, the mesh is compact near the target surface (for specific cases discussed where the fluid +interacts with a target surface), to ensure accurate capture of the phenomena in these regions. Prior +to obtaining the presented numerical results, the density of the grid is varied until a grid +independent solution is obtained. For drop impact in a liquid pool, penetration depth was the +parameter used to compare different meshes for interFoam method for 5cSt liquid. In the liquid jet + +16 + +breakup in stagnant ambience, breakup length of the continuous jet was the parameter used to +compare different meshes for InterFoam method. The same mesh density was used for other cases +studied. In modeling drop impact on thin liquid film, crown diameter was the parameter used to +compare different mesh sizes for InterFoam to obtain grid-independent solutions. + +Figure 3: (a) the wedge-shaped numerical domain (b) typical mesh structure +Performance tests on modified VoF method +All implicit multi-fluid numerical methods rely on interface capturing techniques that are +known to be susceptible to numerical instabilities and can result in unrealistic interface flows. +When the flow phenomena under consideration are dominated by inertia, these numerically +generated interface flows do not lead to unrealistic results. However, when capillary effects +(surface tension forces) are dominant, any errors in calculating interfacial flows can give rise to +inaccurate solutions. To compare and test the efficiency of the two numerical VoF solvers, namely, +Y +X +Nozzle Wall +Axis of Symmetry +Atmospheric Outlet +Axis of symmetry +Y +X +Axis of symmetry +(a) +(b) + +17 + +interFoam and the modified solver, four tests problems are solved and the results obtained are +outlined in the next section. These include the canonical suspended drop problem and three other +cases typically encountered in engineering applications (jet breakup, drop impact on a thin film, +and drop impact on a liquid pool). In all these cases, surface tension forces are significant and that +helps identify the difference in the accuracy of the two methods. + +Stationary drop in stagnant zero-gravity ambience +The native VoF scheme (in interFoam solver), yields an interface curvature calculation +with large gradients, thus contributing to a relatively larger spread of the interface. Consequently, +the error in curvature results in an imbalance between the pressure and surface tension forces that +lead to formation of spurious currents in the domain. The proposed modified VoF solver aims at +creating a sharper interface and thereby improves the curvature estimation. To evaluate the +influence of the modified treatment of interface, a stationary axi-symmetric drop in absence of +gravity in a medium of air is studied. The initial and boundary conditions for this numerical +analysis are shown in Figure 4. A 4-mm diameter liquid drop with surface tension of 72.8 mN/m +and density 1000 kg/m3 is placed in stagnant air. The domain has a uniform grid distribution with +a grid spacing of 0.001 mm. Two different drop viscosities (0.001 Pa.s and 0.01 Pa.s) are tested to +see the influence of viscosity on the numerical generated current. These currents, called parasitic +currents appear as vortices around the interface and, as shown in Figure 5, are present when there +are no external forces acting in the physical domain. This numerical error is not eliminated by grid +refinement. In fact, as noted by Brackbill [22], the magnitude of these spurious currents could get +amplified by a finer grid. Therefore, for unbiased comparison between the two methods, the same +grid size and time stepping were used to check magnitudes of parasitic currents. + +18 + +Figure 5 shows the spurious currents around a stationary drop in air in the absence of +gravity. The observed currents for both interFoam and the modified VoF solver is shown for water +and a more viscous (10 times that of water) liquid drop. It is observed increasing viscosity of the +liquid, increases the intensity of the parasitic currents around interface of the stationary drop. The +results shown in Fig. 5 are at 200 ms. The typical velocities in interFOAM are ~ 0.1 m/s while +those with the modified solver at a tenth of that. The modified VoF solver, does not eliminate these +currents, but the intensity of the same is noted to be greatly reduced. Thus, it is noted that a more +accurate sharper approximation of the interface in the modified VoF solver can help reduce the +spurious currents in the two-phase problem. To estimate the influence of these numerically induced +currents on the domain, the kinetic energy around the drop interface is presented in Figure 6. As +is observed, the kinetic energy at the interface of this viscous drop in the numerical solution +obtained by the modified VoF solver is an order of magnitude lower than that given by the +interFoam solver. + + +Initial drop with zero velocity +and ddrop diameter +Atmosphere +Target Surface +Axis +Outlet +zero gradient velocity, zero +pressure and zero phase +fraction +zero velocity, zero gradient +pressure and zero gradient phase +fraction +zero gradient velocity, +zero gradient pressure and +zero gradient phase +fraction +Zero gravity +Y +X + +19 + +Figure 4: Initial and boundary conditions for numerical solution of stationary drop under no +external forces. + +Figure 5: Comparison of the two VoF solvers: spurious currents around the interface of a +stationary drop in ambient air in the absence of gravity at simulation time of 200 ms. +interFoam VoF solver +Modified VoF solver +𝜈𝑙𝑖𝑞𝑢𝑖𝑑 +𝜈𝑎𝑖𝑟 += 0.07 +(water) +𝜈𝑙𝑖𝑞𝑢𝑖𝑑 +𝜈𝑎𝑖𝑟 += 0.7 + +20 + + +Figure 6: Comparing kinetic energy around the viscous ( +𝜈𝑙𝑖𝑞𝑢𝑖𝑑 +𝜈𝑎𝑖𝑟 = 0.7) stationary drop interface +for interFoam and the modified VoF solver + +Drop impact in a liquid pool +For the second test, experimental findings of Tran et al. [27] on the impact of a viscous +drop on a deep liquid pool with attention to the entrapped air layer is considered. The initial +temporal dynamics is studied for these impact conditions with the different viscous liquids used in +the experimental work. At low impacting velocities, the air layer underneath the impacting drop +prevents merger of the drop and pool surface and the drop could bounce off [28].Increasing the +impact velocity was observed to aid entrapment of the air layer. The centerline depth as the drop +merges with the pool (penetration depth) was studied as a function of time taken for rupture and +was found to be linearly increasing with time. In addition, viscosity was noted to delay rupture +Kinetic Energy (kg.m +2/s +2) +interFoam VoF solver +Modified VoF solver + +3.0E-061.3E-052.3E-053.3E-054.3E-055.3E-056.3E-057.3E-058.3E-059.3E-0521 + +time as higher the viscosity, more the resistance to allow entrapped air to rise and rupture the +interface. These experimental findings are presented in Figure 7. + + +Figure 7: Experimental data showing dimensionless penetration depth (L/R) as a function of +time of impact of drops with same impacting velocity (0.55 m/s) for 3 different viscosities [27] + + +Figure 8: Initial and boundary conditions for numerical solution of drop impact on a deep pool. +5 cSt +10 cSt +20 cSt +Initial drop with velocity +vimpact and diameter ddrop +Atmosphere +Target Surface +Axis +Outlet +zero gradient velocity, zero +pressure and zero phase +fraction +zero velocity, zero gradient +pressure and zero gradient phase +fraction +zero gradient velocity, +zero gradient pressure and +zero gradient phase +fraction + gravity +Deep Pool +Y +X + +1.2 +- +- +1.0 +- +0.8 +L/R 0.6 +t0 +0.2 +0 +-4 +-3 +-2 +-1 +0 +t (ms)22 + + +Figure 9: Performance of interFoam and modified solver against experimental findings [27] for +entrapped air rupture time during viscous drop impact on a deep pool + +To test the effectiveness of the modified VoF solver against the data, a drop impact on a +pool in similar conditions is numerically simulated using both solvers (interFoam and modified +VoF). The initial and boundary conditions for this problem are shown in the schematic in Figure +8. A drop with diameter 1.9 mm and velocity of 0.55 m/s as provided by Tran et al. [27] impacts +a pool of like liquid (10 mm deep) as shown in Figure 8. The mesh for this domain under +consideration contains approximately 390000 cells. While analyzing the effectiveness of the two +solvers, it is observed that interFoam predicts a much slower evolution of air entrapment and +rupture than the modified solver. The findings from both the solvers are compared with +5 cSt +10 cSt +20 cSt + +1.5 +L/R +0.5 +-5 +.4 +-3 +-2 +-1 +0 +t (ms)1.5 +R +0.5 +0 +-5 +-4 +3 +-2 +t (ms)1.5 +L/R +0.5 +0 +.5 +-4 +-3 +2 +-1 +0 +t (ms)Data - Tran et al. (2013) +interFoam Solver +Modified Solver23 + +experimental data provided by Tran et al. [27] in Figure 9. For all the three viscosities tested, the +modified VoF solver is better at predicating the entrapped air dynamics. At lower viscosities (5 +cSt), the modified solver is more precise at predicting the development of penetration depth, than +at higher viscosities (20 cSt) compared to experimental measurements. For all the three viscosities, +penetration depth at rupture (τ = 0) are comparable with experimental data. Comparisons of +dimensionless penetration depth for the three liquids for the two solvers is shown in Table 2. + +Table 2: Comparing dimensionless penetration depths for the three cases in consideration +against interFoam and the modified VoF solvers. +Liquid’s +viscosity +(cSt) +L/R +expt. +L/R +interFoam +L/R +Modified +% +difference +(interFoam) +% +difference +(Modified) +5 +0.54 +1.43 +0.68 +165 +26 +10 +0.77 +1.75 +0.92 +127 +19 +20 +1.16 +1.55 +1.22 +34 +5 + + +While the development time for the rupture is very comparable for 5cSt and 10 cSt drops, +for the 20 cSt drop impact, total time for rupture is shorter than experimentally measured. As +observed in the previous case study with numerically induced spurious currents in stationary drops, +higher viscosity gives rise to stronger parasitic currents. Thus, with increasing viscosity, there is +relatively more error in predicting the pressure and surface tension balance and thus the interface. +It is speculated that this causes the numerical analysis to be more accurate when viscosities are + +24 + +lower. However, comparing with the experimental data, the deviation is significantly less than the +original VoF solver. + +Liquid jet breakup in stagnant ambience +Precise numerical characterization of capillary driven phenomena like liquid jet breakup requires +correct estimation of surface tension and pressure forces on the interface. This becomes more +significant when low velocity liquid jets are taken into consideration. A circular jet breaking up in +stagnant ambience is governed by the interplay of inertial, gravitational, aerodynamic, capillary, +and viscous forces. When the jet is fast moving, though surface tension (capillary force) is a +prominent cause for creation and propagation of surface instabilities, inertial and aerodynamic +forces are most dominant in the interplay [29–31]. This balance however changes when the jet is +slow moving (smaller, lower weber numbers). +To investigate the accuracy of the two solvers in capturing the temporal growth of the +surface instabilities, experimental data from our study of Newtonian low velocity viscous liquid +jet breakup is used [32]. A propylene glycol jet of initial diameter 1.499 mm emerges from a +circular nozzle at We = 5 and disintegrates to generate drops in stagnant ambience under the +presence of gravity. Experimental observations show the phenomena to be symmetric about the +axis and hence an axisymmetric wedge-shaped mesh with non-uniform structured mesh with about +160000 cells is used for this study. The initial and boundary conditions for this problem are shown +in Figure 10. + +25 + + +Figure 10: Initial and boundary conditions for numerical solution of jet breakup. + +The nozzle wall has a no-slip boundary condition imposed on it. To emulate the +experimental test conditions, the inlet has a uniform velocity inlet condition defined based on the +prescribed inlet velocity from the syringe pump in the experimental study [32, 33]. At the outlets, +the pressure is assumed to be atmospheric and zero-gauge pressure is the prescribed boundary +condition. Breakup length calculations and other analyses of results are carried out after 0.2 s of +real time. This provides time for numerical instabilities, if any, to deteriorate. +Figure 11 show the time progression of surface instability on the propylene glycol jet +considered. High-speed experimental images are compared with predictions from the numerical +solvers tested. It is observed that while the original solver (interFoam) is unable to capture the +generation and propagation of surface instabilities unlike the modified VoF solver for high +viscosity liquid jet. For water, this is however not the case and both solvers agree with the +experimental observation. As propylene glycol is 40 times as viscous as water, and the VoF solver +Initial jet with velocity +vjet and diameter djet +Nozzle +inlet +Atmosphere +Axis +Outlet +zero gradient velocity, zero +pressure and zero phase +fraction +zero gradient velocity, zero gradient pressure +and zero gradient phase fraction +gravity +Atmosphere +zero gradient velocity, zero +pressure and zero phase +fraction +Y +X + +26 + +predicts a much longer unperturbed jet surface compared to experimental observations. The +modification on the property averaging helps improve the interface calculation and the modified +solver can be seen to provide results that agree well with experimental data. With increasing +viscosity of the external fluid, jet breakup exhibits different behavior (see for example, Borthakur +et al. (2017) [ ]). The proposed method will be useful in addressing these situations as well. + + +Figure 11: Comparison between (a) experimental [32, 33], and numerical predictions of (b) +interFoam and (c) modified VoF for evolution of surface instabilities during circular jet breakup +of a propylene glycol jet of 1.499 mm diameter and We = 5. The time difference between these +two images is 2.5 ms. + + +(a) +(b) +(c) +(a) +(b) +(c) + +27 + + +Drop impact on thin liquid film +During drop impact on a thin liquid film of the same liquid, viscosity and surface tension +affect the spread on the thin film and the crown growth. Understanding the correct interplay of +these two forces can give insights into the growth and spread of the impacting drop as well as help +enhance or curb splashing as per the requirement of the application such as coating and pesticide +dispersions. A wedge-shaped axisymmetric domain (Figure 3 (a)) with structured mesh is used for +numerical analysis of this problem. In the stagnant ambience, the drop of liquid with known impact +velocity and diameter (from experimental observations) are initialized along with the stagnant thin +film height and a schematic of the same is shown in Figure 12. Figure 13 shows the time +progressions of an ethylene glycol drop of diameter 4.56 mm impacting at a velocity of 1.89 m/s +on a thin film. Successive images are at a gap of 2 ms and the temporal progression of the impact +dynamics as predicted by the two solvers are compared with the experimental observations +reported in [33]. While the initial spread of the drop on the film surface looks similar (at 2 ms and +4 ms), the crown tips are noticed to be thinner in the numerical solution using interFoam. This +thinner crown tip leads to a faster moving crown rim that predicts the pinch-off secondary drop +from the crown rim at 6, 8 and 10 ms after the impact. The modified VoF solver is able to predict +the correct temporal and spatial dynamics of the drop liquid interaction as is seen in Figure 13. +Comparison of the spatial dynamics is done by measurement of the crown diameter as the drop +merges with the thin film (at 10 ms in Figure 13). The maximum difference between the solver +predictions and that from the experiment for different Newtonian viscous liquids are compared in +Table 3. The properties of the working liquid and the impacting drop are also provided in the table. +The predictions using the modified solver are in agreement with both the final crown diameter as + +28 + +well as with the temporal development of the crown growth as shown in Figure 13. We note that +the regimes of drop impact dynamics discussed here, the experimental observations showcase a +very symmetric development of the crown and breakup. This is the reason to consider +axisymmetric 3-D simulation. This is similar to other studies in the literature studying similar drop +impact dynamics (see for example, Ref. [3]). + +Figure 12: Initial and boundary conditions for numerical solution of drop impact dynamics. + +Initial drop with velocity vimpact +and diameter ddrop +Atmosphere +Target Surface +Axis +Outlet +zero gradient velocity, zero +pressure and zero phase fraction +zero velocity, zero gradient pressure +and zero gradient phase fraction +zero gradient velocity, zero +gradient pressure and zero +gradient phase fraction +Initial thin film with height +H +Y +X +gravity + +29 + + +Figure 13: Comparison of numerical prediction from the two solvers with experimental time +progression of drop impact dynamics (Successive image are 2 ms apart) +Table 3: Percentage difference between the experimental and numerical crown diameter +Liquid +Density +(kg/m3) +Viscosity +(Pa.s) +Surface +Tension +(mN/m) +Vimpact +(m/s) +Ddrop +(mm) +% +difference +(interFoam) +% +difference +(Modified) +Ethylene +Glycol +1113.2 +0.0161 +48.4 +1.89 +4.56 +7.67 +1.26 +25% +Propylene +Glycol and +Water +1007.5 +0.00255 +54.1 +1.87 +3.79 +13.53 +1.92 +50% +Propylene +1017.0 +0.005 +45.2 +1.91 +4.68 +9.61 +0.89 +Experiment +interFoam +Modified VoF + +30 + +Glycol and +Water +75% +Propylene +Glycol and +Water +1026.5 +0.012 +41.1 +2.38 +4.93 +8.12 +1.50 + +Conclusions +The volume of fluid or VoF method has been used for numerical analysis of many inertia +dominated as well as surface tension dominated two-phase problems. However, the current study +finds that these predictions of two-phase flow phenomena tend to deviate significantly from +experimental data for liquids with high viscosity. A new exponential property averaging technique +is proposed to address this shortcoming. Computational predictions using the open source VoF +solver interFoam (available as a part of the OpenFOAM computational tool), and those obtained +using the proposed method are compared with experimental data for multiple two-phase +applications. Four different problems, viz., suspended droplet in air, jet breakup, drop impact on +thin films, and air entrapment during drop interaction with liquid pool, are considered to +extensively validate the new method. For all cases, the modified VoF solver is observed to perform +significantly better than original VoF method. For the suspended drop problem, as has been +observed in previous numerical studies, spurious currents are noted to be a key source of numerical +error. The proposed method reduces any spurious currents in simulations of drop suspended in air. +For the cases of drop impacting on a pool and during drop generation from liquid jets, the time +progression of the surface tension governed dynamics is improved from the slower estimate of +interFoam solver. In the case of drop impacting on a thin liquid film, where it is critical to capture + +31 + +the intricate interplay between the surface tension and viscous force, the former effect of surface +tension is exaggerated in the original method but correctly captured with the proposed new method. + +References +[1] +Ghia, U., Ghia, K. N., and Shin, C. T., 1982, “High-Re Solutions for Incompressible Flow +Using the Navier-Stokes Equations and a Multigrid Method,” J. Comput. Phys., 48(3), pp. +387–411. +[2] +Sadhal, S. S., Ayyaswamy, P. S., and Chung, J. N., 1996, “Transport Phenomena with Drops +and Bubbles,” Springer-Verlag New York, (978-1-4612-8470–3), pp. 522–527. +[3] +Berberović, E., Van Hinsberg, N. P., Jakirlić, S., Roisman, I. V., and Tropea, C., 2009, +“Drop Impact onto a Liquid Layer of Finite Thickness: Dynamics of the Cavity Evolution,” +Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys., 79(3). +[4] +Khodaparast, S., Magnini, M., Borhani, N., and Thome, J. R., 2015, “Dynamics of Isolated +Confined Air Bubbles in Liquid Flows through Circular Microchannels: An Experimental +and Numerical Study,” Microfluid. Nanofluidics, 19(1), pp. 209–234. +[5] +Srinivasan, V., Salazar, A. J., and Saito, K., 2011, “Modeling the Disintegration of +Modulated Liquid Jets Using Volume-of-Fluid (VOF) Methodology,” Appl. Math. Model., +35(8), pp. 3710–3730. +[6] +Lewis, S. R., Anumolu, L., and Trujillo, M. F., 2013, “Numerical Simulations of Droplet +Train and Free Surface Jet Impingement,” Int. J. Heat Fluid Flow, 44, pp. 610–623. +[7] +Liu, J., Vu, H., Yoon, S. S., Jepsen, R. a., and Aguilar, G., 2010, “Splashing Phenomena +During Liquid Droplet Impact,” At. Sprays, 20(4), pp. 297–310. + +32 + +[8] +Trujillo, M. F., Alvarado, J., Gehring, E., and Soriano, G. S., 2011, “Numerical Simulations +and Experimental Characterization of Heat Transfer from a Periodic Impingement of +Droplets,” J. Heat Transfer, 133(12), p. 122201. +[9] +Manoharan, S., Deodhar, A. M., Manglik, R. M., and Jog, M. A., 2019, “Computational +Modeling of Adiabatic Bubble Growth Dynamics from Submerged Capillary-Tube Orifices +in Aqueous Solutions of Surfactants,” J. Heat Transfer, 141(5). +[10] Unverdi, S. O., and Tryggvason, G., 1992, “A Front-Tracking Method for Viscous, +Incompressible, Multi-Fluid Flows,” J. Comput. 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P., 2008, “A Hybrid Lagrangian-Eulerian Particle-Level + +33 + +Set Method for Numerical Simulations of Two-Fluid Turbulent Flows,” Int. J. Numer. +Methods Fluids, 56(12), pp. 2271–2300. +[17] Desjardins, O., Moureau, V., and Pitsch, H., 2008, “An Accurate Conservative Level +Set/Ghost Fluid Method for Simulating Turbulent Atomization,” J. Comput. Phys., 227(18), +pp. 8395–8416. +[18] Ménard, T., Tanguy, S., and Berlemont, A., 2007, “Coupling Level Set/VOF/Ghost Fluid +Methods: Validation and Application to 3D Simulation of the Primary Break-up of a Liquid +Jet,” Int. J. Multiph. Flow, 33(5), pp. 510–524. +[19] Farvardin, E., and Dolatabadi, A., 2013, “Numerical Simulation of the Breakup of Elliptical +Liquid Jet in Still Air,” J. Fluids Eng., 135(7), p. 071302. +[20] Ubbink, O., and Issa, R. I., 1999, “A Method for Capturing Sharp Fluid Interfaces on +Arbitrary Meshes,” J. Comput. Phys., 153(1), pp. 26–50. +[21] Deshpande, S. S., Anumolu, L., and Trujillo, M. F., 2012, “Evaluating the Performance of +the Two-Phase Flow Solver InterFoam,” Comput. Sci. Discov., 5(1), p. 014016. +[22] Brackbill, Kothe, and Zemach, 1992, “A Continuum Method for Modeling Surface +Tension,” J. Comput. Phys., 100, pp. 335–354. +[23] Tang, H., and Wrobel, L. C., 2005, “Modelling the Interfacial Flow of Two Immiscible +Liquids in Mixing Processes,” Int. J. Eng. Sci., 43(15–16), pp. 1234–1256. +[24] Beerners, J. C., 2013, “Lubricated Transport of Heavy Oil,” Delft University of Technology. +[25] Bohacek, J., 2010, “Surface Tension Model for High Viscosity Ratios Implemented in VOF +Model,” Annu. Conf. Liq. At. Spray Syst., (September), pp. 1–8. + +34 + +[26] Mooney, K., Menon, S., and Schmidt, D. P., 2010, “A Computational Study of Viscoelastic +Droplet Collisions,” ILASS - Am. 22nd Annu. Conf. Liq. At. Spray Syst., (May). +[27] Tran, T., de Maleprade, H., Sun, C., and Lohse, D., 2013, “Air Entrainment during Impact +of Droplets on Liquid Surfaces,” J. Fluid Mech., 726, p. R3. +[28] Thoraval, M.-J., Takehara, K., Etoh, T. G., and Thoroddsen, S. T., 2013, “Drop Impact +Entrapment of Bubble Rings,” J. Fluid Mech., 724, pp. 234–258. +[29] Lin, S. P., and Reitz, R. D., 1998, “Drop and Spray Formation from a Liquid Jet,” Annu. +Rev. Fluid Mech., 30, pp. 85–105. +[30] Eggers, J., and Villermaux, E., 2008, “Physics of Liquid Jets,” Reports Prog. Phys., 71(3). +[31] van Hoeve, W., Gekle, S., Snoeijer, J. H., Versluis, M., Brenner, M. P., and Lohse, D., 2010, +“Breakup of Diminutive Rayleigh Jets,” Phys. Fluids, 22(12), pp. 1–11. +[32] Rajendran, S., Jog, M. A., and Manglik, R. M., 2017, “Experimental Investigation of Jet +Breakup at Low Weber Number,” At. Sprays, 27(9), pp. 821–834. +[33] Rajendran, S., 2017, “Investigation of Drop Generation from Low Velocity Liquid Jets and +Its Impact Dynamics on Thin Liquid Films,” Ph. D. Thesis, University of Cincinnati +(http://rave.ohiolink.edu/etdc/view?acc_num=ucin1512038966865923). + + diff --git a/p9AzT4oBgHgl3EQfq_2l/content/tmp_files/load_file.txt b/p9AzT4oBgHgl3EQfq_2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41531c6ba2fe23c00645ff5d6d7d591bb72ec44e --- /dev/null +++ b/p9AzT4oBgHgl3EQfq_2l/content/tmp_files/load_file.txt @@ -0,0 +1,759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf,len=758 +page_content='Paper submitted to the Journal of Fluids Engineering – Special Issue in Honor of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Kirti Ghia New Property Averaging Scheme for Volume of Fluid Method for Two-Phase Flows with Large Viscosity Ratios Sucharitha Rajendran (ASME Member) Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221 Raj M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Manglik (ASME Fellow) Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221 Milind A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Jog1 (ASME Fellow) Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221 1 Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Email: Milind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='Jog@uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='edu 2 Abstract To predict liquid-gas two-phase flow phenomena, accurate tracking and prediction of the evolving liquid-gas interface is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Volume-of-Fluid or VoF method has been used in the literature for computationally modeling of such flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the VoF method, a single set of governing equations are solved for both phases along with an advection equation for the volume fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The properties in each computational cell are determined by a linear weighted average of the properties of the two fluids based on the phase fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' While the method predicts water-air flows well, the predictions tend to deviate significantly from experimental data for liquids with high viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A new property averaging technique is proposed in this paper, which is shown to provide accurate results for high viscosity liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Computational predictions using the open source VoF solver interFoam (available as a part of the OpenFOAM computational tool), and those obtained using the proposed method are compared with experimental data for multiple two-phase applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Four different problems, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=', suspended droplet in air, jet breakup, drop impact on thin films, and air entrapment during drop interaction with liquid pool, are considered to extensively validate the new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Experimental data for water and aqueous solutions of propylene and ethylene glycol are used to cover a range of surface tension (72 – 36 mN/m) and viscosities (1 – 40 mPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For all cases, the modified VoF solver is observed to perform significantly better than original VoF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' It reduces any spurious currents in simulations of drop suspended in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For the cases of drop impacting on a pool and during drop generation from liquid jets, the time progression of the surface tension governed dynamics is improved from the slower estimate of interFoam solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the case of drop impacting on a thin liquid film, where it is critical to capture the intricate interplay between the surface tension and viscous force, the former effect of surface tension is exaggerated in the original method but correctly captured with the proposed new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 3 Keywords: Numerical modeling, VoF method, two-phase flow, interfacial phenomena Introduction Our colleague of many years, Professor Kirti “Karman” Ghia, was devoted to the advancement of fundamentals and applications of computational fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' An example is his early work on the lid driven cavity flow [1] that made a seminal contribution to multigrid methods and became a benchmark for work that followed (over 5000 citations to date!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' We enjoyed many technical discussions with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This work is a tribute to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Ghia, and it addresses a limitation of the Volume-of-Fluid (VoF) method in accurately predicting two-phase flows when the viscosity ratio of the two fluids is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Two-phase processes with jets, droplets, and bubbles such as droplet impact on dry surfaces, or on thin films, or on liquid pools, jet breakup and drop formation, and bubble growth from an orifice submerged in liquid, are encountered in a variety of industrial applications [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Numerical modeling of such phenomena has steadily advanced, and along with experiments, it has provided insights into the underlying physics of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For example, Berberovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [3] used numerical techniques along with experimental observations to understand the dynamics of drop impact on liquid surfaces and have characterized formation and propagation of the surface wave during impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Extensive investigation of dynamics of confined bubbles in laminar micro channel flow by Khodaparast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [4] relies on experimental and numerical analysis of two-phase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Similarly, research in jets and atomization characterization [5–7] has advanced with the help of numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In addition to flow characterization, coupled heat transfer solution has been implemented by Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [8] to study drop impact heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Sanjivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [9] have incorporated interfacial surfactant adsorption/desorption kinetics in their computational 4 simulations of bubble growth in aqueous surfactant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' These and many such other studies have used numerical methods for predicting multiphase (or two-phase) flow behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For accurate predictions of the two-phase phenomena encountered in applications associated with liquid breakup, atomization, liquid collisions and entrainment, precise capture of the liquid and gas interface is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Interfacial tracking methods in incompressible fluids include front tracking [10–12] , boundary integral [13], volume of fluid (VoF) [14] and level set (LS) [15,16] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the past decade, the techniques that have received the most attention are the last two – VoF and LS – due to their robustness and ease of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As the interface evolves in a fixed computational mesh, both these methods use a separate parameter to identify the different phases in each cell in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The primary difference in these methods is their treatment of this phase parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the VoF method, an advection equation governs the fraction of volume of the primary phase as the interface moves in time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This makes the interface discontinuous and therefore, reconstruction of the calculated interface is required to obtain a smooth curve for curvature and normal estimations at each interface location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The interface is typically smeared over a few cells owing to the change in volume fraction as it goes from one phase to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In contrast, the LS method defines the interface by a smooth continuous function going from positive in one fluid to negative in the other and having a value of zero at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This eliminates the requirement for reconstruction and creates an interface solution that is sharp unlike in the VoF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, though accurate estimations of the interface topology are achievable, the LS method does not inherently conserve mass by the advection step unlike the VoF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The VoF method has been implemented in many commercial solvers as it inherently conserves mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This is the method under consideration for the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As interface tracking is very sensitive to the reconstruction, VoF technique was seen to yield acceptable results only for certain applications 5 and working liquids and some efforts have been taken to improve the interface tracking for this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This is discussed later in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Prior numerical studies of jets and droplets that use VoF techniques have primarily looked at fluids with viscosities similar to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' They have provided useful results with water, hydrocarbon fuels, and refrigerants – liquids with viscosities close to or less than that of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, liquids used in chemical, pharmaceutical and process industries can have significantly larger viscosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In such cases, with large differences in the viscosities of the two fluids, VoF method was found to produce errors in numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The error in numerical prediction also increases with higher viscosity working fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A few studies have coupled the LS and VoF techniques to improve interface tracking while ensuring mass conservation in the domain [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The improved accuracy of combined LS and VoF (CLSVOF) method can be observed in the work of Ray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (2015) [ ] where the problem of liquid drop impact has been modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Some other studies have focused on improving dynamic meshing methods along with the VoF implementation to refine the numerical calculations at the interface [7,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The current study provides a modification to the VoF method so that accurate predictions can be made for highly viscous liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Extensive validation of the suggested modification to the VoF method is conducted with available experimental data for four different physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The VoF method used for comparison in this study is based on the open source two-phase solver present in OpenFOAM®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This package is well suited to handle complex geometries and is easily parallelizable without the limits of licensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' It is based on C++, where equations can be in a form that has a close resemblance to its mathematical equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This VoF interface tracking solver was first implemented in 1999 by Ubbink & Issa [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A detailed summary of the original solver has been presented by Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Modification to the interface reconstruction is implemented to ensure better predictions for viscous working liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 6 Numerical Procedure The numerical procedure to solve the two-phase incompressible flow problems in this study, and as enabled in the open-source code OpenFOAM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 is discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The implemented solver available for this is interFoam, which, as noted by Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [21], has been reliably employed to solve a number of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, the accuracy of this original solver was noticed to drop when the viscosity of the working fluid is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This VoF solver uses finite volume discretization of the governing equations on a fixed grid in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' All the flow variables are stored in the cell center in the associated finite volume technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A modified method is proposed herein and its efficacy and relatively more precise validity are demonstrated and highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Governing Equations The numerical technique adopted in this study considers two immiscible fluids (a gas and a liquid) that are separated by a sharp interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, the two fluids are not solved separately with their different properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Instead, a single set of equations are solved for the entire fluid region with properties differing continuously from one fluid to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The two fluid media are distinguished by a phase fraction property that has values between 0 and 1, where 1 is identified as liquid and 0 as gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The interface is defined by the region where the value of this phase fraction transitions from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, the interface is smeared in such an evaluation and is not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The definition for this phase fraction (α) is as follows: 7 𝛼 = Volume of primary phase Total control volume where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝛼 = { 1 0 < 𝛼 < 1 0 in primary fluid (liquid) in transition region (interface) in secondary fluid (air) (1) Using this phase fraction identifier for the fluids the respective thermo-physical properties are determined in each computation cell as follows: 𝜑 = 𝛼 𝜑1 + (1 − 𝛼)𝜑2 (2) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝜑 is the property of the fluid and 𝛼 is the phase fraction in the cell and subscripts 1 and 2 represent liquid and air,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Equation (2) therefore allows the properties to be calculated in each cell at each time instant in the computations of the evolving interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The transport of this newly defined fluid bulk property (phase fraction, α) is further defined by an advection equation, 𝜕𝛼 𝜕𝑡 + 𝛻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (v̅𝛼) = 0 (3) Additionally, the finite volume solver uses the continuity and momentum equations to computationally obtain the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In this study, an incompressible flow solution is of interest and hence the energy equation is not added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Continuity: ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (v̅) = 0 (4) Momentum Conservation: ∂(𝜌v̅) ∂t + ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (𝜌v̅ v̅) = −∇𝑝 + ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' T̅ + 𝑓𝑠 + 𝜌g̅ (5) 8 where 𝑇̅ is the viscous stress tensor and 𝑓𝑠 is the volumetric force due to surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The properties such as density and viscosity in these two equations are evaluated based on the phase fraction (𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Equations (3) – (4) are solved numerically by adopting a control volume discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In this process, the surface tension force term in the momentum equation calls for special consideration as it is not inherently a volumetric term like the other components in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' One way to resolve this would be to use this force as a boundary condition on the free surface and the surface pressure is obtained by linear interpolation between the surface pressure required and the fluid pressure inside the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This requires multiple iterations to obtain surface pressures within some tolerance interval with respect to that at previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' After the correction of the surface pressure, the momentum and continuity equations are re-solved before advancing to the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This was the original VoF method suggested by Hirt and Nichols in 1981 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In addition to higher computational time due to the iterative surface pressure correction, this technique required approximate prior knowledge of the interface shape that was to be obtained from upstream and downstream cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Brackbill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [22] in 1992, proposed a means to resolve the problems associated with this surface treatment by converting the surface tension forces to an equivalent volume force that can be added to the Navier-stokes equation as an additional body force term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Their method called continuum surface force (CSF) calculates this equivalent volumetric surface tension force as: 𝑓𝑠 = 𝜎 𝜅 𝒏 𝛿𝑠 (6) where, σ is the surface tension coefficient, κ is the interface curvature, 𝐧 is the surface normal and δs is a Dirac delta function that assists with concentrating this calculation on the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In this 9 model, the interface curvature κ is determined as a function of local gradients of the surface normal, 𝐧 which in turn is a function of the phase fraction (α) given by 𝜅 = 𝛻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝒏̂ and 𝒏 = 𝛻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝛼 (7) Tang and Wrobel [23] have shown how this surface tension model can be written in terms of pressure drop across the interface and expressed in terms of a volume force in the momentum equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' They normalize the interface curvature by using the volume averaged density (ρ) to obtain the following for the surface tension force: 𝑓𝑠 = 𝜎 𝜅 𝜌 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5(𝜌1 + 𝜌2) 𝛻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝛼 (8) It must be noted that the numerical solution would depend on the estimation of the interface normal, 𝐧̂ , which in turn, is a function of the phase fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, correct estimation of these factors determines the accuracy and performance of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Modifications to the VoF Solver One significant cause for incorrect estimation of interface curvature and normal stems from deviations in estimations of fluid properties across the smeared interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Inaccurate interface calculations, in turn, would result in unreliable surface tension force estimation and thus cause failure in predicting correct interface pressure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To resolve this difficulty in interface tracking, some past studies have either looked at correcting the Courant number estimation, such as the case in Beerners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [24], or have proposed alternative methods to estimate interface curvature and therefore surface tension force [25], while still others have looked at better meshing techniques to resolve the computational domain in a dynamic manner [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Other notable 10 improvements to accurately predict surface tension-dominated flows are provided by Puckett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [ ], Popinet & Zelenski [ ], and Gerlach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In our preliminary analysis of the complication in correct interface estimation, it was found that the relative viscosity of the two phases in question was an underlying source of inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, an attempt to correct this is made by proposing a modification to the interface property averaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Multiple experimental measurements have been used to test a generic exponential property averaging equation that is defined as: 𝜑 = 𝜑2 + (𝜑1 − 𝜑2)𝛼𝐂 (9) The specific experimental case employed for this analysis is the dynamic development of the crown on impact of a liquid drop on a thin film of the same liquid [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This case was computationally modeled as well, and numerical solutions duplicating the experimental results obtained with several different working liquid properties (Table 1) are used to identify the correct expression for the constant C in the exponential property averaging of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The results obtained are then correlated with their relative viscosity ratios and the consequent results are graphed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, a property estimation equation that depends on the viscosity of the two working fluids (1 and 2) is determined and is given by 𝜑 = 𝜑2 + (𝜑1 − 𝜑2)𝛼𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='𝟓(𝝂𝟏 𝝂𝟐 ⁄ )+𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='𝟕𝟓 where, 𝜈 is the kinematic viscosity of the fluid in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (10) 11 Table 1: Properties of the Newtonian liquids used to determine the exponent C in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (9) Liquids Density (kg/m3) Viscosity (Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='s) Surface Tension (N/m) Water 998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0728 25% by volume of Propylene Glycol (25% PG/ 75%Water) 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='00255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0541 50% by volume of Propylene Glycol (50% PG/ 50% Water) 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0452 75% by volume of Propylene Glycol (75% PG/ 25% Water) 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0411 Ethylene Glycol (EG) 1113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0484 Though the proposed modification to the VoF method, expressed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (9) and (10), was derived based on an experimental study of drop impact on thin films, its veracity and general applicability has been tested with other two-phase flow problems as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' An example of the significance of property estimation equation is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The plot shows the variation in interface density in the x-axis, between air (\uf072 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='225 kg/m3) and a viscous liquid (ethylene glycol, \uf072 = 1113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 kg/m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As the phase fraction (α) in the x-axis goes from 0 to 1, the interface moves from air to that of the viscous liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As seen, from the two plots, the change in density with the exponential averaging equation (red line) is not monotonically linear as is the case with arithmetic averaging (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This ensures a sharper interface that is not smeared over 12 multiple cells, thereby providing better estimates of the interface curvature and therefore, surface tension force and interface pressure drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 1: Equation for the determination of constant C in the property averaging expression I Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (9), as a function of relative viscosity ratios and as obtained from the mesurements with different air-liquid systems [note: EG – ethylene glycol, and x% PG – x% by volume propylene glcycol and water solution].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝜈1 𝜈2 Τ = 𝜈𝐿𝑖𝑞𝑢𝑖𝑑 𝜈𝐴𝑖𝑟 Τ where, x % PG: x % by volume of propylene glycol and water mixtures EG: ethylene glycol 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 EG 2 75% PG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5x40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 c 50% PG 1 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='PG Water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 113 Figure 2: Comparison of the variation of fluid property (density) at the interface given by the arithmetic weighted averaging (green line) method of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (2) and that by the new proposed method (red line) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In addition to altering the property estimation, the curvature field is smoothened in accordance with the Ubbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [20] technique where the phase fraction smoothness is corrected to influence the smoothness of the interface curvature, as the two are related by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This involves averaging the phase fraction in each computational cell volume by area averaging with respect to the values at each face as expressed in the following: 𝛼[𝑐𝑒𝑙𝑙] = ∑ 𝛼[𝑓𝑎𝑐𝑒] 𝐴𝑟𝑒𝑎[𝑓𝑎𝑐𝑒] ∑ 𝐴𝑟𝑒𝑎[𝑓𝑎𝑐𝑒] (11) Thus, the final set of equations that are solved in the proposed new modified VoF method are the continuity, momentum and phase transport equations along with the surface tension force Phase fraction, 𝜶 Density (kg/m3) 1500 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='814 determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The properties in this one fluid numerical approach (VoF) are estimated as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (10), and the interface curvature is further smoothened by area averaged smoothing of the phase fraction as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Solution Procedure The equations described in the previous section are discretized by using upwind scheme for the spatial terms and solved iteratively on the computational grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' From the computed variables the average values are stored in the cell center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Reconstruction from these averaged values is required to evaluate the cell interface values which need to be used for the subsequent iteration in the solution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Limited linear piecewise reconstruction is used to diminish large variations during this reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The PIMPLE algorithm (combination of PISO and SIMPLE) is used as the iterative procedure for coupling mass and momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For this coupling, within each time solution, this algorithm solves the pressure equation while invoking an explicit correction to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Optionally, each iteration step can begin with a solution to the momentum equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This is called the momentum predictor loop and is set to loop twice for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The linear solvers used in the PIMPLE method are preconditioned Conjugate Gradient for pressure and a preconditioned Bi-Conjugate Gradient for velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Because a transient solution is the goal of these numerical solutions, an implicit Euler time discretization is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This first order implicit scheme is preferred to an explicit scheme as this guarantees boundedness and is unconditionally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For providing good numerical stability, adaptive time stepping is used where at the beginning of each time loop, the time step Δt is calculated dynamically using the following criteria: 15 ∆𝑡 = min {Comax Coo ∆𝑡𝑜;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (1 + 𝜆1 Comax Coo ) ∆𝑡𝑜;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 𝜆2∆𝑡𝑜;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' ∆𝑡𝑚𝑎𝑥 } (12) In the above criteria, ∆𝑡𝑚𝑎𝑥and Comax are predefined values for maximum permissible time step and maximum Courant number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The damping factors, λ1 and, λ2, are defined as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The superscript o refers to values at the previous time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The maximum Courant number is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 for the current numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Numerical domain and mesh discretization The numerical problems used in this study to test and establish the general applicability of the modified VoF for two-phase flows are all considered to be axisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This assumption is based on experimental evidence and helps reduce the computational domain and therefore makes the numerical solution faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The entire physical domain of the problem can be considered as cylindrical with the axis representing the axis of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In OpenFOAM, the computational domain used for these problems is a wedge-shaped thin sliver (with a small angle, < 5o) of this cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This wedge straddles the X-Y plane and runs along the central axis of symmetry as shown in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The typical orthogonal structured non-uniform mesh for this domain is shown in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The flow phenomena under examination are typically around the axis of symmetry in all the problems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Therefore, the mesh is denser in this region than farther away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Likewise, the mesh is compact near the target surface (for specific cases discussed where the fluid interacts with a target surface), to ensure accurate capture of the phenomena in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Prior to obtaining the presented numerical results, the density of the grid is varied until a grid independent solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For drop impact in a liquid pool, penetration depth was the parameter used to compare different meshes for interFoam method for 5cSt liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the liquid jet 16 breakup in stagnant ambience, breakup length of the continuous jet was the parameter used to compare different meshes for InterFoam method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The same mesh density was used for other cases studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In modeling drop impact on thin liquid film, crown diameter was the parameter used to compare different mesh sizes for InterFoam to obtain grid-independent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 3: (a) the wedge-shaped numerical domain (b) typical mesh structure Performance tests on modified VoF method All implicit multi-fluid numerical methods rely on interface capturing techniques that are known to be susceptible to numerical instabilities and can result in unrealistic interface flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' When the flow phenomena under consideration are dominated by inertia, these numerically generated interface flows do not lead to unrealistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, when capillary effects (surface tension forces) are dominant, any errors in calculating interfacial flows can give rise to inaccurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To compare and test the efficiency of the two numerical VoF solvers, namely, Y X Nozzle Wall Axis of Symmetry Atmospheric Outlet Axis of symmetry Y X Axis of symmetry (a) (b) 17 interFoam and the modified solver, four tests problems are solved and the results obtained are outlined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' These include the canonical suspended drop problem and three other cases typically encountered in engineering applications (jet breakup, drop impact on a thin film, and drop impact on a liquid pool).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In all these cases, surface tension forces are significant and that helps identify the difference in the accuracy of the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Stationary drop in stagnant zero-gravity ambience The native VoF scheme (in interFoam solver), yields an interface curvature calculation with large gradients, thus contributing to a relatively larger spread of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Consequently, the error in curvature results in an imbalance between the pressure and surface tension forces that lead to formation of spurious currents in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The proposed modified VoF solver aims at creating a sharper interface and thereby improves the curvature estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To evaluate the influence of the modified treatment of interface, a stationary axi-symmetric drop in absence of gravity in a medium of air is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The initial and boundary conditions for this numerical analysis are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A 4-mm diameter liquid drop with surface tension of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='8 mN/m and density 1000 kg/m3 is placed in stagnant air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The domain has a uniform grid distribution with a grid spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='001 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Two different drop viscosities (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='001 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='01 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='s) are tested to see the influence of viscosity on the numerical generated current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' These currents, called parasitic currents appear as vortices around the interface and, as shown in Figure 5, are present when there are no external forces acting in the physical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This numerical error is not eliminated by grid refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In fact, as noted by Brackbill [22], the magnitude of these spurious currents could get amplified by a finer grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Therefore, for unbiased comparison between the two methods, the same grid size and time stepping were used to check magnitudes of parasitic currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 18 Figure 5 shows the spurious currents around a stationary drop in air in the absence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The observed currents for both interFoam and the modified VoF solver is shown for water and a more viscous (10 times that of water) liquid drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' It is observed increasing viscosity of the liquid, increases the intensity of the parasitic currents around interface of the stationary drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 5 are at 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The typical velocities in interFOAM are ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 m/s while those with the modified solver at a tenth of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The modified VoF solver, does not eliminate these currents, but the intensity of the same is noted to be greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, it is noted that a more accurate sharper approximation of the interface in the modified VoF solver can help reduce the spurious currents in the two-phase problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To estimate the influence of these numerically induced currents on the domain, the kinetic energy around the drop interface is presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As is observed, the kinetic energy at the interface of this viscous drop in the numerical solution obtained by the modified VoF solver is an order of magnitude lower than that given by the interFoam solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Initial drop with zero velocity and ddrop diameter Atmosphere Target Surface Axis Outlet zero gradient velocity, zero pressure and zero phase fraction zero velocity, zero gradient pressure and zero gradient phase fraction zero gradient velocity, zero gradient pressure and zero gradient phase fraction Zero gravity Y X 19 Figure 4: Initial and boundary conditions for numerical solution of stationary drop under no external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 5: Comparison of the two VoF solvers: spurious currents around the interface of a stationary drop in ambient air in the absence of gravity at simulation time of 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' interFoam VoF solver Modified VoF solver 𝜈𝑙𝑖𝑞𝑢𝑖𝑑 𝜈𝑎𝑖𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='07 (water) 𝜈𝑙𝑖𝑞𝑢𝑖𝑑 𝜈𝑎𝑖𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='7 20 Figure 6: Comparing kinetic energy around the viscous ( 𝜈𝑙𝑖𝑞𝑢𝑖𝑑 𝜈𝑎𝑖𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='7) stationary drop interface for interFoam and the modified VoF solver Drop impact in a liquid pool For the second test, experimental findings of Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [27] on the impact of a viscous drop on a deep liquid pool with attention to the entrapped air layer is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The initial temporal dynamics is studied for these impact conditions with the different viscous liquids used in the experimental work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' At low impacting velocities, the air layer underneath the impacting drop prevents merger of the drop and pool surface and the drop could bounce off [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='Increasing the impact velocity was observed to aid entrapment of the air layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The centerline depth as the drop merges with the pool (penetration depth) was studied as a function of time taken for rupture and was found to be linearly increasing with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In addition, viscosity was noted to delay rupture Kinetic Energy (kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='m 2/s 2) interFoam VoF solver Modified VoF solver 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0E 061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='3E 0521 time as higher the viscosity, more the resistance to allow entrapped air to rise and rupture the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' These experimental findings are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 7: Experimental data showing dimensionless penetration depth (L/R) as a function of time of impact of drops with same impacting velocity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='55 m/s) for 3 different viscosities [27] Figure 8: Initial and boundary conditions for numerical solution of drop impact on a deep pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 5 cSt 10 cSt 20 cSt Initial drop with velocity vimpact and diameter ddrop Atmosphere Target Surface Axis Outlet zero gradient velocity, zero pressure and zero phase fraction zero velocity, zero gradient pressure and zero gradient phase fraction zero gradient velocity, zero gradient pressure and zero gradient phase fraction gravity Deep Pool Y X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='8 L/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='6 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 0 4 3 2 1 0 t (ms)22 Figure 9: Performance of interFoam and modified solver against experimental findings [27] for entrapped air rupture time during viscous drop impact on a deep pool To test the effectiveness of the modified VoF solver against the data, a drop impact on a pool in similar conditions is numerically simulated using both solvers (interFoam and modified VoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The initial and boundary conditions for this problem are shown in the schematic in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A drop with diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='9 mm and velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='55 m/s as provided by Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [27] impacts a pool of like liquid (10 mm deep) as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The mesh for this domain under consideration contains approximately 390000 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' While analyzing the effectiveness of the two solvers, it is observed that interFoam predicts a much slower evolution of air entrapment and rupture than the modified solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The findings from both the solvers are compared with 5 cSt 10 cSt 20 cSt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 L/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='4 3 2 1 0 t (ms)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0 5 4 3 2 t (ms)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 L/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 4 3 2 1 0 t (ms)Data Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (2013) interFoam Solver Modified Solver23 experimental data provided by Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [27] in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For all the three viscosities tested, the modified VoF solver is better at predicating the entrapped air dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' At lower viscosities (5 cSt), the modified solver is more precise at predicting the development of penetration depth, than at higher viscosities (20 cSt) compared to experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For all the three viscosities, penetration depth at rupture (τ = 0) are comparable with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Comparisons of dimensionless penetration depth for the three liquids for the two solvers is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Table 2: Comparing dimensionless penetration depths for the three cases in consideration against interFoam and the modified VoF solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Liquid’s viscosity (cSt) L/R expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' L/R interFoam L/R Modified % difference (interFoam) % difference (Modified) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='68 165 26 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='92 127 19 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='22 34 5 While the development time for the rupture is very comparable for 5cSt and 10 cSt drops, for the 20 cSt drop impact, total time for rupture is shorter than experimentally measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As observed in the previous case study with numerically induced spurious currents in stationary drops, higher viscosity gives rise to stronger parasitic currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thus, with increasing viscosity, there is relatively more error in predicting the pressure and surface tension balance and thus the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' It is speculated that this causes the numerical analysis to be more accurate when viscosities are 24 lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, comparing with the experimental data, the deviation is significantly less than the original VoF solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Liquid jet breakup in stagnant ambience Precise numerical characterization of capillary driven phenomena like liquid jet breakup requires correct estimation of surface tension and pressure forces on the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This becomes more significant when low velocity liquid jets are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A circular jet breaking up in stagnant ambience is governed by the interplay of inertial, gravitational, aerodynamic, capillary, and viscous forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' When the jet is fast moving, though surface tension (capillary force) is a prominent cause for creation and propagation of surface instabilities, inertial and aerodynamic forces are most dominant in the interplay [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This balance however changes when the jet is slow moving (smaller, lower weber numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To investigate the accuracy of the two solvers in capturing the temporal growth of the surface instabilities, experimental data from our study of Newtonian low velocity viscous liquid jet breakup is used [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A propylene glycol jet of initial diameter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='499 mm emerges from a circular nozzle at We = 5 and disintegrates to generate drops in stagnant ambience under the presence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Experimental observations show the phenomena to be symmetric about the axis and hence an axisymmetric wedge-shaped mesh with non-uniform structured mesh with about 160000 cells is used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The initial and boundary conditions for this problem are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 25 Figure 10: Initial and boundary conditions for numerical solution of jet breakup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The nozzle wall has a no-slip boundary condition imposed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' To emulate the experimental test conditions, the inlet has a uniform velocity inlet condition defined based on the prescribed inlet velocity from the syringe pump in the experimental study [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' At the outlets, the pressure is assumed to be atmospheric and zero-gauge pressure is the prescribed boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Breakup length calculations and other analyses of results are carried out after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 s of real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This provides time for numerical instabilities, if any, to deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 11 show the time progression of surface instability on the propylene glycol jet considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' High-speed experimental images are compared with predictions from the numerical solvers tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' It is observed that while the original solver (interFoam) is unable to capture the generation and propagation of surface instabilities unlike the modified VoF solver for high viscosity liquid jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For water, this is however not the case and both solvers agree with the experimental observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' As propylene glycol is 40 times as viscous as water, and the VoF solver Initial jet with velocity vjet and diameter djet Nozzle inlet Atmosphere Axis Outlet zero gradient velocity, zero pressure and zero phase fraction zero gradient velocity, zero gradient pressure and zero gradient phase fraction gravity Atmosphere zero gradient velocity, zero pressure and zero phase fraction Y X 26 predicts a much longer unperturbed jet surface compared to experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The modification on the property averaging helps improve the interface calculation and the modified solver can be seen to provide results that agree well with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' With increasing viscosity of the external fluid, jet breakup exhibits different behavior (see for example, Borthakur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (2017) [ ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The proposed method will be useful in addressing these situations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 11: Comparison between (a) experimental [32, 33], and numerical predictions of (b) interFoam and (c) modified VoF for evolution of surface instabilities during circular jet breakup of a propylene glycol jet of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='499 mm diameter and We = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The time difference between these two images is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' (a) (b) (c) (a) (b) (c) 27 Drop impact on thin liquid film During drop impact on a thin liquid film of the same liquid, viscosity and surface tension affect the spread on the thin film and the crown growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Understanding the correct interplay of these two forces can give insights into the growth and spread of the impacting drop as well as help enhance or curb splashing as per the requirement of the application such as coating and pesticide dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A wedge-shaped axisymmetric domain (Figure 3 (a)) with structured mesh is used for numerical analysis of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' In the stagnant ambience, the drop of liquid with known impact velocity and diameter (from experimental observations) are initialized along with the stagnant thin film height and a schematic of the same is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 13 shows the time progressions of an ethylene glycol drop of diameter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='56 mm impacting at a velocity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='89 m/s on a thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Successive images are at a gap of 2 ms and the temporal progression of the impact dynamics as predicted by the two solvers are compared with the experimental observations reported in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' While the initial spread of the drop on the film surface looks similar (at 2 ms and 4 ms), the crown tips are noticed to be thinner in the numerical solution using interFoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This thinner crown tip leads to a faster moving crown rim that predicts the pinch-off secondary drop from the crown rim at 6, 8 and 10 ms after the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The modified VoF solver is able to predict the correct temporal and spatial dynamics of the drop liquid interaction as is seen in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Comparison of the spatial dynamics is done by measurement of the crown diameter as the drop merges with the thin film (at 10 ms in Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The maximum difference between the solver predictions and that from the experiment for different Newtonian viscous liquids are compared in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The properties of the working liquid and the impacting drop are also provided in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The predictions using the modified solver are in agreement with both the final crown diameter as 28 well as with the temporal development of the crown growth as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' We note that the regimes of drop impact dynamics discussed here, the experimental observations showcase a very symmetric development of the crown and breakup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This is the reason to consider axisymmetric 3-D simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' This is similar to other studies in the literature studying similar drop impact dynamics (see for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Figure 12: Initial and boundary conditions for numerical solution of drop impact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Initial drop with velocity vimpact and diameter ddrop Atmosphere Target Surface Axis Outlet zero gradient velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' zero pressure and zero phase fraction zero velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' zero gradient pressure and zero gradient phase fraction zero gradient velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' zero gradient pressure and zero gradient phase fraction Initial thin film with height H Y X gravity 29 Figure 13: Comparison of numerical prediction from the two solvers with experimental time progression of drop impact dynamics (Successive image are 2 ms apart) Table 3: Percentage difference between the experimental and numerical crown diameter Liquid Density (kg/m3) Viscosity (Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='s) Surface Tension (mN/m) Vimpact (m/s) Ddrop (mm) % difference (interFoam) % difference (Modified) Ethylene Glycol 1113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0161 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='26 25% Propylene Glycol and Water 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='00255 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='79 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='92 50% Propylene 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='005 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='68 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='89 Experiment interFoam Modified VoF 30 Glycol and Water 75% Propylene Glycol and Water 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='012 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='50 Conclusions The volume of fluid or VoF method has been used for numerical analysis of many inertia dominated as well as surface tension dominated two-phase problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' However, the current study finds that these predictions of two-phase flow phenomena tend to deviate significantly from experimental data for liquids with high viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A new exponential property averaging technique is proposed to address this shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Computational predictions using the open source VoF solver interFoam (available as a part of the OpenFOAM computational tool), and those obtained using the proposed method are compared with experimental data for multiple two-phase applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Four different problems, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=', suspended droplet in air, jet breakup, drop impact on thin films, and air entrapment during drop interaction with liquid pool, are considered to extensively validate the new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For all cases, the modified VoF solver is observed to perform significantly better than original VoF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For the suspended drop problem, as has been observed in previous numerical studies, spurious currents are noted to be a key source of numerical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' The proposed method reduces any spurious currents in simulations of drop suspended in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' For the cases of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=', and Manglik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=', 2017, “Experimental Investigation of Jet Breakup at Low Weber Number,” At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Sprays, 27(9), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' 821–834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' [33] Rajendran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=', 2017, “Investigation of Drop Generation from Low Velocity Liquid Jets and Its Impact Dynamics on Thin Liquid Films,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content=' Thesis, University of Cincinnati (http://rave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='ohiolink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='edu/etdc/view?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} +page_content='acc_num=ucin1512038966865923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfq_2l/content/2301.01638v1.pdf'} diff --git a/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/2301.01174v1.pdf.txt b/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/2301.01174v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..849435f72d8c11030014071ac277770c41ec456f --- /dev/null +++ b/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/2301.01174v1.pdf.txt @@ -0,0 +1,609 @@ +arXiv:2301.01174v1 [hep-th] 3 Jan 2023 +January 2023 +Should we worry about renormalons in the +ǫ-expansion? +E. Br´ezina) +a) Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e +PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, +e-mail: brezin@lpt.ens.fr +Submitted for publication in a book commemorating +Michael Fisher +(Amnon Aharony, Ora Entin, David Huse, Leo Radzihovsky, editors) +Abstract Turning the divergent ǫ-expansion into a numerically sensible +algorithm, relies on the knowledge of the behaviour of the large order contri- +butions. Two different pictures are known to compete there. The first one +was based on Lipatov’s instantons, which is known to deal with the multi- +plicity of Feynman diagrams which grows factorially at high orders. However +this was challenged by ’t Hooft’s renormalons who pointed out that renormal- +ization could yield a similar growth through one single diagram. We study +here a well-known model, the O(N) model, in the large N limit. The reason +for returning to this familiar model, is that it deals with diagrams known to +give renormalon effects.Through an explicit analytic result, we find no sign +of a non-analyticity of perturbation theory due to these renormalons. + +1 +Introduction +A little more than fifty years ago K.Wilson and M.Fisher introduced the +ǫ-expansion in the celebrated article [1] Critical Exponents in 3.99 Dimen- +sions. This article has had a considerable influence both in the area of critical +phenomena and in quantum field theory in general. However in practice, if +the first two terms of the expansion provided often a reasonable approxima- +tion to the measured indices in dimension three, the situation deteriorated if +one pushed the procedure to higher orders [2]. Clearly the expansion looked +divergent, it is believed to be at best asymptotic, and in fact limited in its ap- +plicability. Of course summation procedure, such as Borel transforms, Pad´e +approximants, could be tried, but in the absence of further indication, it +was a blind shot. Therefore at the time analytic calculations, and computer +simulations, appeared to be limited in their ability of reproducing precision +measurements by the renormalization group approach. +The situation changed significantly in 1977 with Lipatov approach to +the characterization of large orders in pertubation theory [3]. His instanton +method , which he developed for the field theory gφ4 in dimension four, pre- +dicted a large order behaviour of the perturbation expansion of the various +correlation functions, β-function, etc, of the form gkk!(−a)kkbc for large k, +with calculable coefficients a, b, c . I shall recall below why this information, +the explicit knowledge of a, b, c, is essential to extract a numerically sensi- +ble result from such a divergent series. With Le Guillou and Zinn-Justin +[4] we first checked these results in lower dimensions. For instance the one- +dimensional quartic anharmonic oscillator had been extensively studied by +Bender and Wu [5] who had found, on the basis of a WKB method, that +indeed perturbation theory diverged with a k! growth as above. We verified +that the instanton method reproduced exactly what they had found. Ex- +tension to field theory in higher dimensions was similar [4] except that the +instanton solution is not known in analytic form except in dimension four [6]. +In dimensions lower than four the interpretation of these results is simple : +at a given order all Feynman diagrams have the same sign (we are consider- +ing an Euclidean field theory) and their number grows proportionaly to k!. +The situation is less transparent in dimension four since diagrams require +counterterms and subtractions. This is the central point of the investigation +that we try to carry in this paper. +The knowledge of the large order behaviour of the perturbative expansion +was the source of considerable improvements in the ability to get a sensible +answer from those rapidly divergent series [8]. In practice it led to numerically +convergent algorithms developed around Zinn-Justin and collaborators[8, 9], +Borel transform of the series, followed by conformal mappings relying on +the explicit knowledge of the coefficients a.b, c here above, were among the +1 + +techniques that they used and examples, such as the anharmonic oscilla- +tor, revealed that the divergent perturbation series could lead, after those +mappings, to many digits exact results. +In our 1977 paper [4] we tried to carry the large order knowledge of +perturbative series to the ǫ-expansion. In most procedures one has to perform +a double expansion in g and ǫ. Since the fixed point g∗ is of order ǫ one has +to consider at order k, terms of order gk, gk−1ǫ, gk−2ǫ2, · · · . Fortunately the +well-known technique of minimal subtraction allows one to avoid this double- +expansion. +In this renormalization scheme [12] by definition the counter- +terms contain only poles in ǫ without any finite part. As a result one obtains +the renormalization group β and γ functions in dimension d from their four- +dimensional counterpart +βd(g) = −ǫg + β4(g) +γd(g) = γ4(g) +(1) +The fixed point g∗ is then ǫ-expanded from +ǫg∗ = β4(g∗) +(2) +(β4(g) is a series starting at order two), and the critical exponent η, which is +(twice) the conformal anomaly of the φ-field is then expanded in ǫ from +η = γ4(g∗) +(3) +Therefore computing the ǫ-expansion is reduced to four-dimensional calcula- +tions. Using the 4D large-order calculation from Lipatov method we obtained +an estimate of the asymptotic orders in the ǫ-expansion [4] which was again of +the k!-type as before. Then Zinn-Justin and co-workers based a summation +procedure on this large order behaviour [8, 9], and the process looked nicely +convergent : at least adding one more order in the ǫ-expansion improved the +previous result, instead of destroying it as the straight expansion does. +The confidence in this process was severely affected after ’t Hooft’s dis- +covery of a phenomenon [10], now called renormalons, which appears only +in renormalizable theories such as φ4 in dimension four (and not in lower +dimensions). His argument was based on the fact that one single diagram +at order k could be proportional to k! , whereas the k! , for d < 4, resulted +from their multiplicity. In a renormalizable theory a diagram, such as the +(renormalized) bubble diagram, grows logarithmically for a large external +momentum. A repeated insertion of such diagrams leads to an integral with +a log at k-th power, which gives after integration a k!. Clearly this shed +doubt on Lipatov’s estimate for large order, although ’t Hooft’s argument +didn’t show explicitely that it was wrong. It was not really clear either how +those renormalons affected quantitatively the actual calculation. +2 + +I have remained puzzled by this problem since then, wondering whether +the ambitious RG work based on the ǫ-expansion, was simply an approxi- +mation which should not be pushed too far. I want to present here a simple +well-known problem, namely the O(N)-model in the large-N limit. The rea- +son for returning to this familiar model is that it is clearly a candidate for +showing up renormalons effect, whereas this time we do not need an instan- +ton asymptotics since there are only a few diagrams of a given order. I do not +pretend that it solves the instanton-renormalon competition, but I wonder +on the basis of this calculation whereas renormalons affect the perturbation +series as badly as one could think. +2 +The O(N) model in dimension four +This, half a century-old, model consists of an N-component order parameter +φa, a = 1, · · ·, N, with an interaction invariant under O(N) [7]. This model +has been studied by hundreds of authors and elaborate techniques have al- +lowed to compute several orders in a 1/N expansion. We will limit ourselves +here to the leading large N terms for the RG functions β(g) and γ(g) in +the minimal subtraction scheme which allows to make easy contact with the +ǫ-expansion. The reason for our interest is that γ(g) is of order 1/N and the +expansion in powers of g of this leading term involves exactly the diagrams +which have been identified as generating renormalon singularities in the per- +turbative expansion. With the help of previous results on critical indices we +will try to understand what the renormalons do to this expansion. +The model is given by an (Euclidean) action +S = +� +d4x[1 +2∇φa∇φa + 1 +2m2 +oφ2 + g0 +4!N (φ2)2] +(4) +which one could regularize by an ultra-violet cut-off. For the reasons men- +tioned above we prefer here the dimensional regularization by going to 4 − ǫ +dimensions, and then renormalize minimally by a coupling constant renor- +malization Z1 and a field rescaling φ = +√ +Zϕ. We work in the massless theory +(critical temperature) and the action in terms of the renormalized field reads +(omitting mass counterterms) +S = +� +ddx[Z +2 ∇ϕa∇ϕa + µǫ gZ1 +4!N (ϕ2)2] +(5) +Varying µ at fixed bare theory we obtain the standard renormalization group +functions of the Callan-Symanzik equation [11] +ǫg + β(g)(1 + g d +dg ln(Z1/Z2)) = 0 +(6) +3 + +γ(g) = β(g) d +dg ln(Z) +(7) +We are interested in computing the leading terms in a 1/N expansion of Z1 +which is O(N0) and Z since Z = 1 + O(1/N). +2.1 +The β-function at order N0 +The coupling constant involving an explicit factor 1/N, the leading terms +maximize diagrams with internal index-loops which provide a compensating +factor N. For the two-point function the leading diagrams are of order 1/N +and the field renormalization Z = 1 + O(1/N). For the four-point function +the leading diagrams consist of a string of ”bubbles” all of order 1/N and +this yields a vertex renormalization Z1 of order N0 which we now compute. +Let us begin with a bubble with external momentum p +B(p) = +� +ddq +q2(p − q)2 +(8) +where we use the convention of omitting the usual geometric factor +2πd/2 +(2π)dΓ(d/2) +included into a rescaling of g that will be implied ; we take the scale factor +µ as unit of momentum. +r +❅ +❅ +� +� +q +p − q +� +� +❅ +❅ +r +B(p) = +A standard calculation (using a Feynman parameter) yields +B(p) = a(ǫ)p−ǫ +ǫ , +a(ǫ) = 1 − ǫ/2 +1 − ǫ +Γ3(1 − ǫ/2)Γ(1 + ǫ/2) +Γ(1 − ǫ) += 1 + ǫ +2 + O(ǫ2) +(9) +i.e. +B(p) = 1 +ǫ + 1/2 − ln p +(10) +Then the four-point function, for external indices a,a,b,b is given by a +geometric series +.. +❅ +❅ +❅ +❅ +� +� +� +� +· · · +... +... +❅ +❅ +❅ +❅ +� +� +� +� ++ ++ +... +... +... +❅ +❅ +� +� +� +� +❅ +❅+ · · · +Γ(4) = +4 + +−1 +2Γ(4) = +1 +N6gZ1 − g2Z2 +1 +N62 B(p) + g3Z3 +1 +N63 B2(p) + · · · += 1 +N +gZ1/6 +1 + gZ1/6 B(p) = 1 +N +g/6 +1/Z1 + g/6 B(p) + O(1/N2) +(11) +Note that the Z factor omitted here would give only a 1/N2 contribution. +Then taking +1/Z1 = 1 − g +6ǫ +(12) +which satisfies the minimal subtraction rule, we obtain a finite Γ(4) in the +limit ǫ → 0, at order zero in 1/N . The β-function follows immediately +0 = ǫg + β(g)[1 + +g +6ǫ − g] +(13) +i.e. +β(g) = −ǫg + 1 +6g2 + O(1/N) +(14) +One can check this result which is valid to all orders in g , but zeroth order +in 1/N, with the literature. We copy from Zinn-Justin’s book [13] in which +he used the minimal subtraction scheme +β = −ǫg + N + 8 +6N +g2 − 3N + 14 +12N2 g3 + · · · +(15) +and higher terms in g are of order 1/N, which agrees with (14) when N goes +to infinity. +2.2 +The two-point function +We are now considering the diagrams for the inverse two-point function +Γ(2)(p). +p ... +p − q +... +Γ(2)(p) = Zp2− +At order g2 we have one inserted bubble diagram, thus +Γ(2)(p) = Zp2 − 2(gZ1)2 +N62 +� +ddq B(q) +(p − q)2 + O(g3, 1/N2) +(16) +5 + +The integral over q diverges as − 1 +8ǫ, we will compute explicitly the integral +with a string of bubbles of arbitrary length herefafter. So at order g2 +Z = 1 − +1 +144N +g2 +ǫ + O(g3, 1/N2) +(17) +giving +γ(g) = +g2 +72N + O(g3, 1/N2) +(18) +. +Tot all orders in g the string of bubbles gives +Γ(2)(p) = Zp2 − 2 +N +� +k=1 +(−gZ1 +6 +)k+1 +� +ddq Bk(q) +(p − q)2 + O(1/N2) +(19) +We will sum the series later, but it is interesting to study the finite order k +dealing thus with the integral +Ik = +� +ddq Bk(q) +(p − q)2 = ak(ǫ) +ǫk +� +ddq +q−kǫ +(p − q)2 +(20) +with B(q) and a(ǫ) given in (9). +r +r +r +r +r +Ik = +p +p − q +· · · · · · +. +It is interesting to compute Ik explicitly to understand what happens at +higher k’s. Standard techniques give +Ik = +− +k +4(k + 1) +akp2−ǫ(k+1) +ǫk +Γ2(1 − ǫ/2)Γ(1 − ǫ(k + 1)/2)Γ(1 + ǫ(k + 2)/2) +Γ(1 − ǫ(k + 2)/2)Γ(1 + ǫk/2) +× +(1 − ǫ/2) +(1 − ǫ(k + 2)/4)(1 − ǫ(k + 2)/2) += − +k +4(k + 1) +p2−ǫ(k+1) +ǫk +[1 + (5k/4 + 1)ǫ + O(ǫ2)] +(21) +The calculations from thereon are straightforward, we just have to expand in +powers of ǫ the various explicit functions which appear in (20,9) and chose +Z to cancel all the poles in ǫ. For instance at order g3 if we take +Z = 1 − +g2 +N144ǫ − +g3 +N64ǫ2(1 − ǫ/4) + O(g4/N, 1/N2) +(22) +6 + +we verify that Γ(2) is finite up to this order, as renormalization theory implies, +Γ(2)(p) = p2 − +g2 +144N p2(2 ln p − 7/8) − 9g3 +64N p2 ln2 p + O(g4/N). +(23) +From (22) we obtain immediately at leading order 1/N +γ(g) = +1 +72N [g2 − 1 +24g3 + O(g4)] +(24) +which agrees for N large with the result in [13] +γ(g) = N + 2 +72N2 g2[1 − N + 8 +24N g + 5(−N2 + 18N + 100) +576N2 +g2] + · · · +(25) +We can proceed in this fashion to all orders in g, but it is tedious. One +can also sum the series (19) but it is not simple either to extract Z from the +sum. +Γ(2) = Zp2 − g2Z1 +18N +� +ddq +(p − q)2 +1 +1 +Z1B(q) + g/6 +(26) +Fortunately previous results on the 1/N expansion of this model allow +us to recover γ(g), at order 1/N, to all orders in g as explained in the next +section. +3 +Where are the renormalons? +If instead of dimensional regularization we had stayed in four dimensions +with an ultra-violet cut-off Λ we could have computed the same diagrams +The bubble-diagram (8), +� +Λ +d4v +(v − q)2v2 +(27) +behaves as ln q/Λ at small momentum. Inserted in the two-point the k -th +iterated bubble behaves as lnk q/Λ and inserted in the two-point function it +yields the integral +� +Λ d4q[ +1 +(p − q)2 − 1 +q2] lnk q/Λ +(28) +where we have explicited the zero-momentum subtraction of the massless +theory (which automatically vanished in the minimal scheme). The resulting +integral is porportional to p2 and it yields an integral over q which is infra- +red singular in the p small region. Taking ln q/Λ = −x the singular part is +given by a power of a logarithmic singularity in p , with a coefficient which +behaves for large x as +� dxe−x(−x)k, i.e. a factorial growth with alternating +7 + +signs. This is the argument for a perturbation expansion exhibiting (infra- +red) renormalons [7]. +Our goal is to compute the renormalized correlation functions, the scaling +limit of the theory in which distance are much larger than the lattice spacing +Λ−1. +We have seen in the previous section how complex is the interplay +between the diagrams and the counter-terms and this is not transparent in +the above cut-off regularized theory. So let us return to the results of the +previous section within dimensional regularization. Looking back at the k-th +order, i.e. the integral (21), we see explicitly that the k large and ǫ small +limits do not commute. The renormalization procedure is strictly defined as +ǫ goes to zero first, cancelling the poles in ǫ through the poles coming from +Z and Z1. It is only after we have removed those singular terms that we may +examine the asymptotic behavior for large k. In the minimal subtraction +scheme that we have followed here, this is done by cancelling all the poles +and multiple poles in ǫ occuring in the results such as (21) with the poles in +Z and Z1. +Could that produce the same renormalon-k! ? Indeed if we return to (21) +the factor 1 +ǫk p2−ǫ(k+1) will end up expanded k times in powers of ǫ by the time +all the subtractions manage to produce a finite four-dimensional theory (i.e, +ǫ = 0). This might yield a behaviour of the two-point function at order k +with a term p2[−(k + 1) ln p]k and of course (−k)k ≃ (−a)kk!, but this is far +from obvious, the algebra could produce a 1/k! which would kill this would- +be renormalon. However our goal here is to understand the large orders of +the ǫ-expansion and that relies on the expansion of the renormalization group +γ function. There we will see that there is no room for a renormalon large +order behavior. +4 +An explicit solution through earlier results +Since the removal of the poles in ǫ is increasingly more cumbersome when the +order increases, fortunately we can call on previous results on the large N +limit to bypass this long algebra. In fact there are better ways of dealing with +the 1/N expansion, like adding to the action a Lagrange multiplier λ(ϕ2−ψ), +replacing the quartic term in ϕ by ψ2 and tracing out the Gaussian ϕ’s. The +expansion around the saddle-point of the resulting (λ, ψ) action yields the +1/N-expansion [13, 7]. The reason for not following this procedure here is +that we needed to stick to the minimal subtraction scheme. Several terms of +the 1/N expansion of the critical exponents have been computed for arbitrary +dimensions, much more than what we needed here. In particular we find in +8 + +[7] the critical exponent η at leading order, and copy here the result +η = 1 +N [ǫ2 +2 +Γ(1 − ǫ) +Γ3(1 − ǫ/2)Γ(1 + ǫ/2) +1 − ǫ +(1 − ǫ/4)(1 − ǫ/2)] + O( 1 +N2) +(29) +But we know that η = γ(g∗) and g∗ = 6ǫ + O(1/N). Therefore we obtain +the RG function γ(g) by replacing ǫ by g/6 in η providing the result to all +orders in g +γ(g) = +g2 +72N [ +Γ(1 − g/6) +Γ3(1 − g/12)Γ(1 + g/12) +1 − g/6 +(1 − g/24)(1 − g/12)] + O( 1 +N2) (30) +This is the result valid to all orders in g, first order in 1/N, that we were +looking for. One verifies easily that this, expanded in g, reproduces what we +had found before at low order (24). +5 +Concluding remarks +• The result (30) is analytic in g in the neighbourhood of the origin : it +is meromorphic in g with the closest singularity at g = 12. Had the +large orders of the expansion in g be growing factorially that the result +would not be analytic at g = 0. This is reminiscent of what is familiar in +matrix models : the matrix integral is not analytic in g, the coefficient +in gTrM4. However if one considers the large N limit, and the successive +terms of the 1/N-expansion, every term of the expansion is analytic at +g = 0. It has been argued by previous authors that renormalons are +not present in ϕ4 +4 [16] : the explicit calculation performed here confirms +this position. +• The potentially dangerous renormalons do not show up in the final +result +• We have not shown that other correlation functions, other than the +one we have computed here, could not show renormalons, but it seems +likely, in view of what we did, that they are simply absent at first order +in 1/N and we are inclined to believe that this remains true to all orders +in a 1/N expansion. +• We have not shown that renormalons would not show up at fixed N, +but the argument in their favour being a priori operative, but finally +absent, for the case that we have considered here, we see no reason +to believe that they spoil the old result [4] on the large orders of the +ǫ-expansion. +9 + +• The model that we have considered, was considered as a candidate +for infra-red renormalons. We have nothing to say on potential UV +renormalons as in gauge theories. +• Many contemporary scientists probably consider that the problem dis- +cussed here is obsolete ; who needs the ǫ-expansion given the magnif- +icent precision of the conformal bootstrap [14], which defeats earlier +methods in their accuracy at predicting critical exponents ? However +I believe that there are many problems of interest in the scaling region +for which the tools of conformal bootstrap are not (not yet?) available. +For instance, the universal scaling equation of state, still relies on ex- +pansions : it was my first article (with Wallace and Wilson) [15] using +the ǫ-expansion, fifty years ago! +Acknowledgement I thank Giorgio Parisi for a discussion which led me +to reconsider this ancient story. +References +[1] KG. Wilson and ME. Fisher, Phys. Rev. Lett. 28, 240 (1972). Critical +Exponents in 3.99 Dimensions +[2] E. Br´ezin, JC. Le Guillou, J. Zinn-Justin, J. and BG. Nickel, Phys. Lett. +44A (1973) 227 Higher order contributions to critical exponents +This was only O(ǫ3) ; nowadays five or six terms of the ǫ-expansion have +been computed, confirming its numerical divergence. +[3] LN. Lipatov, Sov. Phys. JETP 45, 216 (1977) Divergence of the +perturbation-theory series and the quasi-classical theory +[4] E.Br´ezin, J-C. Le Guillou and J.Zinn-Justin, Phys.Rev. D15, 1544 +(1977), Phys.Rev. D15, 1558 (1977) Perturbation theory at large order; +I and II +[5] CM. Bender and TT. Wu, Phys. Rev. Lett. 27, 461 (1971) Large order +behavior of perturbation theory ; Phys. Bev. D 7, 1620 (1973) +[6] E. Br´ezin and G. Parisi, Journ.Stat.Phys. 19, 269 (1978) Critical expo- +nents and large-order behavior of perturbation theory +[7] M.Moshe and J.Zinn-Justin, Phys. Reports 385, 69 (2003), +Quantum +field theory in the large N limit : a review +10 + +[8] JC Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980) Critical +exponents from field theory +[9] R. Guida and J. Zinn-Justin Journ. Phys.A 31, 8103, (1998) Critical +exponents of the N-vector model +[10] G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics?, +Subnucl. Ser. 15 (1979) 943 +[11] CG. Callan, Phys.Rev. D2, 1541 (1970), Broken Scale Invariance in +Scalar Field Theory +K. Symanzik, Comm. Math. Phys.18, 227 (1970), Small distance be- +haviour in field theory and power counting. +[12] G. ’t Hooft, Nucl. Phys. B. 61, 455(1973) Dimensional regularization +and the renormalization group +S. Weinberg, Phys.Rev.D8, 3497 1973) New Approach to the Renormal- +ization Group +[13] J. Zinn-Justin Quantum Field Theory and Critical Phenomena, Claren- +don Press (Oxford 1989) +[14] S. El-Showk, M. F. Paulos, D. Poland, S.Rychkov and D. Simmons- +Duffin J.Stat.Phys. 157 (2014) 869, Solving the 3d Ising Model with the +Conformal Bootstrap II. c-Minimization and Precise Critical Exponents +and many subsequent work. +[15] E.Br´ezin, DJ Wallace and KG Wilson, Phys.Rev.Lett. 29, 591 (1972), +Feynman-Graph Expansion for the Equation of State near the Critical +Point +[16] IM.Suslov, Sov. Phys. JETP 100, 1188 (2005) Divergent perturbation +series +11 + diff --git a/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/load_file.txt b/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85068f4d2e083913ed7fe61985458a3581f9269b --- /dev/null +++ b/qtAzT4oBgHgl3EQfO_sW/content/tmp_files/load_file.txt @@ -0,0 +1,247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf,len=246 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='01174v1 [hep-th] 3 Jan 2023 January 2023 Should we worry about renormalons in the ǫ-expansion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Br´ezina) a) Laboratoire de Physique de l’Ecole normale sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, e-mail: brezin@lpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='fr Submitted for publication in a book commemorating Michael Fisher (Amnon Aharony, Ora Entin, David Huse, Leo Radzihovsky, editors) Abstract Turning the divergent ǫ-expansion into a numerically sensible algorithm, relies on the knowledge of the behaviour of the large order contri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Two different pictures are known to compete there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The first one was based on Lipatov’s instantons, which is known to deal with the multi- plicity of Feynman diagrams which grows factorially at high orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' However this was challenged by ’t Hooft’s renormalons who pointed out that renormal- ization could yield a similar growth through one single diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We study here a well-known model, the O(N) model, in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The reason for returning to this familiar model, is that it deals with diagrams known to give renormalon effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='Through an explicit analytic result, we find no sign of a non-analyticity of perturbation theory due to these renormalons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 1 Introduction A little more than fifty years ago K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='Wilson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='Fisher introduced the ǫ-expansion in the celebrated article [1] Critical Exponents in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='99 Dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This article has had a considerable influence both in the area of critical phenomena and in quantum field theory in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' However in practice, if the first two terms of the expansion provided often a reasonable approxima- tion to the measured indices in dimension three, the situation deteriorated if one pushed the procedure to higher orders [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Clearly the expansion looked divergent, it is believed to be at best asymptotic, and in fact limited in its ap- plicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Of course summation procedure, such as Borel transforms, Pad´e approximants, could be tried, but in the absence of further indication, it was a blind shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Therefore at the time analytic calculations, and computer simulations, appeared to be limited in their ability of reproducing precision measurements by the renormalization group approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The situation changed significantly in 1977 with Lipatov approach to the characterization of large orders in pertubation theory [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' His instanton method , which he developed for the field theory gφ4 in dimension four, pre- dicted a large order behaviour of the perturbation expansion of the various correlation functions, β-function, etc, of the form gkk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' (−a)kkbc for large k, with calculable coefficients a, b, c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' I shall recall below why this information, the explicit knowledge of a, b, c, is essential to extract a numerically sensi- ble result from such a divergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' With Le Guillou and Zinn-Justin [4] we first checked these results in lower dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For instance the one- dimensional quartic anharmonic oscillator had been extensively studied by Bender and Wu [5] who had found, on the basis of a WKB method, that indeed perturbation theory diverged with a k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' growth as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We verified that the instanton method reproduced exactly what they had found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Ex- tension to field theory in higher dimensions was similar [4] except that the instanton solution is not known in analytic form except in dimension four [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In dimensions lower than four the interpretation of these results is simple : at a given order all Feynman diagrams have the same sign (we are consider- ing an Euclidean field theory) and their number grows proportionaly to k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='. The situation is less transparent in dimension four since diagrams require counterterms and subtractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This is the central point of the investigation that we try to carry in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The knowledge of the large order behaviour of the perturbative expansion was the source of considerable improvements in the ability to get a sensible answer from those rapidly divergent series [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In practice it led to numerically convergent algorithms developed around Zinn-Justin and collaborators[8, 9], Borel transform of the series, followed by conformal mappings relying on the explicit knowledge of the coefficients a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='b, c here above, were among the 1 techniques that they used and examples, such as the anharmonic oscilla- tor, revealed that the divergent perturbation series could lead, after those mappings, to many digits exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In our 1977 paper [4] we tried to carry the large order knowledge of perturbative series to the ǫ-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In most procedures one has to perform a double expansion in g and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Since the fixed point g∗ is of order ǫ one has to consider at order k, terms of order gk, gk−1ǫ, gk−2ǫ2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Fortunately the well-known technique of minimal subtraction allows one to avoid this double- expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In this renormalization scheme [12] by definition the counter- terms contain only poles in ǫ without any finite part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' As a result one obtains the renormalization group β and γ functions in dimension d from their four- dimensional counterpart βd(g) = −ǫg + β4(g) γd(g) = γ4(g) (1) The fixed point g∗ is then ǫ-expanded from ǫg∗ = β4(g∗) (2) (β4(g) is a series starting at order two), and the critical exponent η, which is (twice) the conformal anomaly of the φ-field is then expanded in ǫ from η = γ4(g∗) (3) Therefore computing the ǫ-expansion is reduced to four-dimensional calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Using the 4D large-order calculation from Lipatov method we obtained an estimate of the asymptotic orders in the ǫ-expansion [4] which was again of the k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='-type as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Then Zinn-Justin and co-workers based a summation procedure on this large order behaviour [8, 9], and the process looked nicely convergent : at least adding one more order in the ǫ-expansion improved the previous result, instead of destroying it as the straight expansion does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The confidence in this process was severely affected after ’t Hooft’s dis- covery of a phenomenon [10], now called renormalons, which appears only in renormalizable theories such as φ4 in dimension four (and not in lower dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' His argument was based on the fact that one single diagram at order k could be proportional to k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' , whereas the k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' , for d < 4, resulted from their multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In a renormalizable theory a diagram, such as the (renormalized) bubble diagram, grows logarithmically for a large external momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' A repeated insertion of such diagrams leads to an integral with a log at k-th power, which gives after integration a k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='. Clearly this shed doubt on Lipatov’s estimate for large order, although ’t Hooft’s argument didn’t show explicitely that it was wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' It was not really clear either how those renormalons affected quantitatively the actual calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 2 I have remained puzzled by this problem since then, wondering whether the ambitious RG work based on the ǫ-expansion, was simply an approxi- mation which should not be pushed too far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' I want to present here a simple well-known problem, namely the O(N)-model in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The rea- son for returning to this familiar model is that it is clearly a candidate for showing up renormalons effect, whereas this time we do not need an instan- ton asymptotics since there are only a few diagrams of a given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' I do not pretend that it solves the instanton-renormalon competition, but I wonder on the basis of this calculation whereas renormalons affect the perturbation series as badly as one could think.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 2 The O(N) model in dimension four This, half a century-old, model consists of an N-component order parameter φa, a = 1, · · ·, N, with an interaction invariant under O(N) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This model has been studied by hundreds of authors and elaborate techniques have al- lowed to compute several orders in a 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We will limit ourselves here to the leading large N terms for the RG functions β(g) and γ(g) in the minimal subtraction scheme which allows to make easy contact with the ǫ-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The reason for our interest is that γ(g) is of order 1/N and the expansion in powers of g of this leading term involves exactly the diagrams which have been identified as generating renormalon singularities in the per- turbative expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' With the help of previous results on critical indices we will try to understand what the renormalons do to this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The model is given by an (Euclidean) action S = � d4x[1 2∇φa∇φa + 1 2m2 oφ2 + g0 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='N (φ2)2] (4) which one could regularize by an ultra-violet cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For the reasons men- tioned above we prefer here the dimensional regularization by going to 4 − ǫ dimensions, and then renormalize minimally by a coupling constant renor- malization Z1 and a field rescaling φ = √ Zϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We work in the massless theory (critical temperature) and the action in terms of the renormalized field reads (omitting mass counterterms) S = � ddx[Z 2 ∇ϕa∇ϕa + µǫ gZ1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='N (ϕ2)2] (5) Varying µ at fixed bare theory we obtain the standard renormalization group functions of the Callan-Symanzik equation [11] ǫg + β(g)(1 + g d dg ln(Z1/Z2)) = 0 (6) 3 γ(g) = β(g) d dg ln(Z) (7) We are interested in computing the leading terms in a 1/N expansion of Z1 which is O(N0) and Z since Z = 1 + O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='1 The β-function at order N0 The coupling constant involving an explicit factor 1/N, the leading terms maximize diagrams with internal index-loops which provide a compensating factor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For the two-point function the leading diagrams are of order 1/N and the field renormalization Z = 1 + O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For the four-point function the leading diagrams consist of a string of ”bubbles” all of order 1/N and this yields a vertex renormalization Z1 of order N0 which we now compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Let us begin with a bubble with external momentum p B(p) = � ddq q2(p − q)2 (8) where we use the convention of omitting the usual geometric factor 2πd/2 (2π)dΓ(d/2) included into a rescaling of g that will be implied ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' we take the scale factor µ as unit of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' r ❅ ❅ � � q p − q � � ❅ ❅ r B(p) = A standard calculation (using a Feynman parameter) yields B(p) = a(ǫ)p−ǫ ǫ , a(ǫ) = 1 − ǫ/2 1 − ǫ Γ3(1 − ǫ/2)Γ(1 + ǫ/2) Γ(1 − ǫ) = 1 + ǫ 2 + O(ǫ2) (9) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' B(p) = 1 ǫ + 1/2 − ln p (10) Then the four-point function, for external indices a,a,b,b is given by a geometric series .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='. ❅ ❅ ❅ ❅ � � � � · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' ❅ ❅ ❅ ❅ � � � � + + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' ❅ ❅ � � � � ❅ ❅+ · · · Γ(4) = 4 −1 2Γ(4) = 1 N6gZ1 − g2Z2 1 N62 B(p) + g3Z3 1 N63 B2(p) + · · · = 1 N gZ1/6 1 + gZ1/6 B(p) = 1 N g/6 1/Z1 + g/6 B(p) + O(1/N2) (11) Note that the Z factor omitted here would give only a 1/N2 contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Then taking 1/Z1 = 1 − g 6ǫ (12) which satisfies the minimal subtraction rule, we obtain a finite Γ(4) in the limit ǫ → 0, at order zero in 1/N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The β-function follows immediately 0 = ǫg + β(g)[1 + g 6ǫ − g] (13) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' β(g) = −ǫg + 1 6g2 + O(1/N) (14) One can check this result which is valid to all orders in g , but zeroth order in 1/N, with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We copy from Zinn-Justin’s book [13] in which he used the minimal subtraction scheme β = −ǫg + N + 8 6N g2 − 3N + 14 12N2 g3 + · · · (15) and higher terms in g are of order 1/N, which agrees with (14) when N goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='2 The two-point function We are now considering the diagrams for the inverse two-point function Γ(2)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' p − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Γ(2)(p) = Zp2− At order g2 we have one inserted bubble diagram, thus Γ(2)(p) = Zp2 − 2(gZ1)2 N62 � ddq B(q) (p − q)2 + O(g3, 1/N2) (16) 5 The integral over q diverges as − 1 8ǫ, we will compute explicitly the integral with a string of bubbles of arbitrary length herefafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' So at order g2 Z = 1 − 1 144N g2 ǫ + O(g3, 1/N2) (17) giving γ(g) = g2 72N + O(g3, 1/N2) (18) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Tot all orders in g the string of bubbles gives Γ(2)(p) = Zp2 − 2 N � k=1 (−gZ1 6 )k+1 � ddq Bk(q) (p − q)2 + O(1/N2) (19) We will sum the series later, but it is interesting to study the finite order k dealing thus with the integral Ik = � ddq Bk(q) (p − q)2 = ak(ǫ) ǫk � ddq q−kǫ (p − q)2 (20) with B(q) and a(ǫ) given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' r r r r r Ik = p p − q · · · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' It is interesting to compute Ik explicitly to understand what happens at higher k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Standard techniques give Ik = − k 4(k + 1) akp2−ǫ(k+1) ǫk Γ2(1 − ǫ/2)Γ(1 − ǫ(k + 1)/2)Γ(1 + ǫ(k + 2)/2) Γ(1 − ǫ(k + 2)/2)Γ(1 + ǫk/2) × (1 − ǫ/2) (1 − ǫ(k + 2)/4)(1 − ǫ(k + 2)/2) = − k 4(k + 1) p2−ǫ(k+1) ǫk [1 + (5k/4 + 1)ǫ + O(ǫ2)] (21) The calculations from thereon are straightforward, we just have to expand in powers of ǫ the various explicit functions which appear in (20,9) and chose Z to cancel all the poles in ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For instance at order g3 if we take Z = 1 − g2 N144ǫ − g3 N64ǫ2(1 − ǫ/4) + O(g4/N, 1/N2) (22) 6 we verify that Γ(2) is finite up to this order, as renormalization theory implies, Γ(2)(p) = p2 − g2 144N p2(2 ln p − 7/8) − 9g3 64N p2 ln2 p + O(g4/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' (23) From (22) we obtain immediately at leading order 1/N γ(g) = 1 72N [g2 − 1 24g3 + O(g4)] (24) which agrees for N large with the result in [13] γ(g) = N + 2 72N2 g2[1 − N + 8 24N g + 5(−N2 + 18N + 100) 576N2 g2] + · · · (25) We can proceed in this fashion to all orders in g, but it is tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' One can also sum the series (19) but it is not simple either to extract Z from the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Γ(2) = Zp2 − g2Z1 18N � ddq (p − q)2 1 1 Z1B(q) + g/6 (26) Fortunately previous results on the 1/N expansion of this model allow us to recover γ(g), at order 1/N, to all orders in g as explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 3 Where are the renormalons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' If instead of dimensional regularization we had stayed in four dimensions with an ultra-violet cut-off Λ we could have computed the same diagrams The bubble-diagram (8), � Λ d4v (v − q)2v2 (27) behaves as ln q/Λ at small momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Inserted in the two-point the k -th iterated bubble behaves as lnk q/Λ and inserted in the two-point function it yields the integral � Λ d4q[ 1 (p − q)2 − 1 q2] lnk q/Λ (28) where we have explicited the zero-momentum subtraction of the massless theory (which automatically vanished in the minimal scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The resulting integral is porportional to p2 and it yields an integral over q which is infra- red singular in the p small region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Taking ln q/Λ = −x the singular part is given by a power of a logarithmic singularity in p , with a coefficient which behaves for large x as � dxe−x(−x)k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' a factorial growth with alternating 7 signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This is the argument for a perturbation expansion exhibiting (infra- red) renormalons [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Our goal is to compute the renormalized correlation functions, the scaling limit of the theory in which distance are much larger than the lattice spacing Λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We have seen in the previous section how complex is the interplay between the diagrams and the counter-terms and this is not transparent in the above cut-off regularized theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' So let us return to the results of the previous section within dimensional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Looking back at the k-th order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' the integral (21), we see explicitly that the k large and ǫ small limits do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The renormalization procedure is strictly defined as ǫ goes to zero first, cancelling the poles in ǫ through the poles coming from Z and Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' It is only after we have removed those singular terms that we may examine the asymptotic behavior for large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In the minimal subtraction scheme that we have followed here, this is done by cancelling all the poles and multiple poles in ǫ occuring in the results such as (21) with the poles in Z and Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Could that produce the same renormalon-k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Indeed if we return to (21) the factor 1 ǫk p2−ǫ(k+1) will end up expanded k times in powers of ǫ by the time all the subtractions manage to produce a finite four-dimensional theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='e, ǫ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This might yield a behaviour of the two-point function at order k with a term p2[−(k + 1) ln p]k and of course (−k)k ≃ (−a)kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=', but this is far from obvious, the algebra could produce a 1/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' which would kill this would- be renormalon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' However our goal here is to understand the large orders of the ǫ-expansion and that relies on the expansion of the renormalization group γ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' There we will see that there is no room for a renormalon large order behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 4 An explicit solution through earlier results Since the removal of the poles in ǫ is increasingly more cumbersome when the order increases, fortunately we can call on previous results on the large N limit to bypass this long algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In fact there are better ways of dealing with the 1/N expansion, like adding to the action a Lagrange multiplier λ(ϕ2−ψ), replacing the quartic term in ϕ by ψ2 and tracing out the Gaussian ϕ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The expansion around the saddle-point of the resulting (λ, ψ) action yields the 1/N-expansion [13, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The reason for not following this procedure here is that we needed to stick to the minimal subtraction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Several terms of the 1/N expansion of the critical exponents have been computed for arbitrary dimensions, much more than what we needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' In particular we find in 8 [7] the critical exponent η at leading order, and copy here the result η = 1 N [ǫ2 2 Γ(1 − ǫ) Γ3(1 − ǫ/2)Γ(1 + ǫ/2) 1 − ǫ (1 − ǫ/4)(1 − ǫ/2)] + O( 1 N2) (29) But we know that η = γ(g∗) and g∗ = 6ǫ + O(1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Therefore we obtain the RG function γ(g) by replacing ǫ by g/6 in η providing the result to all orders in g γ(g) = g2 72N [ Γ(1 − g/6) Γ3(1 − g/12)Γ(1 + g/12) 1 − g/6 (1 − g/24)(1 − g/12)] + O( 1 N2) (30) This is the result valid to all orders in g, first order in 1/N, that we were looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' One verifies easily that this, expanded in g, reproduces what we had found before at low order (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 5 Concluding remarks The result (30) is analytic in g in the neighbourhood of the origin : it is meromorphic in g with the closest singularity at g = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Had the large orders of the expansion in g be growing factorially that the result would not be analytic at g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' This is reminiscent of what is familiar in matrix models : the matrix integral is not analytic in g, the coefficient in gTrM4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' However if one considers the large N limit, and the successive terms of the 1/N-expansion, every term of the expansion is analytic at g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' It has been argued by previous authors that renormalons are not present in ϕ4 4 [16] : the explicit calculation performed here confirms this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' The potentially dangerous renormalons do not show up in the final result We have not shown that other correlation functions, other than the one we have computed here, could not show renormalons, but it seems likely, in view of what we did, that they are simply absent at first order in 1/N and we are inclined to believe that this remains true to all orders in a 1/N expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We have not shown that renormalons would not show up at fixed N, but the argument in their favour being a priori operative, but finally absent, for the case that we have considered here, we see no reason to believe that they spoil the old result [4] on the large orders of the ǫ-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 9 The model that we have considered, was considered as a candidate for infra-red renormalons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' We have nothing to say on potential UV renormalons as in gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Many contemporary scientists probably consider that the problem dis- cussed here is obsolete ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' who needs the ǫ-expansion given the magnif- icent precision of the conformal bootstrap [14], which defeats earlier methods in their accuracy at predicting critical exponents ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' However I believe that there are many problems of interest in the scaling region for which the tools of conformal bootstrap are not (not yet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=') available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' For instance, the universal scaling equation of state, still relies on ex- pansions : it was my first article (with Wallace and Wilson) [15] using the ǫ-expansion, fifty years ago!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Acknowledgement I thank Giorgio Parisi for a discussion which led me to reconsider this ancient story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' References [1] KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Wilson and ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Fisher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 28, 240 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Critical Exponents in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='99 Dimensions [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Br´ezin, JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Le Guillou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Zinn-Justin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' and BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Nickel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 44A (1973) 227 Higher order contributions to critical exponents This was only O(ǫ3) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' nowadays five or six terms of the ǫ-expansion have been computed, confirming its numerical divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' [3] LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Lipatov, Sov.' metadata={'source': 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+page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' 29, 591 (1972), Feynman-Graph Expansion for the Equation of State near the Critical Point [16] IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content='Suslov, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} +page_content=' JETP 100, 1188 (2005) Divergent perturbation series 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQfO_sW/content/2301.01174v1.pdf'} diff --git 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AZROUL1, A. BENKIRANE2 AND M. SRATI3 +Abstract. In this paper, first we introduce the s(., .)-fractional Musielak- +Sobolev spaces W s(x,y)LΦx,y(Ω). Next, by means of Ekeland’s variational +principal, we show that there exists λ∗ > 0 such that any λ ∈ (0, λ∗) is an +eigenvalue for the following problem +(Pa) + + + + + +(−∆)s(x,.) +a(x,.) u += +λ|u|q(x)−2u +in +Ω, +u += +0 +in +RN \ Ω, +where Ω is a bounded open subset of RN with C0,1-regularity and bounded +boundary. +Contents +1. +Introduction +1 +2. +Preliminaries results +4 +3. +s(., .)-fractional Musielak-Sobolev spaces +7 +4. +Existence results and proofs +14 +5. +Examples +18 +Disclosure statement +20 +Data Availability Statement +20 +References +20 +1. Introduction +The theory of fractional modular spaces is well developed in the last years. +In particular, in the fractional Orlicz-Sobolev spaces W sLΦ(Ω) (see [5, 6, 7, +8, 9, 16, 14, 17]) and in the fractional Sobolev spaces with variable expo- +nents W s,p(x,y)(Ω) (see [10, 11, 12, 13, 23]). The study of variational problems +where the modular function satisfies nonpolynomial growth conditions instead +of having the usual p-structure arouses much interest in the development of ap- +plications to electrorheological fluids as an important class of non-Newtonian +fluids (sometimes referred to as smart fluids). The electro-rheological fluids +are characterized by their ability to drastically change the mechanical proper- +ties under the influence of an external electromagnetic field. A mathematical +2010 Mathematics Subject Classification. 46E35, 35R11, 35J20, 47G20. +Key words and phrases. +s(., .)-Fractional Musielak-Sobolev spaces, eigenvalue problems, +Ekeland’s variational principal. +1 + +2 +E. AZROUL, A. BENKIRANE, AND M. SRATI +model of electro-rheological fluids was proposed by Rajagopal and Ruzicka (we +refer the reader to [21, 22, 30] for more details). +On the other hand, when we try to integrate both the functional structures +of variable exponent Lebesgue spaces and Orlicz spaces, we are led to the so- +called Musielak-Orlicz spaces. This later functional structure was extensively +studied since the 1950s by Nakano [29] and developed by Musielak and Orlicz +[27, 28]. A natural question has been asked: can we see the same generalization +in the fractional case? The answer to this question is given by Azroul et al in +[2, 3, 4]. That is, the authors have introduced the fractional Musielak-Sobolev +space W sLΦx,y(Ω). This framework is a natural generalization of the above- +mentioned functional spaces. +W s,p(Ω) +W s,p(x,y)(Ω) +W sLΦ(Ω) +W sLΦx,y(Ω) +In present work, we study the existence of the eigenvalues of problem involv- +ing non-local operator (−∆)s(x,.) +a(x,.) with variable exponents s. Here we would like +to emphasize that in our work we have considered the variable growth on the +exponent s as well. Moreover, due to the nonlocality of the operator (−∆)s(x,.) +a(x,.), +we introduce the s(., .)-fractional Musielak-Sobolev space W s(x,y)LΦx,y(Ω). +So, we are interested to study the following eigenvalue problem +(Pa) + + + + + +(−∆)s(x,.) +a(x,.) u += +λ|u|q(x)−2u +in +Ω, +u += +0 +in +RN \ Ω, +where Ω is an open bounded subset in RN, N ⩾ 1, with Lipschitz boundary +∂Ω, q : Ω → (1, ∞) is bounded continuous function, and (−∆)s(x,.) +a(x,.) is the +nonlocal integro-differential operator of elliptic type defined as follows +(−∆)s(x,.) +a(x,.) u(x) = 2 lim +εց0 +� +RN\Bε(x) +a(x,y) +�|u(x) − u(y)| +|x − y|s(x,y) +� u(x) − u(y) +|x − y|s(x,y) +dy +|x − y|N+s(x,y) , +for all x ∈ RN, where: +• s(., .) : Ω × Ω → (0, 1) is a continuous function such that: +s(x, y) = s(y, x) ∀x, y ∈ Ω × Ω, +(1.1) +0 < s− = inf +Ω×Ω +s(x, y) ⩽ s+ = sup +Ω×Ω +s(x, y) < 1. +(1.2) + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +3 +• a(x,y)(t) := a(x, y, t) : Ω × Ω × R −→ R is symmetric function : +a(x, y, t) = a(y, x, t) ∀(x, y, t) ∈ Ω × Ω × R, +(1.3) +and the function : ϕ(., ., .) : Ω × Ω × R −→ R defined by +ϕx,y(t) := ϕ(x, y, t) = + + + +a(x, y, |t|)t +for +t ̸= 0, +0 +for +t = 0, +is increasing homeomorphism from R onto itself. Let +Φx,y(t) := Φ(x, y, t) = +� t +0 +ϕx,y(τ)dτ +for all (x, y) ∈ Ω × Ω, +and all t ⩾ 0. +Then, Φx,y is a Musielak function (see [28]), that is +⋆ Φ(x, y, .) is a Φ-function for every (x, y) ∈ Ω × Ω, i.e., is continuous, +nondecreasing function with Φ(x, y, 0) = 0, Φ(x, y, t) > 0 for t > 0 and +Φ(x, y, t) → ∞ as t → ∞. +⋆ For every t ⩾ 0, Φ(., ., t) : Ω × Ω −→ R is a measurable function. +Also, we take �ax(t) := �a(x, t) = a(x,x)(t) ∀ (x, t) ∈ Ω × R. Then the function +�ϕ(., .) : Ω × R −→ R defined by : +�ϕx(t) := �ϕ(x, t) = + + + +�a(x, |t|)t +for +t ̸= 0, +0 +for +t = 0, +is increasing homeomorphism from R onto itself. If we set +�Φx(t) := �Φ(x, t) = +� t +0 +�ϕx(τ)dτ +for all t ⩾ 0. +(1.4) +Then, �Φx is also a Musielak function. +Note that, when we take ax,y(t) = |t|p(x,y)−2 where p : Ω × Ω −→ (1, +∞) +is a continuous bounded function, then our nonlocal operator (−∆)s(x,.) +a(x,.) which +can be seen as a generalization of the nonlocal operator with variable exponent +(−∆)s(x,.) +p(x,.) (see [15]) defined as +(−∆)s(x,.) +p(x,.)u(x) = 2 lim +εց0 +� +RN\Bε(x) +|u(x) − u(y)|p(x,y)−2(u(x) − u(y)) +|x − y|N+s(x,y)p(x,y) +dy, +for all x ∈ RN, (see also [32, 33]). +Moreover, this work brings us back to introduce the s(., .)-fractional a- +Laplacian (−∆)s(x,.) +a +if ax,y(t) = a(t), i.e. the function a is independent of +variables x, y. +Then, we obtain the following nonlocal operator (−∆)s(x,.) +a +, +defined as +(−∆)s(x,.) +a +u(x) = 2 lim +εց0 +� +RN \Bε(x) +a +�|u(x) − u(y)| +|x − y|s(x,y) +� u(x) − u(y) +|x − y|s(x,y) +dy +|x − y|N+s(x,y) , +for all x ∈ RN. + +4 +E. AZROUL, A. BENKIRANE, AND M. SRATI +This paper is organized as follows, In Section 1, we set the problem (Pa). +Moreover, we are introduced the new nonlocal integro-differential operator +(−∆)s(x,.) +a(x,.). The Section 2, is devoted to recall some properties of fractional +Musielak-Sobolev spaces. In section 3, we introduce the s(., .)-fractional Musielak- +Sobolev spaces and we establish some qualitative properties of these new +spaces. In section 4, by means of Ekeland’s variational principle, we obtain the +existence of λ∗ > 0 such that for any λ ∈ (0, λ∗), is an eigenvalue for the fol- +lowing problem (Pa). In Section 5, we present some examples which illustrate +our results. +2. Preliminaries results +To deal with this situation we define the fractional Musielak-Sobolev space +to investigate Problem (Pa). Let us recall the definitions and some elementary +properties of this spaces. We refer the reader to [2, 3] for further reference and +for some of the proofs of the results in this section. +For the function �Φx given in (1.4), we introduce the Musielak space as follows +L �Φx(Ω) = +� +u : Ω −→ R mesurable : +� +Ω +�Φx(λ|u(x)|)dx < ∞ for some λ > 0 +� +. +The space L �Φx(Ω) is a Banach space endowed with the Luxemburg norm +||u||�Φx = inf +� +λ > 0 : +� +Ω +�Φx +�|u(x)| +λ +� +dx ⩽ 1 +� +. +The conjugate function of Φx,y is defined by Φx,y(t) = +� t +0 ϕx,y(τ)dτ +for all (x, y) ∈ +Ω × Ω +and all t ⩾ 0, where ϕx,y : R −→ R is given by ϕx,y(t) := ϕ(x, y, t) = +sup {s : ϕ(x, y, s) ⩽ t} . Throughout this paper, we assume that there exist +two positive constants ϕ+ and ϕ− such that +1 < ϕ− ⩽ tϕx,y(t) +Φx,y(t) ⩽ ϕ+ < +∞ for all (x, y) ∈ Ω × Ω +and all t ⩾ 0. +(Φ1) +This relation implies that +1 < ϕ− ⩽ t�ϕx(t) +�Φx(t) +⩽ ϕ+ < +∞, for all x ∈ Ω +and all t ⩾ 0. +(2.1) +It follows that Φx,y and �Φx satisfy the global ∆2-condition (see [26]), written +Φx,y ∈ ∆2 and �Φx ∈ ∆2, that is, +Φx,y(2t) ⩽ K1Φx,y(t) +for all (x, y) ∈ Ω × Ω, +and all t ⩾ 0, +(2.2) +and +�Φx(2t) ⩽ K2 �Φx(t) +for any x ∈ Ω, +and all t ⩾ 0, +(2.3) +where K1 and K2 are two positive constants. +Furthermore, we assume that Φx,y satisfies the following condition +the function [0, ∞) ∋ t �→ Φx,y( +√ +t) is convex. +(Φ2) + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +5 +Definition 2.1. Let Ax(t), Bx(t) : R+ ×Ω −→ R+ be two Musielak functions. +Ax is stronger (resp essentially stronger) than Bx, Ax ≻ Bx (resp Ax ≻≻ Bx) +in symbols, if for almost every x ∈ Ω +B(x, t) ⩽ A(x, at), +t ⩾ t0 ⩾ 0, +for some (resp for each) a > 0 and t0 (depending on a). +Remark 2.1 ([1, Section 8.5]). Ax ≻≻ Bx is equivalent to the condition +lim +t→∞ +� +sup +x∈Ω +B(x, λt) +A(x, t) +� += 0, +for all λ > 0. +Now, we define the fractional Musielak-Sobolev space as introduce in [2] as +follows +W sLΦx,y(Ω) = +� +u ∈ L � +Φx(Ω) : +� +Ω +� +Ω +Φx,y +�λ|u(x) − u(y)| +|x − y|s +� +dxdy +|x − y|N < ∞ for some λ > 0 +� +. +This space can be equipped with the norm +||u||s,Φx,y = ||u||�Φx + [u]s,Φx,y, +(2.4) +where [.]s,Φx,y is the Gagliardo seminorm defined by +[u]s,Φx,y = inf +� +λ > 0 : +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +λ|x − y|s +� +dxdy +|x − y|N ⩽ 1 +� +. +Theorem 2.1. ([2]). Let Ω be an open subset of RN, and let s ∈ (0, 1). The +space W sLΦx,y(Ω) is a Banach space with respect to the norm (2.4), and a +separable (resp. reflexive) space if and only if Φx,y ∈ ∆2 (resp. Φx,y ∈ ∆2 and +Φx,y ∈ ∆2). Furthermore, if Φx,y ∈ ∆2 and Φx,y( +√ +t) is convex, then the space +W sLΦx,y(Ω) is an uniformly convex space. +Definition 2.2. ([2]). We say that Φx,y satisfies the fractional boundedness +condition, written Φx,y ∈ Bf, if +sup +(x,y)∈Ω×Ω +Φx,y(1) < ∞. +(Φ3) +Theorem 2.2. ([2]). Let Ω be an open subset of RN, and 0 < s < 1. Assume +that Φx,y ∈ Bf. Then, +C2 +0(Ω) ⊂ W sLΦx,y(Ω). +Lemma 2.1. ([2]) Assume that (Φ1) is satisfied. Then the following inequal- +ities hold true: +Φx,y(σt) ⩾ σϕ−Φx,y(t) +for all t > 0 and any σ > 1, +(2.5) +Φx,y(σt) ⩾ σϕ+Φx,y(t) +for all t > 0 and any σ ∈ (0, 1), +(2.6) +Φx,y(σt) ⩽ σϕ+Φx,y(t) +for all t > 0 and any σ > 1, +(2.7) +Φx,y(t) ⩽ σϕ−Φx,y +� t +σ +� +for all t > 0 and any σ ∈ (0, 1). +(2.8) + +6 +E. AZROUL, A. BENKIRANE, AND M. SRATI +For any u ∈ W sLΦx,y(Ω), we define the modular function on W sLΦx,y(Ω) as +follows +Ψ(u) = +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx. +(2.9) +Proposition 2.1. ([2]). Assume that (Φ1) is satisfied. Then, for any u ∈ +W sLΦx,y(Ω), the following relations hold true: +||u||s,Φx,y > 1 =⇒ ||u||ϕ− +s,Φx,y ⩽ Ψ(u) ⩽ ||u||ϕ+ +s,Φx,y, +(2.10) +||u||s,Φx,y < 1 =⇒ ||u||ϕ+ +s,Φx,y ⩽ Ψ(u) ⩽ ||u||ϕ− +s,Φx,y. +(2.11) +We Define a closed linear subspace of W sLΦx,y(Ω) as follows +W s +0 LΦx,y(Ω) = +� +u ∈ W sLΦx,y(RN) : u = 0 a.e in RN \ Ω +� +. +Theorem 2.3. ([3]) Let Ω be a bounded open subset of RN with C0,1-regularity +and bounded boundary, let s ∈ (0, 1). Then there exists a positive constant γ +such that +||u||�Φx ⩽ γ[u]s,Φx,y for all u ∈ W s +0 LΦx,y(Ω). +We denote by �Φ−1 +x +the inverse function of �Φx which satisfies the following +conditions: +� 1 +0 +�Φ−1 +x (τ) +τ +N+s +N +dτ < ∞ +for all x ∈ Ω, +(2.12) +� ∞ +1 +�Φ−1 +x (τ) +τ +N+s +N +dτ = ∞ +for all x ∈ Ω. +(2.13) +Note that, if ϕx,y(t) = |t|p(x,y)−1, then (2.12) holds precisely when sp(x, y) < N +for all (x, y) ∈ Ω × Ω. +If (2.13) is satisfied, we define the inverse Musielak conjugate function of �Φx +as follows +(�Φ∗ +x,s)−1(t) = +� t +0 +�Φ−1 +x (τ) +τ +N+s +N +dτ. +(2.14) +Theorem 2.4. [3] Let Ω be a bounded open subset of RN with C0,1-regularity +and bounded boundary. If (2.12) and (2.13) hold, then +W sLΦx,y(Ω) ֒→ L �Φ∗x,s(Ω). +(2.15) +Moreover, the embedding +W sLΦx,y(Ω) ֒→ LBx(Ω), +(2.16) +is compact for all Bx ≺≺ �Φ∗ +x,s. +Next, we recall some useful properties of variable exponent spaces. For more +details we refer the reader to [20, 24], and the references therein. +Consider the set +C+(Ω) = +� +q ∈ C(Ω) : q(x) > 1 for all x ∈ Ω +� +. + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +7 +For all q ∈ C+(Ω), we define +q+ = sup +x∈Ω +q(x) +and +q− = inf +x∈Ω +q(x). +For any q ∈ C+(Ω), we define the variable exponent Lebesgue space as +Lq(x)(Ω) = +� +u : Ω −→ R measurable : +� +Ω +|u(x)|q(x)dx < +∞ +� +. +This vector space endowed with the Luxemburg norm, which is defined by +∥u∥Lq(x)(Ω) = inf +� +λ > 0 : +� +Ω +���� +u(x) +λ +���� +q(x) +dx ⩽ 1 +� +is a separable reflexive Banach space. +A very important role in manipulating the generalized Lebesgue spaces with +variable exponent is played by the modular of the Lq(x)(Ω) space, which defined +by +ρq(.) : Lq(x)(Ω) −→ R +u �−→ ρq(.)(u) = +� +Ω +|u(x)|q(x)dx. +Proposition 2.2. Let u ∈ Lq(x)(Ω), then we have +(i) ∥u∥Lq(x)(Ω) < 1 (resp. = 1, > 1) ⇔ ρq(.)(u) < 1 (resp. = 1, > 1), +(ii) ∥u∥Lq(x)(Ω) < 1 ⇒ ∥u∥q+ +Lq(x)(Ω) ⩽ ρq(.)(u) ⩽ ∥u∥q− +Lq(x)(Ω), +(iii) ∥u∥Lq(x)(Ω) > 1 ⇒ ∥u∥q− +Lq(x)(Ω) ⩽ ρq(.)(u) ⩽ ∥u∥q+ +Lq(x)(Ω). +Finally, the proof of our existence result is based on the following Ekeland’s +variational principle theorem. +Theorem 2.5. ([19]) Let V be a complete metric space and F : V −→ R ∪ +{+∞} be a lower semicontinuous functional on V , that is bounded below and +not identically equal to +∞. Fix ε > 0 and a point u ∈ V such that +F(u) ⩽ ε + inf +x∈V F(x). +Then for every γ > 0, there exists some point v ∈ V such that : +F(v) ⩽ F(u), +d(u, v) ⩽ γ +and for all w ̸= v +F(w) > F(v) − ε +γ d(v, w). +3. s(., .)-fractional Musielak-Sobolev spaces +Due to the non-locality of the operator (−∆)s(x,.) +a(x,.), we introduce the s(., .)- +fractional Musielak-Sobolev space as follows +W s(x,y)LΦx,y(Ω) = +� +u ∈ L � +Φx(Ω) : +� +Ω +� +Ω +Φx,y +� λ|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N < ∞ for some λ > 0 +� +. +This space can be equipped with the norm +||u||s(x,y),Φx,y = ||u||�Φx + [u]s(x,y),Φx,y, +(3.1) + +8 +E. AZROUL, A. BENKIRANE, AND M. SRATI +where [u]s(x,y),Φx,y is the Gagliardo seminorm defined by +[u]s(x,y),Φx,y = inf +� +λ > 0 : +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ 1 +� +. +To simplify notations, throughout the rest of this paper, we set +Ds(x,y)u = u(x) − u(y) +|x − y|s(x,y) +and dµ = +dxdy +|x − y|N . +Remark 3.1. +a)− For the case: Φx,y(t) = Φ(t), i.e. Φ is independent of variables x, y, we can +introduce the s(., .)-fractional Orlicz-Sobolev spaces W s(x,y)LΦ(Ω) as follows +W s(x,y)LΦ(Ω) = +� +u ∈ LΦ(Ω) : +� +Ω +� +Ω +Φ +�λ|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N < ∞ for some λ > 0 +� +. +b)− For the case: Φx,y(t) = |t|p(x,y) for all (x, y) ∈ Ω × Ω, where p : Ω × Ω −→ +(1, +∞) is a continuous bounded function such that +1 < p− = +min +(x,y)∈Ω×Ω +p(x, y) ⩽ p(x, y) ⩽ p+ = +max +(x,y)∈Ω×Ω +p(x, y) < +∞, +and +p is symmetric, that is, +p(x, y) = p(y, x) +for all (x, y) ∈ Ω × Ω. +If denoted by ¯p(x) = p(x, x) for all x ∈ Ω. Then, we replace LΦx by Lp(x), and +W s(x,y)LΦx,y by W s(x,y),p(x,y) and we refer them as variable exponent Lebesgue +spaces, and s(., .)-fractional Sobolev spaces with variable exponent respectively, +(see [15, 32, 33]) defined by +Lp(x)(Ω) = +� +u : Ω −→ R measurable : +� +Ω +|u(x)|p(x)dx < +∞ +� +, +and +W = W s(x,y),p(x,y)(Ω) += +� +u ∈ L¯p(x)(Ω) : +� +Ω×Ω +|u(x) − u(y)|p(x,y) +λp(x,y)|x − y|s(x,y)p(x,y)+N dxdy < +∞, +for some +λ > 0 +� +. +with the norm +∥u∥W = ∥u∥L¯p(x)(Ω) + [u]W , +where [.]W is a Gagliardo seminorm with variable exponent given by +[u]W = [u]s(x,y),p(x,y) = inf +� +λ > 0 : +� +Ω×Ω +|u(x) − u(y)|p(x,y) +λp(x,y)|x − y|N+s(x,y)p(x,y) dxdy ⩽ 1 +� +. +Theorem 3.1. Let Ω be an open subset of RN. The space W s(x,y)LΦx,y(Ω) is +a Banach space with respect to the norm (3.1), and a separable (resp. reflexive) +space if and only if Φx,y ∈ ∆2 (resp. Φx,y ∈ ∆2 and Φx,y ∈ ∆2). Furthermore, +if Φx,y ∈ ∆2 and Φx,y( +√ +t) is convex, then the space W s(x,y)LΦx,y(Ω) is an +uniformly convex space. +Proof of this Theorem is similar to [2, Theorem 2.1]. +Theorem 3.2. Let Ω be a bounded open subset of RN. Then +W s+LΦx,y(Ω) ֒→ W s(x,y)LΦx,y(Ω) ֒→ W s−LΦx,y(Ω). + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +9 +Proof. Let u ∈ W s+LΦx,y(Ω) and λ > 0, we have +� +Ω +� +Ω +Φx,y +�|Ds(x,y)u| +λ +� +dxdy +|x − y|N = +� +Ω +� +Ω +Φx,y +� +|Ds+u| +λ +1 +|x − y|s(x,y)−s+ +� +dxdy +|x − y|N +⩽ +� +Ω +� +Ω +Φx,y +� +|Ds+u| +λ +� +dxdy +|x − y|N+p(s(x,y)−s+) +⩽ sup +Ω×Ω +|x − y|p(s+−s(x,y)) +� +Ω +� +Ω +Φx,y +� +|Ds+u| +λ +� +dxdy +|x − y|N . +where p = {ϕ− or ϕ+} is given by Lemma 2.1. This implies that +[u]s(x,y),Φx,y ⩽ sup +Ω×Ω +|x − y|p(s+−s(x,y))[u]s+,Φx,y. +So +∥u∥s(x,y),Φx,y ⩽ c∥u∥s+,Φx,y, +where c = max +� +1, sup +Ω×Ω +|x − y|p(s+−s(x,y)) +� +. +Now, Let u ∈ W s(x,y)LΦx,y(Ω) and λ > 0, we have +� +Ω +� +Ω +Φx,y +� +|Ds−u| +λ +� +dxdy +|x − y|N = +� +Ω +� +Ω +Φx,y +�|Ds(x,y)u| +λ +1 +|x − y|s−−s(x,y) +� +dxdy +|x − y|N +⩽ +� +Ω +� +Ω +Φx,y +�|Ds(x,y)u| +λ +� +dxdy +|x − y|N+p(s−−s(x,y)) +⩽ sup +Ω×Ω +|x − y|p(s(x,y)−s−) +� +Ω +� +Ω +Φx,y +�|Ds(x,y)u| +λ +� +dxdy +|x − y|N . +This implies that +[u]s−,Φx,y ⩽ sup +Ω×Ω +|x − y|p(s(x,y)−s−)[u]s(x,y),Φx,y. +So +∥u∥s−,Φx,y ⩽ c∥u∥s(x,y),Φx,y, +where c = max +� +1, sup +Ω×Ω +|x − y|p(s(x,y)−s−) +� +. +□ +Now, combining Theorem 3.2 and Theorem 2.4, we obtain the following +results. +Corollary 3.1. Let Ω be a bounded open subset of RN with C0,1-regularity and +bounded boundary. If (2.12) and (2.13) hold, then +W s(x,y)LΦx,y(Ω) ֒→ L �Φ∗ +x,s−(Ω). +Also, the embedding +W s(x,y)LΦx,y(Ω) ֒→ LBx(Ω), +is compact for all Bx ≺≺ �Φ∗ +x,s−. + +10 +E. AZROUL, A. BENKIRANE, AND M. SRATI +For any u ∈ W s(x,y)LΦx,y(Ω), we define the modular function on W s(x,y)LΦx,y(Ω) +as follows +J(u) = +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx. +(3.2) +An important role in manipulating the s(., .)-fractional Musielak-Sobolev +spaces is played by the modular function (3.2). It is worth noticing that the +relation between the norm and the modular shows an equivalence between the +topology defined by the norm and that defined by the modular. +Proposition 3.1. Assume that (Φ1) is satisfied. Then, for any u ∈ W s(x,y)LΦx,y(Ω), +the following relations hold true: +||u||s(x,y),Φx,y > 1 =⇒ ||u||ϕ− +s(x,y),Φx,y ⩽ J(u) ⩽ ||u||ϕ+ +s(x,y),Φx,y, +(3.3) +||u||s(x,y),Φx,y < 1 =⇒ ||u||ϕ+ +s(x,y),Φx,y ⩽ J(u) ⩽ ||u||ϕ− +s(x,y),Φx,y. +(3.4) +Proof. To simplify the notation, we take ∥u∥x,y := ||u||s(x,y),Φx,y. First, we +show that if ||u||x,y > 1, then J(u) ⩽ ||u||ϕ+. Indeed, let u ∈ W s(x,y)LΦx,y(Ω) +such that ||u||x,y > 1. Using the definition of the Luxemburg norm and the +relation (2.7), we get +J(u) = +� +Ω +� +Ω +Φx,y +� +||u||x,y +|u(x) − u(y)| +||u||x,y|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx +� +||u||x,y |u(x)| +||u||x,y +� +dx +⩽ ||u||ϕ+ +x,y +� +Ω +� +Ω +Φx,y +� +|u(x) − u(y)| +||u||x,y|x − y|s(x,y) +� +dxdy +|x − y|N + ||u|| � +ϕ+ +x,y +� +Ω +�Φx +� |u(x)| +||u||x,y +� +dx +⩽ ||u||ϕ+ +x,y +�� +Ω +� +Ω +Φx,y +� +|u(x) − u(y)| +||u||x,y|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx +� |u(x)| +||u||x,y +� +dx +� +⩽ ||u||ϕ+ +x,y. +Next, assume that ||u||x,y > 1. Let β ∈ (1, ||u||x,y), by (2.5), we have +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx +⩾ βϕ− � +Ω +� +Ω +Φx,y +� |u(x) − u(y)| +β|x − y|s(x,y) +� +dxdy +|x − y|N + β � +ϕ− � +Ω +�Φx +�|u(x)| +β +� +dx +⩾ βϕ− �� +Ω +� +Ω +Φx,y +� |u(x) − u(y)| +β|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx +�|u(x)| +β +� +dx +� +. +Since β < ||u||x,y, we find +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +β|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx +�|u(x)| +β +� +dx > 1. +Thus, we have +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx ⩾ βϕ−. +Letting β ր ||u||x,y, we deduce that (3.3) holds true. + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +11 +Next, we show that J(u) ⩽ ||u||ϕ− +x,y for all u ∈ W s(x,y)LΦx,y(Ω) with ||u||x,y < +1. Using the definition of the Luxemburg norm and (2.8), we obtain +J(u) ⩽ ||u||ϕ− +x,y +� +Ω +� +Ω +Φx,y +� +|u(x) − u(y)| +||u||x,y|x − y|s(x,y) +� +dxdy +|x − y|N + ||u||�ϕ− +x,y +� +Ω +�Φx +� |u(x)| +||u||x,y +� +dx +⩽ ||u||ϕ− +x,y +�� +Ω +� +Ω +Φx,y +� +|u(x) − u(y)| +||u||x,y|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx +� |u(x)| +||u||x,y +� +dx +� +⩽ ||u||ϕ− +x,y. +Let ξ ∈ (0, ||u||x,y). From (2.6), it follows that +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx +⩾ ξϕ+ � +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +ξ|x − y|s(x,y) +� +dxdy +|x − y|N + ξ �ϕ+ � +Ω +Φ +�|u(x)| +ξ +� +dx +⩾ ξϕ+ �� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +ξ|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +Φ +�|u(x)| +ξ +� +dx +� +. +(3.5) +Defining v(x) = u(x) +ξ +for all x ∈ Ω. Then, ||v||x,y = ||u||x,y +ξ +> 1. Using +relation (2.10), we find +� +Ω +� +Ω +Φx,y +�|v(x) − v(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|v(x)|) dx ⩾ ||v||ϕ− +x,y > 1. (3.6) +Combining (3.5) and (3.6), we deduce that +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N + +� +Ω +�Φx (|u(x)|) dx ⩾ ξϕ−. +Letting ξ ր ||u||x,y in the above inequality, we obtain that relation (3.4) holds +true. +□ +Similar to Proposition 2.1, we obtain the following results. +Proposition 3.2. Assume that (Φ1) is satisfied, Then, for any u ∈ W s(x,y)LΦx,y(Ω), +the following assertions hold true: +[u]s(x,y),Φx,y > 1 =⇒ [u]ϕ− +s(x,y),Φx,y ⩽ φ(u) ⩽ [u]ϕ+ +s(x,y),Φx,y, +[u]s(x,y),Φx,y < 1 =⇒ [u]ϕ+ +s(x,y),Φx,y ⩽ φ(u) ⩽ [u]ϕ− +s(x,y),Φx,y, +where φ(u) = +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N . +Now, we introduce a closed linear subspace of W s(x,y)LΦx,y(Ω) as follows +W s(x,y) +0 +LΦx,y(Ω) = +� +u ∈ W s(x,y)LΦx,y(RN) u = 0 in RN\Ω +� +. +Then we have the following generalized Poincaré type inequality. +Theorem 3.3. Let Ω be a bounded open subset of RN with C0,1-regularity and +bounded boundary. Then there exists a positive constant γ such that +||u||�Φx ⩽ γ[u]s(x,y),Φx,y for all u ∈ W s(x,y) +0 +LΦx,y(Ω). + +12 +E. AZROUL, A. BENKIRANE, AND M. SRATI +Proof. Let u ∈ W s(x,y) +0 +LΦx,y(Ω), by Theorem 3.2, we have +[u]s−,Φx,y ⩽ c[u]s(x,y),Φx,y, +(3.7) +on the other hand, by Theorem 2.3, there exists a positive constant γ′ such +that +||u||�Φx ⩽ γ′[u]s−,Φx,y for all u ∈ W s− +0 LΦx,y(Ω). +(3.8) +Thus, we combining (3.7) with (3.8), we obtain +||u||�Φx ⩽ γ[u]s(x,y),Φx,y for all u ∈ W s(x,y) +0 +LΦx,y(Ω). +with γ = cγ′. +□ +Now, in order to study Problem (Pa), it is important to encode the boundary +condition u = 0 in RN \Ω in the weak formulation. In the scalar case, Servadei +and Valdinoci [31] introduced a new function spaces to study the variational +functionals related to the fractional Laplacian by observing the interaction +between Ω and RN \ Ω. Subsequently, inspired by the work of Servadei and +Valdinoci [31], Azroul et al in [11], have introduced the fractional Sobolev +space with variable exponent, to study the variational functionals related to +the fractional p(x, .)-Laplacian operator by observing the interaction between +Ω and RN \Ω. Motivated by the above papers, and due to the nonlocality of the +operator (−∆)s(x,.) +a(x,.), we introduce the following s(., .)-fractional Orlicz-Sobolev +space as follows +W (x,y)LΦw,y(Q) = +� +u ∈ LΦx,y(Ω) : +� +Q +Φx,y +�λ|u(x) − u(y)| +|x − y|s(x,y) +� +dxdy +|x − y|N < ∞ +for some λ > 0 +� +, +where Q = R2N \ (CΩ × CΩ) with CΩ = RN \ Ω. This spaces are equipped +with the norm, +||u|| = ||u||�Φx + [u], +(3.9) +where [.] is the Gagliardo seminorm, defined by +[u] = inf +� +λ > 0 : +� +Q +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ 1 +� +. +Similar to the spaces (W s(x,y)LΦx,y(Ω), ∥.∥s(x,y),Φx,y) we have that (W s(x,y)LΦx,y(Q), ∥.∥) +is a separable reflexive Banach spaces. +Now, let W s(x,y) +0 +LΦx,y(Q) denotes the following linear subspace of W s(x,y)LΦx,y(Q), +W s(x,y) +0 +LΦx,y(Q) = +� +u ∈ W s(x,y)LΦx,y(Q) : u = 0 a.e in RN \ Ω +� +with the norm +[u] = inf +� +λ > 0 : +� +Q +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ 1 +� +. +In the following theorem, we compare the spaces W s(x,y)LΦx,y(Ω) and W s(x,y)LΦx,y(Q). +Theorem 3.4. The following assertions hold: + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +13 +1) The continuous embedding +W s(x,y)LΦx,y(Q) ⊂ W s(x,y)LΦx,y(Ω) +holds true. +2) If u ∈ W s(x,y) +0 +LΦx,y(Q), then u ∈ W s(x,y)LΦx,y(RN) and +||u||s(x,y),Φx,y ⩽ ||u||W s(x,y)LΦx,y(RN) = ||u||. +Proof. 1) Let u ∈ W s(x,y)LΦx,y(Q), since Ω × Ω ⊊ Q, then for all λ > 0 we +have +� +Ω +� +Ω +Φx,y +� |u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ +� +Q +Φx,y +� |u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N . +(3.10) +We set +As(x,y) +λ,Ω×Ω = +� +λ > 0 : +� +Ω +� +Ω +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ 1 +� +and +As(x,y) +λ,Q += +� +λ > 0 : +� +Q +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N ⩽ 1 +� +. +By (3.10), it is easy to see that As(x,y) +λ,Q +⊂ As(x,y) +λ,Ω×Ω. Hence +[u]s(x,y),Φx,y = inf +λ>0 As(x,y) +λ,Ω×Ω ⩽ [u] = inf +λ>0 As(x,y) +λ,Q . +(3.11) +Consequently, by definitions of the norms ∥u∥s(x,y),Φx,y and ∥u∥, we obtain +∥u∥s(x,y),Φx,y ⩽ ∥u∥ < ∞. +2) Let u ∈ W s(x,y) +0 +LΦx,y(Q), then u = 0 in RN \Ω. So, ∥u∥L � +Φx(Ω) = ∥u∥L � +Φx(RN). +Since +� +R2N Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N = +� +Q +Φx,y +�|u(x) − u(y)| +λ|x − y|s(x,y) +� +dxdy +|x − y|N +for all λ > 0. Then [u]W s(x,y)LΦx,y(RN) = [u]. Thus, we get +||u||s(x,y),Φx,y ⩽ ||u||W s(x,y)LΦx,y(RN) = ||u||. +□ +Corollary 3.2. (Poincaré inequality) Let Ω be a bounded open subset of RN +with C0,1-regularity and bounded boundary. Then there exists a positive con- +stant c such that, +∥u∥�Φx ⩽ c[u], +∀u ∈ W s(x,y) +0 +LΦx,y(Q). +Proof. Let u ∈ W s(x,y) +0 +LΦx,y(Q), by Theorem 3.4, we have u ∈ W s(x,y) +0 +LΦx,y(Ω). +Then by Theorem 3.3, there exists a positive constant γ such that, +∥u∥�Φx ⩽ γ[u]s(x,y),Φx,y. + +14 +E. AZROUL, A. BENKIRANE, AND M. SRATI +Combining the above inequality with (3.11), we obtain that +∥u∥�Φx ⩽ c[u], +∀u ∈ W s(x,y) +0 +LΦx,y(Q). +□ +Remark 3.2. From Corollary 3.2, we deduce that [.] is a norm on W s(x,y) +0 +LΦx,y(Q) +which is equivalent to the norm ∥.∥. +4. Existence results and proofs +In this section, we analyze problem (Pa). under the following basic assump- +tions +q− < ϕ− +(4.1) +and +lim +t→∞ +� +sup +x∈Ω +|t|q+ +(�Φx,s−)∗(kt) +� += 0 ∀k > 0. +(4.2) +The dual space of +� +W s(x,y) +0 +LΦx,y(Q), ||.|| +� +is denoted by +�� +W s(x,y) +0 +LΦx,y(Q) +�∗ +, ||.||∗ +� +. +Definition 4.1. We say that λ ∈ R is an eigenvalue of Problem (Pa) if there +exists u ∈ W s(x,y) +0 +LΦx,y(Q) \ {0} such that +� +Q +ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)vdµ − λ +� +Ω +|u|q(x)−2uvdx = 0 +for all v ∈ W s(x,y) +0 +LΦx,y(Q). +We point that if λ is an eigenvalue of Problem (Pa) then the corresponding +u ∈ W s(x,y) +0 +LΦx,y(Q) \ {0} is a weak solution of (Pa). +Our main results is given by the following theorem. +Theorem 4.1. There exists λ∗ > 0 such that for any λ ∈ (0, λ∗) is an eigen- +value of Problem (Pa). +Remark 4.1. By (4.2), we can apply Theorem 3.4 and Corollary 3.1 we obtain +that W s(x,y) +0 +LΦx,y(Q) is compactly embedded in Lq+(Ω). That fact combined +with the continuous embedding of Lq+(Ω) in Lq(x)(Ω), ensures that W s(x,y) +0 +LΦx,y(Q) +is compactly embedded in Lq(x)(Ω). +Next, for all λ ∈ R, we define the energetic function associated with problem +(Pa) Jλ : W s(x,y) +0 +LΦx,y(Q) → R, as +Jλ(u) = +� +Q +Φx,y +�|u(x) − u(y)| +|x − y|s(x,y) +� +dµ − λ +� +Ω +1 +q(x)|u|q(x)dx. +By a standard argument to [5] and [6], we have Jλ ∈ C1(W s(x,y) +0 +LΦx,y(Q), R), +� +J′ +λ(u), v +� += +� +Q +ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)vdµ − λ +� +Ω +|u|q(x)−2uvdx. + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +15 +Lemma 4.1. Assume that the hypothesis of Theorem 4.1 is fulfilled. Then, +there exists λ∗ > 0 such that for any λ ∈ (0, λ∗), there are ρ, α > 0, such that +Jλ(u) ⩾ α > 0 for any u ∈ W s(x,y) +0 +LΦx,y(Q) with ||u|| = ρ. +Proof. Since W s(x,y) +0 +LΦx,y(Q) is continuously embedded in Lq(x)(Ω), it follows +that there exists a positive constant c1 such that +||u|| ⩾ c1||u||q(x) ∀u ∈ W s(x,y) +0 +LΦx,y(Q) +(4.3) +we fix ρ ∈ (0, 1) such that ρ < 1 +c1 +. Then relation (4.3) implies that +∥u∥q(x) < 1 for all u ∈ W s(x,y) +0 +LΦx,y(Q) with ||u|| = ρ. +Then, we can apply Proposition 2.2, and we have +� +Ω +|u(x)|q(x)dx ⩽ ∥u∥q− +q(x) +for all u ∈ W s(x,y) +0 +LΦx,y(Q) +with ||u|| = ρ. (4.4) +Relation (4.3) and (4.4) implies that +� +Ω +|u(x)|q(x)dx ⩽ cq− +1 ∥u∥q− +for all u ∈ W s(x,y) +0 +LΦx,y(Q) +with ||u|| = ρ. +(4.5) +Taking into account Relation (4.5), we deduce that for any u ∈ W s(x,y) +0 +LΦx,y(Q) +with ||u|| = ρ, the following inequalities hold true: +Jλ(u) ⩾ ∥u∥ϕ+ − λ +q− +� +Ω +|u(x)|q(x)dx +⩾ ∥u∥ϕ+ − λcq− +1 +q− ∥u∥q− += ρq− +� +ρϕ+−q− − λcq− +1 +q− +� +. +Hence, if we define +λ∗ = ρϕ+−q− +2cq− +1 +q−. +(4.6) +Then, for any λ ∈ (0, λ∗) and u ∈ W s(x,y) +0 +LΦx,y(Q) with ||u|| = ρ, we have +Jλ(u) ⩾ α > 0, +such that +α = ρϕ+ +2 . +This completes the proof. +□ +Lemma 4.2. Assume that the hypothesis of Theorem 4.1 is fulfilled. Then, +there exists φ > 0 such that φ ⩾ 0, φ ̸= 0, and Jλ(tφ) < 0 for t > 0 small +enough. +Proof. By assumption (4.1) we can chose ε0 > 0 such that q− + ε0 < ϕ−. On +the other hand, since q ∈ C(Ω), it follows that there exists an open set Ω0 ⊂ Ω +such that |q(x) − q−| < ε0 for all x ∈ Ω0. Thus, q(x) ⩽ q− + ε0 < ϕ− for all + +16 +E. AZROUL, A. BENKIRANE, AND M. SRATI +x ∈ Ω0. Let φ ∈ C∞ +0 (Ω) be such that supp(φ) ⊃ Ω0, φ(x) = 1 for all x ∈ Ω0, +and 0 ⩽ φ ⩽ 1 in Ω0. Then, for any t ∈ (0, 1), we have +Jλ(tφ) = +� +Q +Φx,y +� +t|Ds(x,y)φ| +� +dµ − λ +� +Ω +1 +q(x)tq(x)|φ|q(x)dx +⩽ +� +Q +tϕ−Φx,y +� +|Ds(x,y)φ| +� +dµ − λ +� +Ω0 +tq(x) +q(x)|φ|q(x)dx +⩽ tϕ− � +Q +Φx,y +� +|Ds(x,y)φ| +� +dµ − λtq−+ε0 +q+ +� +Ω0 +|φ|q(x)dx. +Therefore Jλ(tφ) < 0, for t < δ1/(ϕ−−q−−ε0) with +0 < δ < min + + + + + + + +1, +λ +q+ +� +Ω0 +|φ|q(x)dx +� +Q +Φx,y +� +|Ds(x,y)φ| +� +dµ + + + + + + + +. +This is possible, since we claim that +� +Q +Φx,y +� +|Ds(x,y)φ| +� +dµ > 0. +Indeed, it is clear that +� +Ω0 +|φ|q(x)dx ⩽ +� +Ω +|φ|q(x)dx ⩽ +� +Ω +|φ|q−dx. +On the other hand, since W s(x,y) +0 +LΦx,y(Q) is continuously embedded in Lq−(Ω), +it follows that there exists a positive constant c such that +∥φ∥q− ⩽ c||φ||. +The last two inequalities imply that +∥φ∥ > 0 +and combining this fact with Proposition 3.1, the claim follows at once. The +proof of the lemma is now completed. +□ +Proof of Theorem 4.1. Let λ∗ > 0 be defined as in (4.6) and λ ∈ (0, λ∗). By +Lemma 4.1 it follows that on the boundary oh the ball centered in the origin +and of radius ρ in W s(x,y) +0 +LΦx,y(Q), denoted by Bρ(0), we have +inf +∂Bρ(0) Jλ > 0. +On the other hand, by Lemma 4.2, there exists φ ∈ W s(x,y) +0 +LΦx,y(Q) such that +Jλ(tφ) < 0 for all t > 0 small enough. Moreover for any u ∈ Bρ(0), we have +Jλ(u) ⩾ ∥u∥ϕ− − λcq− +1 +q− ∥u∥q−. +It follows that +−∞ < c := inf +Bρ(0) +Jλ < 0. + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +17 +We let now 0 < ε < +inf +∂Bρ(0) Jλ− inf +Bρ(0) Jλ. Applying Theorem 2.5 to the functional +Jλ : Bρ(0) −→ R, we find uε ∈ Bρ(0) such that + + + + + +Jλ(uε) +< inf +Bρ(0) +Jλ + ε, +Jλ(uε) +< Jλ(u) + ε||u − uε||, +u ̸= uε. +Since Jλ(uε) ⩽ inf +Bρ(0) +Jλ + ε ⩽ inf +Bρ(0) Jλ + ε < +inf +∂Bρ(0) Jλ, we deduce uε ∈ Bρ(0). +Now, we define Λλ : Bρ(0) −→ R by +Λλ(u) = Jλ(u) + ε||u − uε||. +It’s clear that uε is a minimum point of Λλ and then +Λλ(uε + tv) − Λλ(uε) +t +⩾ 0 +for small t > 0, and any v ∈ Bρ(0). The above relation yields +Jλ(uε + tv) − Jλ(uε) +t ++ ε||v|| ⩾ 0. +Letting t → it follows that ⟨J′ +λ(uε), v⟩ + ε||v|| > 0 and we infer that +||J′ +λ(uε)||∗ ⩽ ε. +We deduce that there exists a sequence {un} ⊂ Bρ(0) such that +Jλ(un) −→ c and J′ +λ(un) −→ 0. +(4.7) +It is clear that {un} is bounded in W s(x,y) +0 +LΦx,y(Q). Thus, there exists u0 ∈ +W s(x,y) +0 +LΦx,y(Q), such that up to a subsequence {un} converges weakly to u0 +in W s(x,y) +0 +LΦx,y(Q). +On the other hand, since W s(x,y) +0 +LΦx,y(Q) is compactly embedded in Lq(x)(Ω), +it follows that {un} converges strongly to u0 in Lq(x)(Ω). Then by Hölder in- +equality, we have that +lim +n→∞ +� +Ω +|un|q(x)−2un(un − u0)dx = 0. +This fact and relation (4.7), implies that +lim +n→∞ +� +J′ +λ(un), un − u0 +� += 0. +Thus we deduce that +lim +n→∞ +� +Q +ax,y(|Ds(x,y)un|)Ds(x,y)un +� +Ds(x,y)un − Ds(x,y)u0 +� +dµ = 0. +(4.8) +Since {un} converge weakly to u0 in W s(x,y) +0 +LΦx,y(Q), by relation (4.8), we find +that +lim +n→∞ +� +Q +� +ax,y(|Ds(x,y)un|)Ds(x,y)un − ax,y(|Ds(x,y)u0|)Ds(x,y)u0 +� � +Ds(x,y)un − Ds(x,y)u0 +� +dµ = 0. +(4.9) + +18 +E. AZROUL, A. BENKIRANE, AND M. SRATI +Since, Φx,y is convex, we have +Φx,y(|Ds(x,y)u|) ⩽ Φx,y +�|Ds(x,y)u + Ds(x,y)v| +2 +� ++ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)u − Ds(x,y)v +2 +Φx,y(|Ds(x,y)v|) ⩽ Φx,y +�|Ds(x,y)u + Ds(x,y)v| +2 +� ++ax,y(|Ds(x,y)v|)Ds(x,y)v Ds(x,y)v − Ds(x,y)u +2 +for every u, v ∈ W s(x,y) +0 +LΦx,y(Q). Adding the above two relations and integrat- +ing over Q, we find that +1 +2 +� +Q +� +ax,y(|Ds(x,y)u|)Ds(x,y)u − ax,y(|Ds(x,y)v|)Ds(x,y)v +� � +Ds(x,y)u − Ds(x,y)v +� +dµ +⩾ +� +Q +Φx,y(|Ds(x,y)u|)dµ + +� +Q +Φx,y(|Ds(x,y)v|)dµ − 2 +� +Q +Φx,y +� +|Ds(x,y)u − Ds(x,y)v| +2 +� +dµ, +(4.10) +for every u, v ∈ W s(x,y) +0 +LΦx,y(Q). On the other hand, since for each, we know +that Φx,y : [0, ∞) → R is an increasing continuous function, with Φx,y(0) = 0. +Then by the conditions (Φ1) and (Φ2), we can apply [25, Lemma 2.1] in order +to obtain +1 +2 +�� +Q +Φx,y(|Ds(x,y)u|)dµ + +� +Q +Φx,y(|Ds(x,y)v|)dµ +� +⩾ +� +Q +Φx,y +�|Ds(x,y)u + Ds(x,y)v| +2 +� +dµ + +� +Q +Φx,y +�|Ds(x,y)u − Ds(x,y)v| +2 +� +dµ, +(4.11) +for every u, v ∈ W s(x,y) +0 +LΦx,y(Q). By (4.10) and (4.11), we have +� +Q +� +ax,y(|Ds(x,y)u|)Ds(x,y)u − ax,y(|Ds(x,y)v|)Ds(x,y)v +� � +Ds(x,y)u − Ds(x,y)v +� +dµ +⩾ 4 +� +Q +Φi +�|Ds(x,y)u − Ds(x,y)v| +2 +� +dµ +(4.12) +for every u, v ∈ W s(x,y) +0 +LΦx,y(Q). +Relations (4.9) and (4.12) show that {un} converge strongly to u0 in W s(x,y) +0 +LΦx,y(Q). +Then by relation (4.7), we have +Jλ(u0) = c1 > 0 and J′ +λ(u0) = 0. +Then, u0 is a nontrivial weak solution for Problem (Pa). This complete the +proof. +□ +5. Examples +In this section we point certain examples of functions ϕx,y and Φx,y which +illustrate the results of this paper. +Example 5.1. As a first example, we can take +ϕx,y(t) = ϕ1(x, y, t) = p(x, y) |t|p(x,y)−2t +log(1 + |t|) +for all t ⩾ 0, +and thus, +Φx,y(t) = p(x, y) +|t|p(x,y) +log(1 + |t|) + +� |t| +0 +τ p(x,y) +(1 + τ)(log(1 + τ))2 dτ, + +s(., .)-FRACTIONAL MUSIELAK-SOBOLEV SPACES +19 +with p ∈ C(Q) satisfies 2 ⩽ p(x, y) < N for all (x, y) ∈ Q. +Then, in this case problem (Pa) becomes +(P1) + + + +(−∆)s(x,.) +ϕ1 +u += +λ|u|q(x)−2u +in +Ω +u += +0 +in +RN \ Ω, +with +(−∆)s(x,.) +ϕ1 +u(x) = p.v. +� +Ω +p(x, y)|Ds(x,y)u|p(x,y)−2Ds(x,y)u +log(1 + |Ds(x,y)u|)|x − y|N+s(x,y) dy +for all x ∈ Ω. +It easy to see that Φx,y is a Musielak function and satisfy condition (Φ3). +Moreover, for each (x, y) ∈ Q fixed, by Example 3 on p 243 in [18], we have +p(x, y) − 1 ⩽ tϕx,y(t) +Φx,y(t) ⩽ p(x, y) ∀(x, y) ∈ Q, +∀t ⩾ 0. +Thus, (Φ1) holds true with ϕ− = p− − 1 and ϕ+ = p+. +Finally, we point out that trivial computations imply that +d2(Φx,y( +√ +t)) +dt2 +⩾ 0 +for all (x, y) ∈ Q and t ⩾ 0. Thus, relation (Φ2) hold true. +Hence, we derive an existence result for problem (P1) which is given by the +following Remark. +Remark 5.1. If p− − 1 > q−. Then there exists λ∗ > 0 such that for any +λ ∈ (0, λ∗) is an eigenvalue of Problem (P1). +Example 5.2. As a second example, we can take +ϕx,y(t) = ϕ2(x, y, t) = p(x, y) log(1 + α + |t|)|t|p(x,y)−2t for all t ⩾ 0 +and so, +Φx,y(t) = log(1 + |t|)|t|p(x,y) − +� |t| +0 +τ p(x,y) +1 + τ dτ, +where α > 0 is a constant and p ∈ C(Ω × Ω) satisfies 2 ⩽ p(x, y) < N for all +(x, y) ∈ Q. +Then we consider the following fractional p(x, .)-problem +(P2) + + + +(−∆)s(x,y) +ϕ2 +u += +λ|u|q(x)−2u +in +Ω +u += +0 +in +RN \ Ω, +where +(−∆)s(x,y) +ϕ2 +u(x) = p.v. +� +Ω +p(x, y) log(1 + α + |Ds(x,y)u|).|Ds(x,y)u|p(x,y)−2Ds(x,y)u +|x − y|N+s(x,y) +dy +for all x ∈ Ω. +It easy to see that Φx,y is a Musielak function and satisfy condition (Φ3). +Next, we remark that for each (x, y) ∈ Q fixed, we have +p(x, y) ⩽ tϕx,y(t) +Φx,y(t) +for all t ⩾ 0. + +20 +E. AZROUL, A. BENKIRANE, AND M. SRATI +By the above information and taking ϕ− = p−, we have +1 < p− ⩽ t.ϕx,y(t) +Φx,y(t) +for all (x, y) ∈ Q +and all t ⩾ 0. +On the other hand, some simple computations imply +lim +t→∞ +t.ϕx,y(t) +Φx,y(t) = p(x, y) for all (x, y) ∈ Q, +and +lim +t→0 +t.ϕx,y(t) +Φx,y(t) = p(x, y) + 1 for all (x, y) ∈ Q, +Thus, we remark that t.ϕx,y(t) +Φx,y(t) is continuous on Q × [0, ∞). Moreover, +1 < p− ⩽ lim +t→0 +t.ϕx,y(t) +Φx,y(t) ⩽ p+ + 1 < ∞, +and +1 < p− ⩽ lim +t→∞ +t.ϕx,y(t) +Φx,y(t) ⩽ p+ + 1 < ∞. +It follows that +ϕ+ < ∞. +We conclude that relation (Φ1) is satisfied. Finally, we point out that trivial +computations imply that +d2(Φx,y( +√ +t)) +dt2 +⩾ 0 +for all (x, y) ∈ Q and t ⩾ 0. Thus, relation (Φ2) hold true. +Remark 5.2. If p− > q−. 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A class of p1(x, .) and p2(x, .)-fractional Kirchhoff- +type problem with variable s(x, .)-order and without the Ambrosetti-Rabinowitz condition in +RN Open Mathematics, vol. 20, no. 1, 2022, pp. 267-290. https://doi.org/10.1515/math- +2022-0028 +[33] Zuo J, An T, Fiscella A. A critical Kirchhoff-type problem driven by a p(.)- +fractional Laplace operator with variable s(.)-order. Math Meth Appl Sci. 2020;1-15. +https://doi.org/10.1002/mma.6813. +E. Azroul, A. Benkirane and M. Srati +Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, +Laboratory of Mathematical Analysis and Applications, Fez, Morocco. +Email address: 1elhoussine.azroul@gmail.com +Email address: 2abd.benkirane@gmail.com +Email address: 3mohammed.srati@usmba.ac.ma + diff --git a/rNAyT4oBgHgl3EQfmPiH/content/tmp_files/load_file.txt b/rNAyT4oBgHgl3EQfmPiH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..584e928d5232f0759040e0b47e00646e2f08fd75 --- /dev/null +++ b/rNAyT4oBgHgl3EQfmPiH/content/tmp_files/load_file.txt @@ -0,0 +1,852 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf,len=851 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='00467v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='AP] 1 Jan 2023 EIGENVALUE TYPE PROBLEM IN s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE2 AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In this paper, first we introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak- Sobolev spaces W s(x,y)LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Next, by means of Ekeland’s variational principal, we show that there exists λ∗ > 0 such that any λ ∈ (0, λ∗) is an eigenvalue for the following problem (Pa) \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') u = λ|u|q(x)−2u in Ω, u = 0 in RN \\ Ω, where Ω is a bounded open subset of RN with C0,1-regularity and bounded boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Preliminaries results 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak-Sobolev spaces 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Existence results and proofs 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Examples 18 Disclosure statement 20 Data Availability Statement 20 References 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Introduction The theory of fractional modular spaces is well developed in the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In particular, in the fractional Orlicz-Sobolev spaces W sLΦ(Ω) (see [5, 6, 7, 8, 9, 16, 14, 17]) and in the fractional Sobolev spaces with variable expo- nents W s,p(x,y)(Ω) (see [10, 11, 12, 13, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The study of variational problems where the modular function satisfies nonpolynomial growth conditions instead of having the usual p-structure arouses much interest in the development of ap- plications to electrorheological fluids as an important class of non-Newtonian fluids (sometimes referred to as smart fluids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The electro-rheological fluids are characterized by their ability to drastically change the mechanical proper- ties under the influence of an external electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' A mathematical 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 46E35, 35R11, 35J20, 47G20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-Fractional Musielak-Sobolev spaces, eigenvalue problems, Ekeland’s variational principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI model of electro-rheological fluids was proposed by Rajagopal and Ruzicka (we refer the reader to [21, 22, 30] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, when we try to integrate both the functional structures of variable exponent Lebesgue spaces and Orlicz spaces, we are led to the so- called Musielak-Orlicz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This later functional structure was extensively studied since the 1950s by Nakano [29] and developed by Musielak and Orlicz [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' A natural question has been asked: can we see the same generalization in the fractional case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The answer to this question is given by Azroul et al in [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' That is, the authors have introduced the fractional Musielak-Sobolev space W sLΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This framework is a natural generalization of the above- mentioned functional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' W s,p(Ω) W s,p(x,y)(Ω) W sLΦ(Ω) W sLΦx,y(Ω) In present work, we study the existence of the eigenvalues of problem involv- ing non-local operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') with variable exponents s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Here we would like to emphasize that in our work we have considered the variable growth on the exponent s as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover, due to the nonlocality of the operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ), we introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak-Sobolev space W s(x,y)LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' So, we are interested to study the following eigenvalue problem (Pa) \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') u = λ|u|q(x)−2u in Ω, u = 0 in RN \\ Ω, where Ω is an open bounded subset in RN, N ⩾ 1, with Lipschitz boundary ∂Ω, q : Ω → (1, ∞) is bounded continuous function, and (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') is the nonlocal integro-differential operator of elliptic type defined as follows (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') u(x) = 2 lim εց0 � RN\\Bε(x) a(x,y) �|u(x) − u(y)| |x − y|s(x,y) � u(x) − u(y) |x − y|s(x,y) dy |x − y|N+s(x,y) , for all x ∈ RN, where: s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') : Ω × Ω → (0, 1) is a continuous function such that: s(x, y) = s(y, x) ∀x, y ∈ Ω × Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1) 0 < s− = inf Ω×Ω s(x, y) ⩽ s+ = sup Ω×Ω s(x, y) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2) s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 3 a(x,y)(t) := a(x, y, t) : Ω × Ω × R −→ R is symmetric function : a(x, y, t) = a(y, x, t) ∀(x, y, t) ∈ Ω × Ω × R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) and the function : ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') : Ω × Ω × R −→ R defined by ϕx,y(t) := ϕ(x, y, t) = \uf8f1 \uf8f2 \uf8f3 a(x, y, |t|)t for t ̸= 0, 0 for t = 0, is increasing homeomorphism from R onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Φx,y(t) := Φ(x, y, t) = � t 0 ϕx,y(τ)dτ for all (x, y) ∈ Ω × Ω, and all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, Φx,y is a Musielak function (see [28]), that is ⋆ Φ(x, y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') is a Φ-function for every (x, y) ∈ Ω × Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', is continuous, nondecreasing function with Φ(x, y, 0) = 0, Φ(x, y, t) > 0 for t > 0 and Φ(x, y, t) → ∞ as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ⋆ For every t ⩾ 0, Φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', t) : Ω × Ω −→ R is a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Also, we take �ax(t) := �a(x, t) = a(x,x)(t) ∀ (x, t) ∈ Ω × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then the function �ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') : Ω × R −→ R defined by : �ϕx(t) := �ϕ(x, t) = \uf8f1 \uf8f2 \uf8f3 �a(x, |t|)t for t ̸= 0, 0 for t = 0, is increasing homeomorphism from R onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If we set �Φx(t) := �Φ(x, t) = � t 0 �ϕx(τ)dτ for all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) Then, �Φx is also a Musielak function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Note that, when we take ax,y(t) = |t|p(x,y)−2 where p : Ω × Ω −→ (1, +∞) is a continuous bounded function, then our nonlocal operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') which can be seen as a generalization of the nonlocal operator with variable exponent (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') p(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') (see [15]) defined as (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') p(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )u(x) = 2 lim εց0 � RN\\Bε(x) |u(x) − u(y)|p(x,y)−2(u(x) − u(y)) |x − y|N+s(x,y)p(x,y) dy, for all x ∈ RN, (see also [32, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover, this work brings us back to introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional a- Laplacian (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a if ax,y(t) = a(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' the function a is independent of variables x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, we obtain the following nonlocal operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a , defined as (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a u(x) = 2 lim εց0 � RN \\Bε(x) a �|u(x) − u(y)| |x − y|s(x,y) � u(x) − u(y) |x − y|s(x,y) dy |x − y|N+s(x,y) , for all x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI This paper is organized as follows, In Section 1, we set the problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover, we are introduced the new nonlocal integro-differential operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The Section 2, is devoted to recall some properties of fractional Musielak-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In section 3, we introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak- Sobolev spaces and we establish some qualitative properties of these new spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In section 4, by means of Ekeland’s variational principle, we obtain the existence of λ∗ > 0 such that for any λ ∈ (0, λ∗), is an eigenvalue for the fol- lowing problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In Section 5, we present some examples which illustrate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Preliminaries results To deal with this situation we define the fractional Musielak-Sobolev space to investigate Problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let us recall the definitions and some elementary properties of this spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We refer the reader to [2, 3] for further reference and for some of the proofs of the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' For the function �Φx given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4), we introduce the Musielak space as follows L �Φx(Ω) = � u : Ω −→ R mesurable : � Ω �Φx(λ|u(x)|)dx < ∞ for some λ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The space L �Φx(Ω) is a Banach space endowed with the Luxemburg norm ||u||�Φx = inf � λ > 0 : � Ω �Φx �|u(x)| λ � dx ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The conjugate function of Φx,y is defined by Φx,y(t) = � t 0 ϕx,y(τ)dτ for all (x, y) ∈ Ω × Ω and all t ⩾ 0, where ϕx,y : R −→ R is given by ϕx,y(t) := ϕ(x, y, t) = sup {s : ϕ(x, y, s) ⩽ t} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Throughout this paper, we assume that there exist two positive constants ϕ+ and ϕ− such that 1 < ϕ− ⩽ tϕx,y(t) Φx,y(t) ⩽ ϕ+ < +∞ for all (x, y) ∈ Ω × Ω and all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (Φ1) This relation implies that 1 < ϕ− ⩽ t�ϕx(t) �Φx(t) ⩽ ϕ+ < +∞, for all x ∈ Ω and all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1) It follows that Φx,y and �Φx satisfy the global ∆2-condition (see [26]), written Φx,y ∈ ∆2 and �Φx ∈ ∆2, that is, Φx,y(2t) ⩽ K1Φx,y(t) for all (x, y) ∈ Ω × Ω, and all t ⩾ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2) and �Φx(2t) ⩽ K2 �Φx(t) for any x ∈ Ω, and all t ⩾ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) where K1 and K2 are two positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Furthermore, we assume that Φx,y satisfies the following condition the function [0, ∞) ∋ t �→ Φx,y( √ t) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (Φ2) s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ax(t), Bx(t) : R+ ×Ω −→ R+ be two Musielak functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Ax is stronger (resp essentially stronger) than Bx, Ax ≻ Bx (resp Ax ≻≻ Bx) in symbols, if for almost every x ∈ Ω B(x, t) ⩽ A(x, at), t ⩾ t0 ⩾ 0, for some (resp for each) a > 0 and t0 (depending on a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1 ([1, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Ax ≻≻ Bx is equivalent to the condition lim t→∞ � sup x∈Ω B(x, λt) A(x, t) � = 0, for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Now, we define the fractional Musielak-Sobolev space as introduce in [2] as follows W sLΦx,y(Ω) = � u ∈ L � Φx(Ω) : � Ω � Ω Φx,y �λ|u(x) − u(y)| |x − y|s � dxdy |x − y|N < ∞ for some λ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This space can be equipped with the norm ||u||s,Φx,y = ||u||�Φx + [u]s,Φx,y, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) where [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ]s,Φx,y is the Gagliardo seminorm defined by [u]s,Φx,y = inf � λ > 0 : � Ω � Ω Φx,y �|u(x) − u(y)| λ|x − y|s � dxdy |x − y|N ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be an open subset of RN, and let s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The space W sLΦx,y(Ω) is a Banach space with respect to the norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4), and a separable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' reflexive) space if and only if Φx,y ∈ ∆2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Φx,y ∈ ∆2 and Φx,y ∈ ∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Furthermore, if Φx,y ∈ ∆2 and Φx,y( √ t) is convex, then the space W sLΦx,y(Ω) is an uniformly convex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We say that Φx,y satisfies the fractional boundedness condition, written Φx,y ∈ Bf, if sup (x,y)∈Ω×Ω Φx,y(1) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (Φ3) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be an open subset of RN, and 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that Φx,y ∈ Bf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, C2 0(Ω) ⊂ W sLΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([2]) Assume that (Φ1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then the following inequal- ities hold true: Φx,y(σt) ⩾ σϕ−Φx,y(t) for all t > 0 and any σ > 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5) Φx,y(σt) ⩾ σϕ+Φx,y(t) for all t > 0 and any σ ∈ (0, 1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6) Φx,y(σt) ⩽ σϕ+Φx,y(t) for all t > 0 and any σ > 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7) Φx,y(t) ⩽ σϕ−Φx,y � t σ � for all t > 0 and any σ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI For any u ∈ W sLΦx,y(Ω), we define the modular function on W sLΦx,y(Ω) as follows Ψ(u) = � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='9) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that (Φ1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, for any u ∈ W sLΦx,y(Ω), the following relations hold true: ||u||s,Φx,y > 1 =⇒ ||u||ϕ− s,Φx,y ⩽ Ψ(u) ⩽ ||u||ϕ+ s,Φx,y, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10) ||u||s,Φx,y < 1 =⇒ ||u||ϕ+ s,Φx,y ⩽ Ψ(u) ⩽ ||u||ϕ− s,Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='11) We Define a closed linear subspace of W sLΦx,y(Ω) as follows W s 0 LΦx,y(Ω) = � u ∈ W sLΦx,y(RN) : u = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='e in RN \\ Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([3]) Let Ω be a bounded open subset of RN with C0,1-regularity and bounded boundary, let s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then there exists a positive constant γ such that ||u||�Φx ⩽ γ[u]s,Φx,y for all u ∈ W s 0 LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We denote by �Φ−1 x the inverse function of �Φx which satisfies the following conditions: � 1 0 �Φ−1 x (τ) τ N+s N dτ < ∞ for all x ∈ Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) � ∞ 1 �Φ−1 x (τ) τ N+s N dτ = ∞ for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='13) Note that, if ϕx,y(t) = |t|p(x,y)−1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) holds precisely when sp(x, y) < N for all (x, y) ∈ Ω × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='13) is satisfied, we define the inverse Musielak conjugate function of �Φx as follows (�Φ∗ x,s)−1(t) = � t 0 �Φ−1 x (τ) τ N+s N dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='14) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' [3] Let Ω be a bounded open subset of RN with C0,1-regularity and bounded boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='13) hold, then W sLΦx,y(Ω) ֒→ L �Φ∗x,s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='15) Moreover, the embedding W sLΦx,y(Ω) ֒→ LBx(Ω), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='16) is compact for all Bx ≺≺ �Φ∗ x,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Next, we recall some useful properties of variable exponent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' For more details we refer the reader to [20, 24], and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Consider the set C+(Ω) = � q ∈ C(Ω) : q(x) > 1 for all x ∈ Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 7 For all q ∈ C+(Ω), we define q+ = sup x∈Ω q(x) and q− = inf x∈Ω q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' For any q ∈ C+(Ω), we define the variable exponent Lebesgue space as Lq(x)(Ω) = � u : Ω −→ R measurable : � Ω |u(x)|q(x)dx < +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This vector space endowed with the Luxemburg norm, which is defined by ∥u∥Lq(x)(Ω) = inf � λ > 0 : � Ω ���� u(x) λ ���� q(x) dx ⩽ 1 � is a separable reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' A very important role in manipulating the generalized Lebesgue spaces with variable exponent is played by the modular of the Lq(x)(Ω) space, which defined by ρq(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') : Lq(x)(Ω) −→ R u �−→ ρq(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )(u) = � Ω |u(x)|q(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let u ∈ Lq(x)(Ω), then we have (i) ∥u∥Lq(x)(Ω) < 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' = 1, > 1) ⇔ ρq(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )(u) < 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' = 1, > 1), (ii) ∥u∥Lq(x)(Ω) < 1 ⇒ ∥u∥q+ Lq(x)(Ω) ⩽ ρq(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )(u) ⩽ ∥u∥q− Lq(x)(Ω), (iii) ∥u∥Lq(x)(Ω) > 1 ⇒ ∥u∥q− Lq(x)(Ω) ⩽ ρq(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )(u) ⩽ ∥u∥q+ Lq(x)(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Finally, the proof of our existence result is based on the following Ekeland’s variational principle theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ([19]) Let V be a complete metric space and F : V −→ R ∪ {+∞} be a lower semicontinuous functional on V , that is bounded below and not identically equal to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Fix ε > 0 and a point u ∈ V such that F(u) ⩽ ε + inf x∈V F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then for every γ > 0, there exists some point v ∈ V such that : F(v) ⩽ F(u), d(u, v) ⩽ γ and for all w ̸= v F(w) > F(v) − ε γ d(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak-Sobolev spaces Due to the non-locality of the operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ), we introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )- fractional Musielak-Sobolev space as follows W s(x,y)LΦx,y(Ω) = � u ∈ L � Φx(Ω) : � Ω � Ω Φx,y � λ|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N < ∞ for some λ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This space can be equipped with the norm ||u||s(x,y),Φx,y = ||u||�Φx + [u]s(x,y),Φx,y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI where [u]s(x,y),Φx,y is the Gagliardo seminorm defined by [u]s(x,y),Φx,y = inf � λ > 0 : � Ω � Ω Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' To simplify notations, throughout the rest of this paper, we set Ds(x,y)u = u(x) − u(y) |x − y|s(x,y) and dµ = dxdy |x − y|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' a)− For the case: Φx,y(t) = Φ(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Φ is independent of variables x, y, we can introduce the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Orlicz-Sobolev spaces W s(x,y)LΦ(Ω) as follows W s(x,y)LΦ(Ω) = � u ∈ LΦ(Ω) : � Ω � Ω Φ �λ|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N < ∞ for some λ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' b)− For the case: Φx,y(t) = |t|p(x,y) for all (x, y) ∈ Ω × Ω, where p : Ω × Ω −→ (1, +∞) is a continuous bounded function such that 1 < p− = min (x,y)∈Ω×Ω p(x, y) ⩽ p(x, y) ⩽ p+ = max (x,y)∈Ω×Ω p(x, y) < +∞, and p is symmetric, that is, p(x, y) = p(y, x) for all (x, y) ∈ Ω × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If denoted by ¯p(x) = p(x, x) for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, we replace LΦx by Lp(x), and W s(x,y)LΦx,y by W s(x,y),p(x,y) and we refer them as variable exponent Lebesgue spaces, and s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Sobolev spaces with variable exponent respectively, (see [15, 32, 33]) defined by Lp(x)(Ω) = � u : Ω −→ R measurable : � Ω |u(x)|p(x)dx < +∞ � , and W = W s(x,y),p(x,y)(Ω) = � u ∈ L¯p(x)(Ω) : � Ω×Ω |u(x) − u(y)|p(x,y) λp(x,y)|x − y|s(x,y)p(x,y)+N dxdy < +∞, for some λ > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' with the norm ∥u∥W = ∥u∥L¯p(x)(Ω) + [u]W , where [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ]W is a Gagliardo seminorm with variable exponent given by [u]W = [u]s(x,y),p(x,y) = inf � λ > 0 : � Ω×Ω |u(x) − u(y)|p(x,y) λp(x,y)|x − y|N+s(x,y)p(x,y) dxdy ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be an open subset of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The space W s(x,y)LΦx,y(Ω) is a Banach space with respect to the norm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1), and a separable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' reflexive) space if and only if Φx,y ∈ ∆2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Φx,y ∈ ∆2 and Φx,y ∈ ∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Furthermore, if Φx,y ∈ ∆2 and Φx,y( √ t) is convex, then the space W s(x,y)LΦx,y(Ω) is an uniformly convex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proof of this Theorem is similar to [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be a bounded open subset of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then W s+LΦx,y(Ω) ֒→ W s(x,y)LΦx,y(Ω) ֒→ W s−LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let u ∈ W s+LΦx,y(Ω) and λ > 0, we have � Ω � Ω Φx,y �|Ds(x,y)u| λ � dxdy |x − y|N = � Ω � Ω Φx,y � |Ds+u| λ 1 |x − y|s(x,y)−s+ � dxdy |x − y|N ⩽ � Ω � Ω Φx,y � |Ds+u| λ � dxdy |x − y|N+p(s(x,y)−s+) ⩽ sup Ω×Ω |x − y|p(s+−s(x,y)) � Ω � Ω Φx,y � |Ds+u| λ � dxdy |x − y|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' where p = {ϕ− or ϕ+} is given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This implies that [u]s(x,y),Φx,y ⩽ sup Ω×Ω |x − y|p(s+−s(x,y))[u]s+,Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' So ∥u∥s(x,y),Φx,y ⩽ c∥u∥s+,Φx,y, where c = max � 1, sup Ω×Ω |x − y|p(s+−s(x,y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Now, Let u ∈ W s(x,y)LΦx,y(Ω) and λ > 0, we have � Ω � Ω Φx,y � |Ds−u| λ � dxdy |x − y|N = � Ω � Ω Φx,y �|Ds(x,y)u| λ 1 |x − y|s−−s(x,y) � dxdy |x − y|N ⩽ � Ω � Ω Φx,y �|Ds(x,y)u| λ � dxdy |x − y|N+p(s−−s(x,y)) ⩽ sup Ω×Ω |x − y|p(s(x,y)−s−) � Ω � Ω Φx,y �|Ds(x,y)u| λ � dxdy |x − y|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This implies that [u]s−,Φx,y ⩽ sup Ω×Ω |x − y|p(s(x,y)−s−)[u]s(x,y),Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' So ∥u∥s−,Φx,y ⩽ c∥u∥s(x,y),Φx,y, where c = max � 1, sup Ω×Ω |x − y|p(s(x,y)−s−) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Now, combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4, we obtain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be a bounded open subset of RN with C0,1-regularity and bounded boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='13) hold, then W s(x,y)LΦx,y(Ω) ֒→ L �Φ∗ x,s−(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Also, the embedding W s(x,y)LΦx,y(Ω) ֒→ LBx(Ω), is compact for all Bx ≺≺ �Φ∗ x,s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI For any u ∈ W s(x,y)LΦx,y(Ω), we define the modular function on W s(x,y)LΦx,y(Ω) as follows J(u) = � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2) An important role in manipulating the s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Musielak-Sobolev spaces is played by the modular function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It is worth noticing that the relation between the norm and the modular shows an equivalence between the topology defined by the norm and that defined by the modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that (Φ1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, for any u ∈ W s(x,y)LΦx,y(Ω), the following relations hold true: ||u||s(x,y),Φx,y > 1 =⇒ ||u||ϕ− s(x,y),Φx,y ⩽ J(u) ⩽ ||u||ϕ+ s(x,y),Φx,y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) ||u||s(x,y),Φx,y < 1 =⇒ ||u||ϕ+ s(x,y),Φx,y ⩽ J(u) ⩽ ||u||ϕ− s(x,y),Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' To simplify the notation, we take ∥u∥x,y := ||u||s(x,y),Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' First, we show that if ||u||x,y > 1, then J(u) ⩽ ||u||ϕ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Indeed, let u ∈ W s(x,y)LΦx,y(Ω) such that ||u||x,y > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Using the definition of the Luxemburg norm and the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7), we get J(u) = � Ω � Ω Φx,y � ||u||x,y |u(x) − u(y)| ||u||x,y|x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx � ||u||x,y |u(x)| ||u||x,y � dx ⩽ ||u||ϕ+ x,y � Ω � Ω Φx,y � |u(x) − u(y)| ||u||x,y|x − y|s(x,y) � dxdy |x − y|N + ||u|| � ϕ+ x,y � Ω �Φx � |u(x)| ||u||x,y � dx ⩽ ||u||ϕ+ x,y �� Ω � Ω Φx,y � |u(x) − u(y)| ||u||x,y|x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx � |u(x)| ||u||x,y � dx � ⩽ ||u||ϕ+ x,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Next, assume that ||u||x,y > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let β ∈ (1, ||u||x,y), by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5), we have � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx ⩾ βϕ− � Ω � Ω Φx,y � |u(x) − u(y)| β|x − y|s(x,y) � dxdy |x − y|N + β � ϕ− � Ω �Φx �|u(x)| β � dx ⩾ βϕ− �� Ω � Ω Φx,y � |u(x) − u(y)| β|x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx �|u(x)| β � dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Since β < ||u||x,y, we find � Ω � Ω Φx,y �|u(x) − u(y)| β|x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx �|u(x)| β � dx > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, we have � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx ⩾ βϕ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Letting β ր ||u||x,y, we deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 11 Next, we show that J(u) ⩽ ||u||ϕ− x,y for all u ∈ W s(x,y)LΦx,y(Ω) with ||u||x,y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Using the definition of the Luxemburg norm and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8), we obtain J(u) ⩽ ||u||ϕ− x,y � Ω � Ω Φx,y � |u(x) − u(y)| ||u||x,y|x − y|s(x,y) � dxdy |x − y|N + ||u||�ϕ− x,y � Ω �Φx � |u(x)| ||u||x,y � dx ⩽ ||u||ϕ− x,y �� Ω � Ω Φx,y � |u(x) − u(y)| ||u||x,y|x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx � |u(x)| ||u||x,y � dx � ⩽ ||u||ϕ− x,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let ξ ∈ (0, ||u||x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6), it follows that � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx ⩾ ξϕ+ � Ω � Ω Φx,y �|u(x) − u(y)| ξ|x − y|s(x,y) � dxdy |x − y|N + ξ �ϕ+ � Ω Φ �|u(x)| ξ � dx ⩾ ξϕ+ �� Ω � Ω Φx,y �|u(x) − u(y)| ξ|x − y|s(x,y) � dxdy |x − y|N + � Ω Φ �|u(x)| ξ � dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5) Defining v(x) = u(x) ξ for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, ||v||x,y = ||u||x,y ξ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Using relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10), we find � Ω � Ω Φx,y �|v(x) − v(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|v(x)|) dx ⩾ ||v||ϕ− x,y > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6), we deduce that � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N + � Ω �Φx (|u(x)|) dx ⩾ ξϕ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Letting ξ ր ||u||x,y in the above inequality, we obtain that relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Similar to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1, we obtain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that (Φ1) is satisfied, Then, for any u ∈ W s(x,y)LΦx,y(Ω), the following assertions hold true: [u]s(x,y),Φx,y > 1 =⇒ [u]ϕ− s(x,y),Φx,y ⩽ φ(u) ⩽ [u]ϕ+ s(x,y),Φx,y, [u]s(x,y),Φx,y < 1 =⇒ [u]ϕ+ s(x,y),Φx,y ⩽ φ(u) ⩽ [u]ϕ− s(x,y),Φx,y, where φ(u) = � Ω � Ω Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Now, we introduce a closed linear subspace of W s(x,y)LΦx,y(Ω) as follows W s(x,y) 0 LΦx,y(Ω) = � u ∈ W s(x,y)LΦx,y(RN) u = 0 in RN\\Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then we have the following generalized Poincaré type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let Ω be a bounded open subset of RN with C0,1-regularity and bounded boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then there exists a positive constant γ such that ||u||�Φx ⩽ γ[u]s(x,y),Φx,y for all u ∈ W s(x,y) 0 LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let u ∈ W s(x,y) 0 LΦx,y(Ω), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2, we have [u]s−,Φx,y ⩽ c[u]s(x,y),Φx,y, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7) on the other hand, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3, there exists a positive constant γ′ such that ||u||�Φx ⩽ γ′[u]s−,Φx,y for all u ∈ W s− 0 LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8) Thus, we combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8), we obtain ||u||�Φx ⩽ γ[u]s(x,y),Φx,y for all u ∈ W s(x,y) 0 LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' with γ = cγ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Now, in order to study Problem (Pa), it is important to encode the boundary condition u = 0 in RN \\Ω in the weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In the scalar case, Servadei and Valdinoci [31] introduced a new function spaces to study the variational functionals related to the fractional Laplacian by observing the interaction between Ω and RN \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Subsequently, inspired by the work of Servadei and Valdinoci [31], Azroul et al in [11], have introduced the fractional Sobolev space with variable exponent, to study the variational functionals related to the fractional p(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-Laplacian operator by observing the interaction between Ω and RN \\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Motivated by the above papers, and due to the nonlocality of the operator (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') a(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' ), we introduce the following s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-fractional Orlicz-Sobolev space as follows W (x,y)LΦw,y(Q) = � u ∈ LΦx,y(Ω) : � Q Φx,y �λ|u(x) − u(y)| |x − y|s(x,y) � dxdy |x − y|N < ∞ for some λ > 0 � , where Q = R2N \\ (CΩ × CΩ) with CΩ = RN \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This spaces are equipped with the norm, ||u|| = ||u||�Φx + [u], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='9) where [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='] is the Gagliardo seminorm, defined by [u] = inf � λ > 0 : � Q Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Similar to the spaces (W s(x,y)LΦx,y(Ω), ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='∥s(x,y),Φx,y) we have that (W s(x,y)LΦx,y(Q), ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='∥) is a separable reflexive Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Now, let W s(x,y) 0 LΦx,y(Q) denotes the following linear subspace of W s(x,y)LΦx,y(Q), W s(x,y) 0 LΦx,y(Q) = � u ∈ W s(x,y)LΦx,y(Q) : u = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='e in RN \\ Ω � with the norm [u] = inf � λ > 0 : � Q Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' In the following theorem, we compare the spaces W s(x,y)LΦx,y(Ω) and W s(x,y)LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The following assertions hold: s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 13 1) The continuous embedding W s(x,y)LΦx,y(Q) ⊂ W s(x,y)LΦx,y(Ω) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 2) If u ∈ W s(x,y) 0 LΦx,y(Q), then u ∈ W s(x,y)LΦx,y(RN) and ||u||s(x,y),Φx,y ⩽ ||u||W s(x,y)LΦx,y(RN) = ||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 1) Let u ∈ W s(x,y)LΦx,y(Q), since Ω × Ω ⊊ Q, then for all λ > 0 we have � Ω � Ω Φx,y � |u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ � Q Φx,y � |u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10) We set As(x,y) λ,Ω×Ω = � λ > 0 : � Ω � Ω Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ 1 � and As(x,y) λ,Q = � λ > 0 : � Q Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N ⩽ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10), it is easy to see that As(x,y) λ,Q ⊂ As(x,y) λ,Ω×Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Hence [u]s(x,y),Φx,y = inf λ>0 As(x,y) λ,Ω×Ω ⩽ [u] = inf λ>0 As(x,y) λ,Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='11) Consequently, by definitions of the norms ∥u∥s(x,y),Φx,y and ∥u∥, we obtain ∥u∥s(x,y),Φx,y ⩽ ∥u∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 2) Let u ∈ W s(x,y) 0 LΦx,y(Q), then u = 0 in RN \\Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' So, ∥u∥L � Φx(Ω) = ∥u∥L � Φx(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Since � R2N Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N = � Q Φx,y �|u(x) − u(y)| λ|x − y|s(x,y) � dxdy |x − y|N for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then [u]W s(x,y)LΦx,y(RN) = [u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, we get ||u||s(x,y),Φx,y ⩽ ||u||W s(x,y)LΦx,y(RN) = ||u||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (Poincaré inequality) Let Ω be a bounded open subset of RN with C0,1-regularity and bounded boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then there exists a positive con- stant c such that, ∥u∥�Φx ⩽ c[u], ∀u ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let u ∈ W s(x,y) 0 LΦx,y(Q), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4, we have u ∈ W s(x,y) 0 LΦx,y(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3, there exists a positive constant γ such that, ∥u∥�Φx ⩽ γ[u]s(x,y),Φx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI Combining the above inequality with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='11), we obtain that ∥u∥�Φx ⩽ c[u], ∀u ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2, we deduce that [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='] is a norm on W s(x,y) 0 LΦx,y(Q) which is equivalent to the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Existence results and proofs In this section, we analyze problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' under the following basic assump- tions q− < ϕ− (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1) and lim t→∞ � sup x∈Ω |t|q+ (�Φx,s−)∗(kt) � = 0 ∀k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2) The dual space of � W s(x,y) 0 LΦx,y(Q), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='|| � is denoted by �� W s(x,y) 0 LΦx,y(Q) �∗ , ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='||∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We say that λ ∈ R is an eigenvalue of Problem (Pa) if there exists u ∈ W s(x,y) 0 LΦx,y(Q) \\ {0} such that � Q ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)vdµ − λ � Ω |u|q(x)−2uvdx = 0 for all v ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We point that if λ is an eigenvalue of Problem (Pa) then the corresponding u ∈ W s(x,y) 0 LΦx,y(Q) \\ {0} is a weak solution of (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Our main results is given by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' There exists λ∗ > 0 such that for any λ ∈ (0, λ∗) is an eigen- value of Problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2), we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1 we obtain that W s(x,y) 0 LΦx,y(Q) is compactly embedded in Lq+(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' That fact combined with the continuous embedding of Lq+(Ω) in Lq(x)(Ω), ensures that W s(x,y) 0 LΦx,y(Q) is compactly embedded in Lq(x)(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Next, for all λ ∈ R, we define the energetic function associated with problem (Pa) Jλ : W s(x,y) 0 LΦx,y(Q) → R, as Jλ(u) = � Q Φx,y �|u(x) − u(y)| |x − y|s(x,y) � dµ − λ � Ω 1 q(x)|u|q(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By a standard argument to [5] and [6], we have Jλ ∈ C1(W s(x,y) 0 LΦx,y(Q), R), � J′ λ(u), v � = � Q ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)vdµ − λ � Ω |u|q(x)−2uvdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, there exists λ∗ > 0 such that for any λ ∈ (0, λ∗), there are ρ, α > 0, such that Jλ(u) ⩾ α > 0 for any u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Since W s(x,y) 0 LΦx,y(Q) is continuously embedded in Lq(x)(Ω), it follows that there exists a positive constant c1 such that ||u|| ⩾ c1||u||q(x) ∀u ∈ W s(x,y) 0 LΦx,y(Q) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) we fix ρ ∈ (0, 1) such that ρ < 1 c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) implies that ∥u∥q(x) < 1 for all u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, we can apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2, and we have � Ω |u(x)|q(x)dx ⩽ ∥u∥q− q(x) for all u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='4) implies that � Ω |u(x)|q(x)dx ⩽ cq− 1 ∥u∥q− for all u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5) Taking into account Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5), we deduce that for any u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ, the following inequalities hold true: Jλ(u) ⩾ ∥u∥ϕ+ − λ q− � Ω |u(x)|q(x)dx ⩾ ∥u∥ϕ+ − λcq− 1 q− ∥u∥q− = ρq− � ρϕ+−q− − λcq− 1 q− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Hence, if we define λ∗ = ρϕ+−q− 2cq− 1 q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6) Then, for any λ ∈ (0, λ∗) and u ∈ W s(x,y) 0 LΦx,y(Q) with ||u|| = ρ, we have Jλ(u) ⩾ α > 0, such that α = ρϕ+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Assume that the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, there exists φ > 0 such that φ ⩾ 0, φ ̸= 0, and Jλ(tφ) < 0 for t > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1) we can chose ε0 > 0 such that q− + ε0 < ϕ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, since q ∈ C(Ω), it follows that there exists an open set Ω0 ⊂ Ω such that |q(x) − q−| < ε0 for all x ∈ Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, q(x) ⩽ q− + ε0 < ϕ− for all 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI x ∈ Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let φ ∈ C∞ 0 (Ω) be such that supp(φ) ⊃ Ω0, φ(x) = 1 for all x ∈ Ω0, and 0 ⩽ φ ⩽ 1 in Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, for any t ∈ (0, 1), we have Jλ(tφ) = � Q Φx,y � t|Ds(x,y)φ| � dµ − λ � Ω 1 q(x)tq(x)|φ|q(x)dx ⩽ � Q tϕ−Φx,y � |Ds(x,y)φ| � dµ − λ � Ω0 tq(x) q(x)|φ|q(x)dx ⩽ tϕ− � Q Φx,y � |Ds(x,y)φ| � dµ − λtq−+ε0 q+ � Ω0 |φ|q(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Therefore Jλ(tφ) < 0, for t < δ1/(ϕ−−q−−ε0) with 0 < δ < min \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1, λ q+ � Ω0 |φ|q(x)dx � Q Φx,y � |Ds(x,y)φ| � dµ \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This is possible, since we claim that � Q Φx,y � |Ds(x,y)φ| � dµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Indeed, it is clear that � Ω0 |φ|q(x)dx ⩽ � Ω |φ|q(x)dx ⩽ � Ω |φ|q−dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, since W s(x,y) 0 LΦx,y(Q) is continuously embedded in Lq−(Ω), it follows that there exists a positive constant c such that ∥φ∥q− ⩽ c||φ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The last two inequalities imply that ∥φ∥ > 0 and combining this fact with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1, the claim follows at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The proof of the lemma is now completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Let λ∗ > 0 be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='6) and λ ∈ (0, λ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1 it follows that on the boundary oh the ball centered in the origin and of radius ρ in W s(x,y) 0 LΦx,y(Q), denoted by Bρ(0), we have inf ∂Bρ(0) Jλ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2, there exists φ ∈ W s(x,y) 0 LΦx,y(Q) such that Jλ(tφ) < 0 for all t > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover for any u ∈ Bρ(0), we have Jλ(u) ⩾ ∥u∥ϕ− − λcq− 1 q− ∥u∥q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It follows that −∞ < c := inf Bρ(0) Jλ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 17 We let now 0 < ε < inf ∂Bρ(0) Jλ− inf Bρ(0) Jλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='5 to the functional Jλ : Bρ(0) −→ R, we find uε ∈ Bρ(0) such that \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Jλ(uε) < inf Bρ(0) Jλ + ε, Jλ(uε) < Jλ(u) + ε||u − uε||, u ̸= uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Since Jλ(uε) ⩽ inf Bρ(0) Jλ + ε ⩽ inf Bρ(0) Jλ + ε < inf ∂Bρ(0) Jλ, we deduce uε ∈ Bρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Now, we define Λλ : Bρ(0) −→ R by Λλ(u) = Jλ(u) + ε||u − uε||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It’s clear that uε is a minimum point of Λλ and then Λλ(uε + tv) − Λλ(uε) t ⩾ 0 for small t > 0, and any v ∈ Bρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' The above relation yields Jλ(uε + tv) − Jλ(uε) t + ε||v|| ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Letting t → it follows that ⟨J′ λ(uε), v⟩ + ε||v|| > 0 and we infer that ||J′ λ(uε)||∗ ⩽ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We deduce that there exists a sequence {un} ⊂ Bρ(0) such that Jλ(un) −→ c and J′ λ(un) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7) It is clear that {un} is bounded in W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, there exists u0 ∈ W s(x,y) 0 LΦx,y(Q), such that up to a subsequence {un} converges weakly to u0 in W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, since W s(x,y) 0 LΦx,y(Q) is compactly embedded in Lq(x)(Ω), it follows that {un} converges strongly to u0 in Lq(x)(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then by Hölder in- equality, we have that lim n→∞ � Ω |un|q(x)−2un(un − u0)dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This fact and relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7), implies that lim n→∞ � J′ λ(un), un − u0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus we deduce that lim n→∞ � Q ax,y(|Ds(x,y)un|)Ds(x,y)un � Ds(x,y)un − Ds(x,y)u0 � dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8) Since {un} converge weakly to u0 in W s(x,y) 0 LΦx,y(Q), by relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='8), we find that lim n→∞ � Q � ax,y(|Ds(x,y)un|)Ds(x,y)un − ax,y(|Ds(x,y)u0|)Ds(x,y)u0 � � Ds(x,y)un − Ds(x,y)u0 � dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='9) 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI Since, Φx,y is convex, we have Φx,y(|Ds(x,y)u|) ⩽ Φx,y �|Ds(x,y)u + Ds(x,y)v| 2 � +ax,y(|Ds(x,y)u|)Ds(x,y)uDs(x,y)u − Ds(x,y)v 2 Φx,y(|Ds(x,y)v|) ⩽ Φx,y �|Ds(x,y)u + Ds(x,y)v| 2 � +ax,y(|Ds(x,y)v|)Ds(x,y)v Ds(x,y)v − Ds(x,y)u 2 for every u, v ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Adding the above two relations and integrat- ing over Q, we find that 1 2 � Q � ax,y(|Ds(x,y)u|)Ds(x,y)u − ax,y(|Ds(x,y)v|)Ds(x,y)v � � Ds(x,y)u − Ds(x,y)v � dµ ⩾ � Q Φx,y(|Ds(x,y)u|)dµ + � Q Φx,y(|Ds(x,y)v|)dµ − 2 � Q Φx,y � |Ds(x,y)u − Ds(x,y)v| 2 � dµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10) for every u, v ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, since for each, we know that Φx,y : [0, ∞) → R is an increasing continuous function, with Φx,y(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then by the conditions (Φ1) and (Φ2), we can apply [25, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1] in order to obtain 1 2 �� Q Φx,y(|Ds(x,y)u|)dµ + � Q Φx,y(|Ds(x,y)v|)dµ � ⩾ � Q Φx,y �|Ds(x,y)u + Ds(x,y)v| 2 � dµ + � Q Φx,y �|Ds(x,y)u − Ds(x,y)v| 2 � dµ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='11) for every u, v ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='11), we have � Q � ax,y(|Ds(x,y)u|)Ds(x,y)u − ax,y(|Ds(x,y)v|)Ds(x,y)v � � Ds(x,y)u − Ds(x,y)v � dµ ⩾ 4 � Q Φi �|Ds(x,y)u − Ds(x,y)v| 2 � dµ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) for every u, v ∈ W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='12) show that {un} converge strongly to u0 in W s(x,y) 0 LΦx,y(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then by relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='7), we have Jλ(u0) = c1 > 0 and J′ λ(u0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, u0 is a nontrivial weak solution for Problem (Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' This complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Examples In this section we point certain examples of functions ϕx,y and Φx,y which illustrate the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' As a first example, we can take ϕx,y(t) = ϕ1(x, y, t) = p(x, y) |t|p(x,y)−2t log(1 + |t|) for all t ⩾ 0, and thus, Φx,y(t) = p(x, y) |t|p(x,y) log(1 + |t|) + � |t| 0 τ p(x,y) (1 + τ)(log(1 + τ))2 dτ, s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 19 with p ∈ C(Q) satisfies 2 ⩽ p(x, y) < N for all (x, y) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then, in this case problem (Pa) becomes (P1) \uf8f1 \uf8f2 \uf8f3 (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') ϕ1 u = λ|u|q(x)−2u in Ω u = 0 in RN \\ Ω, with (−∆)s(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=') ϕ1 u(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' � Ω p(x, y)|Ds(x,y)u|p(x,y)−2Ds(x,y)u log(1 + |Ds(x,y)u|)|x − y|N+s(x,y) dy for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It easy to see that Φx,y is a Musielak function and satisfy condition (Φ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover, for each (x, y) ∈ Q fixed, by Example 3 on p 243 in [18], we have p(x, y) − 1 ⩽ tϕx,y(t) Φx,y(t) ⩽ p(x, y) ∀(x, y) ∈ Q, ∀t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, (Φ1) holds true with ϕ− = p− − 1 and ϕ+ = p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Finally, we point out that trivial computations imply that d2(Φx,y( √ t)) dt2 ⩾ 0 for all (x, y) ∈ Q and t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, relation (Φ2) hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Hence, we derive an existence result for problem (P1) which is given by the following Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If p− − 1 > q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then there exists λ∗ > 0 such that for any λ ∈ (0, λ∗) is an eigenvalue of Problem (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' As a second example, we can take ϕx,y(t) = ϕ2(x, y, t) = p(x, y) log(1 + α + |t|)|t|p(x,y)−2t for all t ⩾ 0 and so, Φx,y(t) = log(1 + |t|)|t|p(x,y) − � |t| 0 τ p(x,y) 1 + τ dτ, where α > 0 is a constant and p ∈ C(Ω × Ω) satisfies 2 ⩽ p(x, y) < N for all (x, y) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then we consider the following fractional p(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-problem (P2) \uf8f1 \uf8f2 \uf8f3 (−∆)s(x,y) ϕ2 u = λ|u|q(x)−2u in Ω u = 0 in RN \\ Ω, where (−∆)s(x,y) ϕ2 u(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' � Ω p(x, y) log(1 + α + |Ds(x,y)u|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='|Ds(x,y)u|p(x,y)−2Ds(x,y)u |x − y|N+s(x,y) dy for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It easy to see that Φx,y is a Musielak function and satisfy condition (Φ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Next, we remark that for each (x, y) ∈ Q fixed, we have p(x, y) ⩽ tϕx,y(t) Φx,y(t) for all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' AZROUL, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' BENKIRANE, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' SRATI By the above information and taking ϕ− = p−, we have 1 < p− ⩽ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) for all (x, y) ∈ Q and all t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' On the other hand, some simple computations imply lim t→∞ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) = p(x, y) for all (x, y) ∈ Q, and lim t→0 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) = p(x, y) + 1 for all (x, y) ∈ Q, Thus, we remark that t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) is continuous on Q × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Moreover, 1 < p− ⩽ lim t→0 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) ⩽ p+ + 1 < ∞, and 1 < p− ⩽ lim t→∞ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='ϕx,y(t) Φx,y(t) ⩽ p+ + 1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' It follows that ϕ+ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' We conclude that relation (Φ1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Finally, we point out that trivial computations imply that d2(Φx,y( √ t)) dt2 ⩾ 0 for all (x, y) ∈ Q and t ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Thus, relation (Φ2) hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' If p− > q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Then there exists λ∗ > 0 such that for any λ ∈ (0, λ∗) is an eigenvalue of Problem (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Disclosure statement No potential conflict of interest was reported by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Data Availability Statement My manuscript has no associate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Adams, Sobolev Spaces, Academic Press, New York, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Azroul , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Benkirane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Shimi and M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Benkirane , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Shimi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Srati (2020): Embedding and extension results in fractional Musielak-Sobolev spaces, Applicable Analysis, Applicable Analysis, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1080/00036811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1948019.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Azroul, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Benkirane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Srati, Nonlocal eigenvalue type problem in fractional Orlicz- Sobolev space, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theory (2020) doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1007/s43036-020-00067-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' )-FRACTIONAL MUSIELAK-SOBOLEV SPACES 21 [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Azroul, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Benkirane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Srati, Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Theory (2020) doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content='1007/s43036-020-00042- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Azroul, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} +page_content=' Benkirane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAyT4oBgHgl3EQfmPiH/content/2301.00467v1.pdf'} 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mode 100644 index 0000000000000000000000000000000000000000..504035f33f86a04541a01e427c827e81dade1897 --- /dev/null +++ b/xdFKT4oBgHgl3EQf6C78/content/tmp_files/2301.11940v1.pdf.txt @@ -0,0 +1,1596 @@ +LA-UR-23-20730 +Understanding parton evolution in matter from renormalization group analysis +Weiyao Ke∗ and Ivan Vitev† +Theoretical Division, Los Alamos National Laboratory, Los Alamos NM 87545, United States +(Dated: January 31, 2023) +We perform a renormalization group (RG) analysis of collinear hadron production in deep inelastic +scattering on nuclei. We consider the limit where one of the dimensionless in-medium scale ratios +E/(µ2 +DL) ≫ 1, with L, µD, E being the medium size, inverse scattering range and the parton energy +in the nuclear rest frame, while the opacity L/λg remains small. We identify the fixed order and +leading ln[E/(µ2 +DL)] enhanced medium contributions to the semi-inclusive cross sections and derive +RG equations which resum multiple emissions near the x → 0, 1 endpoints of the splitting functions +at first order in opacity.We find that the evolution equations obtained in this work treat the same +type of radiation enhancement in matter as the modified DGLAP approach, but differ in the way one +chooses to regulate the endpoint divergences and provide unique analytic insight into the problem of +resummation. The new RG evolution framework is applied to study fragmentation in eA reactions. +Introduction. A common characteristic of many prob- +lems in science is that microscopic fluctuations in the +system manifest themselves in macroscopic effects. Such +problems arise in fields ranging from social networks [1] +and turbulence [2] to particle [3] and nuclear physics [4, +5]. They are most prevalent in inherently divergent theo- +ries and efficiently addressed using renormalization group +(RG) analysis [6, 7]. Effective theories of quantum chro- +modynamics (QCD) geared toward jet physics [8, 9] have +provided new insights into renormalization and resum- +mation, and given a modern perspective to the problem +of parton production and propagation in nuclear mat- +ter [10–14]. These advances are key to the interpretation +of the data from reactions with nuclei at current and fu- +ture colliders [15]. +Over the past two decades, medium-induced par- +ton showers have been successfully implemented in jet +quenching phenomenology to describe the modification +of hadron and jet cross sections, and jet substructure +in nuclear collisions [16–28]. Still, resummation of QCD +radiation in nuclear matter remains challenging, espe- +cially lacking in analytic insight. We address this long- +standing problem using RG techniques. If we consider +semi-inclusive hadron production in deep inelastic scat- +tering (DIS) on a nuclear target (eA→ h + X), we en- +counter a number of energy and length scales (defined in +the target rest frame) including: 1) the hard scale Q, 2) +energy of the virtual photon/jet ν, 3) the path length L, +4) the mean free path λg and 5) the inverse interaction +range µD. Therefore, observables in eA are functions of +many dimensionless control parameters +Obs ≡ Obs +� Q +Q0 +, +E +µ2 +DL, L +λg +, λgµD, µD +Q0 +, · · · +� +. +(1) +A simplified description with controlled accuracy is of- +ten possible when one (or more) of these dimension- +less ratios become asymptotically large [29]. For exam- +ple, the limit λgµD ≫ 1, implying independent multi- +ple parton-medium scatterings, allows the use of time- +ordered perturbation theory to derive quark and gluon +splitting functions in matter [13, 30]. A partonic trans- +port picture emerges when L/λg ≫ 1 in a thick and dense +medium [31–33]. +In this letter, we compute hadron production in a +particular regime when Q/Q0, E/(µ2 +DL), λgµD become +asymptotically large while L/λg, µD/Q0 stay at or- +der unity/few. +This limit is particularly relevant for +high-energy hadron production in thin, dilute or fast- +expanding media. Still, renormalization is needed to re- +sum large ln[Q/Q0] and ln[E/(µ2 +DL)] enhancements from +the vacuum and medium-induced radiative corrections. +We thus introduce two final-state renormalization scales +µ1 and µ2 in the single-inclusive hadron cross section [34] +dσeA→h +dxBdQ2dzh += 2πα2 +e +Q4 +� +i,j +e2 +q +� � +fi/A⊗ +� +(1 + (1 − y)2)C1 +ij + 2(1 − y)CL +ij +�� +xB ⊗ dh/j +� +zh +, (2) +{h ⊗ g}x ≡ +� 1 +x +h +� x +x′ +� +g(x′)dx′ +x′ . +(3) +Here, y = ν/Ee with Ee and ν = Q2/(2xBMp) be- +ing the energies of the incident electron and the vir- +tual photon, respectively. fi/A(x, Q2), dh/j(z, µ2 +1, µ2 +2) and +C1,L +ij (x, z, Q2, µ2 +1, µ2 +2) are the parton distribution func- +tions, fragmentation functions, and the hard coefficient +functions with fractional electric charge eq. The PDF is +evaluated at scale Q2, and the dependence on Q2 will +not be written explicitly hereafter. +The cross-section +fi/A ⊗ C1,L +ij +⊗ dj/h must not depend on µ1 and µ2, thus +it is sufficient to study scale dependence of F 1,L +ij (z) ≡ +fi/A ⊗ zC1,L +ij +[35]. This quantity can be interpreted as +the invariant distribution of parton j, resolved at scales +µ1, µ2, after the hard process. +This “parton shower” +choice enables us to track the evolving parton energy +E = zν, which is important for the most consistent +implementation of medium-induced splitting functions. +F 1 +ij = zδijfi/A(xB)δ(1 − z) + αs(· · · ), F L +ij = αs(· · · ), and +we use the NLO expression for C1,L +ij +[34, 36] without writ- +ing them explicitly. +arXiv:2301.11940v1 [hep-ph] 27 Jan 2023 + +2 +As will become clear in a moment, the Q/Q0 and +E/(µ2 +DL) enhancements have distinct physics origins. +Therefore, in addition to the vacuum renormalization +that leads to the DGLAP evolution, the “medium bare” +Fij needs to be further renormalized by a medium coef- +ficient Mkj that only depends on µ2, +Fij(z, µ2 +1, µ2 +2) → Fik(y, µ2 +1, µ2 +2) ⊗ Mkj +�z +y , µ2 +2 +� ++ F(z).(4) +Here, the Mkj += M (0) +kj ++ M (1) +kj ++ · · · with M (0) +kj += +yδkjδ(1 − y) and the first non-trivial contribution M (1) +kj . +F(z) stands for medium contributions subleading in +ln[E/(µ2 +DL)], i.e. other fixed order contributions. +Renormalization group analysis of endpoint divergences +in collinear emission spectra. +We consider the correc- +tion to Fij(z) from both vacuum and medium-induced +collinear splittings and find that it has the form +αs(µ2 +1)µ2ϵ +1 +2π2 +� Q2 d2−2ϵk +k2 +Fik ⊗ x[Pkj]+ + ∆F m +ij (z, µ2ϵ +2 ). (5) +We work in d = 4 − 2ϵ dimensions and Pkj are the vac- +uum Altarelli-Parisi splitting functions [37, 38], with the +“+” prescription in Eq. (5) only applied to diagonal con- +tributions. In the first term, µ1 acts as an infrared cutoff +of vacuum emissions. In this case RG analysis leads to +DGLAP evolution of Fij(z) in µ1 that resums ln[Q/Q0], +and we always evolve µ1 from the hard scale Q down to +Q0. Next, we will extract the leading logarithmic and +fixed order contribution from the medium-induced cor- +rection in the second term. +For thin and uniform nuclear matter of length L we use +the medium-induced splitting functions P (1) +ij (x) [13, 14, +30, 39] obtained in the opacity expansion approach us- +ing Soft-Collinear-Effective-Theory with Glauber Gluons +(SCETG) [10–12]. The full expressions involve integra- +tion over both the transverse momentum of the radiated +parton k and the transverse momentum of the Glauber +gluon q that mediates jet-medium interactions, and are +included in the supplementary material for completeness. +They contain transverse momentum propagator terms of +the form Va +V2 +a · Vb +V2 +b , with Va,b being any vectors among +k, k − xq, k − q, k − (1 − x)q. Nevertheless, in the large +parton phase space limit by shifting the integration vari- +able k we can cast P (1) +ij (x) into a generic form, +P (1) +ij (x, E, µ2 +2) = α(0) +s Pij(x) +2π2 +L +� µ2ϵ +2 d2−2ϵk +(2π)−2ϵ +Φ +� +k2L +2x(1−x)E +� +k2 +� +n +� µ2ϵ +2 d2−2ϵq +(2π)−2ϵ +ρG × 4πα(0) +s Cij +n ∆ij +n (x) +(2π)2(q2 + µ2 +D)2 +× q · [k + ∆ij +n (x)q] +[k + ∆ij +n (x)q]2 += α2 +s(µ2 +2)L2ρGPij(x) +8E[x(1 − x)]1+2ϵ +� +n +Cij +n (∆ij +n )2−2ϵ +�eγEµ2 +2L +2E +�2ϵ +ϵΓ(ϵ) +Γ(1 − ϵ) +� wmax +0 +dw 4 +π +Φ(w) +w1+ϵ +� 1 +0 +du +(1 − u)ϵ +−ϵuw + v 1−ϵ +2 (∆ij +n )2 +[uw + v(∆ij +n )2]2+ϵ +≈ A(µ2 +2, E, wmax)Pij(x) +[x(1 − x)]1+2ϵ +� +n +Cij +n (∆ij +n )2−2ϵ +� +µ2 +2L +χ(wmax)E +�2ϵ +(1 + O(ϵ2)) . +(6) +Here, E = zν, α(0) +s +is the bare coupling constant, and +Φ(u) = 1 − sin(u)/u is the Landau-Pomeranchuk-Migdal +interference phase. Cij +n and ∆ij +n (x) are color and kine- +matic factors of jet partons interacting with the Glauber +gluon for different channels i → j, listed in Table I. Be- +cause we focus on the radiative correction to collinear +observables for the jet sector, it is sufficient to represent +target properties by an effective Glauber gluon density +ρG (see supplementary material for detailed definition). +In performing q and k integrals, we introduce v = +µ2 +DL/[2x(1 − x)E] and an integration varaible w = +k2L/[2x(1 − x)E] with wmax = Q2L/(2ν) as bounded +by the maximum virtuality of the parton. Even through +we require Q2, ν/L ≫ µ2 +D, we do not have to assume +any ordering between Q2 and ν/L, so wmax can be an +order one quantity. Because we focus on modifications in +the collinear sector using splitting function obtained in +SCETG, xE, (1−x)E ≫ µD from power counting, which +allows one to take v = 0 [40]. Doing so results in the final +TABLE I. Color (Cij +n ) and kinematic factors (∆ij +n ) in Eq. (6) +i → j +Cij +1 , (∆ij +1 )2 +Cij +2 , (∆ij +2 )2 +Cij +3 , (∆ij +3 )2 +q → q +CA, x2 +CA, 1 +2CF −CA, (1−x)2 +q → g CA, 1 +CA, (1 − x)2 +2CF − CA, x2 +g → q CA, (1 − x)2 +CA, x2 +2CF − CA, 1 +g → g CA, 1 +CA, x2 +CA, (1 − x)2 +expression for the medium-induced branching, where we +denote for brevity +A(µ2 +2, E, wmax) = α2 +s(µ2 +2)L2B(wmax)ρG/(8E) . +(7) +B(wmax) and χ(wmax) depend only weakly on Q2 and +ν/L (see supplementary material). +With the results from Eq. (6), taking the medium- + +3 +induced q → q contribution in Eq. (5) as an example, +∆F m +iq (z, µ2 +2) = +� +Fiq + Fiq ⊗ P (1) +qq +� +⊗ +� +M (0) +qq + M (1) +qq +� +(8) +≈ +� 1 +0 +dx +� +P (1) +qq (x, z +xν, µ2 +2)Fiq +� z +x +� +− P (1) +qq (x, zν, µ2 +2)Fiq (z) +� ++ +� 1 +0 +dx +x Fiq(x)M (1) +qq +� z +x, µ2 +� +, +(9) +where we have used the expression for M (0) +qq , while the +NLO renormalization factor M (1) +qq will be determined af- +ter we identify the relevant poles. To do that, we note +that P (1) +ij (x) in Eq. (6) contain additional [x(1−x)]−1−2ϵ +divergences as compared to the vacuum splitting func- +tions, which do not cancel among real and virtual cor- +rections in Eq. (9). These extra divergences at x = 0, 1 +are consequences of dropping all screening effect in the +collinear sector from power counting. +If we take the +q → q channel as an example (details for other chan- +nels can be found in the supplementary material), we +can isolate these divergences, including the multiplicative +(1 − x)−1 contribution from the vacuum Pqq(x), using +the following decomposition for any well-behaved func- +tion G(x), +� 1 +0 +G(x) +x1+2ϵ(1 − x)2+2ϵ dx = +� 1 +0 +{G(x)}qq +x(1 − x)2 dx +−G(0) +2ϵ ++ G′(1) +2ϵ +− G(1) +� 1 +2ϵ + 2 +� ++ O(ϵ). +(10) +The result has been expanded near ϵ = 0, and the sub- +tracted function is {G(x)}qq = G(x) − (1 − x)2G(0) − +x(2 − x)G(1) − x(x − 1)G′(1). It is straightforward to +check that {G(x)}qq /[x1+2ϵ(1 − x)2+2ϵ] is free from di- +vergences at x = 0, 1, while the second line contains all +singularities. Note that due to the double pole at x = 1 +one needs to subtract both G(x) and the derivative G′(x) +at x = 1, and such derivative subtractions (a higher-order +“plus” prescription) are also used in the study of sublead- +ing power corrections in SCET [41]. +Following Eq. (10), the q → q contribution to the flavor +non-singlet sector ∆FNS = ∆F m +iq − ∆F m +i¯q can be decom- +posed into 1/ϵ poles and ln[Lµ2 +2/(χzν)] enhanced terms +plus fixed-order contributions +∆FNS(z, µ2 +2) = A(µ2 +2, ν, wmax) +� 1 +2ϵ + ln µ2 +2L +χzν +� +2CF +�2CA + CF +z +− 2CA +d +dz +� +FNS(z) + +� 1 +0 +dy +y FNS(y)M (1) +qq +�z +y , µ2, zν +� ++A(µ2 +2, ν, wmax) +� +� +� +� +� +� 1 +0 +�� +n Cqq +n [∆qq +n (x)]2CF (1 + x2) +� +x +z FNS +� z +x +� +− FNS(z) +z +�� +qq +x(1 − x)2 +dx + (4CA − CF )CF +FNS(z) +z +� +� +� +� +� +, +(11) +shown in the first and second lines of Eq. (11), respec- +tively. The medium contribution has a natural scale of +µ2 +2 = χzν/L. Divergences due to the Pqq(x) factor in +P (1) +qq (x) have canceled among the real and virtual terms. +The remaining poles come from extra x → 0, 1 diver- +gences of the medium-induced emission spectra. We can +now define the NLO medium renormalization factor M (1) +qq +such that it cancels the 1/ϵ pole in the first term. Note +that M (1) +qq will contain generalized functions and only de- +pend on µ2 through the coupling constant. +The in-medium RG evolution. +With the 1/ϵ pole ab- +sorbed in the renormazliation factor, we take a derivative +with respect to ln µ2 +2 on both sides of Eq. (11) and keep- +ing leading terms in αs and obtain an evolution equation +for the µ2 dependence of FNS +∂FNS(τ, z) +∂τ += 2CF +� +2CA +∂ +∂z − 2CA + CF +z +� +FNS . (12) +We have defined a new evolution variable τ(z, µ2 +2) = +πB(wmax)ρGL2 +2β0ν +[αs(µ2 +2) − αs( χzν +L )] to take into account the +running coupling effect with β0 = (11−2Nf/3). One can +perform a similar RG analysis on the flavor-singlet sector +and obtain (see supplementary material for details), +∂Ff +∂τ += 2CF +� +2CA +∂ +∂z − 2CA + CF +z +� +Ff + CF +Fg +z , (13) +∂Fg +∂τ = +� +4C2 +A +∂ +∂z − 2NfCF +z +� +Fg + 2C2 +F +� +f +Ff +z , +(14) +where Fg ≡ Fig is the gluon spectrum and Ff = Fiq +Fi¯q +for f = u, d, s are flavor-singlet quark spectra. Eqs. (12), +(13) and (14) are the main results of this letter. Starting +with initial condition at µ2 +2 = χzν/L and evolving down +to µ2 +2 = µ2 +D where screening effects become important, +the non-singlet Eq. (12) has a very elegant traveling wave +solution +FNS(τ, z) = +FNS (0, z + 4CF CAτ) +(1 + 4CF CAτ/z)1+ CF +2CA +. +(15) +The main effect of the µ2 +2 evolution is to shift the dis- +tribution of partons by ∆z = −4CF CAτ. This way, an +effective in-medium energy loss can be directly obtained +from RG analysis. +Neglecting the off-diagonal quark- +gluon coupling terms in the flavor-singlet Eqs. (13), (14), + +4 +similar traveling wave solutions can also be derived. For +applications in this letter, we will solve them numerically +including the off-diagonal coupling terms. +A widely used phenomenological approach for par- +ton evolution in matter is based upon modified DGLAP +(mDGLAP) franework [20, 42–44]. Consider the flavor +non-singlet equation +∂FNS(z) +∂ ln µ2 += +� 1 +0 +k2 d[Pqq(x, k2) + P (1) +qq (x, k2)] +dxdk2 +× +� +FNS +� z +x +� +− FNS(z) +� +dx , +(16) +with µ2 = k2/[x(1−x)] being the virtuality of the parton, +and d[Pqq + P (1) +qq ]/dxdk2 the double differential splitting +function including both vacuum and medium-induced +contributions. +Despite its apparent different form, we +now show that the mDGLAP approach resums the same +medium enhancement as the in-medium RG equation +to leading logarithmic accuracy. +Unlike the RG anal- +ysis that uses dimensional regularization, the mDGLAP +equation evaluates the full splitting functions numerically +and regulates the endpoint divergences of P (1) +ij +such that +x, 1−x ≥ µ2 +D/µ2 [45]. If we focus on the medium-induced +contributions from the x ≈ 1 region and use a fixed cou- +pling A(µfix, ν) = A0 for simplicity, the mDGLAP equa- +tion becomes +∂FNS +∂ ln µ2 = 4CF CAA0 +� 1− +µ2 +D +µ2 +0 +4 +π +Φ(u) +u +x +z FNS( z +x) − FNS(z) +z +(1 − x)2 +dx +≈ 4 +π +Φ(u) +u +4CF CAA0 +�∂FNS +∂z +− FNS +z +� +ln µ2 +µ2 +D +≈ δ +� +µ2 − 2πE +L +� +4CF CAA0 +�∂FNS +∂z +− FNS +z +� +ln µ2 +µ2 +D +, (17) +with u = µ2L/(2E). In the second line, we have per- +formed a Taylor expansion of (x/z)F(z/x) near x = 1 +and omitted subleading terms in ln[E/(Lµ2 +D)]. +The connection to the RG analysis is most easily il- +lustrated by considering a specific case of scale separa- +tion Q2 +0 ≪ E/L ≪ Q2. Because 4 +π +Φ(u) +u +peaks at u = π +and normalizes to unity +� ∞ +0 +4 +π +Φ(u) +u d ln u = 1, one can +make an impulse approximation, as shown in the third +line of Eq. (17). +Then, to leading-log accuracy, one +can perform vacuum DGLAP evolution above and be- +low µ2 = 2πE/L. However, due to medium effects that +sharply peak at µ2 = 2πE/L, the solution below 2πE/L +(F − +NS) and the solution above (F + +NS) are related by +F + +NS(z) = F − +NS(z + 4CF CAτfix) +1 + 4CF CAτfix/z +, +(18) +with τfix = A0 ln 2πE +µ2 +DL. This is the same traveling wave +solution as in Eq. (15), but with fixed coupling and ne- +glecting contributions from the x = 0 endpoint [46]. We, +therefore, conclude that the mDGLAP approach with the +FIG. 1. Top panel: medium modifications to the π+ fragmen- +tation function compare to HERMES data, performed for the +average ν = 12 GeV, Q2 = 2.25 GeV2. Bottom panel: pre- +dictions for the modified pion fragmentation function at EIC +with Pb nucleus for three (xB, Q2) combinations. +chioce µ2 = k2/[x(1 − x)] resums the same medium en- +hanced branchings as the RG equation derived in this +letter. Of course, with a large separation of scales, it be- +comes computationally intensive to evaluate the RHS of +the mDGLAP equation with an explicit cut-off and the +analytic approach that we formulated here is not only +more illuminating, but also easier to implement. +We demonstrate the new method by studying nuclear +effects on pion fragmentation in SIDIS, with the cross +section given in Eq. (2). We implement the fully-coupled +RG evolution Eqs. (12), (13), (14), and the fixed order +terms in Eq. (11). The nuclear modifications is defined +as the ratio of inclusive-normalized cross sections be- +tween electron-nucleus (eA) and electron-deuterium (ed +for HERMES) or electron-proton (ep for EIC) collisions +RA(xB, Q2, zh) = +dσeA→h +dxBdQ2dzh / +dσeA +dxBdQ2 +dσed,ep→h +dxBdQ2dzh / dσed,ep +dxBdQ2 +. +(19) +For parton distribution and fragmentation functions we +use the nNNPDF30nlo [47] and NNFF10lo parametriza- +tions [48] and perform the calculation for the aver- +aged HERMES kinematics Q2 = 2.25 GeV2 and ν = +12 GeV [49]. To numerically solve Eqs. (12) and (14), +we smear the singular hard parton energy spectrum by a +Gaussian with width parameter σz = 0.05 and the evo- +lution from µ1 = Q to Q0 = 1 GeV is performed us- +ing standard vacuum DGLAP. In turn, the in-medium +RG evolves µ2 +2 from χ(wmax)zν/L to µ2 +D. +We take +ΛQCD = 0.16 GeV, the inverse range of the interac- +tion µ2 +D = 0.12 GeV2 and the central value of the effec- +tive medium density parameter ρG = 0.4 fm−3. These +values yield a quark transport parameter ˆqF ≈ 0.053 + +HERMES π + +HERMES ↑ + +HERMES ↑ + +1.0 +0.8 +R +Ne +Kr +Xe +0.6 +FO+RG +RG +0.4 +e+Pb→ +e+Pb→± +e+Pb→ +1.0 +0.8 +R +0.6 +XB = 0.1 +XB = 0.3 +XB = 0.5 +0.4 +2 = 20 GeV2 +Q2 = 20 GeV2 += 20 GeV2 +0.0 0.2 0.4 0.6 0.8 0.0 0.2 +0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 +Zh5 +GeV2/fm at ν = 12 GeV (see supplemental material), +consistent with existing mDGLAP applications [50, 51]. +An average over the geometry of the nucleus of radius +rA = 1.2A1/3 fm is also performed. +The resulting nuclear modification factor RA(zh) is +compared to the HERMES data for 20Ne and 131Xe tar- +gets [49] in the top row of Fig. 1. +Qualitatively, in- +medium evolution shifts hadron spectra towards lower +zh. Results including only RG contributions (red dashed +lines) give a good description of RA from small to inter- +mediate z, but lead to a suppression that is too strong at +large zh. We remark that the region very close to zh = 1 +is dominated by soft emissions, where one should consider +soft power counting and threshold type resummation in +both vacuum [52, 53] and in-medium calculations. There- +fore, we have exclude this region from our comparison. +Blue solid lines include the fixed order (FO) contribution +from Eq. (10) in the initial condition of the RG evolu- +tion, and the bands correspond to the density variation in +the range (ρG/1.5, 1.5ρG). The FO correction improves +the description of HERMES data at large z, but remains +subleading to the RG evolution effect. The nuclear size +dependence of RA for Ne, Kr, and Xe nuclei is naturally +explained with the same set of transport parameters. +Using the same in-medium transport parameters, we +present projections (lower panel of Fig. 1) for modified +pion fragmentation functions at the future electron-ion +collider (EIC) for ePb reactions at fixed Q2 = 20 GeV2 +and various Bjorken xB values. We find that for xB > +0.3, where partons are less energetic in the nuclear rest +frame, modifications become very large, consistent with +existing predictions for heavy flavor and jets [50, 51]. +Summary. +In the limit Q/Q0, E/(µ2 +DL), λgµD ≫ 1 and +to first order in the opacity of QCD matter we performed +a renormalization group (RG) analysis of medium effects +for the SIDIS process on a nuclear target. We derived a +set of in-medium RG equations that resum the leading +ln[E/(µ2 +DL)] terms from multiple medium-induced emis- +sions and identified the corresponding fixed order correc- +tions. We further showed that such resummation is also +contained in the modified DGLAP equations, which dif- +fer in the way of regulating the endpoint divergences of +medium-induced emission spectra. Importantly, the new +RG evolution in matter approach provides analytic in- +sight into the salient features of parton showers respon- +sible for the modification of hadron production in eA +that are not possible with numerical methods alone. It +is a more efficient and systematically improvable way of +treating the logarithmic enhancements in matter as com- +pared to solving mDGLAP. +We applied the new method to study the cold nuclear +matter (CNM) effects on pion fragmentation and found +that it gives a good description of the HERMES SIDIS +data. Predictions for the future EIC were also presented, +where improved theoretical precision is especially impor- +tant [15]. The semi-analytic framework derived here can +be generalized to initial-state CNM effects, such as the +ones observed in Drell-Yan production in proton-nucleus +collisions, and to heavy ion collisions. 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C 75, 064906 (2007), arXiv:hep- +ph/0703002. + +7 +SUPPLEMENTARY MATERIAL +In-medium scattering cross section and jet transport parameter estimate +The elastic cross section between jet and target partons in color representations R and T, respectively, is [13] +dσT R +d2q = +1 +(2π)2 +1 +dA +4παsCT × 4παsCR +(q2 + µ2 +D)2 +, +(20) +with dA = N 2 +c − 1. The collision rate after summing over the medium color sources of representation T with density +ρT then reads +� +T +ρT +dσT R +d2q = +� +T +1 +(2π)2 +ρT +dA +4παsCT × 4παsCR +(q2 + µ2 +D)2 += αsCR +π +1 +(q2 + µ2 +D)2 ρG , +ρG ≡ +� +T +ρT +4παmed +s +CT +dA +. +(21) +In other words, we have chosen to put kinematic factors, the square color charges and coupling to the medium in the +effective medium gluon density ρG. With µ2 +D = 0.12 GeV2, ρG = 0.4 fm−3, the quark jet transport parameter is +ˆqF = +� q2 +max=νµD/2 +0 +d2q q2 αsCF +π +1 +(q2 + µ2 +D)2 ρG ≈ 0.053 GeV2/fm +(22) +for ν = Q2/(2xBMp) = 12 GeV in the nuclear rest frame. Here, the ultraviolet cut off of the q2 integration is chosen to +be νµD/2, as in Ref. [54]. The running coupling is cut-off when αs reaches 2π/β0.The value of ˆq is further consistent +with the analysis of [50, 51]. +Full splitting functions in matter +The splitting functions in nuclear matter induced by final-state interactions are taken from Refs. [13, 14] (in d = 4 +dimension). After performing the path length integration in a medium of uniform density and size L, the splitting +functions become +P (1) +ij (x) = αs +2π2 Pij(x)L +� +d2k +� +T +� +d2q +(2π)2 +ρT +dA +4παsCT × 4παs +(q2 + µ2 +D)2 +Wij(x, k, q, E/L) +≡ αs +2π2 Pij(x)L +� +d2k +� +d2q +(2π)2 +ρG × 4παs +(q2 + µ2 +D)2 Wij(x, k, q, E/L) . +(23) +The continuous parts of the vacuum splitting functions in d = 4 − 2ϵ dimension, arising from real emissions, are +Pqq(x) = CF +�1 + x2 +1 − x − ϵ(1 − x) +� +, Pgq(x) = TR[x2 + (1 − x)2 − 2ϵx(1 − x)], +(24) +Pqg(x) = Pqq(1 − x), Pgg(x) = CA +1 + x4 + (1 − x)4 +x(1 − x) +. +(25) +The diagonal terms also receive virtual corrections, which we determined from flavor and momentum sum rules [44]. +For P (1) +ij (x), we define A⊥ = k, B⊥ = k + (1 − x)q, C⊥ = k − xq, D⊥ = k + q, and interference phase factors +ΦA = Φ +� +A2 +⊥L +2x(1 − x)E +� +, ΦB = Φ +� +B2 +⊥L +2x(1 − x)E +� +, ΦC = Φ +� +C2 +⊥L +2x(1 − x)E +� +, +ΦAD = Φ +�(A2 +⊥ − D2 +⊥)L +2x(1 − x)E +� +, ΦCB = Φ +�(C2 +⊥ − B2 +⊥)L +2x(1 − x)E +� +. +(26) +Then, Wij, including the relevant quadratic Casimirs from the Glauber gluon–hard parton system interactions, are +Wqq = CA +B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− A⊥ +A2 +⊥ +� +ΦB + CA +B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− C⊥ +C2 +⊥ +� +ΦB + (2CF − CA)C⊥ +C2 +⊥ +�C⊥ +C2 +⊥ +− A⊥ +A2 +⊥ +� +ΦC + CA∆W , +(27) +Wgg = CA +B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− A⊥ +A2 +⊥ +� +ΦB + CA +B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− C⊥ +C2 +⊥ +� +ΦB + CA +C⊥ +C2 +⊥ +�C⊥ +C2 +⊥ +− A⊥ +A2 +⊥ +� +ΦC + CA∆W , +(28) +Wgq = CA +B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− A⊥ +A2 +⊥ +� +ΦB + (2CF − CA)B⊥ +B2 +⊥ +�B⊥ +B2 +⊥ +− C⊥ +C2 +⊥ +� +ΦB + CA +C⊥ +C2 +⊥ +�C⊥ +C2 +⊥ +− A⊥ +A2 +⊥ +� +ΦC + (2CF − CA)∆W, (29) + +8 +and Wqg = Wqq(x → 1 − x). For each channel, the first three terms can be cast into the form of the first line of +Eq. (6) by shifting integration variables such that arguments of Φ(·) become k2L/[2x(1 − x)E]. The remaining piece +∆W, +∆W = +�C⊥ +C2 +⊥ +�C⊥ +C2 +⊥ +− B⊥ +B2 +⊥ +� +ΦC + B⊥ +B2 +⊥ +· C⊥ +C2 +⊥ +ΦCB +� +− +�A⊥ +A2 +⊥ +�A⊥ +A2 +⊥ +− D⊥ +D2 +⊥ +� +ΦA + A⊥ +A2 +⊥ +· D⊥ +D2 +⊥ +ΦAD +� +, +(30) +is written as the difference of two terms. Note that the second term can be obtained from the first one by shifting +k → k + xq, causing C⊥ → A⊥ and B⊥ → D⊥. Therefore, ∆W = 0 under dimensional regularized integration of k +and q. If one uses an explicit ultraviolet cut-off ΛUV, the integration of ∆W over q, k (or in general, any differences +caused by a shift of k + ∆ · q of Eqs. (27),(28), and (29)) are further suppressed by Λ−2 +UV and do not contribute to the +medium-induced logarithmic enhancement. +We can account for the virtuality of the collision Eq. (6) by introducing the variable wmax = Q2L/(2ν), which +appears in the functions B and χ defined as +B(wmax) = 4 +π +� wmax +0 +Φ(x)dx +x2 , +χ(wmax) = 2 exp +� +1 +B(wmax) +4 +π +� wmax +0 +Φ(x) ln(x)dx +x2 + γE +� +1 +B(wmax) − 1 +�� +. (31) +For the SIDIS process at moderate xB, wmax ≡ xBMpL ≈ 6.0xBA1/3 is of order few. For fragmentation at mid +rapidity in hadronic collisions, Q2L/(2E) ∼ EL → ∞, and B(∞) = 1, χ(∞) = 2e3/2−γE ≈ 5.0. +The flavor singlet sector +Analogously to the treatment of the flavor non-singlet sector, we provide the details for the subtraction of divergences +and renormalization of the flavor singlet sector. To isolate the extra [x(1 − x)]−1−2ϵ poles, we define the following +decomposition for any well-behaved function G(x). For singularities associate to P (1) +qg (x), +� 1 +0 +G(x) +x2+2ϵ(1 − x)1+2ϵ dx = +� 1 +0 +{G(x)}qg +x2(1 − x)dx − G(0) +2ϵ +− G′(0) +2ϵ +− G(1) +� 1 +2ϵ + 2 +� ++ O(ϵ) , +(32) +with {G(x)}qg = G(x) − x2G(1) − (1 − x2)G(0) − x(1 − x)G′(0). For singularities associate to P (1) +gg (x), +� 1 +0 +G(x) +x2+2ϵ(1 − x)2+2ϵ dx = +� 1 +0 +{G(x)}gg +x2(1 − x)2 dx − G(0) +�1 +ϵ + 2 +� +− G′(0) +2ϵ +− G(1) +�1 +ϵ + 2 +� ++ G′(1) +2ϵ ++ O(ϵ) , +(33) +with {G(x)}gg = G(x) − (1 − x)2 [(1 + 2x)G(0) + xG′(0)] − x2 [(3 − 2x)G(1) + (x − 1)G′(1)]. Finally, for P (1) +gq (x) +� 1 +0 +G(x) +x1+2ϵ(1 − x)1+2ϵ dx = +� 1 +0 +{G(x)}gq +x(1 − x) dx − G(0) +2ϵ +− G(1) +2ϵ ++ O(ϵ) , +(34) +with {G(x)}gq = G(x) − xG(1) − (1 − x)G(0). One can directly and explicitly check that the endpoint divergences +are removed from the integration. With this procedure, the medium-induced NLO contributions from these channels +can be decomposed into log-enhanced and fixed order contributions, +∆Fg +A(µ2 +2, E, wmax) = +� 1 +2ϵ + L +� � +�−4C2 +A +dFg(z) +dz ++ 2CF Nf +Fg(z) +z +− 2C2 +F +� +f +Ff(z) +z +� +� ++ +� +f +� 1 +0 +dx +�� +n Cqg +n (∆qg +n )2 CF [1 + (1 − x)2] x +z Ff( z +x) +� +qg +x2(1 − x) +− Nf +Fg(z) +z +� 1 +0 +dx +�� +i Cgq +n (∆gq +n )2 TR[x2 + (1 − x)2] +� +gq +x(1 − x) ++ +� 1 +0 +dx +�� +i Cgg +n (∆gg +n )2 CA[1 + x4 + (1 − x)4] +� +x +z Fg( z +x) − x Fg(z) +z +�� +gg +x2(1 − x)2 ++ 14C2 +A +Fg(z) +z +− 3C2 +F +� +f +Ff(z) +z +, +(35) + +9 +∆Ff +A(µ2 +2, E, wmax) = +� 1 +2ϵ + L +� � +−4CF CA +dFf(z) +dz ++ 2CF (2CA + CF )Ff(z) +z +− CF +Fg(z) +z +� ++ +� 1 +0 +�� +n Cn∆2 +n(x)CF (1 + x2) +� +x +z Ff +� z +x +� +− Ff (z) +z +�� +qq +x(1 − x)2 +dx + (4CA − CF )CF +Ff(z) +z ++ +� 1 +0 +dx +�� +i Cgq +n (∆gq +n )2 TR[x2 + (1 − x)2] x +z Fg( z +x) +� +gq +x(1 − x) +, +(36) +where L ≡ ln +µ2 +2 +χE/L and {· · · }ij stands for the subtraction for the corresponding channel i → j. +The 1/ϵ poles +are subtracted by the corresponding contribution from M (1) +ij . Then, taking derivatives with respect to τ gives the +in-medium RG Eqs. (13) and (14) for the flavor singlet sector. + diff --git a/xdFKT4oBgHgl3EQf6C78/content/tmp_files/load_file.txt b/xdFKT4oBgHgl3EQf6C78/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..497489405e964d0e78dd4c75066b3075b7837b9b --- /dev/null +++ b/xdFKT4oBgHgl3EQf6C78/content/tmp_files/load_file.txt @@ -0,0 +1,641 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf,len=640 +page_content='LA-UR-23-20730 Understanding parton evolution in matter from renormalization group analysis Weiyao Ke∗ and Ivan Vitev† Theoretical Division, Los Alamos National Laboratory, Los Alamos NM 87545, United States (Dated: January 31, 2023) We perform a renormalization group (RG) analysis of collinear hadron production in deep inelastic scattering on nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We consider the limit where one of the dimensionless in-medium scale ratios E/(µ2 DL) ≫ 1, with L, µD, E being the medium size, inverse scattering range and the parton energy in the nuclear rest frame, while the opacity L/λg remains small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We identify the fixed order and leading ln[E/(µ2 DL)] enhanced medium contributions to the semi-inclusive cross sections and derive RG equations which resum multiple emissions near the x → 0, 1 endpoints of the splitting functions at first order in opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='We find that the evolution equations obtained in this work treat the same type of radiation enhancement in matter as the modified DGLAP approach, but differ in the way one chooses to regulate the endpoint divergences and provide unique analytic insight into the problem of resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The new RG evolution framework is applied to study fragmentation in eA reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' A common characteristic of many prob- lems in science is that microscopic fluctuations in the system manifest themselves in macroscopic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Such problems arise in fields ranging from social networks [1] and turbulence [2] to particle [3] and nuclear physics [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' They are most prevalent in inherently divergent theo- ries and efficiently addressed using renormalization group (RG) analysis [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Effective theories of quantum chro- modynamics (QCD) geared toward jet physics [8, 9] have provided new insights into renormalization and resum- mation, and given a modern perspective to the problem of parton production and propagation in nuclear mat- ter [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' These advances are key to the interpretation of the data from reactions with nuclei at current and fu- ture colliders [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Over the past two decades, medium-induced par- ton showers have been successfully implemented in jet quenching phenomenology to describe the modification of hadron and jet cross sections, and jet substructure in nuclear collisions [16–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Still, resummation of QCD radiation in nuclear matter remains challenging, espe- cially lacking in analytic insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We address this long- standing problem using RG techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' If we consider semi-inclusive hadron production in deep inelastic scat- tering (DIS) on a nuclear target (eA→ h + X), we en- counter a number of energy and length scales (defined in the target rest frame) including: 1) the hard scale Q, 2) energy of the virtual photon/jet ν, 3) the path length L, 4) the mean free path λg and 5) the inverse interaction range µD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Therefore, observables in eA are functions of many dimensionless control parameters Obs ≡ Obs � Q Q0 , E µ2 DL, L λg , λgµD, µD Q0 , · · · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (1) A simplified description with controlled accuracy is of- ten possible when one (or more) of these dimension- less ratios become asymptotically large [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For exam- ple, the limit λgµD ≫ 1, implying independent multi- ple parton-medium scatterings, allows the use of time- ordered perturbation theory to derive quark and gluon splitting functions in matter [13, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' A partonic trans- port picture emerges when L/λg ≫ 1 in a thick and dense medium [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In this letter, we compute hadron production in a particular regime when Q/Q0, E/(µ2 DL), λgµD become asymptotically large while L/λg, µD/Q0 stay at or- der unity/few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This limit is particularly relevant for high-energy hadron production in thin, dilute or fast- expanding media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Still, renormalization is needed to re- sum large ln[Q/Q0] and ln[E/(µ2 DL)] enhancements from the vacuum and medium-induced radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We thus introduce two final-state renormalization scales µ1 and µ2 in the single-inclusive hadron cross section [34] dσeA→h dxBdQ2dzh = 2πα2 e Q4 � i,j e2 q � � fi/A⊗ � (1 + (1 − y)2)C1 ij + 2(1 − y)CL ij �� xB ⊗ dh/j � zh , (2) {h ⊗ g}x ≡ � 1 x h � x x′ � g(x′)dx′ x′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (3) Here, y = ν/Ee with Ee and ν = Q2/(2xBMp) be- ing the energies of the incident electron and the vir- tual photon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' fi/A(x, Q2), dh/j(z, µ2 1, µ2 2) and C1,L ij (x, z, Q2, µ2 1, µ2 2) are the parton distribution func- tions, fragmentation functions, and the hard coefficient functions with fractional electric charge eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The PDF is evaluated at scale Q2, and the dependence on Q2 will not be written explicitly hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The cross-section fi/A ⊗ C1,L ij ⊗ dj/h must not depend on µ1 and µ2, thus it is sufficient to study scale dependence of F 1,L ij (z) ≡ fi/A ⊗ zC1,L ij [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This quantity can be interpreted as the invariant distribution of parton j, resolved at scales µ1, µ2, after the hard process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This “parton shower” choice enables us to track the evolving parton energy E = zν, which is important for the most consistent implementation of medium-induced splitting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' F 1 ij = zδijfi/A(xB)δ(1 − z) + αs(· · · ), F L ij = αs(· · · ), and we use the NLO expression for C1,L ij [34, 36] without writ- ing them explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='11940v1 [hep-ph] 27 Jan 2023 2 As will become clear in a moment, the Q/Q0 and E/(µ2 DL) enhancements have distinct physics origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Therefore, in addition to the vacuum renormalization that leads to the DGLAP evolution, the “medium bare” Fij needs to be further renormalized by a medium coef- ficient Mkj that only depends on µ2, Fij(z, µ2 1, µ2 2) → Fik(y, µ2 1, µ2 2) ⊗ Mkj �z y , µ2 2 � + F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (4) Here, the Mkj = M (0) kj + M (1) kj + · · · with M (0) kj = yδkjδ(1 − y) and the first non-trivial contribution M (1) kj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' F(z) stands for medium contributions subleading in ln[E/(µ2 DL)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' other fixed order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Renormalization group analysis of endpoint divergences in collinear emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We consider the correc- tion to Fij(z) from both vacuum and medium-induced collinear splittings and find that it has the form αs(µ2 1)µ2ϵ 1 2π2 � Q2 d2−2ϵk k2 Fik ⊗ x[Pkj]+ + ∆F m ij (z, µ2ϵ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (5) We work in d = 4 − 2ϵ dimensions and Pkj are the vac- uum Altarelli-Parisi splitting functions [37, 38], with the “+” prescription in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (5) only applied to diagonal con- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In the first term, µ1 acts as an infrared cutoff of vacuum emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In this case RG analysis leads to DGLAP evolution of Fij(z) in µ1 that resums ln[Q/Q0], and we always evolve µ1 from the hard scale Q down to Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Next, we will extract the leading logarithmic and fixed order contribution from the medium-induced cor- rection in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For thin and uniform nuclear matter of length L we use the medium-induced splitting functions P (1) ij (x) [13, 14, 30, 39] obtained in the opacity expansion approach us- ing Soft-Collinear-Effective-Theory with Glauber Gluons (SCETG) [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The full expressions involve integra- tion over both the transverse momentum of the radiated parton k and the transverse momentum of the Glauber gluon q that mediates jet-medium interactions, and are included in the supplementary material for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' They contain transverse momentum propagator terms of the form Va V2 a · Vb V2 b , with Va,b being any vectors among k, k − xq, k − q, k − (1 − x)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Nevertheless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' in the large parton phase space limit by shifting the integration vari- able k we can cast P (1) ij (x) into a generic form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' P (1) ij (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' µ2 2) = α(0) s Pij(x) 2π2 L � µ2ϵ 2 d2−2ϵk (2π)−2ϵ Φ � k2L 2x(1−x)E � k2 � n � µ2ϵ 2 d2−2ϵq (2π)−2ϵ ρG × 4πα(0) s Cij n ∆ij n (x) (2π)2(q2 + µ2 D)2 × q · [k + ∆ij n (x)q] [k + ∆ij n (x)q]2 = α2 s(µ2 2)L2ρGPij(x) 8E[x(1 − x)]1+2ϵ � n Cij n (∆ij n )2−2ϵ �eγEµ2 2L 2E �2ϵ ϵΓ(ϵ) Γ(1 − ϵ) � wmax 0 dw 4 π Φ(w) w1+ϵ � 1 0 du (1 − u)ϵ −ϵuw + v 1−ϵ 2 (∆ij n )2 [uw + v(∆ij n )2]2+ϵ ≈ A(µ2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' wmax)Pij(x) [x(1 − x)]1+2ϵ � n Cij n (∆ij n )2−2ϵ � µ2 2L χ(wmax)E �2ϵ (1 + O(ϵ2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6) Here, E = zν, α(0) s is the bare coupling constant, and Φ(u) = 1 − sin(u)/u is the Landau-Pomeranchuk-Migdal interference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Cij n and ∆ij n (x) are color and kine- matic factors of jet partons interacting with the Glauber gluon for different channels i → j, listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Be- cause we focus on the radiative correction to collinear observables for the jet sector, it is sufficient to represent target properties by an effective Glauber gluon density ρG (see supplementary material for detailed definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In performing q and k integrals, we introduce v = µ2 DL/[2x(1 − x)E] and an integration varaible w = k2L/[2x(1 − x)E] with wmax = Q2L/(2ν) as bounded by the maximum virtuality of the parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Even through we require Q2, ν/L ≫ µ2 D, we do not have to assume any ordering between Q2 and ν/L, so wmax can be an order one quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Because we focus on modifications in the collinear sector using splitting function obtained in SCETG, xE, (1−x)E ≫ µD from power counting, which allows one to take v = 0 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Doing so results in the final TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Color (Cij n ) and kinematic factors (∆ij n ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6) i → j Cij 1 , (∆ij 1 )2 Cij 2 , (∆ij 2 )2 Cij 3 , (∆ij 3 )2 q → q CA, x2 CA, 1 2CF −CA, (1−x)2 q → g CA, 1 CA, (1 − x)2 2CF − CA, x2 g → q CA, (1 − x)2 CA, x2 2CF − CA, 1 g → g CA, 1 CA, x2 CA, (1 − x)2 expression for the medium-induced branching, where we denote for brevity A(µ2 2, E, wmax) = α2 s(µ2 2)L2B(wmax)ρG/(8E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (7) B(wmax) and χ(wmax) depend only weakly on Q2 and ν/L (see supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' With the results from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6), taking the medium- 3 induced q → q contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (5) as an example, ∆F m iq (z, µ2 2) = � Fiq + Fiq ⊗ P (1) qq � ⊗ � M (0) qq + M (1) qq � (8) ≈ � 1 0 dx � P (1) qq (x, z xν, µ2 2)Fiq � z x � − P (1) qq (x, zν, µ2 2)Fiq (z) � + � 1 0 dx x Fiq(x)M (1) qq � z x, µ2 � , (9) where we have used the expression for M (0) qq , while the NLO renormalization factor M (1) qq will be determined af- ter we identify the relevant poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' To do that, we note that P (1) ij (x) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6) contain additional [x(1−x)]−1−2ϵ divergences as compared to the vacuum splitting func- tions, which do not cancel among real and virtual cor- rections in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' These extra divergences at x = 0, 1 are consequences of dropping all screening effect in the collinear sector from power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' If we take the q → q channel as an example (details for other chan- nels can be found in the supplementary material), we can isolate these divergences, including the multiplicative (1 − x)−1 contribution from the vacuum Pqq(x), using the following decomposition for any well-behaved func- tion G(x), � 1 0 G(x) x1+2ϵ(1 − x)2+2ϵ dx = � 1 0 {G(x)}qq x(1 − x)2 dx −G(0) 2ϵ + G′(1) 2ϵ − G(1) � 1 2ϵ + 2 � + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (10) The result has been expanded near ϵ = 0, and the sub- tracted function is {G(x)}qq = G(x) − (1 − x)2G(0) − x(2 − x)G(1) − x(x − 1)G′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' It is straightforward to check that {G(x)}qq /[x1+2ϵ(1 − x)2+2ϵ] is free from di- vergences at x = 0, 1, while the second line contains all singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Note that due to the double pole at x = 1 one needs to subtract both G(x) and the derivative G′(x) at x = 1, and such derivative subtractions (a higher-order “plus” prescription) are also used in the study of sublead- ing power corrections in SCET [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' the q → q contribution to the flavor non-singlet sector ∆FNS = ∆F m iq − ∆F m i¯q can be decom- posed into 1/ϵ poles and ln[Lµ2 2/(χzν)] enhanced terms plus fixed-order contributions ∆FNS(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' µ2 2) = A(µ2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' wmax) � 1 2ϵ + ln µ2 2L χzν � 2CF �2CA + CF z − 2CA d dz � FNS(z) + � 1 0 dy y FNS(y)M (1) qq �z y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' zν � +A(µ2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' wmax) � � � � � � 1 0 �� n Cqq n [∆qq n (x)]2CF (1 + x2) � x z FNS � z x � − FNS(z) z �� qq x(1 − x)2 dx + (4CA − CF )CF FNS(z) z � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (11) shown in the first and second lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (11), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The medium contribution has a natural scale of µ2 2 = χzν/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Divergences due to the Pqq(x) factor in P (1) qq (x) have canceled among the real and virtual terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The remaining poles come from extra x → 0, 1 diver- gences of the medium-induced emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We can now define the NLO medium renormalization factor M (1) qq such that it cancels the 1/ϵ pole in the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Note that M (1) qq will contain generalized functions and only de- pend on µ2 through the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The in-medium RG evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' With the 1/ϵ pole ab- sorbed in the renormazliation factor, we take a derivative with respect to ln µ2 2 on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (11) and keep- ing leading terms in αs and obtain an evolution equation for the µ2 dependence of FNS ∂FNS(τ, z) ∂τ = 2CF � 2CA ∂ ∂z − 2CA + CF z � FNS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (12) We have defined a new evolution variable τ(z, µ2 2) = πB(wmax)ρGL2 2β0ν [αs(µ2 2) − αs( χzν L )] to take into account the running coupling effect with β0 = (11−2Nf/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' One can perform a similar RG analysis on the flavor-singlet sector and obtain (see supplementary material for details), ∂Ff ∂τ = 2CF � 2CA ∂ ∂z − 2CA + CF z � Ff + CF Fg z , (13) ∂Fg ∂τ = � 4C2 A ∂ ∂z − 2NfCF z � Fg + 2C2 F � f Ff z , (14) where Fg ≡ Fig is the gluon spectrum and Ff = Fiq +Fi¯q for f = u, d, s are flavor-singlet quark spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (12), (13) and (14) are the main results of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Starting with initial condition at µ2 2 = χzν/L and evolving down to µ2 2 = µ2 D where screening effects become important, the non-singlet Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (12) has a very elegant traveling wave solution FNS(τ, z) = FNS (0, z + 4CF CAτ) (1 + 4CF CAτ/z)1+ CF 2CA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (15) The main effect of the µ2 2 evolution is to shift the dis- tribution of partons by ∆z = −4CF CAτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This way, an effective in-medium energy loss can be directly obtained from RG analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Neglecting the off-diagonal quark- gluon coupling terms in the flavor-singlet Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (13), (14), 4 similar traveling wave solutions can also be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For applications in this letter, we will solve them numerically including the off-diagonal coupling terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' A widely used phenomenological approach for par- ton evolution in matter is based upon modified DGLAP (mDGLAP) franework [20, 42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Consider the flavor non-singlet equation ∂FNS(z) ∂ ln µ2 = � 1 0 k2 d[Pqq(x, k2) + P (1) qq (x, k2)] dxdk2 × � FNS � z x � − FNS(z) � dx , (16) with µ2 = k2/[x(1−x)] being the virtuality of the parton, and d[Pqq + P (1) qq ]/dxdk2 the double differential splitting function including both vacuum and medium-induced contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Despite its apparent different form, we now show that the mDGLAP approach resums the same medium enhancement as the in-medium RG equation to leading logarithmic accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Unlike the RG anal- ysis that uses dimensional regularization, the mDGLAP equation evaluates the full splitting functions numerically and regulates the endpoint divergences of P (1) ij such that x, 1−x ≥ µ2 D/µ2 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' If we focus on the medium-induced contributions from the x ≈ 1 region and use a fixed cou- pling A(µfix, ν) = A0 for simplicity, the mDGLAP equa- tion becomes ∂FNS ∂ ln µ2 = 4CF CAA0 � 1− µ2 D µ2 0 4 π Φ(u) u x z FNS( z x) − FNS(z) z (1 − x)2 dx ≈ 4 π Φ(u) u 4CF CAA0 �∂FNS ∂z − FNS z � ln µ2 µ2 D ≈ δ � µ2 − 2πE L � 4CF CAA0 �∂FNS ∂z − FNS z � ln µ2 µ2 D , (17) with u = µ2L/(2E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In the second line, we have per- formed a Taylor expansion of (x/z)F(z/x) near x = 1 and omitted subleading terms in ln[E/(Lµ2 D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The connection to the RG analysis is most easily il- lustrated by considering a specific case of scale separa- tion Q2 0 ≪ E/L ≪ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Because 4 π Φ(u) u peaks at u = π and normalizes to unity � ∞ 0 4 π Φ(u) u d ln u = 1, one can make an impulse approximation, as shown in the third line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Then, to leading-log accuracy, one can perform vacuum DGLAP evolution above and be- low µ2 = 2πE/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' However, due to medium effects that sharply peak at µ2 = 2πE/L, the solution below 2πE/L (F − NS) and the solution above (F + NS) are related by F + NS(z) = F − NS(z + 4CF CAτfix) 1 + 4CF CAτfix/z , (18) with τfix = A0 ln 2πE µ2 DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This is the same traveling wave solution as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (15), but with fixed coupling and ne- glecting contributions from the x = 0 endpoint [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We, therefore, conclude that the mDGLAP approach with the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Top panel: medium modifications to the π+ fragmen- tation function compare to HERMES data, performed for the average ν = 12 GeV, Q2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='25 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Bottom panel: pre- dictions for the modified pion fragmentation function at EIC with Pb nucleus for three (xB, Q2) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' chioce µ2 = k2/[x(1 − x)] resums the same medium en- hanced branchings as the RG equation derived in this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Of course, with a large separation of scales, it be- comes computationally intensive to evaluate the RHS of the mDGLAP equation with an explicit cut-off and the analytic approach that we formulated here is not only more illuminating, but also easier to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We demonstrate the new method by studying nuclear effects on pion fragmentation in SIDIS, with the cross section given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We implement the fully-coupled RG evolution Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (12), (13), (14), and the fixed order terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The nuclear modifications is defined as the ratio of inclusive-normalized cross sections be- tween electron-nucleus (eA) and electron-deuterium (ed for HERMES) or electron-proton (ep for EIC) collisions RA(xB, Q2, zh) = dσeA→h dxBdQ2dzh / dσeA dxBdQ2 dσed,ep→h dxBdQ2dzh / dσed,ep dxBdQ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (19) For parton distribution and fragmentation functions we use the nNNPDF30nlo [47] and NNFF10lo parametriza- tions [48] and perform the calculation for the aver- aged HERMES kinematics Q2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='25 GeV2 and ν = 12 GeV [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' To numerically solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (12) and (14), we smear the singular hard parton energy spectrum by a Gaussian with width parameter σz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='05 and the evo- lution from µ1 = Q to Q0 = 1 GeV is performed us- ing standard vacuum DGLAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In turn, the in-medium RG evolves µ2 2 from χ(wmax)zν/L to µ2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We take ΛQCD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='16 GeV, the inverse range of the interac- tion µ2 D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='12 GeV2 and the central value of the effec- tive medium density parameter ρG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 fm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' These values yield a quark transport parameter ˆqF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='053 HERMES π + HERMES ↑ + HERMES ↑ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='8 R Ne Kr Xe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='6 FO+RG RG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 e+Pb→ e+Pb→± e+Pb→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='8 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='6 XB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='1 XB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='3 XB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 2 = 20 GeV2 Q2 = 20 GeV2 = 20 GeV2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='8 Zh5 GeV2/fm at ν = 12 GeV (see supplemental material), consistent with existing mDGLAP applications [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' An average over the geometry of the nucleus of radius rA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='2A1/3 fm is also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The resulting nuclear modification factor RA(zh) is compared to the HERMES data for 20Ne and 131Xe tar- gets [49] in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Qualitatively, in- medium evolution shifts hadron spectra towards lower zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Results including only RG contributions (red dashed lines) give a good description of RA from small to inter- mediate z, but lead to a suppression that is too strong at large zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We remark that the region very close to zh = 1 is dominated by soft emissions, where one should consider soft power counting and threshold type resummation in both vacuum [52, 53] and in-medium calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' There- fore, we have exclude this region from our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Blue solid lines include the fixed order (FO) contribution from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (10) in the initial condition of the RG evolu- tion, and the bands correspond to the density variation in the range (ρG/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='5ρG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The FO correction improves the description of HERMES data at large z, but remains subleading to the RG evolution effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The nuclear size dependence of RA for Ne, Kr, and Xe nuclei is naturally explained with the same set of transport parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Using the same in-medium transport parameters, we present projections (lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' 1) for modified pion fragmentation functions at the future electron-ion collider (EIC) for ePb reactions at fixed Q2 = 20 GeV2 and various Bjorken xB values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We find that for xB > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='3, where partons are less energetic in the nuclear rest frame, modifications become very large, consistent with existing predictions for heavy flavor and jets [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' In the limit Q/Q0, E/(µ2 DL), λgµD ≫ 1 and to first order in the opacity of QCD matter we performed a renormalization group (RG) analysis of medium effects for the SIDIS process on a nuclear target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We derived a set of in-medium RG equations that resum the leading ln[E/(µ2 DL)] terms from multiple medium-induced emis- sions and identified the corresponding fixed order correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We further showed that such resummation is also contained in the modified DGLAP equations, which dif- fer in the way of regulating the endpoint divergences of medium-induced emission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Importantly, the new RG evolution in matter approach provides analytic in- sight into the salient features of parton showers respon- sible for the modification of hadron production in eA that are not possible with numerical methods alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' It is a more efficient and systematically improvable way of treating the logarithmic enhancements in matter as com- pared to solving mDGLAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We applied the new method to study the cold nuclear matter (CNM) effects on pion fragmentation and found that it gives a good description of the HERMES SIDIS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Predictions for the future EIC were also presented, where improved theoretical precision is especially impor- tant [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The semi-analytic framework derived here can be generalized to initial-state CNM effects, such as the ones observed in Drell-Yan production in proton-nucleus collisions, and to heavy ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This work further benefits future QCD studies by providing guidance on incorporating medium effects in Monte-Carlo event gen- erators for the EIC, the Relativistic Heavy Ion Collider and the Large Hadron Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='— The authors would like to thank Duff Neill for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' This work is sup- ported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Anderle, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Ringer, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Vogelsang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' D 87, 034014 (2013), arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='2099 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' [54] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Vitev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' C 75, 064906 (2007), arXiv:hep- ph/0703002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' 7 SUPPLEMENTARY MATERIAL In-medium scattering cross section and jet transport parameter estimate The elastic cross section between jet and target partons in color representations R and T, respectively, is [13] dσT R d2q = 1 (2π)2 1 dA 4παsCT × 4παsCR (q2 + µ2 D)2 , (20) with dA = N 2 c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The collision rate after summing over the medium color sources of representation T with density ρT then reads � T ρT dσT R d2q = � T 1 (2π)2 ρT dA 4παsCT × 4παsCR (q2 + µ2 D)2 = αsCR π 1 (q2 + µ2 D)2 ρG , ρG ≡ � T ρT 4παmed s CT dA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (21) In other words, we have chosen to put kinematic factors, the square color charges and coupling to the medium in the effective medium gluon density ρG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' With µ2 D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='12 GeV2, ρG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='4 fm−3, the quark jet transport parameter is ˆqF = � q2 max=νµD/2 0 d2q q2 αsCF π 1 (q2 + µ2 D)2 ρG ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='053 GeV2/fm (22) for ν = Q2/(2xBMp) = 12 GeV in the nuclear rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Here, the ultraviolet cut off of the q2 integration is chosen to be νµD/2, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The running coupling is cut-off when αs reaches 2π/β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='The value of ˆq is further consistent with the analysis of [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Full splitting functions in matter The splitting functions in nuclear matter induced by final-state interactions are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' [13, 14] (in d = 4 dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' After performing the path length integration in a medium of uniform density and size L, the splitting functions become P (1) ij (x) = αs 2π2 Pij(x)L � d2k � T � d2q (2π)2 ρT dA 4παsCT × 4παs (q2 + µ2 D)2 Wij(x, k, q, E/L) ≡ αs 2π2 Pij(x)L � d2k � d2q (2π)2 ρG × 4παs (q2 + µ2 D)2 Wij(x, k, q, E/L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (23) The continuous parts of the vacuum splitting functions in d = 4 − 2ϵ dimension, arising from real emissions, are Pqq(x) = CF �1 + x2 1 − x − ϵ(1 − x) � , Pgq(x) = TR[x2 + (1 − x)2 − 2ϵx(1 − x)], (24) Pqg(x) = Pqq(1 − x), Pgg(x) = CA 1 + x4 + (1 − x)4 x(1 − x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (25) The diagonal terms also receive virtual corrections, which we determined from flavor and momentum sum rules [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For P (1) ij (x), we define A⊥ = k, B⊥ = k + (1 − x)q, C⊥ = k − xq, D⊥ = k + q, and interference phase factors ΦA = Φ � A2 ⊥L 2x(1 − x)E � , ΦB = Φ � B2 ⊥L 2x(1 − x)E � , ΦC = Φ � C2 ⊥L 2x(1 − x)E � , ΦAD = Φ �(A2 ⊥ − D2 ⊥)L 2x(1 − x)E � , ΦCB = Φ �(C2 ⊥ − B2 ⊥)L 2x(1 − x)E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (26) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Wij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' including the relevant quadratic Casimirs from the Glauber gluon–hard parton system interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' are Wqq = CA B⊥ B2 ⊥ �B⊥ B2 ⊥ − A⊥ A2 ⊥ � ΦB + CA B⊥ B2 ⊥ �B⊥ B2 ⊥ − C⊥ C2 ⊥ � ΦB + (2CF − CA)C⊥ C2 ⊥ �C⊥ C2 ⊥ − A⊥ A2 ⊥ � ΦC + CA∆W ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (27) Wgg = CA B⊥ B2 ⊥ �B⊥ B2 ⊥ − A⊥ A2 ⊥ � ΦB + CA B⊥ B2 ⊥ �B⊥ B2 ⊥ − C⊥ C2 ⊥ � ΦB + CA C⊥ C2 ⊥ �C⊥ C2 ⊥ − A⊥ A2 ⊥ � ΦC + CA∆W ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (28) Wgq = CA B⊥ B2 ⊥ �B⊥ B2 ⊥ − A⊥ A2 ⊥ � ΦB + (2CF − CA)B⊥ B2 ⊥ �B⊥ B2 ⊥ − C⊥ C2 ⊥ � ΦB + CA C⊥ C2 ⊥ �C⊥ C2 ⊥ − A⊥ A2 ⊥ � ΦC + (2CF − CA)∆W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (29) 8 and Wqg = Wqq(x → 1 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For each channel, the first three terms can be cast into the form of the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6) by shifting integration variables such that arguments of Φ(·) become k2L/[2x(1 − x)E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The remaining piece ∆W, ∆W = �C⊥ C2 ⊥ �C⊥ C2 ⊥ − B⊥ B2 ⊥ � ΦC + B⊥ B2 ⊥ C⊥ C2 ⊥ ΦCB � − �A⊥ A2 ⊥ �A⊥ A2 ⊥ − D⊥ D2 ⊥ � ΦA + A⊥ A2 ⊥ D⊥ D2 ⊥ ΦAD � , (30) is written as the difference of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Note that the second term can be obtained from the first one by shifting k → k + xq, causing C⊥ → A⊥ and B⊥ → D⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Therefore, ∆W = 0 under dimensional regularized integration of k and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' If one uses an explicit ultraviolet cut-off ΛUV, the integration of ∆W over q, k (or in general, any differences caused by a shift of k + ∆ · q of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (27),(28), and (29)) are further suppressed by Λ−2 UV and do not contribute to the medium-induced logarithmic enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' We can account for the virtuality of the collision Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (6) by introducing the variable wmax = Q2L/(2ν), which appears in the functions B and χ defined as B(wmax) = 4 π � wmax 0 Φ(x)dx x2 , χ(wmax) = 2 exp � 1 B(wmax) 4 π � wmax 0 Φ(x) ln(x)dx x2 + γE � 1 B(wmax) − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (31) For the SIDIS process at moderate xB, wmax ≡ xBMpL ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0xBA1/3 is of order few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For fragmentation at mid rapidity in hadronic collisions, Q2L/(2E) ∼ EL → ∞, and B(∞) = 1, χ(∞) = 2e3/2−γE ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The flavor singlet sector Analogously to the treatment of the flavor non-singlet sector, we provide the details for the subtraction of divergences and renormalization of the flavor singlet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' To isolate the extra [x(1 − x)]−1−2ϵ poles, we define the following decomposition for any well-behaved function G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For singularities associate to P (1) qg (x), � 1 0 G(x) x2+2ϵ(1 − x)1+2ϵ dx = � 1 0 {G(x)}qg x2(1 − x)dx − G(0) 2ϵ − G′(0) 2ϵ − G(1) � 1 2ϵ + 2 � + O(ϵ) , (32) with {G(x)}qg = G(x) − x2G(1) − (1 − x2)G(0) − x(1 − x)G′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' For singularities associate to P (1) gg (x), � 1 0 G(x) x2+2ϵ(1 − x)2+2ϵ dx = � 1 0 {G(x)}gg x2(1 − x)2 dx − G(0) �1 ϵ + 2 � − G′(0) 2ϵ − G(1) �1 ϵ + 2 � + G′(1) 2ϵ + O(ϵ) , (33) with {G(x)}gg = G(x) − (1 − x)2 [(1 + 2x)G(0) + xG′(0)] − x2 [(3 − 2x)G(1) + (x − 1)G′(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Finally, for P (1) gq (x) � 1 0 G(x) x1+2ϵ(1 − x)1+2ϵ dx = � 1 0 {G(x)}gq x(1 − x) dx − G(0) 2ϵ − G(1) 2ϵ + O(ϵ) , (34) with {G(x)}gq = G(x) − xG(1) − (1 − x)G(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' One can directly and explicitly check that the endpoint divergences are removed from the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' With this procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' the medium-induced NLO contributions from these channels can be decomposed into log-enhanced and fixed order contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' ∆Fg A(µ2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' wmax) = � 1 2ϵ + L � � �−4C2 A dFg(z) dz + 2CF Nf Fg(z) z − 2C2 F � f Ff(z) z � � + � f � 1 0 dx �� n Cqg n (∆qg n )2 CF [1 + (1 − x)2] x z Ff( z x) � qg x2(1 − x) − Nf Fg(z) z � 1 0 dx �� i Cgq n (∆gq n )2 TR[x2 + (1 − x)2] � gq x(1 − x) + � 1 0 dx �� i Cgg n (∆gg n )2 CA[1 + x4 + (1 − x)4] � x z Fg( z x) − x Fg(z) z �� gg x2(1 − x)2 + 14C2 A Fg(z) z − 3C2 F � f Ff(z) z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (35) 9 ∆Ff A(µ2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' wmax) = � 1 2ϵ + L � � −4CF CA dFf(z) dz + 2CF (2CA + CF )Ff(z) z − CF Fg(z) z � + � 1 0 �� n Cn∆2 n(x)CF (1 + x2) � x z Ff � z x � − Ff (z) z �� qq x(1 − x)2 dx + (4CA − CF )CF Ff(z) z + � 1 0 dx �� i Cgq n (∆gq n )2 TR[x2 + (1 − x)2] x z Fg( z x) � gq x(1 − x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (36) where L ≡ ln µ2 2 χE/L and {· · · }ij stands for the subtraction for the corresponding channel i → j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' The 1/ϵ poles are subtracted by the corresponding contribution from M (1) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' Then, taking derivatives with respect to τ gives the in-medium RG Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} +page_content=' (13) and (14) for the flavor singlet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQf6C78/content/2301.11940v1.pdf'} diff --git a/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/2301.01669v1.pdf.txt b/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/2301.01669v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..56ae1e084dbc6edf5c5b9a98f73f355fc977cfde --- /dev/null +++ b/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/2301.01669v1.pdf.txt @@ -0,0 +1,984 @@ +1 + + +Ultra-thin Epitaxial MgB2 on SiC: Substrate Surface Polarity +Dependent Properties +Weibing Yang1†*, Leila Kasaei2†, Hussein Hijazi2, Sylvie Rangan2, Yao-wen Yeh2, Raj K +Sah1, Jay R Paudel1, Ke Chen1, Alexander X. Gray1, Philip Batson2, Leonard C. Feldman2, +and Xiaoxing Xi1 +1Department of Physics, Temple University, Philadelphia, PA 19122, USA +2Department of Physics and Astronomy and Laboratory for Surface Modification, Rutgers +University, New Brunswick, NJ 08901, USA +*Corresponding authors +†These authors contributed equally to this work + +Abstract: High quality, ultrathin, superconducting films are required for advanced devices +such as hot-electron bolometers, superconducting nanowire single photon detectors, and +quantum applications. Using Hybrid Physical-Chemical Vapor Deposition (HPCVD), we show +that MgB2 films as thin as 4 nm can be fabricated on the carbon terminated 6H-SiC (0001) +surface with a superconducting transition temperature above 33K and a rms roughness of 0.7 +nm. Remarkably, the film quality is a function of the SiC surface termination, with the C- +terminated surface preferred to the Si-terminated surface. To understand the MgB2 thin film/ +SiC substrate interactions giving rise to this difference, we characterized the interfacial +structures using Rutherford backscattering spectroscopy/channeling, electron energy loss +spectroscopy, and x-ray photoemission spectroscopy. The MgB2/SiC interface structure is +complex and different for the two terminations. Both terminations incorporate substantial +unintentional oxide layers influencing MgB2 growth and morphology, but with different extent, +intermixing and interface chemistry. In this paper, we report measurements of transport, +resistivity, and critical superconducting temperature of MgB2/SiC that are different for the two +terminations, and link interfacial structure variations to observed differences. The result shows +that the C face of SiC is a preferred substrate for the deposition of ultrathin superconducting +MgB2 films. + +1. Introduction +Ultra-thin superconducting films of MgB2 have attracted much interest [1-4] owing to +their relatively high Tc of 39 K, large coherence lengths, and high critical current density [5,6] +for applications in bolometers, photon detectors, and quantum devices [7-9]. High quality +MgB2 films have been fabricated successfully using Hybrid Physical-Chemical Vapor +Deposition (HPCVD) [10], creating thin films with high Tc and low residual resistivity. + +2 + +However, the Volmer–Weber like island growth mode in the HPCVD process is detrimental to +the production of ultrathin and smooth MgB2 films [11-14], where coalescence of islands often +leads to rough surfaces. For many superconducting electronic applications ultrathin (~5 nm), +smooth (ideally with RMS roughness < 1 nm) and uniform films are required. Previously, we +reported on low-angle ion milling to thin 40 nm HPCVD MgB2 films and fabricated 5 nm +superconducting films with Tc as high as 36 K [2]. Novoselov et al. have reported growth of +10 nm HPCVD MgB2 films directly on the Si terminated SiC (0001) substrate with Tc as high +as 35 K [15]. Recently, we have found that ultrathin MgB2 films grown on C-terminated 6H- +SiC substrate (0001bar) are significantly smoother than those on Si-terminated substrates and +possess high quality electronic properties and high Tc [16]. + Interfacial phenomena are critically important in the design and manipulation of thin +film functional materials[17-20]. Polarity of substrate is one of the important paramaters +determining the structural, electric and magnetic properties of materials grown on polar +materials. The polarity of SiC has been proved to have significant influence on the growth as +well as properties of thin films such as graphene, GaN and AlN thin films [21-23]. Here, we +report a comprehensive characterization of ultrathin HPCVD MgB2 films grown on SiC +substrates with both Si- and C-termination. The interface characteristics are correlated with +measurements of MgB2 transport, resistivity, and critical superconducting temperature +comparing growth on these two principal SiC faces. A significant finding is the existence of a +substantial magnesium oxide layer at the MgB2/SiC interface with thickness and roughness +dependent on the termination of the SiC substrate. The MgOx layer is thinner and smoother on +the C-face than on the Si-face. The smoother MgOx layer leads to a smoother MgB2 ultrathin +film on the C-face. Overall, the result shows that the C face is a preferred substrate for the +deposition of ultrathin superconducting MgB2 films on SiC. To our knowledge these MgB2 +films are at the leading edge of this technology, combining the requirements of high Tc, +thickness, and uniformity for advanced applications. The interface characterization reported +here provides details and possible explanations for the growth habit and suggests procedures +broadly applicable to superconducting thin film growth. + +II. Experimental details + SiC substrates used were the 6H polytype in the (0001) or (0001bar) direction. As +described in Ref. [16], a double-side polished 6H-SiC(0001) substrate is C-terminated on one +side and Si-terminated on the other, leading to surfaces of different polarities. Details of the +HPCVD method for growing MgB2 thin films have been described previously [10]. The +HPCVD growth condition has been optimized to minimize the RMS roughness (a flow rate of +10 sccm diborane gas mixture, 5% B2H6 in H2, was used in this work as compared to 2 sccm +in our previous work [16]), (see section 1 of Supplementary Material [24]). Transport +properties of MgB2 films were characterized by standard four-point measurements [25] where +probes were placed in the four corners of 5×5 mm2 square sample. Resistivity vs. temperature +measurement was carried out by dipping the sample into a liquid helium dewar immediately +after removal from the HPCVD system to minimize air exposure. The measured resistance is +converted into resistivity based on the van der Pauw solution for the square shape sample [26]. + +3 + +Atomic force microscopy (AFM), scanning transmission electron microscopy (STEM), +x-ray photoelectron spectroscopy (XPS), and Rutherford backscattering spectrometry (RBS) +were used to characterize the film and the interface. AFM measurements were performed using +a Veeco atomic force microscope. Imaging and electron energy loss spectroscopy were carried +out using the Rutgers Nion UltraSTEM microscope operated at 60 kV with the convergence +and collection semi-angles at 35 and 16.5 mrad, respectively. XPS measurements were +performed in a Thermo K-alpha system with charge compensation using Al-Kα radiation and +overall energy resolution of 0.7 eV. Under these conditions, the surface hydrocarbons were +found at a binding energy of 284.7 eV. RBS measurements were performed using a General +Ionex tandem accelerator with 1.6 MeV He++ ions and a scattering angle of 130. The sample +was held approximately normal to the ion beam. The estimated depth resolution for Mg is ~ 25 +nm. Ion beam channeling was performed along the (0001) direction of the 6H-SiC substrate +and non-channeling (random) spectra were acquired by randomly varying incident angles. +SIMNRA 7.01 software was used for non-channeling RBS spectra analysis [27]. +III. Results + + +Figure 1. (a),(c), (e) and (g) AFM images for MgB2 films grown on Si-terminated SiC with thicknesses of 4 +nm, 9 nm, 16 nm and 22 nm respectively; (b), (d), (f) and (h) are the corresponding MgB2 films on the C- +terminated SiC that were grown at same conditions as (a), (c), (e) and (g). + +a +4 nm +b +4nm +C +d +9 nm +9 nm +f +16 nm +16 nm +bo. +h +22nm +22 nm +2 μm4 + +i. +Deposition rates and surface roughness +As described in detail in the supplementary information [24] the basic growth method +involves liquified Mg, combined with a flow of diborane gas to form MgB2. In previous work +[16], we used flow rates of 1 sccm and 2 sccm diborane gas mixture in the HPCVD deposition +and obtained an RMS roughness of 1.2 nm and Tc of 34.3 K in a 5.7 nm MgB2 thin film on the +C face of the SiC substrate. From a more recent systematic optimization of the diborane gas +mixture flow rate, we found that 10 sccm produces the smallest roughness with the best +superconducting properties. Figure 1 shows AFM images of MgB2 thin films with nominal +thicknesses of 4 nm, 9 nm, 16 nm, and 22 nm grown on the Si and C faces. Films with the same +thickness but different terminations were grown in the same deposition run to ensure identical +growth conditions. For all thicknesses, films on the C face are smoother than those on the Si +face. Films on the Si face show clusters that are absent on the films on the C face. Energy +dispersive x-ray spectroscopy (EDS) analysis shows that the clusters are MgB2 grains. In +addition, the films on the Si face show taller islands that are not completely connected while +films on the C face show much better connectivity. On the Si face, the MgB2 islands become +larger for thicker depositions, typical for the island growth. Films of the same nominal +thickness on the C face show a much smoother surface and don’t have typical hexagonal MgB2 +grain as the case on Si face. The RMS roughness for the MgB2 films in Fig. 1 is summarized +in Table 1 showing values from 0.5 nm – 0.7 nm for the C face and 2 – 3 nm for the Si face. +The result is a marked improvement from those in Ref. [16] and indicates that MgB2 films +grown on the C can be smoother than those on the Si face. Films composed of grains with a +size comparable to the total film thickness are intuitively rougher. +ii. +Electronic properties +Figure 2 shows corresponding resistivity vs. temperature (ρ -T) curves for the MgB2 +films in Fig. 1. As the film thickness decreases, Tc decreases and the residual resistivity ρ0 +increases for films on both the Si and C faces. The Tc of MgB2 films on the Si-face is slightly +higher than the films on C-face, which is probably due to the biaxial tensile strain between +Table 1. RMS roughness, Tc0, residual resistivity ρ0, and Δρ of MgB2 films on C-SiC and Si-SiC. +Film Thickness (nm) +4 +9 +16 +22 +C face +RMS (nm) +0.7 +0.6 +0.5 +0.7 +Tc0 (K) +33.6 +37.5 +39.4 +39.8 +ρ0 +(µΩ·cm ) +14.9 +4.9 +1.9 +1.3 +Δρ +(µΩ·cm) +14.2 +10.6 +8.3 +7.5 +Si face +RMS (nm) +1.7 +3.2 +2.9 +2.7 +Tc0 (K) +35.4 +37.8 +39.2 +40.6 +ρ0 + (µΩ·cm ) +24.7 +5.3 +2.3 +1.4 +Δρ +(µΩ·cm) +24.3 +10.6 +8.1 +7.7 + + +5 + +MgB2 grains as the MgB2 growth mode in Si-face is more like island growth mode compared +to the C face [28]. The results are summarized in Table 1. While the values of Tc are similar +to our earlier report [16], the ρ0 values of the current films are much lower. The results are +similar for both substrate terminations except for the thinnest films. The residual resistivity is +much higher for the 4 nm film on the Si face than on the C face. +Also included in Table 1 is Δρ, the change in resistivity from room temperature to just +above the superconducting transition temperature. Rowell has shown that this quantity, the +room temperature, and residual resistivity difference can be used to quantify the grain +connectivity in MgB2 samples [29], with larger Δρ indicating poorer connectivity. The +dependence of Δρ on film thickness is shown in Fig. 3a. At 22 nm, the films on both Si and C +faces show Δρ values similar to our thicker, high quality MgB2 films, indicating excellent grain +connectivity. As the nominal thickness of the ultrathin film decreases, the grain connectivity +effect becomes more and more important, reflected as a gradual increase in Δρ. At 4 nm, the +connectivity degrades rapidly, leading to a large Δρ increase for both faces. The film on the Si +face shows much poorer connectivity than that on the C face. The conclusion on the grain +connectivity is corroborated by the residual resistivity data. While Δρ reflects the temperature +dependence of the electron-phonon scattering and grain connectivity, ρ0 is determined by the +grain connectivity and scattering of electrons by impurities, defects, as well as surfaces [29]. +We have shown previously [30] that for clean MgB2 films fabricated by HPCVD, the mean +free path of electron scattering is limited by the film thickness. For example, changing the film +thickness from 22 nm to 4 nm results in a decrease in the mean free path and thus an increase +in electron scattering by a factor of 5.5. Combined with a reduction of grain connectivity, + +Figure 2. Resistivity vs. temperature curves for MgB2 films on the Si face (a and b) and the C face (c +and d). + +50 +a filmsonSiface +25 +b +Resistivity (μuQ cm) +Resistivity (μ cm) +filmson Siface +40 +22nm +30 +16nm +9nm +20 +4nm +10 +5 +O +0 +0 +50 +100150200 +250 +300 +35 +36 +373839 +40 +41 +42 +Temperature (K) +Temperature(K) +15 +50 +C +films on C face +d +Resistivity (μQ cm) +films on Cface +40 +Resistivity (μQ cm) +0 +10 +5 +10 +0 +- +0 +50 +100150200 +250 +300 +30 +323436384042 +44 +Temperature (K) +Temperature(K)6 + +deduced from the increase in Δρ, by a factor of 1.9 in the case of films on the C face, one can +predict the ρ0 value of the 4 nm film from that of the 22 nm film. The result, ρ0 = 1.3 × 5.5 × +1.9 = 13.7 μΩ·cm for the 4 nm film, agrees well with the experimentally measured value. The +results of the same procedure for all the films studied are shown in Figs. 3(b) and 3(c). +iii. +Interface characterization +To understand the influence of the SiC surface termination on the properties of the +ultrathin MgB2 films, we investigated the interfacial structure and chemistry of the MgB2/SiC +interface. The MgB2 samples for interface characterizations are prepared separately using the +same growth conditions as films described above to ensure the properties of MgB2 films are +consistent throughout this work. +iii-a) RBS channeling and interfacial oxides + RBS/channeling measurements were performed to provide depth-dependent +information on both the composition (‘random spectra’) and crystallinity. Channeling, the +reduction of scattering yield when the beam is aligned with a major crystallographic direction, +yields information on crystal quality and identifies the alignment and composition of buried +layers. The results show that for both SiC terminations, there is clearly an interfacial layer +between the MgB2 film and the SiC substrate containing both magnesium and oxygen (see +section 2 of Supplementary Material [24]). The result of a composition analysis at the interface +for the two samples, identified as MgB2/MgOx/SiC, is presented in Table 2, where the +interfacial oxygen is ascribed to a MgOx layer. Within the experimental error, the composition +of the interfacial layer is close to MgO. The thickness of the MgOx layer is estimated to be ~ +Figure 3. The change of resistivity from 300 K to 40 K and residual resistivity as function of +film thickness on both Si and C face. The red curves are values expected based on the +reductions of the electron mean free path and the grain connectivity as the film thickness +decreases. + +25 +a +I-films onSiface +Ap (μQ cm) +20 +-.filmson Cface +10 +b +filmsonSiface +( n) d +20 +15 +Predicted +10 +5 +0 +15 +filmsonCface +2 cm) +ur) +110 +Predicted +5 +0 +5 +10 +15 +20 +25 +Film thickness (nm)7 + +2.3 nm for the Si termination and ~ 0.9 nm for the C termination. Furthermore, there is a +consistent, but small, channeling effect in the MgOx itself, indicating that the oxide is +crystalline. We suggest the Mg-interface peaks result from a partially crystalline MgO layer at +the interface, with orientation affected by the lattice mismatch between MgO and SiC, and then +influenced by the mismatch with overlayer MgB2 and MgOx. (RBS also detected monolayer +scale surface impurities of silicon and carbon at the surface of the MgB2 overlayer that play no +apparent role in the interface formation, but are noted here for completeness). The observation +of a substantial oxide interfacial layer is a major new finding. The “buried oxide” is shown to +be consistent with magnesium oxide by the chemical shift as observed in angular dependent +high energy XPS (HAXPES) analysis (section 3 of Supplementary Material [24], also see the +Ref. [31-37] therein). +The observation of a MgOx interfacial layer raises two questions: 1) the origin of the +oxygen since the HPCVD process, being entirely conducted in a reducing environment, in +principle eliminates the oxygen from the film growth? and 2) the roles of the oxide layer in +determining the properties of the ultrathin MgB2 films associated with different terminations? +To address these questions, we investigated the first stages of the HPCVD process itself by +heating the substrate along with the Mg pieces, without the introduction of the B2H6 gas +mixture. Specifically, Si- and C-terminated SiC substrates were heated in Mg vapor at 740°C +for 1 minute. In Fig. 4, AFM images of these treated Si- and C-terminated SiC substrates are +shown along with those for the pristine substrates as received from the vendor. For the “as +received” material the Si-terminated surface (Fig. 4(a)) shows atomic steps with an RMS +roughness of 0.2 nm whereas the C-terminated surface (Fig. 4(c)) is featureless with an RMS +roughness of ~ 0.3 nm. Following the “Mg only” treatment the samples are essentially +MgO/SiC structures due to oxidation of the air-exposed Mg layer. The roughness measured on +Si-terminated substrate is much higher (RMS roughness ~ 4 nm) than that measured on the C- +terminated substrate (RMS roughness ~ 0.4 nm).. They consistently show an oxygen-rich layer + +Table 2．Atomic composition of MgB2/SiC determined by RBS +MgB2/MgOx/SiC +Ointerface/cm2 +(×1015) +Mginterface/cm2 +(×1015) +Thickness of +MgOx +interface(nm) +Si-Face +(Ch-RBS) +13 ± 2 +12 ± 2 +2.3 ± 0.2 +C-Face +(Ch-RBS) +6 ± 2 +4 ± 2 +0.9 ± 0.2 + + +8 + +with Mg:O ratio of ~ 1:1.3 for both surface terminations and a small channeling effect +indicative of imperfect or misaligned MgO crystallinity. The thickness of the MgOx layer is +thicker on Si-SiC (3.0 ± 0.2 nm as determined from the channeling spectrum) than on C-SiC: +(2.1 ± 0.2 nm). Note that they are both thicker than the MgOx layers detected at the MgB2/SiC +interface: 2.3 ± 0.2 nm on the Si face and 0.9 ± 0.2 nm on the C face, due to further oxidation +upon air exposure. +iii-b STEM Electron energy loss spectroscopy and nm elemental profiling +To further examine the interface between the MgB2 films and the underlying C- and Si- +terminated SiC substrates, cross-section samples were prepared and studied by STEM-EELS. +As shown in the atomic resolution HAADF images in Figure 5, there is an intermediate layer +between the top MgB2 and the bottom SiC for both terminations. This interface layer is about +0.9 nm in both cases. However, while not shown here, the intermediate layer does not have +constant thickness across the observed interface ranges, and it varies from 0.9 to 2.7 nm for the +case of Si-SiC and from 0.9 to 1.8 nm for the case of C-SiC. In addition, the intermediate layer +marked with dash lines between 0 nm to -1 nm in both cases often exhibits a periodic structure +as shown in the figure. Even though the resolution of STEM image for the MgO layer is not +ideal due to the combination of limited resolution of our instrument and the complex and +imperfect thin structure, we can still see that the structure matches well with MgO as viewed +from the (111) direction. The atomic arrangements of MgB2, MgO, and SiC are overlaid in the +figure as visual guides. Note that the image intensity scales with the atomic number due to the +detector arrangement, and it is Mg and Si observed in the MgB2 and SiC, respectively. + +Figure 4. (a) Atomic Force Microscopy (AFM) of Si-SiC as received from the vendor, (b) AFM image +of Si-SiC after annealing in Mg vapor at 740°C for one minute, (c) AFM scan for C-SiC as received +from manufacture, (d) AFM image of C-termination of SiC after annealing in Mg vapor at 740°C for +one minute. + +a +SiFace +b +Si Face +as.received +Mg.treated +500nm +500.nm +C +CFace +d +CFace +asreceived +Mg treated +500nm +500nm9 + +Next, we compare the relative chemical distributions of Mg, Si, B, C, and O along the +eight data points acquired across the interface areas in Figure 6. It is found that oxygen is +mostly confined in the intermediate layer. Si and C are found up to terminate near p the +intermediate layer, as expected, whereas B is found on the thin film/surface side of the +intermediate layer. Finally, Mg is found throughout the intermediate layer,consistent with a +MgO interface. + + + +Figure 6. STEM-EELS data from the same cross section for (a) C-terminated SiC(0001) substrate (b) Si- +terminated SiC(0001) substrate. + +Figure 5. Cross-sectional HAADF STEM imaging of MgB2 on (a) C-terminated SiC(0001) +substrate (b) Si-terminated SiC(0001) substrate. + + +ChemicalDistribution(C-SiC) +ChemicalDistribution(Si-SiC） +1.0 +a +b +0.8 +Intensity +0.8 +Intensit +Mg +Mg +0.6 +0.6 +Si +x +Si +O +B +Normalized +B +C +0.4 +4 +0.4 +1 +0 +人 +0 +0.2 +0.2 +0.0F +00 +0.0 +-3-2 +-1 +0 +1 +2 +3 +4 +5 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +z (nm) +z (nm)a +b +3 nm +2 nm +1 nm +0 nm +-0.5 nm +-1 nm +-2 nm +-3 nm +2nm10 + +iii-c) XPS analysis and interfacial chemistry +More detailed information on interfacial chemistry is revealed by XPS. Figure 7 shows +selected core levels spectra (Mg 1s, Si 2p, O 1s and C 1s) measured on bare SiC substrates +(bottom curves), Mg vapor treated SiC (middle curves), and 7 nm-thick MgB2 grown on SiC +(top curves), for both Si- and C-terminated surfaces. In all cases, the SiC substrate signal is +detectable via the Si 2p and C 1s core levels. Chemical environments attributed to core levels +features are indicated in the figure. On the bare SiC surface, exposed to air after preparation, +the substrate Si 2p and C 1s core levels are found at binding energies of ~100 eV and ~282 eV, +respectively, in good agreement with expected values [38]. The features in the O 1s and C 1s +core levels spectra indicate carbohydrates adsorption, both the results of air exposure of the +bare SiC substrates. + For the Mg vapor treated SiC samples, the O 1s level is split into two peaks. The lower +binding energy component is attributed to MgO while the higher binding energy component is +assigned to Mg(OH)x and Mg carbonates [39]. Of particular interest is that the SiC-related core +level spectra show different binding energies for the two different SiC surface terminations: +the binding energies of both the C 1s and Si 2p core levels are ~1 eV lower for the C face than +for the Si face. For MgB2 films on SiC substrate, exposure to air causes surface oxidation as +well as water and carbohydrates adsorption. As a result, their O 1s and Mg 1s spectra are +affected by both the top surface alteration of MgB2 and the interfacial MgOx layer, and +separating these contributions is not straightforward. However, we again observe a shift of the +Si 2p core level to lower binding energy by ~1 eV in the sample on the C face as compared to +the Si face (a similar energy shift is present for C 1s, but less visible due to the SiC signal +attenuation through the MgB2 layer). This suggests that in the cases of both Mg vapor treated +SiC, for which the stack is effectively MgO/SiC, and for MgB2 films on SiC, for which the +stack is likely MgB2/MgO/SiC, there is a similar energy alignment related to the MgO/SiC +interface that is dependent on the SiC surface termination. Similar behavior has been reported +in the case of intentionally grown epitaxial MgO films on SiC surfaces [40,41]: binding energy +offsets of the order of eVs, have been measured for different MgO/SiC interfaces, supported +by electronic structure calculations of atomically different interfaces. + +Figure 7. XPS spectrum of Mg 1s, Si 2p, O 1s and C 1s for bare SiC, Mg vapor treated SiC, and thin MgB2 +film on SiC for both C and Si terminations. + +Mg1s +MgB2/MgO/SiC +Si2p +MgB,/MgO/Sic +01s +C1s +C-Sic +C-SiC +Si-SiC +C-SiC +C-SiC +Si-Sic +Si-Sic +Si-Sic +MgB,/MgO/SiC +MgB,/MgO/SiC +Units) +Units) +Units) +Units) +Si-C +Intensity (Arb. +Mg(OH)2 +Mgo +Intensity (Arb. +Intensity (Arb. +/MgCO3 +Intensity(Arb. +cO +C-C/C-H +MgO/Sic +MgO/Sic +MgO/Sic +MgO/Sic +BareSic +BareSic +BareSic +BareSic +-1308 +-1304 +-1300 +-104 +-102 +-100 +-98 +-96 +-536 +532 +-528 +-292 +288 +-284 +-280 +Binding energy (eV) +Binding energy (eV) +Bindingenergy (eV) +Binding energy (eV)11 + +IV. Discussion +In the following, we consider specific aspects of these analyses and how they might influence +the film morphology. +- Effect of native oxide +It is clear from the different interface analysis results above, that a buried MgO layer +exists at the interface of the MgB2 film and the SiC substrate. It is likely that this oxide is the +result of the reaction of Mg with the “native oxide” that exists on the SiC surface. The reaction +of Mg with SiO2 has been reported by a number of authors suggesting that the reaction of Mg +with SiO2 can result in formation of MgO and possibly Mg silicates [42, 43]. +“Native oxide” as a thin film is not necessarily well defined, as the resulting oxide +thickness, usually only 1-2 nm at most, is a result of environmental variables, time and crystal +face. In recent work using contact angle measurements Park et al [44] showed that the native +oxide growth is greater on the Si face than on the carbon face, consistent with the reports in +Table 2 of a Si face MgO layer of 2.3 nm compared to a C face of 0.9 nm. For calibration, we +note that 1 nm of SiO2 corresponds to 5.3×1015 /cm2 of oxygen, consistent with the existence +of very thin native oxides yielding nanometer MgO. Furthermore, Nagai et al [45] characterized +the roughness in very thin oxides on SiC as a function of crystal face. In this work, the authors +show that the RMS roughness is proportional to oxide thickness and the rate of increase of +roughness with film thickness is the same for the two crystal faces. Therefore, the Si face oxide +roughness is greater than the C face. These reports allow some mechanistic conclusions as +follows. Native oxide thickness on the C face is less than that of the Si face [44], consistent +with the reports in Table 2. The roughness of very thin oxides is proportional to oxide thickness +[45]. In short, oxide roughness is proportional to oxide thickness, this oxide roughness is +transferred to a MgO layer and then reflected in the overlayer MgB2 film uniformity. Since the +C-face oxide is substantially less than the Si face the net roughness is reduced for the C-face +resulting in a more uniform thin film. +Relevant to that point, it is interesting to note that there are reports of growth of MgO +on SiC by MBE for MOS systems [46, 47], a good lattice match, and there are reports of the +growth of MgB2 on MgO [48], also a reasonable lattice match resulting in high quality MgB2 +films. Therefore, an MgB2/MgO/SiC epitaxial structure may be realized. Nevertheless the +properties of the resulting film but may depend on the SiC termination: if the starting (oxidized) +surface of the C face of SiC is less rough than the Si face, a smoother MgO/MgB2 structure is +expected. +Finally, among the interesting remaining questions is the “necessity” for magnesium- +based oxide layer to achieve higher quality epitaxy and crystallinity. We note this point was +explicitly raised in the MBE work of Laloe et al [49], for MgB2 on Si where a Mg starting layer +was explicitly added to enhance growth. Possibly the “native oxide” on the Si face is just the +correct amount to achieve a high-quality epitaxial film. In that regard, we note that some +preliminary experiments on HF treated SiC (presumably minimal oxide) in our laboratory did +not produce quality films. + +V. Conclusion + +12 + +The growth of ultrathin MgB2 films on the different surface terminations of SiC has +been studied, seeking the conditions for optimum superconducting properties and film +uniformity. It has been shown that the best conditions are associated with 10 sccm of 5% +diborane gas flow rate on the C terminated face of SiC for our specific system. A significant +difference has been identified between growth on the Si face and the C face, with the latter +producing higher quality films. This difference has been explored by various interface probes. +RBS/channeling measurements indicated that the samples consisted of MgB2/MgO/SiC +stacks, in which the thickness MgOx layer was SiC surface termination dependent: is 2.3 ± 0.2 +nm on the Si face and 0.9 ± 0.2 nm on the C face. High resolution EELS and TEM confirmed +this structural difference, indicating that different interfacial constituents on the two surfaces +may control the final morphology. XPS analysis indicated a similar energy band offset at the +MgO/SiC interfaces, for both MgB2 films grown on SiC and for MgO films on SiC, but highly +dependent on the SiC surface termination. High energy, grazing exit angle XPS confirmed the +presence of a buried, thin MgOx layer at the MgB2/SiC interface. +This MgO layer in turn may govern the MgB2 film quality: a thicker and rougher MgOx +layer on the Si face of the SiC substrate is the cause of the rougher ultrathin MgB2 films as +compared to the films on the C face of SiC. 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Lee, Journal of the Korean Institute of Electrical +and Electronic Material Engineers 33, 3 (2020). +45. R. Nagai, R. Hasunuma, and K. Yamabe, Japanese Journal of Applied Physics 55, 08PC07 (2016). +46. T. L. Goodrich, J. Parisi, Z. Cai, and K. S. Ziemer, Appl. Phys. Lett. 90, 042910 (2007). +47. T. L. Goodrich, Z. Cai, and K. S. Ziemer, Applied Surface Science 254, 10 (2008). +48. Y. Harada, M. Uduka, Y. Nakanishi, N. Yoshimoto, and M. Yoshizawa, Physica C: +Superconductivity 412, 1383-1386 (2004). +49. J.- B. Laloë, T. H. Kim, and J. S. Moodera, Advances in Condensed Matter Physics 2011, 989732 (2011). + + + +15 + +Supplementary Information + + + +Ultra-thin Epitaxial MgB2 on SiC: Substrate Surface Polarity Dependent Properties + + +Weibing Yang1†*, Leila Kasaei2†, Hussein Hijazi2, Sylvie Rangan2, Yao-wen Yeh2, Raj K +Sah1, Jay R Paudel1, Ke Chen1, Alexander X. Gray1, Philip Batson2, Leonard C. Feldman2, +and Xiaoxing Xi1 +1Department of Physics, Temple University, Philadelphia, PA 19122, USA +2Department of Physics and Astronomy, Rutgers University, New Brunswick, NJ 08901, USA +*Corresponding author + + + + + + + + + + + + + + + + + +16 + +1. Hybrid Physical Chemical Vapor Deposition of epitaxial MgB2 thin films. +The HPCVD reactor was first purged with ultra-high purity H2 gas. The substrate and +Mg pieces placed nearby on the same heater were then heated to 740C in the H2 ambient when +Mg began to evaporate. After about 10 seconds, a mixture of 5% B2H6 in H2 was introduced +into the reactor to initiate growth. The film deposition rate can be controlled by adjusting the +flow rates of the diborane gas mixture. In this work, a flow rate of 10 sccm was used, the result +of parameter optimization for ultrathin MgB2 film with the best film quality. The corresponding +deposition rate was ~ 0.23 nm/s, determined by a linear fitting of the thicknesses-flow rate data +from a series of calibration runs. The thickness of MgB2 ultrathin films was then controlled by +the deposition time. The deposition temperature was 740 C. +In addition to figure 3 in the main text, we present in figure S1 the residual resistivity +(ρ0) of MgB2 films as a function of thickness for many more samples, measured in unpatterned +as-grown films, to demonstrate the reproducibility of our deposition process. The additional +data points are derived from the measurements of MgB2 films grown using 5, 10, and 20 sccm +gas mixture of 5% B2H6 in H2. It is clear that the flow rate of B2H6 gas mixture does not change +the trend of residual resistivity as a function of film thickness. The residual resistivity is about +the same for thick films on the Si and C faces, but it is much smaller on the C face than on the +Si face when the film is only a few nanometers, indicating much better gain connectivity of +ultrathin MgB2 on the C face. + +Figure S4. Residual resistivity as a function of MgB2 film thickness on the Si and C faces. The MgB2 films were +deposited using 5, 10 and 20 sccm gas mixture of 5% B2H6 in hydrogen. + + + +17 + +2. Comparison of Rutherford Back Scattering on MgB2 thin films grown on two +terminations of SiC. +Figure S2(a) shows a SIMNRA simulation of a random (non-channeling) RBS +spectrum for an ideal 80 nm MgB2 film grown epitaxially on SiC substrate. The thickness and +scattering geometry was chosen such that signals from the film surface and film/substrate +interface are clearly distinguishable. Figures S2(b) and S2(c) show RBS channeling and +random spectra for two 80 nm MgB2 films on Si- and C-terminated SiC substrates, respectively. +The experimental random RBS spectra agree well with the SIMNRA fitting shown in Fig. S2(a). +The minimum yield χmin (the ratio between the channeling and random yields), evaluated using +a width of 3 channels at around channel number 665, with contributions from both the MgB2 +and the underlying SiC, was ~5%, indicating excellent crystallinity in MgB2 films on both the +Si and C faces. +The strong channeling effect allows us to probe the properties of the MgB2/SiC +interface as an alignment discontinuity at the interface inevitably leads to a weaker channeling +effect. In Figs. S2(b) and S2(c), the channeling yields are multiplied by 5x for clarity. For both +SiC terminations, there is an interfacial layer between the MgB2 film and the SiC substrate +containing both magnesium and oxygen, marked as “Mg interface” and “O interface”, +respectively. We identify this interfacial layer as MgOx. The degraded channeling effect in the +interfacial layer may be the result of partially crystalline MgOx and/or lattice mismatch between +MgB2 and MgOx. + +18 + + +3. Angle Resolved Hard X-ray Photoelectron Spectroscopy on MgB2 thin films on two +terminations of SiC substrates +To investigate the chemical bonding and the depth profile of the Mg species associated +with the pristine MgB2 and the oxidized film, we utilized angle-resolved bulk-sensitive hard x- +ray photoelectron spectroscopy (HAXPES) [1]. The measurements were carried out using a +lab-based HAXPES system equipped with a monochromatized Cr Kα x-ray source with the +photon energy of 5.4 keV and a wide acceptance angle hemispherical electrostatic analyzer +ScientaOmicron EW4000. +We measured two MgB2 films, nominally 4 nm thick, deposited simultaneously and side- +by-side on the C- and Si-terminated SiC substrates. At the photon energy of 5.4 keV, the +inelastic mean-free path (IMFP) for the Mg 1s photoelectrons (Ekin = 4.1 keV) in MgB2 is +estimated to be approximately 7 nm [2], which ensures that the entire film and the interface +with the substrate are being probed. Measurements were carried out at four different +photoelectron take-off angles, 4°, 30°, 45°, and 60°, facilitating different average probing +depths varying from approximately 7 nm (at 4°) to approximately 3.5 nm at 60°, making the +latter more surface sensitive and less sensitive to the buried SiC/MgB2 interface [3]. + +Figure S2. (a) Simulated random RBS spectrum for an ideal 80 nm MgB2 films grown on SiC. (b) and (c) +RBS channeling and random spectra for 80 nm MgB2 films on Si- and C-terminated SiC(0001) substrates, +respectively. The channeling yields multiplied by 5x are also shown for clarity. + +16 wev He+ Ras sttafon of +8Cnm MgBSiC +Si +Mg +C +B +C surface +oIrterface +Osutace +Wafm +Random +thannelng +Random +chxs +Channeing19 + +The results of the measurements for the films on the C-terminated and Si-terminated SiC +substrates are presented in Figures S3(a) and S3(b), respectively. The most intense peak at the +binding energy of 1303.3 eV in both plots corresponds to the Mg 1s core-level photoemission +originating from the MgB2 film [4]. The higher-binding-energy feature at approximately +1305.2 eV corresponds to the chemically shifted state originating, most likely, from the Mg +oxide species, such as MgO [5]. Normalization of the photoemission intensities to the +maximum of the main MgB2 peak reveals two different trends in the angle-dependent +evolutions of the photoemission intensities of the higher-binding-energy (MgO) component. +For the film on the C-terminated SiC substrate [Fig. S3(a)] the intensity of the MgO +component increases with increasing surface sensitivity, as shown using blue circular markers +in Figure S3(c). Such a trend generally corresponds to the angle-resolved measurement of the +surface oxide species [6]. +Conversely, for the film on the Si-terminated substrate [Fig. S3(b)] two major differences +are observed. First, the average intensity of the MgO component is increased relative to the +main MgB2 peak, suggesting a higher oxide content in the sample. Secondly, the angle- +dependent evolution of the MgO component exhibits a flatter trend, with the most surface- +Figure S3. (a) Mg 1s core-level peak measured on the MgB2 film deposited on the C-terminated SiC +substrate at four photoelectron take-off angles in the experimental geometry shown on the right. (b) Same +angle-resolved measurement carried out on the MgB2 film deposited on the Si-terminated SiC substrate. +(c) Peak intensities of the higher-binding-energy MgO peak component normalized to the height of the +main MgB2 peak (circular markers) and the corresponding simulated intensities (diamond-shaped markers) +for the best-fit film structures shown in the figure legend. + +analyzer +C termination +Mg +1S +60° +45° +30° +MgB2 +Mgo +1kev +Si termination +Mg +0.8 +0.7 +0.6 +MgB2 +0.5 +Mgo +0.4 +0.3 +0 +20 +40 +6020 + +sensitive measurement at 60° exhibiting a slight decrease in intensity. Such a trend, shown +using red circular markers in Figure S3(c), suggests the presence of a buried Mg oxide layer at +a depth that is larger compared to the probing depth at the take-off angle of 60° (3.5 nm). +In order to confirm the presence of the buried Mg oxide layer in the sample on the Si- +terminated SiC substrate, we carried out angle-resolved simulations using the SESSA +simulation package, which quantitatively predicts photoemission peak intensities by taking into +account relevant parameters such as IMFP, elastic-scattering cross-sections, photoionization +asymmetry parameters, and the photoelectron take-off angles [7]. The best-fit results for both +samples are shown using diamond-shaped markers in Figure S3(c). For both samples, the +thickness of the surface oxide is predicted to be approximately 0.65 – 0.7 nm. The thickness of +the pristine MgB2 layer (3.5-3.6 nm) is close to the target thickness of 4 nm. The main +difference between the two samples is the extracted thickness of the buried interfacial oxide, +which is mainly responsible for the difference in the two observed intensity trends. Specifically, +the best-fit thickness of the buried Mg oxide at the interface between MgB2 and the Si- +terminated substrate is 0.75 nm, while the same value for the C-terminated substrate is only 0.3 +nm. The presence of an interfacial oxide was necessary for obtaining reasonable fits for both +datasets. As explained in the discussion the interfacial oxide on these samples is determined by +the "native oxide" on the SiC surface. Such native oxides are not well defined and depend on +numerous environmental factors. Nevertheless, in agreement with the trend reported in Table +1, the MgO interfacial layer is greater on the Si face than on the C face. +Slight discrepancies between the experimental and simulated intensities are observed for +both samples at the lowest take-off angle (4°) due to the limitations of the experimental +geometry, which features a fixed 90° angle between the x-ray incidence direction and the +analyzer orientation. In this geometry, the x-ray incidence angle is so grazing (4°) that the +effects of the x-ray beam cone-angle (18°), as well as the total external reflection, may play a +significant role. + +References: +1. A. X. Gray et al., Nature Mater. 10, 759 (2011). +2. A. Jablonski and C. J. Powell, J. Vac. Sci. Technol. A 27, 253 (2009). +3. C. S. Fadley, Prog. Surf. Sci. 16, 275 (1984). +4. H. Li, J. Zhu, Z. Wang, Z. Song, and H. Chen, Optical Mat. Express 3, 546 (2013). +5. M. Niwa, A. Yasui, E. Ikenaga, H. Honjo, S. Ikeda, T. Nakamura, and T. Endoh, J. +Appl. Phys. 125, 203903 (2019). +6. G. Panaccione and K. Kobayashi, Surf. Sci. 606, 125 (2012). +7. W. Smekal,W. S.M.Werner, and C. J. Powell, Surf. Interface Anal. 37, 1059 (2005). + + + + diff --git a/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/load_file.txt b/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2aa8eab7b3afacdb3d592d0e5c35001104e7d2b --- /dev/null +++ b/xtAzT4oBgHgl3EQfs_2h/content/tmp_files/load_file.txt @@ -0,0 +1,981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf,len=980 +page_content='1 Ultra-thin Epitaxial MgB2 on SiC: Substrate Surface Polarity Dependent Properties Weibing Yang1†*, Leila Kasaei2†, Hussein Hijazi2, Sylvie Rangan2, Yao-wen Yeh2, Raj K Sah1, Jay R Paudel1, Ke Chen1, Alexander X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Gray1, Philip Batson2, Leonard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Feldman2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' and Xiaoxing Xi1 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Temple University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Philadelphia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' PA 19122,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' USA 2Department of Physics and Astronomy and Laboratory for Surface Modification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Rutgers University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' New Brunswick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' NJ 08901,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' USA Corresponding authors †These authors contributed equally to this work Abstract: High quality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' ultrathin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' superconducting films are required for advanced devices such as hot-electron bolometers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' superconducting nanowire single photon detectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' and quantum applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Using Hybrid Physical-Chemical Vapor Deposition (HPCVD), we show that MgB2 films as thin as 4 nm can be fabricated on the carbon terminated 6H-SiC (0001) surface with a superconducting transition temperature above 33K and a rms roughness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Remarkably, the film quality is a function of the SiC surface termination, with the C- terminated surface preferred to the Si-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' To understand the MgB2 thin film/ SiC substrate interactions giving rise to this difference, we characterized the interfacial structures using Rutherford backscattering spectroscopy/channeling, electron energy loss spectroscopy, and x-ray photoemission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The MgB2/SiC interface structure is complex and different for the two terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Both terminations incorporate substantial unintentional oxide layers influencing MgB2 growth and morphology, but with different extent, intermixing and interface chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In this paper, we report measurements of transport, resistivity, and critical superconducting temperature of MgB2/SiC that are different for the two terminations, and link interfacial structure variations to observed differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The result shows that the C face of SiC is a preferred substrate for the deposition of ultrathin superconducting MgB2 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Introduction Ultra-thin superconducting films of MgB2 have attracted much interest [1-4] owing to their relatively high Tc of 39 K, large coherence lengths, and high critical current density [5,6] for applications in bolometers, photon detectors, and quantum devices [7-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' High quality MgB2 films have been fabricated successfully using Hybrid Physical-Chemical Vapor Deposition (HPCVD) [10], creating thin films with high Tc and low residual resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 2 However, the Volmer–Weber like island growth mode in the HPCVD process is detrimental to the production of ultrathin and smooth MgB2 films [11-14], where coalescence of islands often leads to rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For many superconducting electronic applications ultrathin (~5 nm), smooth (ideally with RMS roughness < 1 nm) and uniform films are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Previously, we reported on low-angle ion milling to thin 40 nm HPCVD MgB2 films and fabricated 5 nm superconducting films with Tc as high as 36 K [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Novoselov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' have reported growth of 10 nm HPCVD MgB2 films directly on the Si terminated SiC (0001) substrate with Tc as high as 35 K [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Recently, we have found that ultrathin MgB2 films grown on C-terminated 6H- SiC substrate (0001bar) are significantly smoother than those on Si-terminated substrates and possess high quality electronic properties and high Tc [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Interfacial phenomena are critically important in the design and manipulation of thin film functional materials[17-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Polarity of substrate is one of the important paramaters determining the structural, electric and magnetic properties of materials grown on polar materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The polarity of SiC has been proved to have significant influence on the growth as well as properties of thin films such as graphene, GaN and AlN thin films [21-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Here, we report a comprehensive characterization of ultrathin HPCVD MgB2 films grown on SiC substrates with both Si- and C-termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The interface characteristics are correlated with measurements of MgB2 transport, resistivity, and critical superconducting temperature comparing growth on these two principal SiC faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A significant finding is the existence of a substantial magnesium oxide layer at the MgB2/SiC interface with thickness and roughness dependent on the termination of the SiC substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The MgOx layer is thinner and smoother on the C-face than on the Si-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The smoother MgOx layer leads to a smoother MgB2 ultrathin film on the C-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Overall, the result shows that the C face is a preferred substrate for the deposition of ultrathin superconducting MgB2 films on SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' To our knowledge these MgB2 films are at the leading edge of this technology, combining the requirements of high Tc, thickness, and uniformity for advanced applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The interface characterization reported here provides details and possible explanations for the growth habit and suggests procedures broadly applicable to superconducting thin film growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Experimental details SiC substrates used were the 6H polytype in the (0001) or (0001bar) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' [16], a double-side polished 6H-SiC(0001) substrate is C-terminated on one side and Si-terminated on the other, leading to surfaces of different polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Details of the HPCVD method for growing MgB2 thin films have been described previously [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The HPCVD growth condition has been optimized to minimize the RMS roughness (a flow rate of 10 sccm diborane gas mixture, 5% B2H6 in H2, was used in this work as compared to 2 sccm in our previous work [16]), (see section 1 of Supplementary Material [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Transport properties of MgB2 films were characterized by standard four-point measurements [25] where probes were placed in the four corners of 5×5 mm2 square sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' temperature measurement was carried out by dipping the sample into a liquid helium dewar immediately after removal from the HPCVD system to minimize air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The measured resistance is converted into resistivity based on the van der Pauw solution for the square shape sample [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 3 Atomic force microscopy (AFM), scanning transmission electron microscopy (STEM), x-ray photoelectron spectroscopy (XPS), and Rutherford backscattering spectrometry (RBS) were used to characterize the film and the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' AFM measurements were performed using a Veeco atomic force microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Imaging and electron energy loss spectroscopy were carried out using the Rutgers Nion UltraSTEM microscope operated at 60 kV with the convergence and collection semi-angles at 35 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 mrad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' XPS measurements were performed in a Thermo K-alpha system with charge compensation using Al-Kα radiation and overall energy resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Under these conditions, the surface hydrocarbons were found at a binding energy of 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' RBS measurements were performed using a General Ionex tandem accelerator with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 MeV He++ ions and a scattering angle of 130\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The sample was held approximately normal to the ion beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The estimated depth resolution for Mg is ~ 25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Ion beam channeling was performed along the (0001) direction of the 6H-SiC substrate and non-channeling (random) spectra were acquired by randomly varying incident angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' SIMNRA 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='01 software was used for non-channeling RBS spectra analysis [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Results Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (a),(c), (e) and (g) AFM images for MgB2 films grown on Si-terminated SiC with thicknesses of 4 nm, 9 nm, 16 nm and 22 nm respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (b), (d), (f) and (h) are the corresponding MgB2 films on the C- terminated SiC that were grown at same conditions as (a), (c), (e) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' a 4 nm b 4nm C d 9 nm 9 nm f 16 nm 16 nm bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' h 22nm 22 nm 2 μm4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Deposition rates and surface roughness As described in detail in the supplementary information [24] the basic growth method involves liquified Mg, combined with a flow of diborane gas to form MgB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In previous work [16], we used flow rates of 1 sccm and 2 sccm diborane gas mixture in the HPCVD deposition and obtained an RMS roughness of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm and Tc of 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 K in a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 nm MgB2 thin film on the C face of the SiC substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' From a more recent systematic optimization of the diborane gas mixture flow rate, we found that 10 sccm produces the smallest roughness with the best superconducting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 1 shows AFM images of MgB2 thin films with nominal thicknesses of 4 nm, 9 nm, 16 nm, and 22 nm grown on the Si and C faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Films with the same thickness but different terminations were grown in the same deposition run to ensure identical growth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For all thicknesses, films on the C face are smoother than those on the Si face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Films on the Si face show clusters that are absent on the films on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Energy dispersive x-ray spectroscopy (EDS) analysis shows that the clusters are MgB2 grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In addition, the films on the Si face show taller islands that are not completely connected while films on the C face show much better connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' On the Si face, the MgB2 islands become larger for thicker depositions, typical for the island growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Films of the same nominal thickness on the C face show a much smoother surface and don’t have typical hexagonal MgB2 grain as the case on Si face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The RMS roughness for the MgB2 films in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 1 is summarized in Table 1 showing values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 nm – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 nm for the C face and 2 – 3 nm for the Si face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The result is a marked improvement from those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' [16] and indicates that MgB2 films grown on the C can be smoother than those on the Si face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Films composed of grains with a size comparable to the total film thickness are intuitively rougher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Electronic properties Figure 2 shows corresponding resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' temperature (ρ -T) curves for the MgB2 films in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As the film thickness decreases, Tc decreases and the residual resistivity ρ0 increases for films on both the Si and C faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The Tc of MgB2 films on the Si-face is slightly higher than the films on C-face, which is probably due to the biaxial tensile strain between Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' RMS roughness, Tc0, residual resistivity ρ0, and Δρ of MgB2 films on C-SiC and Si-SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Film Thickness (nm) 4 9 16 22 C face RMS (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 Tc0 (K) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 ρ0 (µΩ·cm ) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 Δρ (µΩ·cm) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 Si face RMS (nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 Tc0 (K) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 ρ0 (µΩ·cm ) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 Δρ (µΩ·cm) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 5 MgB2 grains as the MgB2 growth mode in Si-face is more like island growth mode compared to the C face [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The results are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' While the values of Tc are similar to our earlier report [16], the ρ0 values of the current films are much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The results are similar for both substrate terminations except for the thinnest films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The residual resistivity is much higher for the 4 nm film on the Si face than on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Also included in Table 1 is Δρ, the change in resistivity from room temperature to just above the superconducting transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Rowell has shown that this quantity, the room temperature, and residual resistivity difference can be used to quantify the grain connectivity in MgB2 samples [29], with larger Δρ indicating poorer connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The dependence of Δρ on film thickness is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' At 22 nm, the films on both Si and C faces show Δρ values similar to our thicker, high quality MgB2 films, indicating excellent grain connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As the nominal thickness of the ultrathin film decreases, the grain connectivity effect becomes more and more important, reflected as a gradual increase in Δρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' At 4 nm, the connectivity degrades rapidly, leading to a large Δρ increase for both faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The film on the Si face shows much poorer connectivity than that on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The conclusion on the grain connectivity is corroborated by the residual resistivity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' While Δρ reflects the temperature dependence of the electron-phonon scattering and grain connectivity, ρ0 is determined by the grain connectivity and scattering of electrons by impurities, defects, as well as surfaces [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' We have shown previously [30] that for clean MgB2 films fabricated by HPCVD, the mean free path of electron scattering is limited by the film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For example, changing the film thickness from 22 nm to 4 nm results in a decrease in the mean free path and thus an increase in electron scattering by a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Combined with a reduction of grain connectivity, Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' temperature curves for MgB2 films on the Si face (a and b) and the C face (c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 50 a filmsonSiface 25 b Resistivity (μuQ cm) Resistivity (μ cm) filmson Siface 40 22nm 30 16nm 9nm 20 4nm 10 5 O 0 0 50 100150200 250 300 35 36 373839 40 41 42 Temperature (K) Temperature(K) 15 50 C films on C face d Resistivity (μQ cm) films on Cface 40 Resistivity (μQ cm) 0 10 5 10 0 0 50 100150200 250 300 30 323436384042 44 Temperature (K) Temperature(K)6 deduced from the increase in Δρ, by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 in the case of films on the C face, one can predict the ρ0 value of the 4 nm film from that of the 22 nm film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The result, ρ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 μΩ·cm for the 4 nm film, agrees well with the experimentally measured value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The results of the same procedure for all the films studied are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 3(b) and 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Interface characterization To understand the influence of the SiC surface termination on the properties of the ultrathin MgB2 films, we investigated the interfacial structure and chemistry of the MgB2/SiC interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The MgB2 samples for interface characterizations are prepared separately using the same growth conditions as films described above to ensure the properties of MgB2 films are consistent throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' iii-a) RBS channeling and interfacial oxides RBS/channeling measurements were performed to provide depth-dependent information on both the composition (‘random spectra’) and crystallinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Channeling, the reduction of scattering yield when the beam is aligned with a major crystallographic direction, yields information on crystal quality and identifies the alignment and composition of buried layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The results show that for both SiC terminations, there is clearly an interfacial layer between the MgB2 film and the SiC substrate containing both magnesium and oxygen (see section 2 of Supplementary Material [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The result of a composition analysis at the interface for the two samples, identified as MgB2/MgOx/SiC, is presented in Table 2, where the interfacial oxygen is ascribed to a MgOx layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Within the experimental error, the composition of the interfacial layer is close to MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The thickness of the MgOx layer is estimated to be ~ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The change of resistivity from 300 K to 40 K and residual resistivity as function of film thickness on both Si and C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The red curves are values expected based on the reductions of the electron mean free path and the grain connectivity as the film thickness decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 25 a I-films onSiface Ap (μQ cm) 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='filmson Cface 10 b filmsonSiface ( n) d 20 15 Predicted 10 5 0 15 filmsonCface 2 cm) ur) 110 Predicted 5 0 5 10 15 20 25 Film thickness (nm)7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 nm for the Si termination and ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 nm for the C termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Furthermore, there is a consistent, but small, channeling effect in the MgOx itself, indicating that the oxide is crystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' We suggest the Mg-interface peaks result from a partially crystalline MgO layer at the interface, with orientation affected by the lattice mismatch between MgO and SiC, and then influenced by the mismatch with overlayer MgB2 and MgOx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (RBS also detected monolayer scale surface impurities of silicon and carbon at the surface of the MgB2 overlayer that play no apparent role in the interface formation, but are noted here for completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The observation of a substantial oxide interfacial layer is a major new finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The “buried oxide” is shown to be consistent with magnesium oxide by the chemical shift as observed in angular dependent high energy XPS (HAXPES) analysis (section 3 of Supplementary Material [24], also see the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' [31-37] therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The observation of a MgOx interfacial layer raises two questions: 1) the origin of the oxygen since the HPCVD process, being entirely conducted in a reducing environment, in principle eliminates the oxygen from the film growth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' and 2) the roles of the oxide layer in determining the properties of the ultrathin MgB2 films associated with different terminations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' To address these questions, we investigated the first stages of the HPCVD process itself by heating the substrate along with the Mg pieces, without the introduction of the B2H6 gas mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Specifically, Si- and C-terminated SiC substrates were heated in Mg vapor at 740°C for 1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 4, AFM images of these treated Si- and C-terminated SiC substrates are shown along with those for the pristine substrates as received from the vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For the “as received” material the Si-terminated surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 4(a)) shows atomic steps with an RMS roughness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm whereas the C-terminated surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 4(c)) is featureless with an RMS roughness of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Following the “Mg only” treatment the samples are essentially MgO/SiC structures due to oxidation of the air-exposed Mg layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The roughness measured on Si-terminated substrate is much higher (RMS roughness ~ 4 nm) than that measured on the C- terminated substrate (RMS roughness ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='. They consistently show an oxygen-rich layer Table 2．Atomic composition of MgB2/SiC determined by RBS MgB2/MgOx/SiC Ointerface/cm2 (×1015) Mginterface/cm2 (×1015) Thickness of MgOx interface(nm) Si-Face (Ch-RBS) 13 ± 2 12 ± 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 C-Face (Ch-RBS) 6 ± 2 4 ± 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 8 with Mg:O ratio of ~ 1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 for both surface terminations and a small channeling effect indicative of imperfect or misaligned MgO crystallinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The thickness of the MgOx layer is thicker on Si-SiC (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm as determined from the channeling spectrum) than on C-SiC: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Note that they are both thicker than the MgOx layers detected at the MgB2/SiC interface: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm on the Si face and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm on the C face, due to further oxidation upon air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' iii-b STEM Electron energy loss spectroscopy and nm elemental profiling To further examine the interface between the MgB2 films and the underlying C- and Si- terminated SiC substrates, cross-section samples were prepared and studied by STEM-EELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As shown in the atomic resolution HAADF images in Figure 5, there is an intermediate layer between the top MgB2 and the bottom SiC for both terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' This interface layer is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 nm in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' However, while not shown here, the intermediate layer does not have constant thickness across the observed interface ranges, and it varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 nm for the case of Si-SiC and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 nm for the case of C-SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In addition, the intermediate layer marked with dash lines between 0 nm to -1 nm in both cases often exhibits a periodic structure as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Even though the resolution of STEM image for the MgO layer is not ideal due to the combination of limited resolution of our instrument and the complex and imperfect thin structure, we can still see that the structure matches well with MgO as viewed from the (111) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The atomic arrangements of MgB2, MgO, and SiC are overlaid in the figure as visual guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Note that the image intensity scales with the atomic number due to the detector arrangement, and it is Mg and Si observed in the MgB2 and SiC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (a) Atomic Force Microscopy (AFM) of Si-SiC as received from the vendor, (b) AFM image of Si-SiC after annealing in Mg vapor at 740°C for one minute, (c) AFM scan for C-SiC as received from manufacture, (d) AFM image of C-termination of SiC after annealing in Mg vapor at 740°C for one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' a SiFace b Si Face as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='received Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='treated 500nm 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='nm C CFace d CFace asreceived Mg treated 500nm 500nm9 Next, we compare the relative chemical distributions of Mg, Si, B, C, and O along the eight data points acquired across the interface areas in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' It is found that oxygen is mostly confined in the intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Si and C are found up to terminate near p the intermediate layer, as expected, whereas B is found on the thin film/surface side of the intermediate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Finally, Mg is found throughout the intermediate layer,consistent with a MgO interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' STEM-EELS data from the same cross section for (a) C-terminated SiC(0001) substrate (b) Si- terminated SiC(0001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Cross-sectional HAADF STEM imaging of MgB2 on (a) C-terminated SiC(0001) substrate (b) Si-terminated SiC(0001) substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' ChemicalDistribution(C-SiC) ChemicalDistribution(Si-SiC） 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='0 a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 Intensit Mg Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 Si x Si O B Normalized B C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 1 0 人 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='0F 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='0 3-2 1 0 1 2 3 4 5 3 2 1 0 1 2 3 4 5 z (nm) z (nm)a b 3 nm 2 nm 1 nm 0 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 nm 1 nm 2 nm 3 nm 2nm10 iii-c) XPS analysis and interfacial chemistry More detailed information on interfacial chemistry is revealed by XPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 7 shows selected core levels spectra (Mg 1s, Si 2p, O 1s and C 1s) measured on bare SiC substrates (bottom curves), Mg vapor treated SiC (middle curves), and 7 nm-thick MgB2 grown on SiC (top curves), for both Si- and C-terminated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In all cases, the SiC substrate signal is detectable via the Si 2p and C 1s core levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Chemical environments attributed to core levels features are indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' On the bare SiC surface, exposed to air after preparation, the substrate Si 2p and C 1s core levels are found at binding energies of ~100 eV and ~282 eV, respectively, in good agreement with expected values [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The features in the O 1s and C 1s core levels spectra indicate carbohydrates adsorption, both the results of air exposure of the bare SiC substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For the Mg vapor treated SiC samples, the O 1s level is split into two peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The lower binding energy component is attributed to MgO while the higher binding energy component is assigned to Mg(OH)x and Mg carbonates [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Of particular interest is that the SiC-related core level spectra show different binding energies for the two different SiC surface terminations: the binding energies of both the C 1s and Si 2p core levels are ~1 eV lower for the C face than for the Si face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For MgB2 films on SiC substrate, exposure to air causes surface oxidation as well as water and carbohydrates adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As a result, their O 1s and Mg 1s spectra are affected by both the top surface alteration of MgB2 and the interfacial MgOx layer, and separating these contributions is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' However, we again observe a shift of the Si 2p core level to lower binding energy by ~1 eV in the sample on the C face as compared to the Si face (a similar energy shift is present for C 1s, but less visible due to the SiC signal attenuation through the MgB2 layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' This suggests that in the cases of both Mg vapor treated SiC, for which the stack is effectively MgO/SiC, and for MgB2 films on SiC, for which the stack is likely MgB2/MgO/SiC, there is a similar energy alignment related to the MgO/SiC interface that is dependent on the SiC surface termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Similar behavior has been reported in the case of intentionally grown epitaxial MgO films on SiC surfaces [40,41]: binding energy offsets of the order of eVs, have been measured for different MgO/SiC interfaces, supported by electronic structure calculations of atomically different interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' XPS spectrum of Mg 1s, Si 2p, O 1s and C 1s for bare SiC, Mg vapor treated SiC, and thin MgB2 film on SiC for both C and Si terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Mg1s MgB2/MgO/SiC Si2p MgB,/MgO/Sic 01s C1s C-Sic C-SiC Si-SiC C-SiC C-SiC Si-Sic Si-Sic Si-Sic MgB,/MgO/SiC MgB,/MgO/SiC Units) Units) Units) Units) Si-C Intensity (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Mg(OH)2 Mgo Intensity (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Intensity (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' /MgCO3 Intensity(Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' cO C-C/C-H MgO/Sic MgO/Sic MgO/Sic MgO/Sic BareSic BareSic BareSic BareSic 1308 1304 1300 104 102 100 98 96 536 532 528 292 288 284 280 Binding energy (eV) Binding energy (eV) Bindingenergy (eV) Binding energy (eV)11 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Discussion In the following, we consider specific aspects of these analyses and how they might influence the film morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Effect of native oxide It is clear from the different interface analysis results above, that a buried MgO layer exists at the interface of the MgB2 film and the SiC substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' It is likely that this oxide is the result of the reaction of Mg with the “native oxide” that exists on the SiC surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The reaction of Mg with SiO2 has been reported by a number of authors suggesting that the reaction of Mg with SiO2 can result in formation of MgO and possibly Mg silicates [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' “Native oxide” as a thin film is not necessarily well defined, as the resulting oxide thickness, usually only 1-2 nm at most, is a result of environmental variables, time and crystal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In recent work using contact angle measurements Park et al [44] showed that the native oxide growth is greater on the Si face than on the carbon face, consistent with the reports in Table 2 of a Si face MgO layer of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 nm compared to a C face of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For calibration, we note that 1 nm of SiO2 corresponds to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3×1015 /cm2 of oxygen, consistent with the existence of very thin native oxides yielding nanometer MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Furthermore, Nagai et al [45] characterized the roughness in very thin oxides on SiC as a function of crystal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In this work, the authors show that the RMS roughness is proportional to oxide thickness and the rate of increase of roughness with film thickness is the same for the two crystal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Therefore, the Si face oxide roughness is greater than the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' These reports allow some mechanistic conclusions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Native oxide thickness on the C face is less than that of the Si face [44], consistent with the reports in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The roughness of very thin oxides is proportional to oxide thickness [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In short, oxide roughness is proportional to oxide thickness, this oxide roughness is transferred to a MgO layer and then reflected in the overlayer MgB2 film uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Since the C-face oxide is substantially less than the Si face the net roughness is reduced for the C-face resulting in a more uniform thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Relevant to that point, it is interesting to note that there are reports of growth of MgO on SiC by MBE for MOS systems [46, 47], a good lattice match, and there are reports of the growth of MgB2 on MgO [48], also a reasonable lattice match resulting in high quality MgB2 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Therefore, an MgB2/MgO/SiC epitaxial structure may be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Nevertheless the properties of the resulting film but may depend on the SiC termination: if the starting (oxidized) surface of the C face of SiC is less rough than the Si face, a smoother MgO/MgB2 structure is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Finally, among the interesting remaining questions is the “necessity” for magnesium- based oxide layer to achieve higher quality epitaxy and crystallinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' We note this point was explicitly raised in the MBE work of Laloe et al [49], for MgB2 on Si where a Mg starting layer was explicitly added to enhance growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Possibly the “native oxide” on the Si face is just the correct amount to achieve a high-quality epitaxial film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In that regard, we note that some preliminary experiments on HF treated SiC (presumably minimal oxide) in our laboratory did not produce quality films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Conclusion 12 The growth of ultrathin MgB2 films on the different surface terminations of SiC has been studied, seeking the conditions for optimum superconducting properties and film uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' It has been shown that the best conditions are associated with 10 sccm of 5% diborane gas flow rate on the C terminated face of SiC for our specific system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A significant difference has been identified between growth on the Si face and the C face, with the latter producing higher quality films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' This difference has been explored by various interface probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' RBS/channeling measurements indicated that the samples consisted of MgB2/MgO/SiC stacks, in which the thickness MgOx layer was SiC surface termination dependent: is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm on the Si face and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 nm on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' High resolution EELS and TEM confirmed this structural difference, indicating that different interfacial constituents on the two surfaces may control the final morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' XPS analysis indicated a similar energy band offset at the MgO/SiC interfaces, for both MgB2 films grown on SiC and for MgO films on SiC, but highly dependent on the SiC surface termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' High energy, grazing exit angle XPS confirmed the presence of a buried, thin MgOx layer at the MgB2/SiC interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' This MgO layer in turn may govern the MgB2 film quality: a thicker and rougher MgOx layer on the Si face of the SiC substrate is the cause of the rougher ultrathin MgB2 films as compared to the films on the C face of SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The achievement of such high-quality superconducting films, and the knowledge of the parameters that control their growth, maybe a precursor to new devices and device configurations employing their unique electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Acknowledgment Research at Temple University was supported by NASA’s Astrophysics Research and Analysis Program through a contract from JPL (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 1632463).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Research at Rutgers was supported by Laboratory for Surface Modification (LSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=', and J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Express 6, 023101 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Zeng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 22, 125015 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Wolak, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Acharya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Tan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Cunnane, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Karasik and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Xi, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 25, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Zhuang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Chen, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Kobayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Kobata, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' De Groot, Nano letters, 17, 2 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Gray, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Papp, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Ueda, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Balke, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Yamashita, L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Pickett, Nature Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 10, 759 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Jablonski and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Yoshizawa, Physica C: Superconductivity 412, 1383-1386 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='- B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Laloë, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Kim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Moodera, Advances in Condensed Matter Physics 2011, 989732 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 15 Supplementary Information Ultra-thin Epitaxial MgB2 on SiC: Substrate Surface Polarity Dependent Properties Weibing Yang1†*, Leila Kasaei2†, Hussein Hijazi2, Sylvie Rangan2, Yao-wen Yeh2, Raj K Sah1, Jay R Paudel1, Ke Chen1, Alexander X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Gray1, Philip Batson2, Leonard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Feldman2, and Xiaoxing Xi1 1Department of Physics, Temple University, Philadelphia, PA 19122, USA 2Department of Physics and Astronomy, Rutgers University, New Brunswick, NJ 08901, USA Corresponding author 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Hybrid Physical Chemical Vapor Deposition of epitaxial MgB2 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The HPCVD reactor was first purged with ultra-high purity H2 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The substrate and Mg pieces placed nearby on the same heater were then heated to 740\uf0b0C in the H2 ambient when Mg began to evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' After about 10 seconds, a mixture of 5% B2H6 in H2 was introduced into the reactor to initiate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The film deposition rate can be controlled by adjusting the flow rates of the diborane gas mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In this work, a flow rate of 10 sccm was used, the result of parameter optimization for ultrathin MgB2 film with the best film quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The corresponding deposition rate was ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='23 nm/s, determined by a linear fitting of the thicknesses-flow rate data from a series of calibration runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The thickness of MgB2 ultrathin films was then controlled by the deposition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The deposition temperature was 740 \uf0b0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In addition to figure 3 in the main text, we present in figure S1 the residual resistivity (ρ0) of MgB2 films as a function of thickness for many more samples, measured in unpatterned as-grown films, to demonstrate the reproducibility of our deposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The additional data points are derived from the measurements of MgB2 films grown using 5, 10, and 20 sccm gas mixture of 5% B2H6 in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' It is clear that the flow rate of B2H6 gas mixture does not change the trend of residual resistivity as a function of film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The residual resistivity is about the same for thick films on the Si and C faces, but it is much smaller on the C face than on the Si face when the film is only a few nanometers, indicating much better gain connectivity of ultrathin MgB2 on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Residual resistivity as a function of MgB2 film thickness on the Si and C faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The MgB2 films were deposited using 5, 10 and 20 sccm gas mixture of 5% B2H6 in hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Comparison of Rutherford Back Scattering on MgB2 thin films grown on two terminations of SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure S2(a) shows a SIMNRA simulation of a random (non-channeling) RBS spectrum for an ideal 80 nm MgB2 film grown epitaxially on SiC substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The thickness and scattering geometry was chosen such that signals from the film surface and film/substrate interface are clearly distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figures S2(b) and S2(c) show RBS channeling and random spectra for two 80 nm MgB2 films on Si- and C-terminated SiC substrates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The experimental random RBS spectra agree well with the SIMNRA fitting shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The minimum yield χmin (the ratio between the channeling and random yields), evaluated using a width of 3 channels at around channel number 665, with contributions from both the MgB2 and the underlying SiC, was ~5%, indicating excellent crystallinity in MgB2 films on both the Si and C faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The strong channeling effect allows us to probe the properties of the MgB2/SiC interface as an alignment discontinuity at the interface inevitably leads to a weaker channeling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S2(b) and S2(c), the channeling yields are multiplied by 5x for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For both SiC terminations, there is an interfacial layer between the MgB2 film and the SiC substrate containing both magnesium and oxygen, marked as “Mg interface” and “O interface”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' We identify this interfacial layer as MgOx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The degraded channeling effect in the interfacial layer may be the result of partially crystalline MgOx and/or lattice mismatch between MgB2 and MgOx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Angle Resolved Hard X-ray Photoelectron Spectroscopy on MgB2 thin films on two terminations of SiC substrates To investigate the chemical bonding and the depth profile of the Mg species associated with the pristine MgB2 and the oxidized film, we utilized angle-resolved bulk-sensitive hard x- ray photoelectron spectroscopy (HAXPES) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The measurements were carried out using a lab-based HAXPES system equipped with a monochromatized Cr Kα x-ray source with the photon energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 keV and a wide acceptance angle hemispherical electrostatic analyzer ScientaOmicron EW4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' We measured two MgB2 films, nominally 4 nm thick, deposited simultaneously and side- by-side on the C- and Si-terminated SiC substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' At the photon energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 keV, the inelastic mean-free path (IMFP) for the Mg 1s photoelectrons (Ekin = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='1 keV) in MgB2 is estimated to be approximately 7 nm [2], which ensures that the entire film and the interface with the substrate are being probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Measurements were carried out at four different photoelectron take-off angles, 4°, 30°, 45°, and 60°, facilitating different average probing depths varying from approximately 7 nm (at 4°) to approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 nm at 60°, making the latter more surface sensitive and less sensitive to the buried SiC/MgB2 interface [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (a) Simulated random RBS spectrum for an ideal 80 nm MgB2 films grown on SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (b) and (c) RBS channeling and random spectra for 80 nm MgB2 films on Si- and C-terminated SiC(0001) substrates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The channeling yields multiplied by 5x are also shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 16 wev He+ Ras sttafon of 8Cnm MgBSiC Si Mg C B C surface oIrterface Osutace Wafm Random thannelng Random chxs Channeing19 The results of the measurements for the films on the C-terminated and Si-terminated SiC substrates are presented in Figures S3(a) and S3(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The most intense peak at the binding energy of 1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 eV in both plots corresponds to the Mg 1s core-level photoemission originating from the MgB2 film [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The higher-binding-energy feature at approximately 1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='2 eV corresponds to the chemically shifted state originating, most likely, from the Mg oxide species, such as MgO [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Normalization of the photoemission intensities to the maximum of the main MgB2 peak reveals two different trends in the angle-dependent evolutions of the photoemission intensities of the higher-binding-energy (MgO) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For the film on the C-terminated SiC substrate [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S3(a)] the intensity of the MgO component increases with increasing surface sensitivity, as shown using blue circular markers in Figure S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Such a trend generally corresponds to the angle-resolved measurement of the surface oxide species [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Conversely, for the film on the Si-terminated substrate [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S3(b)] two major differences are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' First, the average intensity of the MgO component is increased relative to the main MgB2 peak, suggesting a higher oxide content in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Secondly, the angle- dependent evolution of the MgO component exhibits a flatter trend, with the most surface- Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (a) Mg 1s core-level peak measured on the MgB2 film deposited on the C-terminated SiC substrate at four photoelectron take-off angles in the experimental geometry shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (b) Same angle-resolved measurement carried out on the MgB2 film deposited on the Si-terminated SiC substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' (c) Peak intensities of the higher-binding-energy MgO peak component normalized to the height of the main MgB2 peak (circular markers) and the corresponding simulated intensities (diamond-shaped markers) for the best-fit film structures shown in the figure legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' analyzer C termination Mg 1S 60° 45° 30° MgB2 Mgo 1kev Si termination Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 MgB2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 Mgo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 0 20 40 6020 sensitive measurement at 60° exhibiting a slight decrease in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Such a trend, shown using red circular markers in Figure S3(c), suggests the presence of a buried Mg oxide layer at a depth that is larger compared to the probing depth at the take-off angle of 60° (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In order to confirm the presence of the buried Mg oxide layer in the sample on the Si- terminated SiC substrate, we carried out angle-resolved simulations using the SESSA simulation package, which quantitatively predicts photoemission peak intensities by taking into account relevant parameters such as IMFP, elastic-scattering cross-sections, photoionization asymmetry parameters, and the photoelectron take-off angles [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The best-fit results for both samples are shown using diamond-shaped markers in Figure S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' For both samples, the thickness of the surface oxide is predicted to be approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='65 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='7 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The thickness of the pristine MgB2 layer (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='6 nm) is close to the target thickness of 4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The main difference between the two samples is the extracted thickness of the buried interfacial oxide, which is mainly responsible for the difference in the two observed intensity trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Specifically, the best-fit thickness of the buried Mg oxide at the interface between MgB2 and the Si- terminated substrate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='75 nm, while the same value for the C-terminated substrate is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' The presence of an interfacial oxide was necessary for obtaining reasonable fits for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' As explained in the discussion the interfacial oxide on these samples is determined by the "native oxide" on the SiC surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Such native oxides are not well defined and depend on numerous environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Nevertheless, in agreement with the trend reported in Table 1, the MgO interfacial layer is greater on the Si face than on the C face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Slight discrepancies between the experimental and simulated intensities are observed for both samples at the lowest take-off angle (4°) due to the limitations of the experimental geometry, which features a fixed 90° angle between the x-ray incidence direction and the analyzer orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' In this geometry, the x-ray incidence angle is so grazing (4°) that the effects of the x-ray beam cone-angle (18°), as well as the total external reflection, may play a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Niwa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Yasui, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Ikenaga, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Honjo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Ikeda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Nakamura, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Endoh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 125, 203903 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Panaccione and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Kobayashi, Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 606, 125 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Smekal,W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content='Werner, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Powell, Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' Interface Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} +page_content=' 37, 1059 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAzT4oBgHgl3EQfs_2h/content/2301.01669v1.pdf'} diff --git 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Zambello,𝑎,∗ D. A. Clarke,𝑐 P. Dimopoulos,𝑏 F. Di Renzo,𝑏 J. Goswami,𝑑 G. +Nicotra,𝑐 C. Schmidt𝑐 and S. Singh𝑏 +𝑎Dipartimento di Fisica, Università di Pisa and INFN, Sezione di Pisa, Pisa, Italy +𝑏Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and INFN, Gruppo +Collegato di Parma, Parma, Italy +𝑐Fakultät für Physik, Universität Bielefeld, D-33615, Bielefeld, Germany +𝑑RIKEN Center for Computational Science, Kobe 650-0047, Japan +E-mail: kevin.zambello@pi.infn.it, clarke.davida@gmail.com, +petros.dimopoulos@unipr.it, francesco.direnzo@unipr.it, +jishnu@physik.uni-bielefeld.de, gnicotra@physik.uni-bielefeld.de, +schmidt@physik.uni-bielefeld.de, simran.singh@unipr.it +We report updated results on the determination of Lee-Yang edge (LYE) singularities in 𝑁 𝑓 = 2+1 +QCD using highly improved staggered quarks (HISQ) with physical masses on 𝑁𝜏 = 4, 6, 8 +lattices. The singularity structure of QCD in the complex 𝜇𝐵 plane is probed using conserved +charges calculated at imaginary 𝜇𝐵. The location of the singularities is determined by studying the +(uncancelled) poles of multi-point Padé approximants. We show that close to the Roberge-Weiss +(RW) transition, the location of the LYE singularities scales according to the 3-𝑑 𝑍(2) universality +class. By combining the new 𝑁𝜏 = 6 data with the 𝑁𝜏 = 4 data from our previous analysis +we extract a rough estimate for the RW temperature in the continuum limit. We also discuss +some preliminary results for the singularities close to the chiral phase transition obtained from +simulations on 𝑁𝜏 = 6, 8 lattices. +The 39th International Symposium on Lattice Field Theory, +8th-13th August, 2022, +Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.03952v1 [hep-lat] 10 Jan 2023 + +Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +1. +Introduction +A detailed knowledge of the phase diagram of QCD is necessary for a complete understanding +of the physics of strongly interacting matter at finite temperature and density. At zero chemical +potentials the QCD transition is known to be a crossover. The transition is conjectured to become +a first-order transition at large chemical potentials; then one expects the presence of a second order +critical endpoint (CEP), whose exact location however is still unknown. Much progress has been +done in investigating the QCD phase diagram at small chemical potentials (for a review see ref. [1]). +Unfortunately at large chemical potentials the QCD phase diagram remains inaccessible to direct +lattice simulations because of the sign problem. Some methods have been developed to tackle the +sign problem, such as Taylor expansion [2][3] and analytic continuation from imaginary 𝜇 [4][5]. +These techniques exploit the absence of the sign problem at zero and purely imaginary chemical +potentials. With the former method physical quantities are Taylor expanded around 𝜇 = 0, and the +series coefficients are calculated directly on the lattice. With the latter method physical quantities +are calculated at imaginary 𝜇 and then analytically continued to real 𝜇. Here we follow a new +method which can be regarded as a combination of these two approaches. For any given observable +we calculate the Taylor series coefficients at zero and purely imaginary chemical potentials. The +Taylor series are merged by using multi-point Padé approximants. Not only these can be analytically +continued to real 𝜇, but information about the analytical structure of the observables can also be +obtained by studying the poles of the approximants [6]. +2. +Lee-Yang edge singularities +The main focus of this work is the study of the critical points of QCD in the complex 𝜇𝐵 plane. +Notably there is a deep connection between singularities in the complex plane and phase transitions. +Consider for instance the Ising model (see [7][8][9]). At 𝑇 > 𝑇𝑐 the free energy has branch cuts +on the imaginary axis for the magnetic field ℎ. The branch points are known as Yang-Lee edge +singularities. As 𝑇 → 𝑇𝑐 the branch cuts pinch the real axis, signaling the presence of a real phase +transition. Lee-Yang singularities are particularly relevant for QCD. From their trajectory one can +infer the location of a physical phase transition. Since their presence implies a finite radius of +convergence for Taylor expansions, information can be obtained about the range of validity of the +results obtained in the literature by the Taylor expansion method. Finally from the universal scaling +properties of the LYE singularities, information can be obtained about the non-universal parameters +of the theory. +3. +The Roberge-Weiss transition region +We have investigated by lattice simulations two temperature regimes, one at high temperature +(𝑇 = 179.5 - 195.0 𝑀𝑒𝑉) and one at low temperature (𝑇 = 136.1 - 166.6 𝑀𝑒𝑉). In the high +temperature regime we have studied the LYE singularities associated with the RW critical point. +We have performed numerical simulations for 𝑁 𝑓 = 2 + 1 QCD using highly-improved staggered +quarks (HISQ) on 363 × 6 lattices. We have calculated the baryon number and electric charge +2 + +Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +cumulants, +𝜒1 +𝐵,𝑄 ≡ +1 +𝑉𝑇3 +1 +𝑍 +𝜕𝑍 +𝜕 ˆ𝜇𝐵,𝑄 +, +and their first derivative with respect to ˆ𝜇𝐵 = 𝜇𝐵 +𝑇 for 𝑂(10) imaginary chemical potentials. +The calculations were made at four different temperatures 𝑇 = 195.0, 190.0, 185.0 and 179.5 +𝑀𝑒𝑉. The highest temperature is close to the RW temperature that we expect given our choice for +the action discretization [10]. The numerical results for the baryon number density are shown in +fig. 1. +0 +1 +2 +3 +4 +5 +6 +Im[µB/T] +−0.4 +−0.2 +0.0 +0.2 +0.4 +Im[χ1 +B] +T = 179.50 MeV +T = 185.00 MeV +T = 190.00 MeV +T = 195.00 MeV +0 +1 +2 +3 +4 +5 +6 +Im[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +−Re[χ2 +B] +T = 179.50 MeV +T = 185.00 MeV +T = 190.00 MeV +T = 195.00 MeV +Figure 1: Imaginary part of 𝜒1 +𝐵 (left) and real part of 𝜒2 +𝐵 (right) as a function of ˆ𝜇𝐵 for different temperatures +(high temperature regime). +Figure 2: Singularity structure of the rational approximants at 𝑇 =179.5 𝑀𝑒𝑉 (top left), 185.0 𝑀𝑒𝑉 (top +right), 190.0 𝑀𝑒𝑉 (bottom left) and 195.0 𝑀𝑒𝑉 (bottom right). +The left picture shows the imaginary part of the baryon number density and the right picture +shows the real part of its derivative with respect to ˆ𝜇𝐵. Different colors correspond to different +temperatures and we can see a divergence emerging at the highest temperature for 𝜒2 +𝐵. These data +have been approximated by multi-point Padé approximants. Fig. 2 shows the singularity structure +3 + +5 +Zeros of the numerator +Zeros of the denominator +4 : +2. +1 +-4 +-2 +2 +Re[μB/T5 +Zeros of the numerator +Zeros of the denominator +4* +2. +1 +0 +-4 +-2 +2 +Re[μB/T5 +Zeros of the numerator +Zeros of the denominator +4 +3. +2. +1 +0 +-4 +-2 +2 +Re[μB/T5 +Zeros of the numerator +Zeros of the denominator +4 : +2. +1 +0 +-4 +-2 +2 +Re[μB/TDetermination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +of the Padé approximants. The zeros of the numerator and denominator are displayed respectively +in blue and red. At 𝐼𝑚( ˆ𝜇𝐵) = 𝜋 we observe an alternation of zeros of the numerator and zeros of +the denominator signaling the presence of branch cuts. The branch points pinch the real axis as the +temperature is increased. +3.1 Scaling analysis +In the top picture of fig. 3 we display for each temperature the closest singularity to the +imaginary axis. Different colors denote different temperatures. Different symbols denote different +approximants, respectively the approximants for 𝜒1 +𝐵 and 𝜒1 +𝑄. The singularities located by these two +observables are in agreement within errors. +0 +2 +4 +6 +8 +Re[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Im[µB/T] +195.00 MeV +195.00 MeV +190.00 MeV +190.00 MeV +185.00 MeV +185.00 MeV +179.50 Mev +179.50 Mev +Figure 3: LYE singularities for different temperatures (top) and scaling fit (bottom). +Next we checked whether these singularities could be identified as the LYE singularities +associated with the RW endpoint and whether they follow the critical behaviour expected for the +3𝑑, 𝑍(2) universality class. The location of the LYE singularities can be expressed in terms of the +scaling field 𝑧 = 𝑡/ℎ1/𝛽𝛿 ≡ |𝑧𝑐|𝑒𝑖𝜋/2𝛽𝛿, where 𝑡 is the reduced temperature and ℎ is the symmetry +breaking field. This relation makes a connection between the universal parameters 𝑧𝑐, 𝛽, 𝛿 and the +non-universal parameters embedded in 𝑡, ℎ [11] [12] [6]. +4 + +2.5 +I +Nt = 6 +2.0 +1.5 +Re[μB/' +1.0 +0.5 - +0.0 +0.0 +0.1 +0.2 +0 +0.4 +0.5 +(T - Trw)/TrwDetermination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +For the Roberge-Weiss transition we may introduce the reduced temperature 𝑡 = 𝑡−1 +0 +𝑇𝑅𝑊 −𝑇 +𝑇𝑅𝑊 +and +the symmetry breaking field ℎ = ℎ−1 +0 +ˆ𝜇𝐵−𝑖 𝜋 +𝑖𝜋 +. One obtains the following scaling law, +ˆ𝜇𝐿𝑌 𝐸 = ±𝜋 +� 𝑧0 +|𝑧𝑐| +�𝛽 𝛿 �𝑇𝑅𝑊 − 𝑇 +𝑇𝑅𝑊 +�𝛽 𝛿 +± 𝑖𝜋 , +where 𝛽, 𝛿 are the universal critical exponents and 𝑧0 = ℎ1/𝛽 𝛿 +0 +/𝑡0, 𝑇𝑅𝑊 are non-universal +parameters. +In the top picture of fig. +3 it can be recognized that the imaginary part of the +singularities we have found by our Padé analysis is indeed trivially 𝑖𝜋 within errors. For the real +part we can fit the data using the ansatz +ˆ𝜇𝑅 +𝐿𝑌 𝐸 = 𝑎 +�𝑇𝑅𝑊 − 𝑇 +𝑇𝑅𝑊 +�𝛽 𝛿 +. +By fitting with the 3𝑑, 𝑍(2) critical exponents (𝛽𝛿 = 1.5635) we obtain a good fit (𝜒2/𝑑𝑜 𝑓 = 0.25). +From the fit we estimate 𝑇𝑅𝑊 (𝑁𝜏 = 6) = 206.67(59) 𝑀𝑒𝑉. Using |𝑧𝑐| = 2.42(4) from ref. [12] +we also find 𝑧0 = 10.0-10.9. +3.2 Continuum limit +We previously conducted a similar analysis using the data obtained from 𝑁𝜏 = 4 lattice +simulations [6]. Using these results and the results from our current analysis we made a crude +estimate for the continuum limit. The result is displayed in fig. 4. We obtained 𝑇 𝑐𝑜𝑛𝑡 +𝑅𝑊 += 207.1(2.4) +𝑀𝑒𝑉. This is only a preliminary result and a proper continuum extrapolation would require data +from 𝑁𝜏 = 8 lattice simulations. +Still it is in nice agreement with a previous determination +obtained by a different method using a different discretization in ref. [13], where the authors found +𝑇 𝑐𝑜𝑛𝑡 +𝑅𝑊 += 208(5) 𝑀𝑒𝑉. +Figure 4: Estimate for the Roberge-Weiss temperature in the continuum limit. +4. +The chiral transition and CEP regions +In the low temperature regime we have been hunting for the LYE singularities associated with +the chiral transition (or possibly the critical endpoint of QCD). We ran a series of simulations at +5 + +230 +220 +210 +T +190 +180 +- +170 +00'0 +- +0.01 +0.02 +0'0 +0.04 +90'0 +90'0 +0.07 +80'0 +1 / N?Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +𝑇 = 166.59, 157.50, 145.00 and 136.1 𝑀𝑒𝑉. The numerical results for the baryon number density +are shown in fig. 5. +0 +1 +2 +3 +4 +5 +6 +Im[µB/T] +−0.4 +−0.2 +0.0 +0.2 +0.4 +Im[χ1 +B] +T = 136.10 MeV +T = 145.00 MeV +T = 157.50 MeV +T = 166.59 MeV +T = 179.50 MeV +T = 185.00 MeV +T = 190.00 MeV +T = 195.00 MeV +0 +1 +2 +3 +4 +5 +6 +Im[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +−Re[χ2 +B] +T = 136.10 MeV +T = 145.00 MeV +T = 157.50 MeV +T = 166.59 MeV +T = 179.50 MeV +T = 185.00 MeV +T = 190.00 MeV +T = 195.00 MeV +Figure 5: Imaginary part of 𝜒1 +𝐵 (left) and real part of 𝜒2 +𝐵 (right) as a function of ˆ𝜇𝐵 for different temperatures +(all temperatures). +The low temperature data have been approximated by rational functions, just as we did for the +high temperature data. However in this case we observed a strong interval dependence. This is +exemplified in the left picture of fig. 6, where we plot the singularities resulting from the Padé +analysis of the 𝑇 = 145.00 𝑀𝑒𝑉 data. Different colors denote fits over different intervals. We +can recognize two distinct clusters. The results in the bottom cluster are from the fits over (small +variations of) the [0, 𝑖𝜋] interval, the results in the top cluster are from the fits over (small variations +of) the full [0, 2𝑖𝜋] interval. Being located at 𝐼𝑚( ˆ𝜇𝐵) ≈ 𝑖𝜋, the top cluster is likely related to the +RW endpoint. In the following we focus on the bottom cluster and we study how it moves as we +change the temperature. +0 +2 +4 +6 +8 +Re[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Im[µB/T] +136.10MeV +145.00 MeV +157.50 MeV +166.59 MeV +195.00 MeV +195.00 MeV +190.00 MeV +190.00 MeV +185.00 MeV +185.00 MeV +179.50 Mev +179.50 Mev +Figure 6: LYE singularities for different temperatures (right) and scatter plot of the singularities resulting +from Padé fits over different intervals for 𝑇 = 145 𝑀𝑒𝑉 (left). +4.1 Scaling analysis +In the right picture of fig. 6 we summarize the singularities that we have determined in this +work. The uncertainty in the determinations of the singularities in the low temperature regime are +given by the 1𝜎 confidence ellipses. These singularities apparently move towards the real axis as +the temperature is decreased. Their locations imply a radius of convergence R ≈ 3 - 4 for Taylor +series expansions around ˆ𝜇𝐵 = 0 at 𝑇 = 136 - 166 𝑀𝑒𝑉. +These singularities might be identified as the LYE singularities associated with the chiral +transition or possibly with the critical endpoint of QCD. +6 + +A +Im[μB/' +2 +1 +2 +0 +Re[μB/T]Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +Figure 7: Comparison with the predictions for the chiral singularities from HotQCD data at temperatures +𝑇 = 166.59 𝑀𝑒𝑉 (top left), 𝑇 = 157.50 𝑀𝑒𝑉 (top right), 𝑇 = 145.0 𝑀𝑒𝑉 (bottom left) and 𝑇 = 136.10 +𝑀𝑒𝑉 (bottom right). +0 +1 +2 +3 +4 +5 +6 +Re[ +B/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Im[ +B/T] +ˆµLY [T = 136.10 MeV ] +ˆµLY [T = 145.00 MeV ] +ˆµLY [T = 157.50 MeV ] +ˆµLY [T = 166.59 MeV ] +Figure 8: Overall picture of the comparison with the predictions for the chiral singularities from HotQCD +data (left) and summary of the expected scaling for the LYE singularities associated with the RW endpoint, +with the chiral transition and with the CEP (right, figure from ref. [14]). +7 + +3.5 +3.0 +2.5 +pBW(N = 4) +W(N = 6) +川2.0 +ImμB/ +T=166.6 +T=157.5 +1.5 +T=145.0 +T=136.1 +1.0 +RW scaling +0.5 +chiral scaling +CEP scaling +0.0 +0 +1 +2 +3 +4 +5 +Re[μB/T]4.0 +μLY [T = 166.59 MeV +3.5 +3.0 +2.5 +[L/a]] +2.0 +1.5 +1.0 +0.5 +0.0 - +0 +1 +2 +3 +4 +5 +6 +Re[μB/T]4.0 +aLY [T = 157.50 MeV +3.5 +3.0 +95 +2.5 +[L/a]] +2.0 +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +5 +6 +Re[μB/T]4.0 +aLY [T = 145.00 MeV +3.5 +3.0 +2.5 +[L/a]] +2.0 +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +5 +6 +Re[μB/T]4.0 +aLy [T = 136.10 MeV +3.5 +3.0 +2.5 +[L/a]] +2.0 +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +5 +6 +Re[μB/T]Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +The critical behaviour close to the chiral transition region is expected to follow the behaviour +of a theory belonging to the 3𝑑, 𝑂(2) universality class. Following ref. [15] the critical behaviour +can be studied by introducing the scaling fields +𝑡 = 𝑡−1 +0 +�𝑇 − 𝑇𝑐 +𝑇𝑐 ++ 𝑘 𝐵 +2 +� 𝜇𝐵 +𝑇 +�2� +, +ℎ = ℎ−1 +0 +𝑚𝑙 +𝑚 𝑝ℎ𝑦𝑠 +𝑠 +, +where 𝑇𝑐 is the critical temperature, 𝑘 𝐵 +2 is the curvature coefficient of the critical line 𝑇𝑐( ˆ𝜇𝐵) and +the symmetry breaking field ℎ is expressed in terms of the light-to-strange quark mass ratio +𝑚𝑙 +𝑚𝑝ℎ𝑦𝑠 +𝑠 +. +One can derive the scaling law for the chiral singularities +ˆ𝜇𝐿𝑌 𝐸 = +������ +1 +𝑘 𝐵 +2 +𝑧𝑐 +𝑧0 +� +𝑚𝑙 +𝑚 𝑝ℎ𝑦𝑠 +𝑠 +� +1 +𝛽 𝛿 +− 𝑇 − 𝑇𝑐 +𝑇𝑐 +������ +1 +2 +. +In fig. 7 the singularities resulting from the Padé analysis are compared with the predictions +obtained by setting +𝑚𝑙 +𝑚𝑝ℎ𝑦𝑠 +𝑠 += 1/27, by using the non-universal parameters 𝑇𝑐 = 147(6) 𝑀𝑒𝑉, +𝑘 𝐵 +2 = 0.012(2), 𝑧0 = 2.35(20) from the HotQCD data, and by using |𝑧𝑐| = 2.032 from ref. [11]. +The results are in agreement within errors, but if we look at the overall picture of fig. 8 (left) we +observe that the singularities follow a steeper curve than the one predicted using the best estimates +for the non-universal parameters (the dashed line in the picture). +A possibility remains open that the singularities that we observed are the LYE singularities +associated with the critical endpoint of QCD. In this case the mapping from the non-universal +parameters to the universal theory in unknown. An approach one may try is to use the linear ansatz +[16] +𝑡 = 𝛼𝑡 (𝑇 − 𝑇𝐶𝐸𝑃) + 𝛽𝑡 (𝜇𝐵 − 𝜇𝐶𝐸𝑃) +, +ℎ = 𝛼ℎ(𝑇 − 𝑇𝐶𝐸𝑃) + 𝛽ℎ(𝜇𝐵 − 𝜇𝐶𝐸𝑃) +to derive the scaling law +𝜇𝐿𝑌 𝐸 ∼ 𝜇𝐶𝐸𝑃 − 𝑐1(𝑇 − 𝑇𝐶𝐸𝑃) + 𝑖𝑐2|𝑧𝑐|−𝛽 𝛿(𝑇 − 𝑇𝐶𝐸𝑃)𝛽 𝛿 . +Using some reasonable estimates for the non-universal parameters one obtains the qualitative +prediction shown as a red band in the right picture of fig. 8. The red band seems to better describe +our data than the expected scaling for the chiral singularities (the green band in the same picture). +This conjecture will be explored in future work. A different approach in which LYE singularities +are located by studying the Fourier coefficients of the baryon number density is also being explored +[17]. +4.2 Comparison with 𝑁𝜏 = 8 data +Finally we ran a separate set of simulations at 𝑇 = 156.5 𝑀𝑒𝑉 using 323 × 8 lattices. The +singularity resulting from the Padé analysis is shown in fig. 9. Also shown is the singularity +obtained at a similar temperature (𝑇 = 157.5 𝑀𝑒𝑉) for 𝑁𝜏 = 6 lattices. The results are in very +good agreement despite the different volume. This suggests that both the UV-cutoff effects and the +finite-size effects are negligible within the accuracy of our results. +8 + +Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +0 +2 +4 +6 +8 +Re[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Im[µB/T] +157.50 MeV [Nt = 6] +156.50 MeV [Nt = 8] +Figure 9: Comparison between 𝑁𝜏 = 8 and 𝑁𝜏 = 6 lattices results for the LYE singularities at 𝑇 ≈ 157 +𝑀𝑒𝑉. +5. +Conclusions +We have studied the complex singularities of QCD by a multi-point Padé analysis. In the +high temperature regime we have identified the LYE singularities associated to the Roberge-Weiss +endpoint. These singularities have the expected critical behaviour for a transition belonging to the +3𝑑, 𝑍(2) universality class. By combining these new results and the results from a previous analysis +we have calculated an estimate for the Roberge-Weiss temperature in the continuum limit. +In the low temperature regime we have found singularities that may be identified as the LYE +singularities associated with either the chiral transition or the critical endpoint of QCD. Their +location imply a radius of convergence R = 3 - 4 for 𝑇 = 136 - 166 𝑀𝑒𝑉, which constrains the +validity of the Taylor series expansions at ˆ𝜇𝐵 = 0. +For future work we plan to extend the study to 𝑁𝑡 = 8 lattices in order to make a proper +continuum extrapolation for the RW temperature. We also plan to generate data at lower temperatures +in order to get a better understanding of the nature of the complex singularities that we have found +at low temperatures. +6. +Acknowledgements +This work was supported by European Union Horizon 2020 research and innovation programme +under the Marie Sklodowska-Curie grant agreement No 813942 (EuroPLEx) and by the I.N.F.N. +under the research project i.s. QCDLAT. This research used computing resources made available +(i) by CINECA on Marconi and Marconi 100 under both the I.N.F.N-CINECA agreement and the +ISCRA B program, (ii) by the Gauss Centre for Supercomputing on the Juwels GPU nodes at the +Jülich Supercomputing Centre and (iii) by the Bielefeld University on the Bielefeld GPU-Cluster. +References +[1] J. N. Guenther, “An overview of the QCD phase diagram at finite 𝑇 and 𝜇,” PoS LATTICE2021 (2022), +013 doi:10.22323/1.396.0013 [arXiv:2201.02072 [hep-lat]]. +9 + +Determination of Lee-Yang edge singularities in QCD by rational approximations +K. Zambello +[2] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt and L. Scorzato, +“The QCD thermal phase transition in the presence of a small chemical potential,” Phys. Rev. D 66 +(2002), 074507 doi:10.1103/PhysRevD.66.074507 [arXiv:hep-lat/0204010 [hep-lat]]. +[3] R. V. Gavai and S. 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Schmidt, “Fourier coefficients of the net-baryon number density,” PoS LATTICE2022 (2023), 159. +10 + diff --git a/zdE2T4oBgHgl3EQfhwfG/content/tmp_files/load_file.txt b/zdE2T4oBgHgl3EQfhwfG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..313044ef621dce5a36bc0164c9f3bcf979c50434 --- /dev/null +++ b/zdE2T4oBgHgl3EQfhwfG/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf,len=587 +page_content='Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello,𝑎,∗ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Clarke,𝑐 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Dimopoulos,𝑏 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Di Renzo,𝑏 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Goswami,𝑑 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Nicotra,𝑐 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Schmidt𝑐 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Singh𝑏 𝑎Dipartimento di Fisica, Università di Pisa and INFN, Sezione di Pisa, Pisa, Italy 𝑏Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and INFN, Gruppo Collegato di Parma, Parma, Italy 𝑐Fakultät für Physik, Universität Bielefeld, D-33615, Bielefeld, Germany 𝑑RIKEN Center for Computational Science, Kobe 650-0047, Japan E-mail: kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='zambello@pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='it, clarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='davida@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='com, petros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='dimopoulos@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='it, francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='direnzo@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='it, jishnu@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='de, gnicotra@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='de, schmidt@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='de, simran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='singh@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='it We report updated results on the determination of Lee-Yang edge (LYE) singularities in 𝑁 𝑓 = 2+1 QCD using highly improved staggered quarks (HISQ) with physical masses on 𝑁𝜏 = 4, 6, 8 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The singularity structure of QCD in the complex 𝜇𝐵 plane is probed using conserved charges calculated at imaginary 𝜇𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The location of the singularities is determined by studying the (uncancelled) poles of multi-point Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We show that close to the Roberge-Weiss (RW) transition, the location of the LYE singularities scales according to the 3-𝑑 𝑍(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' By combining the new 𝑁𝜏 = 6 data with the 𝑁𝜏 = 4 data from our previous analysis we extract a rough estimate for the RW temperature in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We also discuss some preliminary results for the singularities close to the chiral phase transition obtained from simulations on 𝑁𝜏 = 6, 8 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='03952v1 [hep-lat] 10 Jan 2023 Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Introduction A detailed knowledge of the phase diagram of QCD is necessary for a complete understanding of the physics of strongly interacting matter at finite temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' At zero chemical potentials the QCD transition is known to be a crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The transition is conjectured to become a first-order transition at large chemical potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' then one expects the presence of a second order critical endpoint (CEP), whose exact location however is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Much progress has been done in investigating the QCD phase diagram at small chemical potentials (for a review see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Unfortunately at large chemical potentials the QCD phase diagram remains inaccessible to direct lattice simulations because of the sign problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Some methods have been developed to tackle the sign problem, such as Taylor expansion [2][3] and analytic continuation from imaginary 𝜇 [4][5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' These techniques exploit the absence of the sign problem at zero and purely imaginary chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' With the former method physical quantities are Taylor expanded around 𝜇 = 0, and the series coefficients are calculated directly on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' With the latter method physical quantities are calculated at imaginary 𝜇 and then analytically continued to real 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Here we follow a new method which can be regarded as a combination of these two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' For any given observable we calculate the Taylor series coefficients at zero and purely imaginary chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The Taylor series are merged by using multi-point Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Not only these can be analytically continued to real 𝜇, but information about the analytical structure of the observables can also be obtained by studying the poles of the approximants [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Lee-Yang edge singularities The main focus of this work is the study of the critical points of QCD in the complex 𝜇𝐵 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Notably there is a deep connection between singularities in the complex plane and phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Consider for instance the Ising model (see [7][8][9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' At 𝑇 > 𝑇𝑐 the free energy has branch cuts on the imaginary axis for the magnetic field ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The branch points are known as Yang-Lee edge singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' As 𝑇 → 𝑇𝑐 the branch cuts pinch the real axis, signaling the presence of a real phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Lee-Yang singularities are particularly relevant for QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' From their trajectory one can infer the location of a physical phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Since their presence implies a finite radius of convergence for Taylor expansions, information can be obtained about the range of validity of the results obtained in the literature by the Taylor expansion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Finally from the universal scaling properties of the LYE singularities, information can be obtained about the non-universal parameters of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The Roberge-Weiss transition region We have investigated by lattice simulations two temperature regimes, one at high temperature (𝑇 = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 - 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 𝑀𝑒𝑉) and one at low temperature (𝑇 = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 - 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='6 𝑀𝑒𝑉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In the high temperature regime we have studied the LYE singularities associated with the RW critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We have performed numerical simulations for 𝑁 𝑓 = 2 + 1 QCD using highly-improved staggered quarks (HISQ) on 363 × 6 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We have calculated the baryon number and electric charge 2 Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello cumulants, 𝜒1 𝐵,𝑄 ≡ 1 𝑉𝑇3 1 𝑍 𝜕𝑍 𝜕 ˆ𝜇𝐵,𝑄 , and their first derivative with respect to ˆ𝜇𝐵 = 𝜇𝐵 𝑇 for 𝑂(10) imaginary chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The calculations were made at four different temperatures 𝑇 = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0, 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0, 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 and 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The highest temperature is close to the RW temperature that we expect given our choice for the action discretization [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The numerical results for the baryon number density are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 0 1 2 3 4 5 6 Im[µB/T] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4 Im[χ1 B] T = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 0 1 2 3 4 5 6 Im[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 −Re[χ2 B] T = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV Figure 1: Imaginary part of 𝜒1 𝐵 (left) and real part of 𝜒2 𝐵 (right) as a function of ˆ𝜇𝐵 for different temperatures (high temperature regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Figure 2: Singularity structure of the rational approximants at 𝑇 =179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 𝑀𝑒𝑉 (top left), 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 𝑀𝑒𝑉 (top right), 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 𝑀𝑒𝑉 (bottom left) and 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 𝑀𝑒𝑉 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The left picture shows the imaginary part of the baryon number density and the right picture shows the real part of its derivative with respect to ˆ𝜇𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Different colors correspond to different temperatures and we can see a divergence emerging at the highest temperature for 𝜒2 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' These data have been approximated by multi-point Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 2 shows the singularity structure 3 5 Zeros of the numerator Zeros of the denominator 4 : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 1 4 2 2 Re[μB/T5 Zeros of the numerator Zeros of the denominator 4* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 1 0 4 2 2 Re[μB/T5 Zeros of the numerator Zeros of the denominator 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 1 0 4 2 2 Re[μB/T5 Zeros of the numerator Zeros of the denominator 4 : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 1 0 4 2 2 Re[μB/TDetermination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello of the Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The zeros of the numerator and denominator are displayed respectively in blue and red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' At 𝐼𝑚( ˆ𝜇𝐵) = 𝜋 we observe an alternation of zeros of the numerator and zeros of the denominator signaling the presence of branch cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The branch points pinch the real axis as the temperature is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 Scaling analysis In the top picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 3 we display for each temperature the closest singularity to the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Different colors denote different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Different symbols denote different approximants, respectively the approximants for 𝜒1 𝐵 and 𝜒1 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The singularities located by these two observables are in agreement within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 0 2 4 6 8 Re[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 Im[µB/T] 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 Mev 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 Mev Figure 3: LYE singularities for different temperatures (top) and scaling fit (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Next we checked whether these singularities could be identified as the LYE singularities associated with the RW endpoint and whether they follow the critical behaviour expected for the 3𝑑, 𝑍(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The location of the LYE singularities can be expressed in terms of the scaling field 𝑧 = 𝑡/ℎ1/𝛽𝛿 ≡ |𝑧𝑐|𝑒𝑖𝜋/2𝛽𝛿, where 𝑡 is the reduced temperature and ℎ is the symmetry breaking field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This relation makes a connection between the universal parameters 𝑧𝑐, 𝛽, 𝛿 and the non-universal parameters embedded in 𝑡, ℎ [11] [12] [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 I Nt = 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content="5 Re[μB/' 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 (T - Trw)/TrwDetermination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello For the Roberge-Weiss transition we may introduce the reduced temperature 𝑡 = 𝑡−1 0 𝑇𝑅𝑊 −𝑇 𝑇𝑅𝑊 and the symmetry breaking field ℎ = ℎ−1 0 ˆ𝜇𝐵−𝑖 𝜋 𝑖𝜋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' One obtains the following scaling law, ˆ𝜇𝐿𝑌 𝐸 = ±𝜋 � 𝑧0 |𝑧𝑐| �𝛽 𝛿 �𝑇𝑅𝑊 − 𝑇 𝑇𝑅𝑊 �𝛽 𝛿 ± 𝑖𝜋 , where 𝛽, 𝛿 are the universal critical exponents and 𝑧0 = ℎ1/𝛽 𝛿 0 /𝑡0, 𝑇𝑅𝑊 are non-universal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In the top picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 3 it can be recognized that the imaginary part of the singularities we have found by our Padé analysis is indeed trivially 𝑖𝜋 within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' For the real part we can fit the data using the ansatz ˆ𝜇𝑅 𝐿𝑌 𝐸 = 𝑎 �𝑇𝑅𝑊 − 𝑇 𝑇𝑅𝑊 �𝛽 𝛿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' By fitting with the 3𝑑, 𝑍(2) critical exponents (𝛽𝛿 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5635) we obtain a good fit (𝜒2/𝑑𝑜 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' From the fit we estimate 𝑇𝑅𝑊 (𝑁𝜏 = 6) = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='67(59) 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Using |𝑧𝑐| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='42(4) from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [12] we also find 𝑧0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 Continuum limit We previously conducted a similar analysis using the data obtained from 𝑁𝜏 = 4 lattice simulations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Using these results and the results from our current analysis we made a crude estimate for the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The result is displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We obtained 𝑇 𝑐𝑜𝑛𝑡 𝑅𝑊 = 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4) 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This is only a preliminary result and a proper continuum extrapolation would require data from 𝑁𝜏 = 8 lattice simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Still it is in nice agreement with a previous determination obtained by a different method using a different discretization in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [13], where the authors found 𝑇 𝑐𝑜𝑛𝑡 𝑅𝑊 = 208(5) 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Figure 4: Estimate for the Roberge-Weiss temperature in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The chiral transition and CEP regions In the low temperature regime we have been hunting for the LYE singularities associated with the chiral transition (or possibly the critical endpoint of QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=" We ran a series of simulations at 5 230 220 210 T 190 180 170 00'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content="02 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content="04 90'0 90'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content="07 80'0 1 / N?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello 𝑇 = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59, 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50, 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 and 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The numerical results for the baryon number density are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 0 1 2 3 4 5 6 Im[µB/T] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='4 Im[χ1 B] T = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10 MeV T = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 MeV T = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 0 1 2 3 4 5 6 Im[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 −Re[χ2 B] T = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10 MeV T = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 MeV T = 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV T = 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV T = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV Figure 5: Imaginary part of 𝜒1 𝐵 (left) and real part of 𝜒2 𝐵 (right) as a function of ˆ𝜇𝐵 for different temperatures (all temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The low temperature data have been approximated by rational functions, just as we did for the high temperature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' However in this case we observed a strong interval dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This is exemplified in the left picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 6, where we plot the singularities resulting from the Padé analysis of the 𝑇 = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 𝑀𝑒𝑉 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Different colors denote fits over different intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We can recognize two distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The results in the bottom cluster are from the fits over (small variations of) the [0, 𝑖𝜋] interval, the results in the top cluster are from the fits over (small variations of) the full [0, 2𝑖𝜋] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Being located at 𝐼𝑚( ˆ𝜇𝐵) ≈ 𝑖𝜋, the top cluster is likely related to the RW endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In the following we focus on the bottom cluster and we study how it moves as we change the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 0 2 4 6 8 Re[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 Im[µB/T] 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10MeV 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 MeV 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 Mev 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 Mev Figure 6: LYE singularities for different temperatures (right) and scatter plot of the singularities resulting from Padé fits over different intervals for 𝑇 = 145 𝑀𝑒𝑉 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 Scaling analysis In the right picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 6 we summarize the singularities that we have determined in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The uncertainty in the determinations of the singularities in the low temperature regime are given by the 1𝜎 confidence ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' These singularities apparently move towards the real axis as the temperature is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Their locations imply a radius of convergence R ≈ 3 - 4 for Taylor series expansions around ˆ𝜇𝐵 = 0 at 𝑇 = 136 - 166 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' These singularities might be identified as the LYE singularities associated with the chiral transition or possibly with the critical endpoint of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=" 6 A Im[μB/' 2 1 2 0 Re[μB/T]Determination of Lee-Yang edge singularities in QCD by rational approximations K." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello Figure 7: Comparison with the predictions for the chiral singularities from HotQCD data at temperatures 𝑇 = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 𝑀𝑒𝑉 (top left), 𝑇 = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 𝑀𝑒𝑉 (top right), 𝑇 = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 𝑀𝑒𝑉 (bottom left) and 𝑇 = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10 𝑀𝑒𝑉 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 0 1 2 3 4 5 6 Re[ B/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 Im[ B/T] ˆµLY [T = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10 MeV ] ˆµLY [T = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV ] ˆµLY [T = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV ] ˆµLY [T = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 MeV ] Figure 8: Overall picture of the comparison with the predictions for the chiral singularities from HotQCD data (left) and summary of the expected scaling for the LYE singularities associated with the RW endpoint, with the chiral transition and with the CEP (right, figure from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 pBW(N = 4) W(N = 6) 川2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 ImμB/ T=166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='6 T=157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 T=145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 T=136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 RW scaling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 chiral scaling CEP scaling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0 1 2 3 4 5 Re[μB/T]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 μLY [T = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='59 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 [L/a]] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 - 0 1 2 3 4 5 6 Re[μB/T]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 aLY [T = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 [L/a]] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0 1 2 3 4 5 6 Re[μB/T]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 aLY [T = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='00 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 [L/a]] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0 1 2 3 4 5 6 Re[μB/T]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 aLy [T = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='10 MeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 [L/a]] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0 1 2 3 4 5 6 Re[μB/T]Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello The critical behaviour close to the chiral transition region is expected to follow the behaviour of a theory belonging to the 3𝑑, 𝑂(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Following ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [15] the critical behaviour can be studied by introducing the scaling fields 𝑡 = 𝑡−1 0 �𝑇 − 𝑇𝑐 𝑇𝑐 + 𝑘 𝐵 2 � 𝜇𝐵 𝑇 �2� , ℎ = ℎ−1 0 𝑚𝑙 𝑚 𝑝ℎ𝑦𝑠 𝑠 , where 𝑇𝑐 is the critical temperature, 𝑘 𝐵 2 is the curvature coefficient of the critical line 𝑇𝑐( ˆ𝜇𝐵) and the symmetry breaking field ℎ is expressed in terms of the light-to-strange quark mass ratio 𝑚𝑙 𝑚𝑝ℎ𝑦𝑠 𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' One can derive the scaling law for the chiral singularities ˆ𝜇𝐿𝑌 𝐸 = ������ 1 𝑘 𝐵 2 𝑧𝑐 𝑧0 � 𝑚𝑙 𝑚 𝑝ℎ𝑦𝑠 𝑠 � 1 𝛽 𝛿 − 𝑇 − 𝑇𝑐 𝑇𝑐 ������ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 7 the singularities resulting from the Padé analysis are compared with the predictions obtained by setting 𝑚𝑙 𝑚𝑝ℎ𝑦𝑠 𝑠 = 1/27, by using the non-universal parameters 𝑇𝑐 = 147(6) 𝑀𝑒𝑉, 𝑘 𝐵 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='012(2), 𝑧0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='35(20) from the HotQCD data, and by using |𝑧𝑐| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='032 from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The results are in agreement within errors, but if we look at the overall picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 8 (left) we observe that the singularities follow a steeper curve than the one predicted using the best estimates for the non-universal parameters (the dashed line in the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' A possibility remains open that the singularities that we observed are the LYE singularities associated with the critical endpoint of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In this case the mapping from the non-universal parameters to the universal theory in unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' An approach one may try is to use the linear ansatz [16] 𝑡 = 𝛼𝑡 (𝑇 − 𝑇𝐶𝐸𝑃) + 𝛽𝑡 (𝜇𝐵 − 𝜇𝐶𝐸𝑃) , ℎ = 𝛼ℎ(𝑇 − 𝑇𝐶𝐸𝑃) + 𝛽ℎ(𝜇𝐵 − 𝜇𝐶𝐸𝑃) to derive the scaling law 𝜇𝐿𝑌 𝐸 ∼ 𝜇𝐶𝐸𝑃 − 𝑐1(𝑇 − 𝑇𝐶𝐸𝑃) + 𝑖𝑐2|𝑧𝑐|−𝛽 𝛿(𝑇 − 𝑇𝐶𝐸𝑃)𝛽 𝛿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Using some reasonable estimates for the non-universal parameters one obtains the qualitative prediction shown as a red band in the right picture of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The red band seems to better describe our data than the expected scaling for the chiral singularities (the green band in the same picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This conjecture will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' A different approach in which LYE singularities are located by studying the Fourier coefficients of the baryon number density is also being explored [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='2 Comparison with 𝑁𝜏 = 8 data Finally we ran a separate set of simulations at 𝑇 = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 𝑀𝑒𝑉 using 323 × 8 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The singularity resulting from the Padé analysis is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Also shown is the singularity obtained at a similar temperature (𝑇 = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 𝑀𝑒𝑉) for 𝑁𝜏 = 6 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' The results are in very good agreement despite the different volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This suggests that both the UV-cutoff effects and the finite-size effects are negligible within the accuracy of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 8 Determination of Lee-Yang edge singularities in QCD by rational approximations K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Zambello 0 2 4 6 8 Re[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='5 Im[µB/T] 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV [Nt = 6] 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='50 MeV [Nt = 8] Figure 9: Comparison between 𝑁𝜏 = 8 and 𝑁𝜏 = 6 lattices results for the LYE singularities at 𝑇 ≈ 157 𝑀𝑒𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Conclusions We have studied the complex singularities of QCD by a multi-point Padé analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In the high temperature regime we have identified the LYE singularities associated to the Roberge-Weiss endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' These singularities have the expected critical behaviour for a transition belonging to the 3𝑑, 𝑍(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' By combining these new results and the results from a previous analysis we have calculated an estimate for the Roberge-Weiss temperature in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' In the low temperature regime we have found singularities that may be identified as the LYE singularities associated with either the chiral transition or the critical endpoint of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Their location imply a radius of convergence R = 3 - 4 for 𝑇 = 136 - 166 𝑀𝑒𝑉, which constrains the validity of the Taylor series expansions at ˆ𝜇𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' For future work we plan to extend the study to 𝑁𝑡 = 8 lattices in order to make a proper continuum extrapolation for the RW temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' We also plan to generate data at lower temperatures in order to get a better understanding of the nature of the complex singularities that we have found at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' Acknowledgements This work was supported by European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 813942 (EuroPLEx) and by the I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' under the research project i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' QCDLAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' This research used computing resources made available (i) by CINECA on Marconi and Marconi 100 under both the I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content='N-CINECA agreement and the ISCRA B program, (ii) by the Gauss Centre for Supercomputing on the Juwels GPU nodes at the Jülich Supercomputing Centre and (iii) by the Bielefeld University on the Bielefeld GPU-Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE2T4oBgHgl3EQfhwfG/content/2301.03952v1.pdf'} +page_content=' N.' metadata={'source': 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b/zdE5T4oBgHgl3EQfOA4d/content/tmp_files/2301.05493v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a82e201334d161e80963ad14e33d2f22be566311 --- /dev/null +++ b/zdE5T4oBgHgl3EQfOA4d/content/tmp_files/2301.05493v1.pdf.txt @@ -0,0 +1,1027 @@ +Spin and current transport in the robust half-metallic magnet c-CoFeGe +Vikrant Chaudhary,1 Sapna Singh,1 Deepak Gujjar,1 Tashi Nautiyal,2 +Tulika Maitra,2 Jeroen van den Brink,3, 4 and Hem C. Kandpal1, ∗ +1Indian Institute of Technology Roorkee, Department of Chemistry, Roorkee 247667, Uttarakhand, India +2Indian Institute of Technology Roorkee, Department of Physics, Roorkee 247667, Uttarakhand, India +3Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany +4Institute for Theoretical Physics and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Technische Universit¨at Dresden, 01069 Dresden, Germany +(Dated: January 16, 2023) +Spintronics is an emerging form of electronics based on the electrons’ spin degree of freedom for +which materials with robust half-metallic ferromagnet (HMF) character are very attractive. Here +we determine the structural stability, electronic, magnetic, and mechanical properties of the half- +Heusler (hH) compound CoFeGe, in particular also in its cubic form. The first-principles calculations +suggest that the electronic structure is robust with 100 % spin polarization at the Fermi level under +hydrostatic pressure and uni-axial strain. Both the longitudinal and Hall current polarization are +calculated and the longitudinal current polarization (PL) is found to be > 99% and extremely robust +under uniform pressure and uni-axial strain. The anomalous Hall conductivity (AHC) and Spin Hall +conductivity (SHC) of hH cubic CoFeGe (c-CoFeGe) are found to be ∼ −100 S/cm and ∼ 39 ¯h/e +S/cm, respectively. Moreover, the Curie temperature of the alloy is calculated to be ∼524 K with +a 3 µB magnetic moment. Lastly, the calculated mechanical properties indicate that c-CoFeGe is +ductile and mechanically stable with a bulk modulus of ≈ 154 GPa. Overall, this analysis reveals +that cubic CoFeGe is a robust half-metallic ferromagnet and an interesting material for spintronic +applications. +I. +INTRODUCTION +The discovery of giant magnetoresistance (GMR) is of- +ten regarded as the beginning of spintronics. Spintronic +applications focus primarily on information storage, us- +ing GMR and tunnelling magnetoresistance (TMR) ef- +fects. A typical GMR device is a trilayer structure con- +sisting of ferromagnetic (FM) and nonmagnetic (NM) +layers. Up to 20 % GMR was achieved in spin valve sys- +tems, devised using NiFeTa and CoFe/Ru/CoFe layered +structure and these materials have been used in GMR +read head devices [1]. Materials that can be grown on the +commonly used substrates in the layered structure and +have robust electronic and magnetic structure, high Curie +temperature, and half-metallic behaviour are preferred +for such applications. A class of materials showing great +promise in this area is the Heusler alloys[2–6]. +These +alloys are predicted to become half-metal at room tem- +perature. In their electronic structure, one spin channel +indicates metallic conduction while the other spin chan- +nel is insulating or semiconducting. These half-metal fer- +romagnets can intrinsically provide single spin channel +electrons, with spin polarization reaching unity. +Heusler alloys also gained attention owing to the abil- +ity to control their unique electrical and magnetic prop- +erties by making changes to their crystalline structures. +The Fermi level and band gaps of Heusler alloys can be +tuned by substituting other transition elements in par- +ent alloys. Interestingly, it is not difficult to qualitatively +∗ Corresponding author: hem.kandpal[at]cy.iitr.ac.in +predict the behaviour of Heusler alloys from their compo- +sition. The magnetic moment of these compounds can be +predicted with the help of the Slater-Pauling rule where, +the magnetic moment of half and full Heusler (hH and +fH) compounds are given as M = Z−18 and M = Z−24, +respectively [7–11]. In addition, the HMF behaviour has +been studied in detail in NiMnSb [8, 12–15], FeMnSb +[8, 11, 16], PtMnSb [12, 13], NiTiSb [8, 17, 18], FeVSb +[8, 19] etc. The discovery of NiMnSb with ≈ 99 % spin +polarization [15] is a significant breakthrough establish- +ing that Heusler alloys are potential candidates for spin- +tronic applications. The HMF hH NiMnSb is known for +its role as a spin filter for spin-polarized current injection +in a semiconductor [20]. The HMF Heusler alloys and +their significance in spintronic devices have been widely +investigated[21–23]. +Among the known Heusler alloys, the Co-based com- +pounds in particular have received a lot of attention due +to their potential in spintronics in the last two decades. +A few of the Co-based fH alloys such as Co2CrAl [24], +Co2MnSi [25], and Co2FeSi [25–27] are well known and +have been thoroughly studied for their electronic and +magnetic structures. A device based on Co2FeGa0.5Ge0.5 +[28] was reported to have the largest GMR (∼ 82 %) +at room temperature. CoMnSb [8, 13, 14, 16], CoTiSb +[8, 17, 29], and CoZrSb [8] are some of the well-explored +half Heusler alloys on which many spintronic devices are +based. +Amongst these, CoMnSb is reported to have +∼ 99% spin polarization and a band gap ≈1 eV in one +of the minority channels. Additionally, a device made of +CoFeB was reported to meet the criteria of 1 Gbit magne- +toresistive random access memory (MRAM) with a 124 % +Tunnel magnetoresistance (TMR) at room temperature +arXiv:2301.05493v1 [cond-mat.mtrl-sci] 13 Jan 2023 + +2 +[30]. Much work has been done on CoFeB based devices +including some recent works [31, 32] that have helped in +understanding the dynamical spin injection [33]. +In line with the recent studies on Co-based Heusler +alloys for spintronics, our focus is the ternary ferromag- +netic material CoFeGe with hexagonal Ni2In-type crystal +structure and a magnetic moment of 2.2 µB per formula +unit [34], where Szytu�la et al. observed additional cu- +bic phase signatures along with hexagonal peaks in their +X-ray diffraction (XRD) pattern. This presence of cu- +bic phase poses the question on the favourable synthe- +sis conditions and the existence of c-CoFeGe. +On the +other hand, L. Feng et al. +performed first-principles +calculations for many compounds, including CoFeGe in +hH structure and investigated the HMF properties [35]. +However, their work does not address in detail the struc- +tural phase transition and spin and current transport in +CoFeGe, which is important for spintronics applications. +Thus, our interest in Co-based hH alloys as potential +spintronic material led us to perform a thorough study +of the electronic and anomalous transport and magnetic +aspects of CoFeGe. +In this work, utilizing the ab-initio approach, we sys- +tematically investigated the structural, electronic, and +magnetic properties of CoFeGe. These properties were +also studied under external pressure and uni-axial strain. +The uni-axial strain breaks the cubic symmetry and +CoFeGe adapts the tetragonal crystal structure. +The +static, dynamic, and mechanical stability of CoFeGe was +investigated and the thermodynamic stability was con- +firmed from data available at AFLOW [36] and OQMD +[37, 38]. The spin polarization from the density of states +(PDOS), longitudinal current polarization (PL), Hall cur- +rent polarization (PHall), anomalous Hall conductivity +(AHC), and spin Hall conductivity (SHC) have also been +evaluated. Further, the mechanical properties have also +been reported from first principles calculations. +II. +COMPUTATIONAL DETAILS +We have made use of the first-principles density func- +tional theory (DFT) under the projector augmented +wave (PAW) formalism based Vienna ab-initio Simula- +tion Package (VASP) [39–43] within the generalized gra- +dient approximation (GGA) developed by J. P. Perdew, +K. Burke, and M. Ernzerhof (PBE) [44]. +The crystal +structure was optimized on a 41×41×41 k-mesh using 550 +eV cutoff energy and with a charge density convergence +criteria of 1.0 × 10−7. The energy convergence criteria in +the self-consistent field (SCF) cycles was set to 1.0×10−8 +eV. The phonon band structures were obtained with the +help of Phonopy package [45] interfacing it with VASP. +We also used VASPKIT [46] for various pre-processing +and post-processing tasks for VASP calculations. +The Wannier90 package [47] was used with VASP to +generate maximally localized Wannier functions (ML- +WFs). These MLWFs were generated on a 9×9×9 DFT +10 +12 +14 +16 +Volume (Å +3/Atom) +0.00 +0.05 +0.10 +0.15 +0.20 +Rel. Energy (eV/Atom) +P63/mmc +F43m +FIG. 1. The calculated energy per atom on a relative scale +as a function of volume per atom for cubic and hexagonal +CoFeGe. +k-mesh and were converged up to an order of 10−11 and +10−10 during disentanglement and spread calculations, +respectively. +We obtained a total spread of 37.46 ˚A2. +These well-converged and localized MLWFs were further +used to calculate AHC and SHC on a 200 × 200 × 200 +Berry mesh with an adaptive refinement of 7 × 7 × 7 +whenever Berry curvature exceeded 100 ˚A2. +The electrical transport properties at different pres- +sure and strain conditions were calculated using Boltz- +mann theory and relaxation time approximation as im- +plemented in the Boltztrap code [48]. The electrical con- +ductivity has been calculated with respect to relaxation +time (τ) which was further used in the evaluation of the +PL. +III. +RESULTS AND DISCUSSION +A. +Structural Analysis +As discussed in section I, the ternary inter-metallic +CoFeGe exists in the hexagonal phase, space group +P63/mmc (No. +194), with a signature of minor cubic +phase in the XRD pattern. To understand the possibil- +ity of the synthesis of cubic phase, space group F¯43m +(No. 216), we investigated the static, dynamic, and ther- +modynamic stability of CoFeGe as discussed ahead. +The static stability requires structure optimization on +a sufficiently dense k-mesh, thus, we have used a fine +k-mesh as discussed in section II. The energy vs vol- +ume optimization was done by calculating energies in +the volume range from −10 % to 10 % in a step of 2 % +change. In order to obtain the optimized lattice param- +eters, energy vs volume data was fitted with the Birch- +Murnaghan (BM) equation of state [49–51] as shown in + +3 +TABLE I. The lattice parameter, magnetic moment, and band gap for CoFeGe in F¯43m and P63/mmc space group of the +CoFeGe. +Space Group +a +c/a +Magnetic Moment +Minority band gap +(˚A) +(µB/f.u.) +(E↓ +g, eV) +F43m +5.4967 +1 +3.0000 +0.4399 +P63/mmc +4.0821 +1.2352 +2.8862 +0.00 +Fig. 1. All the non-magnetic (NM), ferromagnetic (FM), +and anti-ferromagnetic (AFM) phases of both the struc- +tures were optimized and it was found that the FM phase +is favourable over the other two for both the structures. +Thus, only the FM phase of both the structures are shown +in Fig. +1. +Table I lists the lattice parameters, mag- +netic moment, and band gap of both the structures. The +ground state energy difference (per atom) between hexag- +onal and c-CoFeGe is 50 meV, which indicates that the +hexagonal phase is energetically favourable at ambient +conditions. However, this small energy difference leads +to chances of achieving cubic phase e. g. by inducing +internal pressure in the system by methods such as sub- +stituting larger size atoms or synthesis on a substrate +etc. These methods can lead to an increase in the vol- +ume of the system, making the cubic phase energetically +favourable over the hexagonal phase (Fig.1). +Γ +X U|K +Γ +L +W X +0 +2 +4 +6 +8 +10 +Frequency (THz) +(a) +Γ +M K +Γ +A +L H +0 +2 +4 +6 +8 +10 +Frequency (THz) +(b) +FIG. 2. The phonon band structure of (a) the F¯43m and (b) +P63/mmc phases of CoFeGe. +Next, we checked for the dynamical stability of the cu- +bic and the hexagonal analogues of CoFeGe with the help +of phonon dispersion curves. +The phonon calculations +are performed in accordance with the density functional +perturbation theory (DFPT) implemented in VASP and +Phonopy. For the cubic structure, a 2 × 2 × 2 supercell +was generated and the calculations were performed on a +5 × 5 × 5 k-mesh in the phonon Brillouin zone. Similarly, +for the h-CoFeGe, we used a k-mesh 4×4×5 for a 3×3×2 +supercell. The force constants were extracted from these +calculations using Phonopy and were used for obtaining +the phonon dispersion curves. Phonons are basically the +normal modes or quanta of vibrations in a crystal which +help in governing the stability of a system. For a sys- +tem to be dynamically stable, it should have only real +phonon frequencies and not imaginary ones. Fig. 2 (a) +and (b) show 9 (3 acoustical and 6 optical) and 18 (3 +acoustical and 15 optical) phonon branches of the cubic +and the hexagonal structures, respectively, of CoFeGe. +The absence of the imaginary modes asserts the dynamic +stability of both the structures. +In addition to static and dynamic stabilities, ther- +modynamic stability is also important for predicting +new materials. +Interestingly the P63/mmc structure +of CoFeGe is thermodynamically unstable according to +the data available at OQMD [37, 38] and AFLOW [36], +whereas it has been realized experimentally [34]. Simi- +larly, there are many thermodynamically unstable mate- +rials which have been successfully synthesized by exper- +imental groups. For example, the fH Ni2CuSn lies ≈ 32 +meV/Atom above the convex hull (available at OQMD) +but it has been experimentally realized and studied ex- +tensively [52–54]. +Another example is the W doped +Fe2VAl, which exhibited the best thermoelectric perfor- +mance [55]. The Fe2V0.8W0.2Al was found metastable +through the DFT calculations but was successfully syn- +thesized in the form of thin film Heusler alloy. Cu3N and +the bulk NdNiO2 are also the examples of metastable +and unstable materials, of which Cu3N [56] was synthe- +sized and a significant reduction was achieved in the in- +stability of NdNiO2 [57] was achieved. NdNiO2 has an +energy +176 meV/atom [57], while c-CoFeGe is +120 +meV/atom above the hull as per the data available on +OQMD. +These findings suggest that though the thermodynam- +ical stability gives an idea about the structural stability +of the system, it does not conclusively predict the pos- +sibility of synthesis of a material. The formation energy +of hH CoFeGe is negative (data available on OQMD) +with respect to the constituting elements. +On consid- +ering other binary and ternary decompositions, the for- +mation energy is -67 meV/Atom and the energy above + +4 +Γ +XU|K +Γ +L +W +X +-2.0 +-1.0 +0.0 +1.0 +2.0 +Energy (eV) +Γ +XU|K +Γ +L +W +X +-2.0 +-1.0 +0.0 +1.0 +2.0 +Energy (eV) +Γ +XU|K +Γ +L +W +X +-2.0 +-1.0 +0.0 +1.0 +2.0 +Energy (eV) +(a) +(b) +(c) +FIG. 3. The bandstructure of c-CoFeGe at (a) -10 GPa, (b) +ambient, and (c) 10 GPa pressure. +The solid and dotted +curves are spin-down and spin-up bands, respectively. +the Hull is 120 meV/Atom. The hexagonal phase has +-123 meV/Atom formation energy and 64 meV/Atom +energy above the convex Hull, as per the data avail- +able on OQMD. Many experimentally known thermo- +dynamically metastable compounds have less than 100 +meV/atom energy above the convex hull. There are also +some experimentally known compounds that have more +than 100 meV/atom energy above the hull [58]. Since +both h-CoFeGe and c-CoFeGe are unstable as per first- +principles data on OQMD, but the h-CoFeGe does exist +experimentally, hence even the cubic structure may be +realized experimentally. +Γ +X +P +N +Γ +M S +-2.0 +-1.0 +0.0 +1.0 +2.0 +Energy (eV) +Γ +X +P +N +Γ +M S +-2.0 +-1.0 +0.0 +1.0 +2.0 +Energy (eV) +(a) +(b) +FIG. 4. The bandstructure of the tetragonal CoFeGe, when +(a) c = 0.98a0 and (b) c = 1.02a0, where a0 is the lattice +constant of the relaxed c-CoFeGe. +B. +Electronic and Magnetic Properties +The electronic band structure of HMF c-CoFeGe was +calculated using GGA-PBE approximation as imple- +mented in the VASP code. As discussed in section III A, +the ground state is FM for both (F¯43m and P63/mmc) +structures. The cubic structure is half metallic, whereas +the hexagonal is metallic as both spin (up and down) +bands are present at the Fermi level. This confirms the +HMF nature of CoFeGe important for applications in +spintronics. Next, we also checked for the robustness of +this HMF character. Also, an impact of negative pressure +was observed in the alloy for a better understanding of +the durability under extreme conditions. A uniform pres- +sure was applied to the c-CoFeGe and the half-metallic +behaviour was found to remains intact within a pres- +sure range of -10 GPa to 10 GPa as shown in Fig. 3. +In order to further support the robustness of the HMF +behaviour, the electronic structure was studied under a +compressive as well as elongative strain of 2 % along the +c-direction keeping the volume fixed. As shown in Fig. +4, the CoFeGe remains an HMF with a ≈ 0.43 eV band +gap in the minority channel. To conclude, our electronic + +5 +-2 -1 +0 +1 +2 +E (eV) +0 +50 +100 +150 +200 +σe/τ (x10 +19 Sm +-1s +-1) +-2 -1 +0 +1 +2 +E (eV) +0 +50 +100 +150 +200 +σe/τ (x10 +19 Sm +-1s +-1) +-2 -1 +0 +1 +2 +E (eV) +0 +50 +100 +150 +200 +σe/τ (x10 +19 Sm +-1s +-1) +-2 +-1 +0 +1 +2 +E (eV) +0 +50 +100 +150 +200 +σe/τ (x10 +19 Sm +-1s +-1) +-2 +-1 +0 +1 +2 +E (eV) +0 +50 +100 +150 +200 +σe/τ (x10 +19 Sm +-1s +-1) +↑ +↓ +(a) +(c) +(d) +(e) +(b) +FIG. 5. +The spin decomposed electrical conductivity of c-CoFeGe at (a) 10 GPa, (b) ambient, and (c) -10 GPa pressure. Fig. +(d) and (e) show the electrical conductivity under 2 % compressive and equally elongative strain, respectively, along the c-axis. +The solid (dotted) curve represents the spin-up (spin-down) component of the electrical conductivity and the Fermi level has +been shifted to 0 eV. +structure investigations suggest that the HMF behaviour +of c-CoFeGe is extremely robust, and the spin polariza- +tion remains 100 % at the Fermi level, which was con- +firmed with the help of the equation, +PDOS = +�N↑ − N↓ +N↑ + N↓ +� +EF +, +(1) +where N↑ and N↓ are spin-up and spin-down electronic +states present at the Fermi level. +In most of the experiments, currents are studied and +the PDOS may not give us the important information re- +quired for the spintronic applications. Therefore, we also +calculated the longitudinal current polarization (PL) and +the Hall current polarization (PHall). For PL calculation, +we need electrical conductivity of spin up (σ↑) and spin +down (σ↓) states and corresponding relaxation times (τ↑ +and τ↓) as given by +PL = σ↑/τ↑ − σ↓/τ↓ +σ↑/τ↑ + σ↓/τ↓ +. +(2) +Since it is challenging to calculate τ↑ and τ↓, we used +an approximation that the relaxation time does not de- +pend on the k-points, energy, and the direction of spin. +This assumption allows us to use same relaxation time +(τ↑ = τ↓) for both spins. We used BoltzTrap2 [48] code +based on semi-classical Boltzmann transport theory for +the calculation of longitudinal electrical conductivity (σ↑ +and σ↓) divided by the corresponding relaxation times. +We employed a 41×41×41 k-mesh for obtaining ground +state charge density and energies and used those as input +in BoltzTrap2 code for further calculations. +Fig. 5 shows that the trends in spin decomposed elec- +trical conductivity at -10 GPa, 10 GPa, and after uni- +axial strain match well with the electrical conductivity +at the ambient pressure. The electrical conductivity be- +low the valance band maximum (VBM) is low in both +spin channels; as soon as VBM is crossed, σ↑ increases +rapidly while σ↓ picks up slowly and approaches a neg- +ligible value with respect to the σ↑. Our results show +that the PL is > 99 % at the Fermi level in all the cases. +Above the conduction band minimum, the σ↓ also in- +creases rapidly. +Another crucial property from the spintronic viewpoint +is the Hall current (spin) polarization given as [59, 60] +PHall = σHC↑ +xy +− σHC↓ +xy +σHC↑ +xy ++ σHC↓ +xy +. +(3) + +6 +6 +6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 +Energy (eV) +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +σxy (S/cm) +AHC +SHC * h/2πe +-1 +0 +1 +2 +3 +PHall +PHall +VBM +CBM +EF +FIG. 6. The AHC, SHC, and PHall for c-CoFeGe. The VBM, +CBM, and Fermi level (EF ) are indicated using vertical lines. +The σHC↑ +xy +(σHC↓ +xy +) is the spin up (down) Hall conductivity +and can be obtained with the help of AHC and SHC. The +simplified expression for PHall is +PHall = 2e +¯h +σSHC +xy +σAHC +xy +, +(4) +where σSHC +xy +and σAHC +xy +are spin Hall conductivity (SHC) +and anomalous Hall conductivity (AHC), respectively +[61, 62]. +The AHC has been studied extensively for the Heusler +class and the fH compounds show large AHC (e. +g. +Co2MnAl ∼ 1420 S/cm) [63], arising due to the pres- +ence of Weyl points, nodal lines, and band crossings near +the Fermi level [64, 65]. On the other hand, the hH al- +loys show a relatively lower AHC value. The well-known +hH alloys, GdPtBi [66], TbPtBi [67] and HoPtBi [68] are +reported to have an AHC close to 60 S/cm, 100 S/cm +and -75 S/cm, respectively. +The AHC in topological +semimetal TbPtBi was tuned up to ≈ 125 S/cm with +the help of magnetic field and temperature. +Our calculations show that in comparison to many +popular hH alloys discussed above, the hH CoFeGe has +a large AHC (≈ 100 S/cm) in the minority channel band +gap, as shown in Fig.6. The largest value of AHC and +SHC is found to be -164.67 S/cm at 0.37 eV and 143.23 +e/¯h S/cm at 0.31 eV below the Fermi level, respectively. +The AHC and SHC change rapidly below the VBM and +become almost constant within the energy gap, as shown +in Fig. 6, resulting in an absolute PHall value between +0.79 and 1.00. The maximum of AHC and SHC occurs +a little below the VBM possibly resulting from the many +band crossings seen at/around -0.37 eV and -0.31 eV as +shown in Fig. 3(b). These crossings split up after switch- +ing on the spin-orbit coupling, leading to large Berry cur- +vature and a large AHC and SHC. The current (spin), +static DOS polarization and AHC calculations suggest +that the c-CoFeGe may be a promising candidate for +spintronic devices. The magnetic structure is also of im- +TABLE II. Mechanical properties of hH CoTiSb and hH +CoFeGe. The modulus, hardness, and Cauchy pressure are +in GPa, whereas the Poisson’s ratio and Pugh’s ration are di- +mensionless. The experimentally measured quantities are in +paranthesis. +Mechanical Properties +hH CoTiSb [29, 73] +hH CoFeGe +Bulk Modulus (B) +142 (166) +154 +Young’s Modulus (Y) +224 +148 +Shear Modulus (G) +91 +55 +Poisson’s Ration (v) +0.24 +0.34 +Vickers Hardness (V) +4.89 +Pugh’s Ratio (B/G) +1.56 +2.80 +Cauchy Pressure (PC) +62.7 +portance in the study of spin transport and is discussed +in the following section. +The Curie temperature (TC) of c-CoFeGe was cal- +culated within the mean field approximation (MFA) +[69] using the spin-polarized relativistic Korringa-Kohn- +Rostoker (SPR-KKR) code [70, 71]. The predicted value +of TC as 528 K for c-CoFeGe is quite large when com- +pared with the measured value of 370 K [34] for h- +CoFeGe. We found the Curie temperature of c-CoFeGe +as ≈ 528 K with a magnetic moment of 3 µB per formula +unit. The Fe and Co atoms contribute approximately 2.5 +µB and 0.5 µB to the total magnetic moment, while the +contribution of Ge is close to zero. The value of TC as +528 K for c-CoFeGe is comparable with that for other +well-known hH alloys, i.e. PtMnSb (582K)[13], CoMnSb +(490K)[13, 14], NiTiSb (330 K)[18] etc. The fH alloys +show relatively higher TC, for example, the TC and mag- +netic moment of Co2FeSi have been experimentally re- +ported to be 1100 K and 6 µB [? ], supposed to be the +highest known values for a Heusler HMF alloy. In gen- +eral, the Curie temperature of the Heusler alloys mostly +falls between 200 K and 1200 K. Hence c-CoFeGe is pre- +dicted to have a high TC, making it a promising candidate +in spin transport applications. +C. +Mechanical Properties +Having investigated c-CoFeGe for applications in spin +transport, it is worthwhile to explore its mechanical prop- +erties. Heusler alloys are generally ductile and their me- +chanical properties are compiled in a review article [72]. +For a better insight into this study, we used hH al- +loy TiCoSb as a reference system in order to adjudge +our results as shown in Table II. The bulk modulus of +TiCoSb has been experimentally measured to be ≈ 166 +GPa with no structural phase transition up to 115 GPa +external pressure [73]. The calculated bulk modulus (154 +GPa) of c-CoFeGe is close to the bulk modulus of hH +TiCoSb, suggesting that a large external pressure would +be needed for the structural phase transition. The calcu- +lated Young’s modulus (≈ 148 GPa) of c-CoFeGe is also + +7 +fairly high (from Voigt [74] methods). +The calculated value of Pugh’s ratio, one of the impor- +tant properties to understand the mechanical nature of +the materials, is 2.80 for c-CoFeGe. This value is > 1.75, +indicating that the compound is ductile. The stiffness +tensor of any cubic structure has primarily three me- +chanical constants; C11, C12, and C44. These constants +are used to calculate the mechanical properties and un- +derstand the mechanical stability [75]. +The c-CoFeGe +meets the elastic stability criteria (C11 − C12 > 0, +C11 + 2C12 > 0, and C44 > 0), making it mechanically +stable. +IV. +CONCLUSIONS +We have thoroughly studied CoFeGe from spintronic +viewpoint. The hexagonal phase of CoFeGe is already +known but the signature of a cubic phase in the XRD +pattern of the compound prompted us to check the pos- +sibility of structural phase transition from hexagonal +(P63/mmc) to cubic (F¯43m) analogue. The small energy +difference of the order of ≈ 51 eV/Atom between the +hexagonal and cubic phases can be overcome to realize +the cubic phase. In its ground state, the c-CoFeGe is fer- +romagnetic with half-metallic behaviour. The HMF char- +acter is preserved with 100 % spin polarization within +extreme condition of the pressure range of -10 GPa to 10 +GPa, and under 2 % compressive and elongative strain. +Further, the longitudinal and Hall current spin polariza- +tion show promising values with the PL staying > 99 % +spin-polarized and PHall value changes from 80 % to +100 % between the VBM and CBM of minority spin +channel. The calculated TC and magnetic moment of c- +CoFeGe are ≈ 524 K and 3 µB, respectively. Lastly, we +investigated the mechanical properties of the c-CoFeGe +and observed that the structure is ductile and mechani- +cally stable. Thus c-CoFeGe is predicted to be a robust +HMF with large PL and moderately high TC values. 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B 90, 224104 +(2014). + diff --git a/zdE5T4oBgHgl3EQfOA4d/content/tmp_files/load_file.txt b/zdE5T4oBgHgl3EQfOA4d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1adfae73c93f1a34900fa9aad66cbb8abcf27b93 --- /dev/null +++ b/zdE5T4oBgHgl3EQfOA4d/content/tmp_files/load_file.txt @@ -0,0 +1,1041 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf,len=1040 +page_content='Spin and current transport in the robust half-metallic magnet c-CoFeGe Vikrant Chaudhary,1 Sapna Singh,1 Deepak Gujjar,1 Tashi Nautiyal,2 Tulika Maitra,2 Jeroen van den Brink,3, 4 and Hem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Kandpal1, ∗ 1Indian Institute of Technology Roorkee, Department of Chemistry, Roorkee 247667, Uttarakhand, India 2Indian Institute of Technology Roorkee, Department of Physics, Roorkee 247667, Uttarakhand, India 3Institute for Theoretical Solid State Physics, IFW Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany 4Institute for Theoretical Physics and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='qmat, Technische Universit¨at Dresden, 01069 Dresden, Germany (Dated: January 16, 2023) Spintronics is an emerging form of electronics based on the electrons’ spin degree of freedom for which materials with robust half-metallic ferromagnet (HMF) character are very attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Here we determine the structural stability, electronic, magnetic, and mechanical properties of the half- Heusler (hH) compound CoFeGe, in particular also in its cubic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The first-principles calculations suggest that the electronic structure is robust with 100 % spin polarization at the Fermi level under hydrostatic pressure and uni-axial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Both the longitudinal and Hall current polarization are calculated and the longitudinal current polarization (PL) is found to be > 99% and extremely robust under uniform pressure and uni-axial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The anomalous Hall conductivity (AHC) and Spin Hall conductivity (SHC) of hH cubic CoFeGe (c-CoFeGe) are found to be ∼ −100 S/cm and ∼ 39 ¯h/e S/cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Moreover, the Curie temperature of the alloy is calculated to be ∼524 K with a 3 µB magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Lastly, the calculated mechanical properties indicate that c-CoFeGe is ductile and mechanically stable with a bulk modulus of ≈ 154 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Overall, this analysis reveals that cubic CoFeGe is a robust half-metallic ferromagnet and an interesting material for spintronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' INTRODUCTION The discovery of giant magnetoresistance (GMR) is of- ten regarded as the beginning of spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Spintronic applications focus primarily on information storage, us- ing GMR and tunnelling magnetoresistance (TMR) ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' A typical GMR device is a trilayer structure con- sisting of ferromagnetic (FM) and nonmagnetic (NM) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Up to 20 % GMR was achieved in spin valve sys- tems, devised using NiFeTa and CoFe/Ru/CoFe layered structure and these materials have been used in GMR read head devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Materials that can be grown on the commonly used substrates in the layered structure and have robust electronic and magnetic structure, high Curie temperature, and half-metallic behaviour are preferred for such applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' A class of materials showing great promise in this area is the Heusler alloys[2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These alloys are predicted to become half-metal at room tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In their electronic structure, one spin channel indicates metallic conduction while the other spin chan- nel is insulating or semiconducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These half-metal fer- romagnets can intrinsically provide single spin channel electrons, with spin polarization reaching unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Heusler alloys also gained attention owing to the abil- ity to control their unique electrical and magnetic prop- erties by making changes to their crystalline structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The Fermi level and band gaps of Heusler alloys can be tuned by substituting other transition elements in par- ent alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Interestingly, it is not difficult to qualitatively ∗ Corresponding author: hem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='kandpal[at]cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='in predict the behaviour of Heusler alloys from their compo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The magnetic moment of these compounds can be predicted with the help of the Slater-Pauling rule where, the magnetic moment of half and full Heusler (hH and fH) compounds are given as M = Z−18 and M = Z−24, respectively [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In addition, the HMF behaviour has been studied in detail in NiMnSb [8, 12–15], FeMnSb [8, 11, 16], PtMnSb [12, 13], NiTiSb [8, 17, 18], FeVSb [8, 19] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The discovery of NiMnSb with ≈ 99 % spin polarization [15] is a significant breakthrough establish- ing that Heusler alloys are potential candidates for spin- tronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The HMF hH NiMnSb is known for its role as a spin filter for spin-polarized current injection in a semiconductor [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The HMF Heusler alloys and their significance in spintronic devices have been widely investigated[21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Among the known Heusler alloys, the Co-based com- pounds in particular have received a lot of attention due to their potential in spintronics in the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' A few of the Co-based fH alloys such as Co2CrAl [24], Co2MnSi [25], and Co2FeSi [25–27] are well known and have been thoroughly studied for their electronic and magnetic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' A device based on Co2FeGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='5Ge0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='5 [28] was reported to have the largest GMR (∼ 82 %) at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' CoMnSb [8, 13, 14, 16], CoTiSb [8, 17, 29], and CoZrSb [8] are some of the well-explored half Heusler alloys on which many spintronic devices are based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Amongst these, CoMnSb is reported to have ∼ 99% spin polarization and a band gap ≈1 eV in one of the minority channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Additionally, a device made of CoFeB was reported to meet the criteria of 1 Gbit magne- toresistive random access memory (MRAM) with a 124 % Tunnel magnetoresistance (TMR) at room temperature arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='05493v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='mtrl-sci] 13 Jan 2023 2 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Much work has been done on CoFeB based devices including some recent works [31, 32] that have helped in understanding the dynamical spin injection [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In line with the recent studies on Co-based Heusler alloys for spintronics, our focus is the ternary ferromag- netic material CoFeGe with hexagonal Ni2In-type crystal structure and a magnetic moment of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='2 µB per formula unit [34], where Szytu�la et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' observed additional cu- bic phase signatures along with hexagonal peaks in their X-ray diffraction (XRD) pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' This presence of cu- bic phase poses the question on the favourable synthe- sis conditions and the existence of c-CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' On the other hand, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' performed first-principles calculations for many compounds, including CoFeGe in hH structure and investigated the HMF properties [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' However, their work does not address in detail the struc- tural phase transition and spin and current transport in CoFeGe, which is important for spintronics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Thus, our interest in Co-based hH alloys as potential spintronic material led us to perform a thorough study of the electronic and anomalous transport and magnetic aspects of CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In this work, utilizing the ab-initio approach, we sys- tematically investigated the structural, electronic, and magnetic properties of CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These properties were also studied under external pressure and uni-axial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The uni-axial strain breaks the cubic symmetry and CoFeGe adapts the tetragonal crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The static, dynamic, and mechanical stability of CoFeGe was investigated and the thermodynamic stability was con- firmed from data available at AFLOW [36] and OQMD [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The spin polarization from the density of states (PDOS), longitudinal current polarization (PL), Hall cur- rent polarization (PHall), anomalous Hall conductivity (AHC), and spin Hall conductivity (SHC) have also been evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Further, the mechanical properties have also been reported from first principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' COMPUTATIONAL DETAILS We have made use of the first-principles density func- tional theory (DFT) under the projector augmented wave (PAW) formalism based Vienna ab-initio Simula- tion Package (VASP) [39–43] within the generalized gra- dient approximation (GGA) developed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Perdew, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Burke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Ernzerhof (PBE) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The crystal structure was optimized on a 41×41×41 k-mesh using 550 eV cutoff energy and with a charge density convergence criteria of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 × 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The energy convergence criteria in the self-consistent field (SCF) cycles was set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0×10−8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The phonon band structures were obtained with the help of Phonopy package [45] interfacing it with VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We also used VASPKIT [46] for various pre-processing and post-processing tasks for VASP calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The Wannier90 package [47] was used with VASP to generate maximally localized Wannier functions (ML- WFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These MLWFs were generated on a 9×9×9 DFT 10 12 14 16 Volume (Å 3/Atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='20 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Energy (eV/Atom) P63/mmc F43m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The calculated energy per atom on a relative scale as a function of volume per atom for cubic and hexagonal CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' k-mesh and were converged up to an order of 10−11 and 10−10 during disentanglement and spread calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We obtained a total spread of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='46 ˚A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These well-converged and localized MLWFs were further used to calculate AHC and SHC on a 200 × 200 × 200 Berry mesh with an adaptive refinement of 7 × 7 × 7 whenever Berry curvature exceeded 100 ˚A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The electrical transport properties at different pres- sure and strain conditions were calculated using Boltz- mann theory and relaxation time approximation as im- plemented in the Boltztrap code [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The electrical con- ductivity has been calculated with respect to relaxation time (τ) which was further used in the evaluation of the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Structural Analysis As discussed in section I, the ternary inter-metallic CoFeGe exists in the hexagonal phase, space group P63/mmc (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 194), with a signature of minor cubic phase in the XRD pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' To understand the possibil- ity of the synthesis of cubic phase, space group F¯43m (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 216), we investigated the static, dynamic, and ther- modynamic stability of CoFeGe as discussed ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The static stability requires structure optimization on a sufficiently dense k-mesh, thus, we have used a fine k-mesh as discussed in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The energy vs vol- ume optimization was done by calculating energies in the volume range from −10 % to 10 % in a step of 2 % change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In order to obtain the optimized lattice param- eters, energy vs volume data was fitted with the Birch- Murnaghan (BM) equation of state [49–51] as shown in 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The lattice parameter, magnetic moment, and band gap for CoFeGe in F¯43m and P63/mmc space group of the CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Space Group a c/a Magnetic Moment Minority band gap (˚A) (µB/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=') (E↓ g, eV) F43m 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='4967 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='4399 P63/mmc 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0821 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='2352 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='8862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='00 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' All the non-magnetic (NM), ferromagnetic (FM), and anti-ferromagnetic (AFM) phases of both the struc- tures were optimized and it was found that the FM phase is favourable over the other two for both the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Thus, only the FM phase of both the structures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Table I lists the lattice parameters, mag- netic moment, and band gap of both the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The ground state energy difference (per atom) between hexag- onal and c-CoFeGe is 50 meV, which indicates that the hexagonal phase is energetically favourable at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' However, this small energy difference leads to chances of achieving cubic phase e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' by inducing internal pressure in the system by methods such as sub- stituting larger size atoms or synthesis on a substrate etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These methods can lead to an increase in the vol- ume of the system, making the cubic phase energetically favourable over the hexagonal phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Γ X U|K Γ L W X 0 2 4 6 8 10 Frequency (THz) (a) Γ M K Γ A L H 0 2 4 6 8 10 Frequency (THz) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The phonon band structure of (a) the F¯43m and (b) P63/mmc phases of CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Next, we checked for the dynamical stability of the cu- bic and the hexagonal analogues of CoFeGe with the help of phonon dispersion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The phonon calculations are performed in accordance with the density functional perturbation theory (DFPT) implemented in VASP and Phonopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' For the cubic structure, a 2 × 2 × 2 supercell was generated and the calculations were performed on a 5 × 5 × 5 k-mesh in the phonon Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Similarly, for the h-CoFeGe, we used a k-mesh 4×4×5 for a 3×3×2 supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The force constants were extracted from these calculations using Phonopy and were used for obtaining the phonon dispersion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Phonons are basically the normal modes or quanta of vibrations in a crystal which help in governing the stability of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' For a sys- tem to be dynamically stable, it should have only real phonon frequencies and not imaginary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 2 (a) and (b) show 9 (3 acoustical and 6 optical) and 18 (3 acoustical and 15 optical) phonon branches of the cubic and the hexagonal structures, respectively, of CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The absence of the imaginary modes asserts the dynamic stability of both the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In addition to static and dynamic stabilities, ther- modynamic stability is also important for predicting new materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Interestingly the P63/mmc structure of CoFeGe is thermodynamically unstable according to the data available at OQMD [37, 38] and AFLOW [36], whereas it has been realized experimentally [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Simi- larly, there are many thermodynamically unstable mate- rials which have been successfully synthesized by exper- imental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' For example, the fH Ni2CuSn lies ≈ 32 meV/Atom above the convex hull (available at OQMD) but it has been experimentally realized and studied ex- tensively [52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Another example is the W doped Fe2VAl, which exhibited the best thermoelectric perfor- mance [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The Fe2V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='8W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='2Al was found metastable through the DFT calculations but was successfully syn- thesized in the form of thin film Heusler alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Cu3N and the bulk NdNiO2 are also the examples of metastable and unstable materials, of which Cu3N [56] was synthe- sized and a significant reduction was achieved in the in- stability of NdNiO2 [57] was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' NdNiO2 has an energy +176 meV/atom [57], while c-CoFeGe is +120 meV/atom above the hull as per the data available on OQMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These findings suggest that though the thermodynam- ical stability gives an idea about the structural stability of the system, it does not conclusively predict the pos- sibility of synthesis of a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The formation energy of hH CoFeGe is negative (data available on OQMD) with respect to the constituting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' On consid- ering other binary and ternary decompositions, the for- mation energy is -67 meV/Atom and the energy above 4 Γ XU|K Γ L W X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 Energy (eV) Γ XU|K Γ L W X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 Energy (eV) Γ XU|K Γ L W X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 Energy (eV) (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The bandstructure of c-CoFeGe at (a) -10 GPa, (b) ambient, and (c) 10 GPa pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The solid and dotted curves are spin-down and spin-up bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' the Hull is 120 meV/Atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The hexagonal phase has 123 meV/Atom formation energy and 64 meV/Atom energy above the convex Hull, as per the data avail- able on OQMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Many experimentally known thermo- dynamically metastable compounds have less than 100 meV/atom energy above the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' There are also some experimentally known compounds that have more than 100 meV/atom energy above the hull [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Since both h-CoFeGe and c-CoFeGe are unstable as per first- principles data on OQMD, but the h-CoFeGe does exist experimentally, hence even the cubic structure may be realized experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Γ X P N Γ M S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 Energy (eV) Γ X P N Γ M S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='0 Energy (eV) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The bandstructure of the tetragonal CoFeGe, when (a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='98a0 and (b) c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='02a0, where a0 is the lattice constant of the relaxed c-CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Electronic and Magnetic Properties The electronic band structure of HMF c-CoFeGe was calculated using GGA-PBE approximation as imple- mented in the VASP code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' As discussed in section III A, the ground state is FM for both (F¯43m and P63/mmc) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The cubic structure is half metallic, whereas the hexagonal is metallic as both spin (up and down) bands are present at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' This confirms the HMF nature of CoFeGe important for applications in spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Next, we also checked for the robustness of this HMF character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Also, an impact of negative pressure was observed in the alloy for a better understanding of the durability under extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' A uniform pres- sure was applied to the c-CoFeGe and the half-metallic behaviour was found to remains intact within a pres- sure range of -10 GPa to 10 GPa as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In order to further support the robustness of the HMF behaviour, the electronic structure was studied under a compressive as well as elongative strain of 2 % along the c-direction keeping the volume fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 4, the CoFeGe remains an HMF with a ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='43 eV band gap in the minority channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' To conclude, our electronic 5 2 -1 0 1 2 E (eV) 0 50 100 150 200 σe/τ (x10 19 Sm 1s 1) 2 -1 0 1 2 E (eV) 0 50 100 150 200 σe/τ (x10 19 Sm 1s 1) 2 -1 0 1 2 E (eV) 0 50 100 150 200 σe/τ (x10 19 Sm 1s 1) 2 1 0 1 2 E (eV) 0 50 100 150 200 σe/τ (x10 19 Sm 1s 1) 2 1 0 1 2 E (eV) 0 50 100 150 200 σe/τ (x10 19 Sm 1s 1) ↑ ↓ (a) (c) (d) (e) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The spin decomposed electrical conductivity of c-CoFeGe at (a) 10 GPa, (b) ambient, and (c) -10 GPa pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' (d) and (e) show the electrical conductivity under 2 % compressive and equally elongative strain, respectively, along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The solid (dotted) curve represents the spin-up (spin-down) component of the electrical conductivity and the Fermi level has been shifted to 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' structure investigations suggest that the HMF behaviour of c-CoFeGe is extremely robust, and the spin polariza- tion remains 100 % at the Fermi level, which was con- firmed with the help of the equation, PDOS = �N↑ − N↓ N↑ + N↓ � EF , (1) where N↑ and N↓ are spin-up and spin-down electronic states present at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In most of the experiments, currents are studied and the PDOS may not give us the important information re- quired for the spintronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Therefore, we also calculated the longitudinal current polarization (PL) and the Hall current polarization (PHall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' For PL calculation, we need electrical conductivity of spin up (σ↑) and spin down (σ↓) states and corresponding relaxation times (τ↑ and τ↓) as given by PL = σ↑/τ↑ − σ↓/τ↓ σ↑/τ↑ + σ↓/τ↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' (2) Since it is challenging to calculate τ↑ and τ↓, we used an approximation that the relaxation time does not de- pend on the k-points, energy, and the direction of spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' This assumption allows us to use same relaxation time (τ↑ = τ↓) for both spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We used BoltzTrap2 [48] code based on semi-classical Boltzmann transport theory for the calculation of longitudinal electrical conductivity (σ↑ and σ↓) divided by the corresponding relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We employed a 41×41×41 k-mesh for obtaining ground state charge density and energies and used those as input in BoltzTrap2 code for further calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 5 shows that the trends in spin decomposed elec- trical conductivity at -10 GPa, 10 GPa, and after uni- axial strain match well with the electrical conductivity at the ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The electrical conductivity be- low the valance band maximum (VBM) is low in both spin channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' as soon as VBM is crossed, σ↑ increases rapidly while σ↓ picks up slowly and approaches a neg- ligible value with respect to the σ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Our results show that the PL is > 99 % at the Fermi level in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Above the conduction band minimum, the σ↓ also in- creases rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Another crucial property from the spintronic viewpoint is the Hall current (spin) polarization given as [59, 60] PHall = σHC↑ xy − σHC↓ xy σHC↑ xy + σHC↓ xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' (3) 6 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='8 Energy (eV) 200 150 100 50 0 50 100 150 200 σxy (S/cm) AHC SHC * h/2πe 1 0 1 2 3 PHall PHall VBM CBM EF FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The AHC, SHC, and PHall for c-CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The VBM, CBM, and Fermi level (EF ) are indicated using vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The σHC↑ xy (σHC↓ xy ) is the spin up (down) Hall conductivity and can be obtained with the help of AHC and SHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The simplified expression for PHall is PHall = 2e ¯h σSHC xy σAHC xy , (4) where σSHC xy and σAHC xy are spin Hall conductivity (SHC) and anomalous Hall conductivity (AHC), respectively [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The AHC has been studied extensively for the Heusler class and the fH compounds show large AHC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Co2MnAl ∼ 1420 S/cm) [63], arising due to the pres- ence of Weyl points, nodal lines, and band crossings near the Fermi level [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' On the other hand, the hH al- loys show a relatively lower AHC value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The well-known hH alloys, GdPtBi [66], TbPtBi [67] and HoPtBi [68] are reported to have an AHC close to 60 S/cm, 100 S/cm and -75 S/cm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The AHC in topological semimetal TbPtBi was tuned up to ≈ 125 S/cm with the help of magnetic field and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Our calculations show that in comparison to many popular hH alloys discussed above, the hH CoFeGe has a large AHC (≈ 100 S/cm) in the minority channel band gap, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The largest value of AHC and SHC is found to be -164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='67 S/cm at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='37 eV and 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='23 e/¯h S/cm at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='31 eV below the Fermi level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The AHC and SHC change rapidly below the VBM and become almost constant within the energy gap, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 6, resulting in an absolute PHall value between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='79 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The maximum of AHC and SHC occurs a little below the VBM possibly resulting from the many band crossings seen at/around -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='37 eV and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='31 eV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These crossings split up after switch- ing on the spin-orbit coupling, leading to large Berry cur- vature and a large AHC and SHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The current (spin), static DOS polarization and AHC calculations suggest that the c-CoFeGe may be a promising candidate for spintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The magnetic structure is also of im- TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Mechanical properties of hH CoTiSb and hH CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The modulus, hardness, and Cauchy pressure are in GPa, whereas the Poisson’s ratio and Pugh’s ration are di- mensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The experimentally measured quantities are in paranthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Mechanical Properties hH CoTiSb [29, 73] hH CoFeGe Bulk Modulus (B) 142 (166) 154 Young’s Modulus (Y) 224 148 Shear Modulus (G) 91 55 Poisson’s Ration (v) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='34 Vickers Hardness (V) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='89 Pugh’s Ratio (B/G) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='80 Cauchy Pressure (PC) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='7 portance in the study of spin transport and is discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The Curie temperature (TC) of c-CoFeGe was cal- culated within the mean field approximation (MFA) [69] using the spin-polarized relativistic Korringa-Kohn- Rostoker (SPR-KKR) code [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The predicted value of TC as 528 K for c-CoFeGe is quite large when com- pared with the measured value of 370 K [34] for h- CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We found the Curie temperature of c-CoFeGe as ≈ 528 K with a magnetic moment of 3 µB per formula unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The Fe and Co atoms contribute approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='5 µB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='5 µB to the total magnetic moment, while the contribution of Ge is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The value of TC as 528 K for c-CoFeGe is comparable with that for other well-known hH alloys, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' PtMnSb (582K)[13], CoMnSb (490K)[13, 14], NiTiSb (330 K)[18] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The fH alloys show relatively higher TC, for example, the TC and mag- netic moment of Co2FeSi have been experimentally re- ported to be 1100 K and 6 µB [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' ], supposed to be the highest known values for a Heusler HMF alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In gen- eral, the Curie temperature of the Heusler alloys mostly falls between 200 K and 1200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Hence c-CoFeGe is pre- dicted to have a high TC, making it a promising candidate in spin transport applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Mechanical Properties Having investigated c-CoFeGe for applications in spin transport, it is worthwhile to explore its mechanical prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Heusler alloys are generally ductile and their me- chanical properties are compiled in a review article [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' For a better insight into this study, we used hH al- loy TiCoSb as a reference system in order to adjudge our results as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The bulk modulus of TiCoSb has been experimentally measured to be ≈ 166 GPa with no structural phase transition up to 115 GPa external pressure [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The calculated bulk modulus (154 GPa) of c-CoFeGe is close to the bulk modulus of hH TiCoSb, suggesting that a large external pressure would be needed for the structural phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The calcu- lated Young’s modulus (≈ 148 GPa) of c-CoFeGe is also 7 fairly high (from Voigt [74] methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The calculated value of Pugh’s ratio, one of the impor- tant properties to understand the mechanical nature of the materials, is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='80 for c-CoFeGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' This value is > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content='75, indicating that the compound is ductile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The stiffness tensor of any cubic structure has primarily three me- chanical constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' C11, C12, and C44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' These constants are used to calculate the mechanical properties and un- derstand the mechanical stability [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The c-CoFeGe meets the elastic stability criteria (C11 − C12 > 0, C11 + 2C12 > 0, and C44 > 0), making it mechanically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' CONCLUSIONS We have thoroughly studied CoFeGe from spintronic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The hexagonal phase of CoFeGe is already known but the signature of a cubic phase in the XRD pattern of the compound prompted us to check the pos- sibility of structural phase transition from hexagonal (P63/mmc) to cubic (F¯43m) analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The small energy difference of the order of ≈ 51 eV/Atom between the hexagonal and cubic phases can be overcome to realize the cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In its ground state, the c-CoFeGe is fer- romagnetic with half-metallic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The HMF char- acter is preserved with 100 % spin polarization within extreme condition of the pressure range of -10 GPa to 10 GPa, and under 2 % compressive and elongative strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Further, the longitudinal and Hall current spin polariza- tion show promising values with the PL staying > 99 % spin-polarized and PHall value changes from 80 % to 100 % between the VBM and CBM of minority spin channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' The calculated TC and magnetic moment of c- CoFeGe are ≈ 524 K and 3 µB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Lastly, we investigated the mechanical properties of the c-CoFeGe and observed that the structure is ductile and mechani- cally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Thus c-CoFeGe is predicted to be a robust HMF with large PL and moderately high TC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' In nutshell, c-CoFeGe, when realized in hH structure, will be an interesting candidate for spin transport applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work used the Supercomputing facility of IIT Roorkee established under National Supercomputing Mission (NSM), Government of India and supported by Centre for Development of Advanced Computing (CDAC), Pune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' We have also used other computational facilities provided by Institute Computer Center (ICC), IIT Roorkee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' VC wish to acknowledge the financial sup- port received from Ministry of Education, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Araki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Sano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Tsuchiya, O.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Fecher, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Felser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' & Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' B 32, 020802 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K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Kobayashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' B 86, 045116 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Ikeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Miura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Allayarov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Koplak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Morgunov, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Mangin, ACS Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 6, 4315 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Yamamoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Ichinose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Uzuhashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Nozaki, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' D: Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 55, 275003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Obinata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Iimori, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Ohnishi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Kimura, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' 12, 3467 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [34] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} +page_content=' [35] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE5T4oBgHgl3EQfOA4d/content/2301.05493v1.pdf'} 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