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+ARXIV VERSION, 2022
+1
+Automatically Prepare Training Data for YOLO
+Using Robotic In-Hand Observation and Synthesis
+Hao Chen1, Weiwei Wan1∗, Masaki Matsushita2, Takeyuki Kotaka2 and Kensuke Harada13
+Abstract—Deep learning methods have recently exhibited im-
+pressive performance in object detection. However, such methods
+needed much training data to achieve high recognition accuracy,
+which was time-consuming and required considerable manual
+work like labeling images. In this paper, we automatically
+prepare training data using robots. Considering the low efficiency
+and high energy consumption in robot motion, we proposed
+combining robotic in-hand observation and data synthesis to
+enlarge the limited data set collected by the robot. We first used a
+robot with a depth sensor to collect images of objects held in the
+robot’s hands and segment the object pictures. Then, we used a
+copy-paste method to synthesize the segmented objects with rack
+backgrounds. The collected and synthetic images are combined to
+train a deep detection neural network. We conducted experiments
+to compare YOLOv5x detectors trained with images collected
+using the proposed method and several other methods. The
+results showed that combined observation and synthetic images
+led to comparable performance to manual data preparation.
+They provided a good guide on optimizing data configurations
+and parameter settings for training detectors. The proposed
+method required only a single process and was a low-cost way to
+produce the combined data. Interested readers may find the data
+sets and trained models from the following GitHub repository:
+github.com/wrslab/tubedet
+Note to Practitioners—The background of this study is a
+requirement in lab automation – Using robots to arrange
+randomly placed tubes automatically. Before sending test tubes
+to an examination machine for gradient tests, humans need to
+categorize and organize the tubes into specific patterns to fit the
+machine’s internal design. Employing humans is difficult as the
+tube arrangement requirements are time-varying. A preferred
+solution is using robots to replace humans. The robots should
+have a vision system to detect the tubes and a manipulation
+system to perform physical arranging actions. They will be
+used in busy seasons while deployed for other tasks in leisure
+time. Deep neural networks like YOLO are effective for the
+tube detection task. However, preparing the training data is
+challenging and unsuitable for lab end users. Pre-trained neural
+networks are options but have limited tube detection ability
+and cannot deal with newly included tube types. The method
+developed in this work helps solve the training data preparation
+problem. With its support, the robot can automatically prepare
+training data that has comparable quality to manually labeled
+ones in a single-process and low-cost way.
+Index Terms—Robotic data preparation, data synthesis, test
+tube detection
+I. INTRODUCTION
+Recent advances in deep learning have led to a revolution
+in object detection. Deep learning-based methods use deep
+1Department of System Innovation, Graduate School of Engineering Sci-
+ence, Osaka University, Toyonaka, Osaka, Japan.
+2H.U. Group Research Inst. G. K., Japan.
+3National Inst. of AIST, Japan.
+∗Contact: Weiwei Wan, [email protected]
+Fig. 1: Several examples of in-rack test tube detection. Each
+grid includes two images. The left image is captured by a
+vision sensor. The right image is the recognition result. The
+data used for training the detection neural network is prepared
+using the proposed method.
+neural networks to learn features from training data. They
+outperform traditional hand-crafted features with impressive
+results. Despite these advantages, deep learning-based object
+detection requires collecting a large amount of labeled data
+for training, which is time-consuming and labor-intensive, and
+has significantly hindered the scalability and flexibility of deep
+learning-based applications.
+Previously, researchers have developed several methods to
+reduce data collection costs. For example, data augmentation
+[1] enriched existing training data sets by applying random
+transformations like image rotation or scaling. Data synthesis
+[2][3] generated previously unseen data using simulation or
+adversarial neural networks. The main challenge of the aug-
+mentation or synthesis methods was the “domain gap” [4][5]:
+arXiv:2301.01441v1  [cs.CV]  4 Jan 2023
+
+ARXIV VERSION, 2022
+2
+Augmented data had less varied visual contexts. Synthesized
+data was prone to discrepancies with the real world. Recently,
+researchers have revisited using the copy-paste method [6] to
+increase data. The method was effective in compensating for
+the “domain gap” problem, exhibiting impressive performance.
+There is no clear boundary between augmentation and synthe-
+sis when using the copy-paste method to generate data. It was
+mainly classified as an synthesis method [7][8], although some
+studies considered it to be augmentation [9]. This paper calls
+it a synthesis method to avoid confusion with transformation
+and scale-based data generation.
+The most tiring aspect of the copy-and-paste method is
+how to neatly cut a large variety of target regions and paste
+them onto a new background. Previously, researchers work-
+ing on robotic manipulation have developed robotic methods
+to segment novel objects from backgrounds. For example,
+Florence et al. [10], Boerdijk et al. [11], and Pathak et
+al. [12] respectively used robotic in-hand or non-prehensile
+manipulation to change objects’ observation viewpoints and
+segmented the objects based on the robot motion. Such sys-
+tems could replace humans to segment goal object regions
+under various conditions. Very recent studies [11][13] has
+noticed the advantage, and increased data size and contextual
+variety by pasting the objects segmented by robotic systems
+onto random backgrounds. Despite their seminal proposals,
+the need for copy-and-paste synthesis and the impact of data
+volume and ratios remain undiscussed.
+Based on the current research status, this paper further
+delves into using robots to collect training data automatically.
+Considering the low efficiency and high energy consumption
+in robotic data collection, we propose combining robotic ob-
+servation and copy-paste synthesis to reduce costs. We assume
+a test tube detection task shown in Fig. 1 and use a robot with
+a depth sensor to move and observe tubes. The robot collects
+observation images and, at the same time, segments tubes
+from the images for copy-paste synthesis. The observation and
+synthetic images are used as training data for deep detection
+neural networks. Especially for the synthesis routine, we value
+the co-occurrence of tubes and racks, and paste tubes inside a
+rack area to obtain contextual consistency. Also, we take into
+account factors like tube-to-tube occlusions and foreground
+changes caused by environment or visual difference to reduce
+unrealistic synthetic results. The proposed method helps enrich
+the data set and resolve the “domain gap”. It does not need
+heavy robotic effort.
+In experiments, we trained several YOLOv5x networks to
+understand the performance of the proposed method. The
+training data was collected using the proposed and several
+other methods. The results confirmed data collected using the
+proposed method do have claimed advantages. We also con-
+ducted multiple ablation studies to look into the impact of data
+volumes and ratios when training detection neural networks
+using data collected with the proposed method. The results
+provided a good guide on optimizing data configurations and
+parameter settings for training detectors.
+The contributions of this work are as follows. (1) We
+develop an automatic data-collection method in which a robot
+holds target objects and observes them. The method yields
+observation images and target regions segmented from the
+images. (2) We develop a copy-paste image synthesis method
+to enrich the training data. The method pastes object regions on
+various rack backgrounds to balance “domain randomization”
+and “domain gap”. The rack backgrounds are also automati-
+cally collected by the robot. (3) We examined combinations
+of the observation and synthetic images and compared them
+with other data sets to understand the impact of data volume
+and ratios.
+The remaining part of this paper is organized as follows:
+Section II reviews related work. Section III presents the
+hardware system and the proposed method’s workflow. Section
+IV delivers technical details. Section V shows experiments and
+analysis. Section VI draws conclusions.
+II. RELATED WORK
+We review the related work considering robotic data collec-
+tion and data synthesis, respectively.
+A. Automatic Data Collection Using Robots
+Segmenting the object regions from a picture is the basis of
+automatic data collection. Conventional methods used simple
+backgrounds [14], known environments [15], or designed eas-
+ily identifiable gadgets [16][17] to simplify object extraction.
+The methods required careful preparation about scenes and
+objects.
+Robot-based methods leverage actuated robots to simplify
+object segmentation. They can be traced back to early studies
+in object recognition and 3D object modeling [18][19][20][21].
+These work took advantages of robotic manipulation sequence
+to perceive objects from different viewpoints and segment
+the objects from the background. From the robotic manipu-
+lation perspective, such segmentation can be divided into two
+categories: In-hand object segmentation and Interaction-based
+segmentation.
+1) In-hand object segmentation: Previous work on in-hand
+object segmentation used known robot models and handcrafted
+visual features to isolate in-hand objects from background
+environments and robot hands. For example, Krainin et al.
+[21] isolated in-hand objects’ point clouds by examining
+the Euclidean distance to the robot model. Welke et al.
+[19] segmented in-hand objects from images based on Eigen
+background subtraction, disparity map, and hand localization.
+These methods required manually preparing detectors for
+various targets considering their visual features.
+More recent studies used deep learning to reduce the
+reliance on hand-crafted visual features for in-hand object
+segmentation. For instance, Florence et al. [10] proposed a
+self-supervised framework to segment in-hand objects. The
+framework involved two steps that used the same training
+and learning routine. In the first step, the authors generated
+masks for the robot by considering combined depth and RGB
+information, and trained a neural network model based on
+the masks to differentiate the robot from the background.
+In the second step, the authors masked the grasped object
+and train neural network models to isolate the object from
+the robot hand. Boerdijk et al. [13] used optical flow to
+
+ARXIV VERSION, 2022
+3
+respectively segment manipulators that were holding and not
+holding objects. The segmented data set were used to train a
+neural network for isolating manipulators and grasped objects.
+2) Robot-object interaction: On the other hand, some re-
+searchers took advantages of non-prehensile robot manipu-
+lation like push to change object perspectives and segment
+them based on robot motion cues [12][22][23]. For example,
+Pathak et al. [12] designed a framework to continuously refine
+a neural network model that generates object segmentation
+masks through robot interaction. The model initially generated
+hypothesis segmentation masks for objects. The masks were
+refined based on the pixel differences of the images captured
+before and after robotic interactions. The generating model
+was updated along with the refined masks. Singh et al. [24]
+proposed to segment unknown objects in a cluttered scene
+while repeatedly using robotic nudge motions to interact with
+objects and induce geometric constraints. Robotic interactive
+segmentation often requires a static scene or surface to permit
+interaction between robots and objects [25][26]. It is more
+complicated compared with the in-hand object segmentation
+as the object poses needs to be controlled and changed through
+robotic manipulation.
+A critical problem of the robotic methods is that they are
+unsuitable for preparing a large amount of training data as
+robots consume much time and energy to perform the physical
+motion. Conducting thousands of robotic motion trajectories
+to collect data is impractical. Also, the robots in the systems
+are fixed, have limited views, and can only collect data in a
+narrow range of scenarios. Neural networks trained using the
+collected data may suffer from contextual (background) bias
+and have bad generalizability [27][28].
+This study focuses on robotic data collection while con-
+sidering leveraging data synthesis to reduce robotic usage.
+We first ask the robot to hold a single tube and annotate the
+tube’s bounding polyhedron by extracting in-hand point cloud
+according to the robot’s tool center point (TCP) and hand
+model. Then, we map the annotated bounding polyhedron to
+2D image regions in the robot’s camera view for extracting the
+tube region. The robot moves the tube to different positions
+and rotations to obtain many varieties of 2D images and tube
+regions. The images and tube regions are respectively used for
+training and synthesizing new data in a later stage.
+B. Data Augmentation and Synthesis
+Data augmentation and synthesis are the two most well-
+used methods to enrich training data. Data augmentation
+generates new data by transforming the existing training data
+with specific rules or learning-based methods. Data synthesis
+generates synthetic data by merging existing data with others
+or using computer simulations. Concurrent publications tend
+to mix these nomenclatures. Therefore we conduct a uniform
+literature review of them below without differentiation.
+The copy-paste method is widely used for generating syn-
+thetic data. It segments foreground objects from existing
+images, possibly modifies them, and pastes them onto new
+backgrounds [8][7][9]. The copy-paste method is easy to
+implement and shows notable performance over using pure
+real data. Previous studies showed that it was important to
+carefully select the backgrounds when pasting objects. For
+example, Divvala et al. [29] experimentally showed visual
+context benefited object detection performance and reduced
+detection errors. Dvornik et al. [30] showed that the correct
+visual context when pasting object can improve prediction
+performance while inappropriate visual context led to negative
+results. Wang et al. [31] swapped objects of the same class
+in different images to ensure contextual consistency between
+objects and backgrounds and showed using the exsiting back-
+grounds had better performance than random ones. Also,
+the copy-paste method requires a data set containing many
+possible views of the object that are easy to be cut out. It is
+burdensome for humans to prepare them.
+Graphical simulation is another popular method for synthe-
+sizing training data. The benefits of simulation is that it allows
+freely changing light conditions and materials to increase
+variation. It also allows capturing many views of objects by
+simply transforming virtual camera poses. For example, Hodaˇn
+et al. [32] and Richter et al. [33] respectively used photo-
+realistic rendering to synthesize images of 3D object models
+and scenes. The methods required a lot of computational
+resources to narrow down the domain gap between synthetic
+and realistic data. Tobin et al. [34] proposed the concept of
+domain randomization (DR). They randomized a simulator to
+expose models to a wide range of environments and obtain
+varied training data. Instead of photo-realistic rendering, the
+method only required low-fidelity rendering results to reach
+satisfying accuracy for medium-size objects. Carlson et al.
+[35], Hinterstoisser et al. [4], Prakash et al. [36], and Tremblay
+et al. [37] respectively used DR to narrow down the domain
+gap. The authors randomly changed the context in simulation
+so that “the real data was made to be just like another
+simulation” [38]. Yang et al. [39] and Sundermeyer et al. [40]
+respectively sampled viewpoints of 3D object models using
+simulation and mixed the samples with real backgrounds to
+reduce the human effort for preparing scenes with rich domain
+randomness. Besides DR, Generative Adversarial Networks
+(GANs) were also promising to reduce domain gap. For
+example, Chatterjee et al. [41] designed a lightweight-GAN
+to synthesize data for training plastic bottle detectors.
+In this study, we leverage data synthesis to enrich the
+training data. We develop a copy-paste based method to
+attach tube cap regions separated from robotic observation
+images to rack backgrounds and thus synthesize new images.
+Various constraints like rack dimensions and tube occlusions
+can be considered during the synthesis to reduce the domain
+gap. The synthetic data is mixed with real-world data to
+promote the performance of YOLO-based tube recognition
+neural networks. It is also compared with other data collection
+methods to understand the influence of data volume and data
+combination ratio.
+III. ROBOT SYSTEM AND WORKFLOW
+A. Configurations of the Robot System
+Fig. 2(a) shows our robot system used for preparing the
+training data. A Photoneo Phoxi M 3D Scanner is used for
+
+ARXIV VERSION, 2022
+4
+Fig. 2: (a) The system configuration. (b) The test tubes and
+rack in the view of the Phoxi M 3D Scanner.
+capturing objects on the flat table. An ABB Yumi dual-arm
+robot with a two-finger gripper is used to manipulate objects
+in the system. A flat table is set up in the front of the Yumi
+robot. The in-rack test tubes to be recognized are placed on
+the surface. The Phoxi scanner is a structured-light based
+depth sensor. It can capture gray images and point clouds
+simultaneously. Each data point of a point cloud captured by
+the Phoxi scanner have a one-to-one correspondence to a pixel
+in a gray image. We can segment an object in the gray image
+by considering its point cloud.
+Especially, we install the Phoxi scanner on top of the robot
+to obtain a top view of the racks and tubes. When recognizing
+tubes in the rack, we select the tube caps as the primary
+identifiers. There are two reasons why we prefer using the
+tube caps for identification. The first one is that obtaining the
+point cloud of a translucent or crystal test tube fails easily
+due to limitations of the structured-light based depth sensors.
+The second one is that the tube bodies are blocked by the
+caps and also occluded by surrounding tubes when placed
+in the rack and viewed from a top position. They are less
+visible. However, despite the reasons and their merits, there is
+a problem that different types of tubes may share a same cap
+type. In this work, we assume the test tubes with the same
+caps can be identified by their heights in the rack and analyze
+the point cloud to differentiate them.
+B. Workflow for Data Preparation
+We prepare the training data using the robot system follow-
+ing the workflow shown in Fig. 3. There are four dashed boxes
+in the chart, where (a.1) and (a.2) have a blue background color
+and represent the data collection component, (b) has an orange
+background color and represents the data synthesis component,
+(c) has a gray background and represents the resulted data.
+The first blue dashed box (Fig. 3(a.1)) comprises three steps.
+First, a human hands over an unknown test tube to the robot.
+The tube is assumed to be grasped vertically by the robot after
+handover, with the tube cap left above the robotic fingertips.
+Second, the robot moves the test tube to the observation poses
+prepared offline while considering avoiding self-occlusions.
+The Phoxi sensor will capture the test tube’s gray image
+and point cloud at each observation pose. Third, the system
+segments the cap region out of the captured image based on
+a mapping from its counterpart point cloud. The segmentation
+result only includes the cap. The background will be removed
+thanks to the point cloud mapping. The output of this dashed
+box includes many cap region pictures. They are observed
+from different views and thus have different illumination and
+visual conditions.
+The second blue dashed box (Fig. 3(a.2)) is similar to the
+first one and also comprises three steps. First, a person places
+a rack in the environment. Then, the robot pushes the rack to
+random poses, capturing the rack’s gray image and point cloud
+at each pose. Third, the system segments the rack region out
+of the captured image based on the mapping from the rack’s
+counterpart point cloud. The result of this dashed box includes
+many rack region pictures. Like the caps, the rack region
+pictures also have different illumination and visual conditions
+since the data is captured from different view positions.
+The orange dashed box shows the data synthesis process,
+where the cap region pictures obtained in the first “Data
+Collection” dashed box are pasted onto the rack region pictures
+obtained in the second “Data Collection” dashed box for
+synthesizing new images. Constraints like rack boundaries and
+overlapping caused by perspective projection are considered
+during the synthesis. The output of the dashed box will
+be racks filled with many tube caps. The “Copy-paste data
+synthesis” sub-block illustrates several examples of the output.
+The final data preparation results include the images ob-
+tained during collecting the tube cap data (observation images)
+and the synthetic images. They are illustrated in the gray
+dashed box (Fig. 3(c)).
+Note that the above workflow is not completely automatic.
+The sub-blocks with texts highlighted in a green color involve
+human intervention. Also, before data collection, we need to
+prepare the camera calibration matrix and test tube observation
+poses. The camera calibration matrix transforms the point
+cloud captured in the camera’s local coordinate system into
+the robot coordinate system. Many existing methods exist for
+obtaining the calibration matrix [42]. To avoid repetition, we
+don’t discuss the details in this manuscript. The test tube
+observation poses are a set of tube positions and rotations
+for the robot to hold and capture observation images. The
+developed method will generate robot joint configurations
+considering the robot grasping and tube observation poses.
+Section IV will present detailed algorithms on the generation.
+IV. IMPLEMENTATION DETAILS
+A. Observation Poses for Collecting Tube Caps
+When collecting the tube cap data, the robot moves the
+tube held in its hand to different poses for observation. The
+observation poses are generated considering two constraints:
+(1) Diversity of the captured cap data; (2) Occlusions by robot
+links. Taking into account these two constraints allow us to
+include the tube caps from many viewpoints and thus cover
+lots of illumination and visual conditions. Meanwhile, they
+help to prevent the robot links from occluding the grasped
+test tubes and make sure the tubes are visible to the vision
+sensor.
+Fig. 4 illustrates the observation pose generation process and
+how the two constraints are taken into account in it. First, we
+sample the positions and rotations of a tube held by the robot
+hand uniformly in the Phoxi depth sensor’s visible range. Tube
+
+(b)
+_Tube
+(a)
+Phoxi M
+Holder
+3D Scanner
+Rack
+Yumi Robot
+Purple
+Tube
+Flat Table
+AB
+Purple
+Ring
+Blue
+White
+In-rack
+Tube
+Tube
+Tube
+Test TubesARXIV VERSION, 2022
+5
+Fig. 3: Workflow of the proposed method. (a.1,2) Data collection component. (b) Data synthesis component. (c) Resulted data.
+data captured under the sampled poses will have rich light
+conditions and a large variety of visible tube edges for training
+a recognition neural network. Especially, the tube rotations are
+sampled according to the vertices of a level-four icosphere
+[43]. An icosphere is a spherical polyhedron with regularly
+distributed vertices. The vectors pointing to the vertices of an
+icosphere help to define the rotations of a tube1. A level-four
+icosphere has 642 vertices and thus leads to 642 vectors and
+test tube rotation poses. Thanks to the visibility constraints,
+we do not move a test tube to all of the rotation poses for
+capturing data as the tube caps facing downward will not
+be seen by the Phoxi sensor. We filter the 642 vectors by
+considering their angles with the normal of the table surface
+for placing a rack. The vectors with large angles from the
+surface normal cannot be seen and will not be considered. The
+spherical polyhedron in Fig. 4(b.1) illustrates the level-four
+icosphere. Vectors pointing to the red vertices have more than
+θ angles from the surface normal and are removed. The green
+vertices are the remaining candidates. The purple tube bouquet
+on the right side of Fig. 4(b.1) illustrate the tube poses implied
+by vectors pointing to the remaining candidate vertices.
+Next, we plan the robot motion to move the test tube held in
+a robot hand to the sampled tube positions and rotations. We
+assume a test tube is vertically grasped at the finger center of a
+robot hand. Since a tube is central symmetric, many grasping
+poses meet the assumption. The grasping hand may rotate
+freely around the symmetry axis of the test tube, as shown
+in Fig. 4(b.2). The rotation is compact and forms a SO(2)
+group. For numerical analysis, we sample the rotation in the
+SO(2) group with a rotation interval hyperparameter named
+ω to obtain a series of discretized grasping poses. The hand
+1A tube is centeral symmetric. We do not need to consider its rotation
+around the central axis. The vectors pointing to the vertices of an icosphere
+can thus define a tube pose.
+illustrations in Fig. 4(b.2) are the grasping poses obtained with
+ω = 60◦. The sampled grasping poses provide many candidate
+goals for robot motion planning and thus increase the chances
+of successfully moving and observing the tube.
+When determining which exact candidate goal to move to,
+we examine the occlusions from the robot arm links and
+avoid choosing the grasping poses that lead to invisible tubes.
+In detail, examining the occlusion is done by checking the
+collision between a visual polyhedron and the robot arm links.
+The visual polyhedron is computed using the camera origin
+and vertices of the robot hand model, as illustrated in Fig.
+4(c.1). The robot arm may occlude the tube and the vision
+sensor fails to capture it when there is collision between
+the visual polyhedron and the robot arm links. Fig. 4(c.2)
+exemplifies such a case.
+B. Using Annotation Masks to Segment Cap Pictures
+Since the tube is handed over from a human and the
+Phoxi sensor captures the cap data from many different
+views, the captured tube point clouds change dynamically
+and have noises. It is unstable to extract cap point clouds by
+autonomously detecting them. Thus, instead of autonomous
+detection, we prepare an annotation mask in the robot hand’s
+local coordinate system to help extract the test tube cap’s point
+clouds. The extracted point clouds will be back-projected to
+the corresponding 2D grey image for segmenting a picture
+of the cap region. Fig. 5 shows the details of this mask and
+how it helps to segment the cap regions. The mask and back
+projection enable us to precisely segment the cap regions while
+avoiding including backgrounds.
+To prepare an annotation mask, we move the robot hand
+that holds a test tube to a fixed position under the Phoxi
+sensor and trigger the sensor to capture a point cloud. We can
+
+(a.1) Collecting the tube cap data
+(i) A human
+(ii) The robot moves
+Many pictures of the tube caps
+(iii) Crop the tube cap
+hands over
+the test tube to new
+based on the mapping
+a test tube
+observation poses
+between the captured
+to the robot
+for capturing data
+point cloud and image
+(a.2) Collecting the rack data
+(i) A human
+(i) The robot
+(iii) Crop the
+Many pictures of the racks
+places a
+pushes the rack
+rack from the
+rack in the
+to new poses for
+environmental
+environment
+capturing data
+background
+(b) Data synthesis
+(c) Resulted data (Images with known tube caps)
+Pictures of tube caps
+ Pictures of racks
+Images obtained
+during collecting
+Copy-paste data synthesis
+the tube cap data
+(Top view pictures
+with tubes held in
+Synthesized racks with
+the robotic hands)
+different backgroundsARXIV VERSION, 2022
+6
+Fig. 4: (a) Sampling observation positions. The green region
+is the visible area of the Phoxi scanner. The red points are
+the sampled positions. (b.1) Sampling rotations based on a
+level-four icosphere. The left spherical polyhedron illustrates
+the icosphere. The green vertices are the ends of feasible
+vectors that have less than θ = 60◦ angles with the surface
+normal. They imply the tube rotation poses shown on the right.
+(b.2) The grasping poses for each sampled tube pose form a
+SO(2) group. They are sampled considering an interval ω for
+numerical analysis. (c.1) A visual polyhedron computed using
+the camera origin and vertices of the robot hand model. (c.2)
+The grasped object has a risk of being occluded by the robot
+arm when there is a collision between the visual polyhedron
+and the robot arm links.
+Fig. 5: Workflow for extracting the cap picture using an
+annotation mask. (a) Applying a mask described in the local
+coordinate system of the holding robot hand to the captured
+point cloud. (b) The extract point cloud is projected back to
+the 2D grey image for segmenting a picture of the cap region.
+(b.1) The back-projected results might be disconnected pixels.
+(b.2) A bounding convex hull of the disconnected pixels is
+computed. (b.3,4) The cap region is segmented based on the
+bounding convex hull.
+easily get the cap’s point cloud data by examining the area
+on top of the holding fingers and obtain an annotation mask
+by considering a bounding polyhedron of the data. However,
+a single bounding polyhedron may not be general for others
+since the captured point cloud is susceptible to light reflection
+or perspective projection (self-occlusion). Thus, instead of a
+single point cloud and polyhedron, we collect point clouds
+from multiple views, merge them under the robot hand’s local
+coordinate system, and compute a bounding box of the merged
+result as an annotation mask. Fig. 6 shows an example. The
+multiple views are sampled the same way as the observation
+poses mentioned in the previous subsection. However, we do
+not need to change the observation positions since we aim to
+Fig. 6: (a) Capture data from different views. The tube cap’s
+point clouds are obtained by examining the area on top
+of the holding fingers. They are high lighted with colored
+polyhedrons. (b) Merge the cap’s point clouds in (a) under
+the robot hand’s local coordinate system, and compute a
+bounding box of the merged result as an annotation mask. (b.1)
+Raw bounding box. (b.2) The bounding box can be adjusted
+interactively if needed.
+obtain a bounding box mask in the hand’s coordinate system.
+The views under various rotations could provide enough
+superficial point cloud data to meet the requirements. Note
+that the merged result may include noise point data induced by
+reflections from the transparent tube body and lead to a mask
+larger than the cap. We provide an interactive user interface
+for manually adjusting the bounding box sizes and minimizing
+the negative influences caused by the noises. The adjustment
+is optional and may be performed when precisely segmenting
+the cap region is demanded.
+C. Copy-Paste Synthesis
+We apply random scaling, blurring, brightness, and contrast
+to the segmented tube caps and then paste them onto the
+segmented rack background for data synthesis. During pasting,
+we permit the overlap among the cap regions to approximate
+tube-to-tube occlusion. After pasting, we randomize the en-
+vironmental background (background of the rack) to narrow
+further the domain gap between synthetic images and images
+captured in the real world.
+A critical maneuver here is that we consider the co-
+occurrence of the test tubes and the rack and paste the tube
+cap pictures onto a rack instead of random backgrounds like
+[11]. We randomly sample positions inside rack pictures for
+pasting tube caps and use a pasting number T to control the
+clutter. Note that there is no need to exactly paste a tube cap
+near the hole centers of a rack as the tubes tilt randomly inside
+the rack holes. The visible cap regions may reasonably overlap
+with a hole boundary or other holes.
+For tube-to-tube occlusion, we consider the perspective
+projection of a vision sensor and define an occlusion threshold
+t to permit overlap among the visible cap regions. A vision
+sensor’s perspective projection leads to mutual occlusions in
+the rack at certain viewpoints. The occlusion threshold helps to
+simulate the occlusion and defines the maximum percentage
+that segmented cap pictures can overlap or occlude. Fig. 7
+shows how the t threshold works. It adds a constraint to
+pasting, where a previously pasted cap picture “A” must have
+less than t percentage overlap with the union of caps pasted
+later. The B ∪ C ∪ ... component in the nominator of Fig.
+7 implies the union of caps pasted after “A”. When a new
+cap is randomized, it must be unioned with this component
+
+(a)
+(b.1)
+0 = 60°
+Surface
+Test
+normal
+tube
+Positions
+Rotations
+(b.2)
+(c.1)
+(c.2)
+Visual
+Gripper
+polygon
+.09 = 3(a)
+(b)
+Annotation Mask
+(b.1)
+(b.2)
+ZH
+(b.3)
+(b.4)
+ZH(b)
+(b.1)
+(b.2)ARXIV VERSION, 2022
+7
+to ensure the t constraint on all previous “A” is not violated.
+There are two noticeable points for t. First, its value could be
+devised respectively considering the heights of specific tube
+types. Second, its value is correlated with the pasting number
+T. The maximum number of pasted tube caps in a rack that
+meet the t threshold may be less than a given T. In that case,
+we constrain the maximum number of pasted tube caps to the
+smaller value to ensure t is not invalidated.
+Fig. 7: (a) Using a threshold t to simulate cap occlusions.
+“A” represents a previously pasted cap region. “B”, “C”, ...
+represent the caps pasted after “A”. (b) Results with different
+t values.
+For the environmental background, we use the BG-20k data
+set [44] to obtain high-resolution random background images
+and change the background of a synthetic image with a 0.5
+probability.
+V. EXPERIMENTS AND ANALYSIS
+We carried out experiments to compare YOLOv5x [45]
+detectors trained using data sets collected with the proposed
+method and several other methods to understand the per-
+formance. Table II shows the methods. The SR (Synthesis
+by pasting to Racks) method pastes randomly selected cap
+pictures onto rack backgrounds to synthesize training data.
+It represents the synthesizing method used in this work.
+The SB (Synthesis by pasting to BG-20k) method is an
+alternative synthesis method. Instead of being pasted onto a
+rack, randomly selected cap pictures are pasted to random
+backgrounds selected from the BG-20k data set. The RO
+(Robotic Observation) method is a byproduct of robotic cap
+segmentation, where the robot holds test tubes for data col-
+lection. We considered RO an independent method because
+we wondered if the hand-held observation was enough for
+training. We also combined RO, SR, and SB methods (the **
+row) to see if they help achieve a satisfying performance. The
+RO+SR combination is exactly our proposed method in this
+work. We especially proposed it since RO is a pre-process
+of robotic cap segmentation. Using combined RO+SR does
+not increase effort. Combining RO+SB or RO+SR+SB are
+also candidate choices. They have the same cost as using
+independent SR or SB data2. Finally, the CL (Crowd-source
+Labeling) method is a conventional one that requires humans
+to place racks with tubes under the robot and label the captured
+images manually. Fig. 8 shows exemplary images collected
+using the different methods.
+2Synthesizing data is considered to be free as it only require computational
+work. Thus, the costs of SR and SB depend on the RO process.
+TABLE I: Summary of the data collection methods
+Abbr.
+Full Name
+Description
+SR
+Synthesis by pasting to Racks
+Caps on racks
+SB
+Synthesis by pasting to BG-20k
+Caps on random background
+RO
+Robotic Observation
+Tubes held in robotic hands
+**
+Combinations of above methods
+SR+SB is the proposed one
+CL
+Crowd-source Labeling
+Tubes in a rack on the table
+Fig. 8: Exemplary images collected using the various methods.
+(a) RO. (b) SR. (c) SB (d) CL.
+A. Performance of Various Data sets
+We collected various data sets with the methods and their
+combinations, used the data sets to train YOLOv5x detectors,
+and examined the performance of the trained detectors using
+a testing data set for comparison.
+The first data set is CL200. It is considered a baseline for
+comparison. In collecting the data set, we collected 200 images
+with random tube and rack states and labeled the tube regions
+manually using LabelImg3. There are, in total, 5916 labeled
+instances in the 200 images.
+The second data set is SR1600. In order to collect it, we
+first prepared many cap pictures using robotic observation. As
+shown in Fig. 2(b), we assumed four different test tubes and
+took advantage of the Yumi robot’s both arms to collect cap
+data quickly. For each tube type, we handed over two same
+ones to the two robotic arms for observation. Each arm moved
+its held tube to 400 observation poses for data collection. See
+Fig. 9(a) for example. Here, we set the hyperparameter θ and
+ω to 30◦ and 360◦ (single grasping pose) and set the positions
+to be evenly sampled on the table with a granularity of 0.1m
+for generating the observation poses. In total, more than 400
+observation poses were obtained under the parameter setting
+for each arm, and we used the first 400 for collecting images.
+As a result, we obtained 400 observation images (800 cap
+pictures since there are two tubes in each image, see Fig.
+9(b) for example) for a single tube type and 1600 observation
+images for all tube types. We segmented 3200 pictures of
+cap regions from the observation images considering point
+cloud mapping. Fig. 9(c) shows the collected point clouds with
+highlighted caps (green). Fig. 9(d) shows the segmented cap
+regions. Besides the cap regions, we collected 15 images with
+racks (a single rack in each image) and segmented 15 pictures
+of racks. We synthesized a data set of 1600 images by pasting
+caps randomly selected from the 3200 cap pictures to racks
+randomly selected from the 15 rack pictures (SR method).
+During synthesis, we set the pasting number to be T = 30, and
+set the occlusion threshold for the “Blue Tube” to be tblue =
+0.4 and other tubes to be tothers = 0.15. We chose these
+parameter settings because the “Blue Tube” was shorter and
+susceptible to occlusion. We increased its occlusion threshold
+3https://github.com/heartexlabs/labelImg
+
+(a)
+IAN(BUCU.
+Overlap(A, B, C, ...)
+[A|
+(b)
+10
+t = 0.3 T = 20
+t = 0.6 T = 40a
+dARXIV VERSION, 2022
+8
+to mimic frequent visual blockage from other tubes. Also,
+we increased the variety of the segmented cap pictures by
+applying random scaling (0.9 ∼ 1.1 of original picture size,
+0.5 probability), random blur (3 × 3 kernel, 0.5 probability),
+random brightness (0.9 ∼ 1.1 of original brightness, 0.5
+probability), and random contrast (0.9 ∼ 1.1 of the original
+contrast value, 0.5 probability) using the Albumentations4
+library. The background of the rack was randomly chosen from
+the BG-20k data set with a 0.5 changing probability.
+Fig. 9: (a) The robot moves test tubes for observation. Both
+arms are used. (b) Observation Image. (c) Point clouds cap-
+tured by the Phoxi sensor. (d) Cap pictures segmented from
+the observation image.
+The third data set is RO1600. It is a semi product of robotic
+cap segmentation and comprises the 1600 observation images
+obtained during robotic observation.
+The fourth data set is SB1600. In contrast with the SR1600
+data set, we pasted randomly selected caps directly to images
+from the BG-20k data set for obtaining data. The pasted
+caps might freely distribute on the image background. The
+segmented racks were not used. The pasting number T and
+occlusion threshold t are 35 and 0.15 respectively. There was
+no difference on t for different tubes. The randomization were
+performed in the same way as obtaining SR1600.
+We also used combined methods to collect data sets
+and study if the combination led to better results. The
+combined data sets include RO1600+SR800, RO1600+SB800,
+RO1600+SR400+SB400, SR800+SB800. Here, the superscript
+number on the upper-right of a method name means the
+number of images collected using the method. The “+” sym-
+bol indicates that the data sets comprise data collected using
+different methods. The RO1600+SR800 data set represents the
+data collected using the proposed method.
+The left part of Table II summarizes the various data sets.
+They are used to train YOLOv5x detectors for comparison.
+Before training, the YOLOv5x detectors for all data sets were
+initialized with weights pre-trained using the COCO data set.
+The images in all data sets were regulated into a resolution of
+1376 × 1376. Each data set is divided into a training subset
+and a validation subset according to a 4 to 1 data ratio. During
+learning, the training subset was fed to the training program
+with a batch size of 2, and the training program performed
+validation per episode. The training process was stopped
+when the mAPs (mean Average Precision) [46] for all objects
+reached higher than 99.0% under a 0.5 IoU (Intersection over
+Union). Here, we defined a detected bounding box to be
+correct when its IoU with a ground truth cap bounding box
+was larger than 0.5.
+4https://albumentations.ai/
+For evaluating the performance of YOLOv5x detectors
+trained using the various data sets, we collected a testing
+data set with 100 images and labeled their ground truth
+using the same method as CL. We used the trained detectors
+to detect tubes in the testing data set. Like validation, we
+defined a detected bounding box as correct when its IoU
+with a ground truth cap bounding box is larger than 0.5.
+We used the AP (Average Precision) metric to measure the
+detection performance of a single object class and used the
+mAP for all objects. Since the detector that met a single
+satisfying validation was not necessarily the best, we trained
+each detector twice and took the higher precision value on the
+testing data set as the final evaluation result.
+Table II shows the evaluation results. We obtained the
+following observations and speculations from them.
+i) Using the data set collected by robotic observation for
+training exhibited the worst performance, as shown by
+the 2nd row (RO1600).
+Speculation: All images in the data set had a similar
+robotic background. They suffered from a domain shift.
+ii) The synthetic data sets do not necessarily lead to a good
+AP, as shown by the 3rd (SR1600) and 4th (SB1600) rows.
+The SR1600 data set exhibited higher performance than
+the SB1600 data set.
+Speculation: The copy-paste synthesis failed to cover
+certain visual contexts; Pasting onto racks (SR) provided
+more effective visual contexts and benefited the neural
+network more than pasting onto random backgrounds
+(SB).
+iii) Combining the synthetic data sets with robotic observa-
+tions is effective. It can be concluded by comparing the
+5th, 6th, and 7th rows (RO1600+SR800, RO1600+SB800,
+RO1600+SR400+SB400) with the 2nd, 3rd, and 4th rows
+(RO1600, SR1600, and SB1600). The former rows had
+higher mAP than the latter.
+Speculation: The robotic observation data set additionally
+provided helpful visual contexts.
+iv) The 5th row (RO1600+SR800) had a 2.4% higher mAP
+than the 6th row (RO1600+SB800). Especially, the AP of
+the “Blue Tube” on the 5th row was 8.7% higher than
+that on the 6th row. The AP of other tubes also had
+0.1% ∼ 0.7% performance increase.
+Speculation: Considering the rack as a local context
+helped improve domain-specific performance; The short
+“Blue Tube” could be easily blocked. The data set
+collected using the SR method had more simulated oc-
+clusions. They were important for recognizing the short
+“Blue Tube”.
+v) The 7th row (RO1600+SR400+SB400) exhibited slightly
+higher mAP (0.3%) than the 5th row (RO1600+SR800).
+Speculation: Pasting onto racks (SR) provided better
+domain-specific features. Random backgrounds for the
+tubes slightly benefited the neural network and were less
+necessary if the goal context was limited.
+vi) The 5th row (RO1600+SR800) is competitive compared
+with the 1st row (CL200). The mAP was 0.7% lower. The
+AP of the “Blue Tube” and “White Tube” were 2.5% and
+1.1% lower, respectively. The AP of the “Purple Tube”
+
+(c) Point clouds
+(d) Cap
+(b)
+picturesARXIV VERSION, 2022
+9
+TABLE II: Comparison of detectors trained using different data sets
+AP
+ID
+Data Set Names
+# Caps
+Remark
+Blue
+Purple
+White
+Purple Ring
+mAP
+1
+CL200
+5916
+Multiple tubes / image
+0.993
+0.995
+0.989
+0.984
+0.990
+2
+RO1600
+3200
+Two tubes / image
+0.380
+0.923
+0.695
+0.630
+0.657
+3
+SR1600
+40000
+tblue=0.4 & tothers=0.15, T = 25
+0.955
+0.979
+0.871
+0.953
+0.940
+4
+SB1600
+56000
+t=0.15 (same for all tubes), T = 35
+0.808
+0.978
+0.812
+0.897
+0.874
+5
+RO1600+SR800
+23200
+See note 2
+0.968
+0.995
+0.978
+0.992
+0.983
+6
+RO1600+SB800
+31200
+See note 2
+0.881
+0.994
+0.971
+0.992
+0.959
+7
+RO1600+SR400+SB400
+27200
+See note 2
+0.969
+0.994
+0.986
+0.993
+0.986
+8
+SR800+SB800
+48000
+See note 2
+0.973
+0.993
+0.969
+0.985
+0.980
+Note 1: Largest AP and mAP values are highlighted in bold.
+Note 2: The combined data sets are collected using the same parameters as respective ones.
+was the same. The AP of the “Purple Ring” tube was
+0.8% higher.
+Speculation: The robotic observation and paste-to-rack
+synthesis compensated for each other’s shortcomings;
+There remained extreme cases that could be labeled
+manually but failed to be covered by robotic observation
+or synthesis, especially for the “Blue Tube”.
+Several failure cases are visualized in Fig. 10 to provide
+the readers an insight into our observations and speculations.
+Fig. 10(a) and (b) exemplify the recognition results of detec-
+tors trained using the 5th (RO1600+SR800) and 6th data sets
+(RO1600+SR800). The latter one failed to recognize occluded
+tubes as the training data set had fewer simulated occlusions.
+The example is consistent with the observation and speculation
+in iv). Fig. 10(c) and (d) exemplify cases that the detectors
+trained using the 5th (RO1600+SB800) data set failed. In the
+first case, shadows from other test tubes were cast on a blue
+test tube cap. The detector failed to recognize the tube. In the
+second case, the detector misrecognized a crystal tube body
+as the “Blue Tube” cap due to the illusion caused by body-
+and-rack overlap. The two failure examples are consistent with
+the observation and speculation in vi). The synthetic data sets
+do not involve shadows or tube bodies. The detectors trained
+using them had worse performance in these cases than the one
+trained using the crowd-sourced real-world data.
+Fig. 10: (a) Detector trained using the 5th data set successfully
+recognized all tubes. (b) Detector trained using the 6th data
+set failed to recognize the occluded tube in the red circle. (c)
+Detector trained using the 5th data set failed to recognize the
+shadowed tube in the red circle. (d) Detector trained using the
+5th data set misrecognized the tube body in the red circle as
+a “Blue Tube”.
+In summary, the results of the various training data sets
+showed that combining data collected using the RO and SR
+methods was effective. The conclusion was satisfying as the
+RO method is a subset of the SR method. The workflow for
+collecting them is simple and clean. However, we wonder if
+the number of images in the RO data set could be reduced,
+as it needs much manual handover to collect them. This
+query prompted us to carry out the studies in the following
+subsection.
+B. Ablation Study
+In this subsection, we conduct multiple ablation studies on
+the combined RO+SR data set to further understand 1) the
+influence of the data combination ratio and 2) the influence
+of pasting number T and occlusion threshold t used for
+generating synthetic data.
+1) Influence of data combination ratio: The experiments for
+studying the influence of data combination ratio are divided
+into two parts. In the first part, we set the number of images
+collected using the RO method to 800 and varied the number
+of images collected using the SR method from 200 to 1600
+in a 2-fold ratio to understand the importance of the SR data.
+The upper section of Table III shows the precision of detectors
+trained using the varied data. The results indicate that the mAP
+improved when the SR image numbers increased from 200 to
+1600. The second part is similar to the first one. In this part, we
+fixed the number of images collected using the SR method to
+800 and varied the number of images collected using the RO
+method from 200 to 1600 in a two-fold ratio to understand
+the importance of the RO data. The lower section of Table
+III shows the precision of detectors trained using the varied
+data. The result indicates that the mAP improved when the
+RO image numbers increased from 200 to 1600.
+2) Influence of hyperparameters: Besides the data combi-
+nation ratio, we also studied the influence of pasting number
+T and occlusion threshold t used in the SR method. We set
+both the RO and SR image numbers to 800 and observed
+the performance of detectors trained with data sets collected
+using different T and t values. Although we previously used a
+different t value for the “Blue Tube”, we did not differentiate
+the tubes here. Like the study on different data combination
+ratios, this study also comprised two parts. In the first part,
+we fixed t to be 0.1 and increased T from 10 to 40 with
+a step length of 10. The upper section of Table IV shows
+the precision changes under the parameter variations. The
+
+ring
+ng0.94
+(d)
+a
+b
+cARXIV VERSION, 2022
+10
+TABLE III: In��uence of data combination ratio
+AP
+Data Set Names
+Blue
+Purple
+White
+Purple Ring
+mAP
+RO800+SR200
+0.958
+0.994
+0.973
+0.987
+0.978
+RO800+SR400
+0.964
+0.992
+0.975
+0.985
+0.979
+RO800+SR800
+0.966
+0.995
+0.979
+0.986
+0.981
+RO800+SR1600
+0.970
+0.994
+0.987
+0.987
+0.985
+RO200+SR800
+0.962
+0.992
+0.952
+0.978
+0.971
+RO400+SR800
+0.965
+0.992
+0.979
+0.978
+0.979
+RO800+SR800
+0.966
+0.995
+0.979
+0.986
+0.981
+RO1600+SR800
+0.968
+0.995
+0.978
+0.992
+0.983
+Note 1 Largest AP and mAP values are highlighted in bold.
+Note 2 We used the following hyper-parameter setting tblue
+=
+0.4 &
+tothers = 0.15, T = 30 to collect the SB data sets. The values were
+the same as the experiments in Section V.A.
+results exhibited a significant increase from 10 to 30. However,
+an even larger T had little influence on the recognition
+performance. In the second part, we set T to be 30 and varied t
+from 0.20 to 0.80 with a step length of 0.2. The lower section
+of Table IV shows the precision changes under the parameter
+variations. The results exhibited a clear precision increase on
+the “Blue Tube”. We speculate that the reason was that the
+“Blue Tube” was shorter and vulnerable to occlusions. A larger
+t helped provide more occlusion cases in the training data set,
+leading to a higher detection rate. The results also indicated
+that the precision of the ”White Tube” and ”Purple Ring Tube”
+irregularly changed as the t increased. They were taller and did
+not suffer from occlusions. Adding occlusions for them caused
+unexpected errors. For a complete observation, we recommend
+interested readers to compare with the third row of the table’s
+upper section to catch the changes starting from t = 0.1. The
+T value of the upper section’s third row was the same as the
+rows in the lower section.
+TABLE IV: Influence of parameters used for synthesis
+AP
+Params. (T, t)
+Blue
+Purple
+White
+Purple Ring
+mAP
+(10, 0.10)
+0.904
+0.995
+0.971
+0.973
+0.961
+(20, 0.10)
+0.915
+0.995
+0.976
+0.985
+0.968
+(30, 0.10)
+0.939
+0.995
+0.987
+0.992
+0.978
+(40, 0.10)
+0.934
+0.994
+0.972
+0.983
+0.970
+(30, 0.20)
+0.945
+0.995
+0.985
+0.989
+0.978
+(30, 0.40)
+0.969
+0.995
+0.985
+0.994
+0.986
+(30, 0.60)
+0.985
+0.995
+0.965
+0.967
+0.978
+(30, 0.80)
+0.987
+0.995
+0.984
+0.988
+0.988
+* Largest AP and mAP values are highlighted in bold.
+C. Further Analysis on Synthetic Data
+We also studied the influence of cap variation and combina-
+tion ratio on synthetic data sets (the data sets collected using
+the SR, SB, or SR+SB methods). The goal was to understand
+the best performance we could reach with synthesis.
+First, we fixed the number of images collected by the SR
+and SB methods to 800, respectively. We changed the number
+of cap region pictures (equals to the number of observation
+images multiplied by two) used for synthesis from 400 to 3200
+in a 2-fold ratio to study the influence of cap variation. The
+previsions YOLOv5x detectors using the changing data sets
+are shown in Table V. The results showed that the 400 row
+had competitive precision compared to the 1600 or 3200 rows.
+The number was enough to support a satisfying detector. The
+cap variations were thus considered to have a low influence
+on learning.
+Second, we fix the number of cap region pictures to 3200
+and change the number of images collected using the SR
+and SB methods, respectively, to study the influence of the
+combination ratio. Like the ablation study in Section V-B1,
+we divided the experiment here into two parts. In the first
+part, we set the number of images collected by the SR method
+to 800 and varied the number of images collected by the SB
+method from 200 to 1600 in a 2-fold ratio to understand the
+importance of the SB data. The upper section of Table VI
+shows the precision of detectors trained using the varied data.
+The number of SB images did not appear to be positively
+correlated with the final detector’s precision, although the
+largest mAP was observed when the number of SB images
+was 800. In the second part, we fixed the number of images
+collected by the SB method to 800 and varied the number
+of images collected by the SR method to understand the
+importance of the SR data. The lower section of Table VI
+shows the precision of detectors trained using the varied data.
+The result indicated that the mAP improved as the SR image
+number increased to 800. There was no significant difference
+when the image number increased from 800 to 1600.
+TABLE V: The influence of #caps to synthesis
+AP
+#Caps
+Blue
+Purple
+White
+Purple Ring
+mAP
+400
+0.970
+0.994
+0.969
+0.984
+0.979
+800
+0.971
+0.993
+0.954
+0.976
+0.973
+1600
+0.971
+0.992
+0.980
+0.985
+0.982
+3200
+0.973
+0.993
+0.969
+0.985
+0.980
+TABLE VI: Influence of the SR and SB ratio
+AP
+Data Set Names
+Blue
+Purple
+White
+Purple Ring
+mAP
+SB200 + SR800
+0.975
+0.994
+0.943
+0.957
+0.967
+SB400 + SR800
+0.973
+0.990
+0.960
+0.970
+0.973
+SB600 + SR800
+0.967
+0.988
+0.951
+0.968
+0.969
+SB800 + SR800
+0.973
+0.993
+0.969
+0.985
+0.980
+SB1600 + SR800
+0.952
+0.978
+0.926
+0.972
+0.957
+SB800 + SR200
+0.951
+0.982
+0.925
+0.963
+0.955
+SB800 + SR400
+0.932
+0.969
+0.914
+0.952
+0.942
+SB800 + SR600
+0.966
+0.990
+0.930
+0.893
+0.945
+SB800 + SR800
+0.973
+0.993
+0.969
+0.985
+0.980
+SB800 + SR1600
+0.975
+0.993
+0.967
+0.986
+0.980
+Note 1 Largest AP and mAP values are highlighted in bold.
+Note 2 We used the following hyper-parameter setting tblue
+=
+0.4 &
+tothers = 0.15, T = 30 to collect the SB data sets, and used the following
+hyper-parameter setting T = 30, t = 0.15 (same for all tubes) to collect
+the SR data sets. The values were the same as the experiments in Section
+V.A.
+Note 2 We used 3200 segmented cap region pictures for both methods.
+
+ARXIV VERSION, 2022
+11
+VI. CONCLUSIONS
+In this paper, we proposed an integrated robot observation
+and data synthesis framework for data preparation. The pro-
+posed framework can significantly reduce the human effort in
+data preparation. It required only a single process and was a
+low-cost way to produce the combined data. The experimental
+result showed that combined observation and synthetic images
+led to comparable performance to manual data preparation.
+The ablation studies provided a good guide on optimizing data
+configurations and parameter settings for training detectors
+using the combined data.
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+

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+arXiv:2301.13423v1  [math.OA]  31 Jan 2023
+ANALYSIS FOR IDEMPOTENT STATES ON QUANTUM
+PERMUTATION GROUPS
+J.P. MCCARTHY
+Abstract. Woronowicz proved the existence of the Haar state for compact quantum
+groups under a separability assumption later removed by Van Daele in a new existence
+proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-
+empty weak∗-compact convolution-closed convex subset of the state space. Such subsets,
+and their associated idempotent states, are studied in the case of quantum permutation
+groups.
+Contents
+Introduction
+1
+1.
+Compact quantum groups
+5
+2.
+Pal sets and quasi-subgroups
+10
+3.
+Stabiliser quasi-subgroups
+17
+4.
+Exotic quasi-subgroups of the quantum permutation group
+23
+5.
+Convolution dynamics
+29
+6.
+Integer fixed points quantum permutations
+35
+References
+38
+Introduction
+It is sometimes quipped that quantum groups are neither quantum nor groups. What-
+ever about compact quantum groups not being quantum, compact quantum groups are,
+of course, not in general classical groups. On the other hand, compact Hausdorff groups
+are compact quantum groups. Furthermore, the classical theorems of the existence of
+the Haar measure, Peter–Weyl, Tannaka–Krein duality, etc., can all be viewed as special
+cases of the quantum analogues proved by Woronowicz [29, 30], and thus naturally the
+theory of compact quantum groups has many commonalities with the theory of compact
+groups.
+2020 Mathematics Subject Classification. 46L30,46L67.
+Key words and phrases. quantum permutations, idempotent states.
+1
+
+2
+J.P. MCCARTHY
+Not all classical theorems generalise so nicely:
+Theorem 0.1. (Kawada–Itˆo Theorem, [14], Th. 3) Let G be a compact separable group.
+Then a probability distribution on G is idempotent with respect to convolution if and only
+if it is the uniform distribution on a closed subgroup H ⊆ G.
+The quantum analogue of a closed subgroup, H ⊆ G, is given by a comultiplication-
+respecting surjective *-homomorphism π : C(G) → C(H), and the direct quantum ana-
+logue of the Kawada–Itˆo theorem would be that each state idempotent with respect to
+convolution is a Haar idempotent, that is a state on C(G) of the form hC(H) ◦ π (where
+hC(H) is the Haar state on C(H)). However in 1996 Pal discovered non-Haar idempotents
+in the Kac–Paljutkin quantum group [20], and thus the direct quantum analogue of the
+Kawada–Itˆo theorem is false (in fact there are counterexamples in the dual of S3, an even
+‘smaller’ quantum group [8]).
+The null-spaces of Pal’s idempotent states are only one-sided ideals. Starting with [8],
+Franz, Skalski and coauthors undertook a general study of idempotent states on compact
+quantum groups, and, amongst other results, showed that the null-space being a one-
+sided rather than two-sided ideal is the only obstruction to an idempotent being Haar
+(Proposition 2.21). In the case of quantum permutation groups, interpreting elements
+of the state space as quantum permutations, called the Gelfand–Birkhoff picture in [17],
+leads to the consideration of distinguished subsets of the state space. In [17], using the
+fact that idempotent states in the case of finite quantum groups have group-like support
+([8], Cor. 4.2), subsets of the state space are associated to idempotent states. The current
+work generalises this point of view: the subset associated to an idempotent state φ on
+the algebra of continuous functions on a quantum permutation group G is called a quasi-
+subgroup (after [12]), and given by the set of states absorbed by the idempotent:
+Sφ = {ϕ ∈ S(C(G)): ϕ ⋆ φ = φ = φ ⋆ ϕ}.
+Whenever a quasi-subgroup is given by a (universal) Haar idempotent, it is stable under
+wave-function collapse (see Definition 2.14). There is an obvious relationship between
+ideals and wave-function collapse: that all classical quasi-subgroups are subgroups is just
+another way of saying that there are no one-sided ideals in the commutative case. An
+equivalence between Haar idempotent states and the stability of the associated quasi-
+subgroup under wave-function collapse is not proven here, but there is a partial result
+(Theorem 2.23).
+The other theme of the study of Franz, Skalski and coauthors is the relationship be-
+tween idempotent states and group-like projections, and culminates in a comprehensive
+statement about idempotent states being group-like projections in the multiplier algebra
+of the dual discrete quantum group [8]. This work contains no such comprehensive state-
+ment, but does extend the definition of continuous group-like projections p ∈ C(G) to
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+3
+group-like projections p ∈ C(G)∗∗, the bidual. Idempotent states with group-like support
+projection are particularly well-behaved, however it is shown that in the non-coamenable
+case the support projection of the Haar state is not group-like.
+The consideration of subsets of the state space leads directly to the key observation in
+this work that non-empty weak∗-compact convolution-closed convex subsets S of the state
+space, which are termed Pal sets, contain S-invariant idempotent states φS:
+ϕ ⋆ φS = φS = φS ⋆ ϕ
+(ϕ ∈ S).
+This observation is via Van Daele’s proof of the existence of the Haar state [26] (os-
+tensibly for the apparently esoteric and pathological non-separable case). This observa-
+tion yields new examples of (generally) non-Haar idempotent states in the case of quan-
+tum permutation groups: namely from the stabiliser quasi-subgroups of Section 3. Pal
+sets, through their idempotent state, generate quasi-subgroups. Consider S3 ⊂ S+
+4 via
+C(S+
+4 ) → C(S+
+4 )/⟨u44 = 1⟩: this study yields the interesting example of an intermediate
+quasi-subgroup
+S3 ⊊ (S+
+4 )4 ⊊ S+
+4 .
+Where h is the Haar state on C(S+
+4 ), the (non-Haar) idempotent in (S+
+4 )4 is given by:
+φ(f) = h(u44fu44)
+h(u44)
+(f ∈ C(S+
+4 )).
+The quasi-subgroup shares many properties of the state space of C(S3), namely it is
+closed under convolution, closed under reverses ([17], (5.1)), and contains an identity for
+the convolution (i.e. the counit). Moreover, if any quantum permutation ϕ ∈ (S+
+4 )4 is
+measured with u44 ∈ C(S+
+4 ) (in the sense of the Gelfand–Birkhoff picture), it gives one
+with probability one (i.e. it fixes label four). However, while it contains states non-zero on
+the commutator ideal of C(S+
+4 ), this isn’t a quantum permutation group on three labels
+because (S+
+4 )4 is not closed under wave-function collapse (the null-space of φ is one-sided).
+A famous open problem in the theory of quantum permutation groups is the maxi-
+mality conjecture: that the classical permutation group SN ⊆ S+
+N is a maximal quantum
+subgroup. Following on from Section 6.3 of [17], the current work considers the possibil-
+ity of an exotic intermediate quasi-subgroup strictly between the classical and quantum
+permutation groups. An attack on the maximality conjecture via such methods is not a
+priori particularly promising, but some basic analysis of the support projections of the
+characters might be useful in the future. This analysis shows that the support projec-
+tion of the Haar idempotent associated with SN ⊂ S+
+N is a group-like projection in the
+bidual. One consequence of this is Theorem 4.8 which says that hSN and any “genuinely
+quantum permutation” generates a quasi-subgroup strictly bigger than SN, i.e. an idem-
+potent state between hSN and the Haar state on C(S+
+N). It isn’t hSN, but it could be
+(1) a non-Haar idempotent; or, for some N ≥ 6, (2) the Haar idempotent from an exotic
+
+4
+J.P. MCCARTHY
+quantum subgroup SN ⊊ GN ⊊ S+
+N; or (3) the Haar state on C(S+
+N). If it is always
+(3), a strictly stronger statement than the maximality conjecture, then the maximality
+conjecture holds.
+Using the Gelfand–Birkhoff picture, this particular analysis allows us to consider the
+(classically) random and truly quantum parts of a quantum permutation, and there are
+some basic rules governing the convolution of (classically) random quantum permutations
+and truly quantum permutations. Some consequences of these are explored: for example,
+an idempotent state on C(S+
+N) is either random, or “less than half” random (Corollary
+5.11).
+The paper is organised as follows. Section 1 introduces compact quantum groups, and
+discusses Van Daele’s proof of the existence of the Haar state. Key in this work is the
+restriction to universal algebras of continuous functions, and the reasons for this restriction
+are explained. A further restriction to quantum permutation groups is made, and finally
+some elementary properties of the bidual are summarised. Section 2 introduces Pal sets,
+and asserts that they contain idempotent states. Quasi-subgroups are defined to fix the
+non-injectivity of the association of a Pal set to its idempotent state.
+The definition
+of a group-like projection is extended to include group-like projections in the bidual,
+and the interplay between such group-like projections and idempotent states is explored.
+Wave-function collapse is defined, and the question of stability of a quasi-subgroup under
+wave-function collapse studied. In Section 3, stabiliser quasi-subgroups are defined, and
+it is shown that there is a strictly intermediate quasi-subgroup between S+
+N−1 ⊂ S+
+N
+and S+
+N. In Section 4, exotic quasi-subgroups of S+
+N are considered (and by extension
+exotic quantum subgroups). Necessarily this section talks about the classical version of a
+quantum permutation group. The support projections of characters are studied, and it is
+proved that the sum of these is a group-like projection in the bidual. In the case of S+
+N, this
+group-like projection is used to define the (classically) random and truly quantum parts of
+a quantum permutation, and it is proven that the Haar idempotent coming from SN ⊂ S+
+N
+together with a quantum permutation with non-zero truly quantum part generates a non-
+classical quasi-subgroup in S+
+N that is strictly bigger than SN (but possibly equal to S+
+N).
+In Section 5 the convolution of random and truly quantum permutations is considered,
+and as a corollary a number of quantitative and qualitative results around the random
+and truly quantum parts of convolutions. In Section 6 there is a brief study of the number
+of fixed points of a quantum permutation, and it is shown that as a corollary of never
+having an integer number of fixed points, the Haar state is truly quantum.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+5
+1. Compact quantum groups
+1.1. Definition and the Haar state.
+Definition 1.1. An algebra of continuous functions on a (C∗-algebraic) compact quan-
+tum group G is a C∗-algebra C(G) with unit 1G together with a unital ∗-homomorphism
+∆ : C(G) → C(G) ⊗ C(G) into the minimal tensor product that satisfies coassociativity
+and Baaj–Skandalis cancellation:
+∆(C(G))(1G ⊗ C(G)) = ∆(C(G))(C(G) ⊗ 1G) = C(G) ⊗ C(G).
+Woronowicz defined compact matrix quantum groups [28], and extended this definition
+to compact quantum groups [30]. In order to establish the existence of a Haar state,
+Theorem 1.2 below, Woronowicz assumed that the algebra of functions was separable.
+Shortly afterwards Van Daele removed this condition [26], and established the existence
+of a Haar state in the non-separable case. The quantum groups in the current work are
+compact matrix quantum groups, which are separable, however, a careful study of Van
+Daele’s proof suggests further applications. Therefore, Van Daele’s proof will be teased
+out in some detail, and then adapted in Section 2. Note that while Lemmas 1.3 and 1.4
+are attributed here to Van Daele, it is pointed out by Van Daele that the techniques of
+their proofs were largely present in the work of Woronowicz.
+Define the convolution of states ϕ1, ϕ2 on C(G):
+ϕ1 ⋆ ϕ2 := (ϕ1 ⊗ ϕ2)∆.
+Theorem 1.2 ([26, 30]). The algebra of continuous functions C(G) on a compact quantum
+group admits a unique invariant state h, such that for all states ϕ on C(G):
+h ⋆ ϕ = h = ϕ ⋆ h.
+Lemma 1.3 ([26], Lemma 2.1). Let ϕ be a state on C(G). There exists a state φϕ on
+C(G) such that
+ϕ ⋆ φϕ = φϕ = ϕ ⋆ φϕ.
+Proof. Define
+ϕn = 1
+n(ϕ + ϕ⋆2 + · · · + ϕ⋆n).
+As the state space S(C(G)) is convex and closed under convolution, (ϕn)n≥1 ⊂ S(C(G)).
+Via the weak*-compactness of the state space, Van Daele shows that φϕ, a weak*-limit
+point of (ϕn)n≥1, is ϕ-invariant.
+□
+Lemma 1.4 ([26], Lemma 2.2). Let ϕ and φ be states on C(G) such that ϕ ⋆ φ = φ. If
+ρ ∈ C(G)∗ and 0 ≤ ρ ≤ ϕ, then also ρ ⋆ φ = ρ(1G)φ.
+
+6
+J.P. MCCARTHY
+Proof of Theorem 1.2. Where S(C(G)) is the state space of C(G), for each positive linear
+functional ω on C(G), define:
+Kω := {ϕ ∈ S(C(G)) : ω ⋆ ϕ = ω(1G)ϕ}.
+As per Van Daele, Kω is closed and thus compact with respect to the weak*-topology.
+It is non-empty because ω can be normalised to a state �ω on C(G), and by Lemma 1.3,
+there exists φω ∈ K�ω and thus φω ∈ Kω.
+Let φ ∈ Kω1+ω2. Note that both ω1, ω2 ≤ ω1 +ω2, and so by Lemma 1.4, φ ∈ Kω1 ∩Kω2
+so that:
+Kω1+ω2 ⊂ Kω1 ∩ Kω2.
+Assume that the intersection of the Kω over the positive linear functionals on C(G) is
+empty. Thus, where the complement is with respect to S(C(G)):
+�
+ω pos. lin. func.
+Kc
+ω = S(C(G)),
+is an open cover of a compact set, and thus admits a finite subcover {Kc
+ωi : i = 1, . . . , n}
+such that
+n�
+i=1
+Kc
+ωi = S(C(G)) =⇒
+n�
+i=1
+Kωi = ∅.
+Let ψ = �n
+i=1 ωi: the set Kψ is non-empty. It is also a subset of:
+n�
+i=1
+Kωi = ∅,
+an absurdity, and so the intersection of all the Kω is non-empty, and thus there is a state
+h that is left-invariant for all positive linear functionals and thus for S(C(G)).
+□
+1.2. The universal and reduced versions. A reference for this section is Timmer-
+mann [24]. A compact quantum group has a dense Hopf*-algebra of regular functions,
+O(G). The algebra of regular functions has a minimal norm-completion, the reduced
+algebra of continuous functions, Cr(G), the image of the GNS representation associated
+to the Haar state; and a maximal norm-completion, the universal algebra of continuous
+functions, Cu(G). The compact quantum group G is coamenable if O(G) has a unique
+norm-completion to an algebra of continuous functions on a compact quantum group,
+and so in particular Cr(G) ∼= Cu(G). The Haar state is faithful on O(G) and Cr(G), but
+Cr(G) does not in general admit a character. On the other hand, Cu(G) does admit a
+character, but the Haar state is no longer faithful in general.
+After an abelianisation πab : C(G) → C(G)/Nab, and via Gelfand’s theorem, the algebra
+of continuous functions on the classical version of a compact quantum group is given by
+the algebra of continuous function on the set of characters. However, not every completion
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+7
+Cα(G) of O(G) admits a classical version: in particular, when G is not coamenable the
+abelianisation of Cr(G) is zero, and Cr(G) admits no characters. This work includes a
+study of the classical versions of quantum permutation groups G ⊆ S+
+N, and working at
+the universal level ensures that talking about the classical version G ⊆ G makes sense.
+The quantum subgroup relation H ⊆ G will be given at the universal level: a quantum
+subgroup is given by a surjective *-homomorphism π : Cu(G) → Cu(H) that respects the
+comultiplication in the sense that:
+∆Cu(H) ◦ π = (π ⊗ π) ◦ ∆.
+Every such morphism of algebras of continuous function Cu(G) → Cu(H) restricts to a
+morphism on the level of regular functions O(G) → O(H); and every morphism O(G) →
+O(H) extends to the level of universal algebras of continuous functions [6].
+Key in this work is the notion of a quasi-subgroup Sφ ⊆ S(Cu(G)), defined as the set
+of states ϕ that are absorbed by a given idempotent state φ on Cu(G):
+ϕ ⋆ φ = φ = φ ⋆ ϕ.
+If hH := hCα(H) ◦ π is a Haar idempotent associated with π : C(G) → Cα(H), it is the case
+that
+{ϕ ◦ π : ϕ ∈ S(Cα(H))} ⊆ ShH.
+Remark 1.5. As explained by Stefaan Vaes1 [25], in general this is not an equality. In
+particular the Haar state of Cr(G) in Cu(G),
+hr := hCr(G) ◦ πr,
+is in fact equal to the Haar state on Cu(G). Thus the quasi-subgroup generated by hr
+is the whole state space of Cu(G), but in the non-coamenable case there are states on
+Cu(G), such as the counit, that do not factor through πr, and thus in this case:
+{ϕ ◦ πr : ϕ ∈ S(Cr(G))} ⊊ Shr.
+Vaes goes on to prove that in the universal case of π : Cu(G) → Cu(H) that indeed:
+(1)
+{ϕ ◦ π : ϕ ∈ H} = ShH,
+and this is more satisfactory for a theory of quasi-subgroups. Note that Vaes’s observation
+yields Theorem 4.6 as a special case.
+There are issues related to the non-faithfulness of the Haar state on Cu(G). For example,
+suppose that π : Cr(G) → Cr(H) is a comultiplication-preserving quotient map and
+consider the Haar idempotent:
+φ := hCr(H) ◦ π.
+1it is believed that (1) is not in the literature, however as its proof requires representation theory, not
+used in the current work, Vaes’s proof is omitted
+
+8
+J.P. MCCARTHY
+As the Haar state is faithful on Cr(H), the null-space Nφ of φ coincides with ker π, and the
+support projection pφ ∈ Cr(G)∗∗ gives a nice direct sum structure to the bidual Cr(G)∗∗.
+For a non-coamenable compact quantum group H, and a quotient π : Cu(G) → Cu(H),
+the inclusion ker π ⊂ Nφ can be proper:
+Cu(G) → Cu(H) → Cu(H)/Nφ,
+with the final algebra of continuous functions isomorphic to Cr(H) ̸∼= Cu(H) [6].
+From this point on, all algebras of continuous functions will be assumed
+universal, C(G) ∼= Cu(G). Careful readers can extract results which hold more generally.
+1.3. Quantum Permutation Groups. Let C(X) be a C∗-algebra with unit 1X.
+A
+(finite) partition of unity is a (finite) set of projections {pi}N
+i=1 ⊂ C(X) that sum to the
+identity:
+N
+�
+i=1
+pi = 1X.
+Definition 1.6. A matrix u ∈ MN(C(X)) is a magic unitary if the rows and columns are
+partitions of unity:
+N
+�
+k=1
+uik = 1X =
+N
+�
+k=1
+ukj
+(1 ≤ i, j ≤ N).
+Consider the universal unital C∗-algebra:
+C(S+
+N) := C∗(uij : u an N × N magic unitary).
+Define
+(2)
+∆(uij) =
+N
+�
+k=1
+uik ⊗ ukj.
+Using the universal property, Wang [27] shows that ∆ is a *-homomorphism, and S+
+N is
+a compact quantum group, called the quantum permutation group on N symbols. Note
+S+
+N is not coamenable for N ≥ 5.
+Definition 1.7. Let G be a compact quantum group. A magic unitary u ∈ MN(C(G))
+whose entries generate C(G) as a C∗-algebra, and such that ∆(uij) is given by (2), is
+called a magic fundamental representation. A compact quantum group that admits such
+a magic fundamental representation is known as a quantum permutation group, and by
+the universal property G ⊆ S+
+N.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+9
+The relation G ⊆ S+
+N yields a specific fundamental magic representation u ∈ MN(C(G)),
+and whether uij is a generator of C(G) or of C(S+
+N) should be clear from context. From
+this point on, all quantum groups G will be assumed to be quantum permu-
+tations groups G ⊆ S+
+N. Again, careful readers can extract results which hold more
+generally.
+The antipode is given by:
+S(uij) = uji =⇒ S2(uij) = uij,
+that is quantum permutation groups are Kac, where the antipode is a bounded linear
+map satisfying S2 = IC(G).
+Proposition 1.8. Let ϕ1, ϕ2 be states on C(G):
+(ϕ1 ⋆ ϕ2) ◦ S = (ϕ2 ◦ S) ⋆ (ϕ1 ◦ S).
+Proof. Where τ is the flip, f ⊗ g �→ g ⊗ f, in O(G):
+∆ ◦ S = (S ⊗ S) ◦ τ ◦ ∆.
+If f ∈ O(G), then using the antipodal property
+((ϕ1 ⋆ ϕ2) ◦ S)(f) = ((ϕ2 ◦ S) ⋆ (ϕ1 ◦ S))(f).
+The same holds for all f ∈ C(G) because the antipode is bounded, and the comultiplica-
+tion is a *-homomorphism, and thus both are norm-continuous.
+□
+Lemma 1.9 ([8], Section 3). If a state φ on C(G) is idempotent, φ⋆φ = ��, then φ◦S = φ.
+1.4. The Bidual. In the sequel the bidual C(X)∗∗ of a unital C∗-algebra C(X) will be
+utilised. Here some of its properties are summarised from Takesaki, Vol. I. [23]. The
+bidual admits C(G)∗ as a predual, and so is a von Neumann algebra. States ϕ on C(X)
+have extensions to states ωϕ on C(X)∗∗. Where
+Nϕ = {f ∈ C(X) : ϕ(|f|2) = 0},
+the σ-weak-closure is a σ-weakly-closed left ideal in a von Neumann algebra, and so of
+the form C(X)∗∗qϕ for some projection qϕ. The support projection of a state ϕ on C(X)
+is pϕ = 1X − qϕ. It has the property that:
+ϕ(f) = ωϕ(fpϕ) = ωϕ(pϕf) = ωϕ(pϕfpϕ)
+(f ∈ C(X)),
+and it is the smallest projection p ∈ C(X)∗∗ such that ωϕ(p) = 1 (if ωϕ(p) = 1 then ϕ is
+said to be supported on p, and pϕ ≤ p). If N ⊆ C(X) is an ideal, then N∗∗ ⊆ C(X)∗∗ is
+σ-weakly closed, and so equal to C(X)∗∗q for a central projection q ∈ C(X)∗∗. Then, as
+C∗-algebras:
+(3)
+C(X)∗∗ ∼= (C(X)/N)∗∗ ⊕ N∗∗.
+
+10
+J.P. MCCARTHY
+The embedding C(X) ⊂ C(X)∗∗ is an isometry, so that C(X) is norm closed, and
+the norm closure of a norm dense *-subalgebra O(X) ⊆ C(X) in C(X)∗∗ is C(X). In
+addition, the σ-weak closures of O(X) and C(X) are both C(X)∗∗. A *-homomorphism
+T : C(X) → C(Y) extends to a σ-weakly continuous *-homomorphism:
+T ∗∗ : C(X)∗∗ → C(Y)∗∗.
+In particular, the extension of a character on C(X) is a character on C(X)∗∗, and thus
+the support projections of characters in C(X) are minimal projections in C(X)∗∗.
+The product on the bidual is separately σ-weakly continuous:
+�
+lim
+λ fλ
+�
+f = lim
+λ (fλf)
+(fλ, f ∈ C(X)∗∗).
+Via the Sherman–Takeda Theorem [21, 22], projections p1, . . . , pN ∈ C(X) may be
+viewed as Hilbert space projections. Then
+(4)
+lim
+n→∞[(p1 · · · pN)n] = p1 ∧ · · · ∧ pN,
+strongly [11]. The powers of products of projections are in the unit ball. The strong and
+σ-strong coincide on the unit ball, and σ-strong convergence implies σ-weak convergence
+of (4). Finally, for any Borel set E ⊆ σ(f) of self-adjoint f ∈ C(X), the spectral projection
+1E(f) ∈ C(X)∗∗.
+2. Pal sets and quasi-subgroups
+2.1. Pal sets. The following notation/terminology is outlined in [17] and used hereafter:
+Definition 2.1. Given a quantum permutation group G, the Gelfand–Birkhoff picture
+interprets elements of the state-space as quantum permutations, so that ϕ ∈ G means ϕ
+is a state on C(G), and a subset of the state space S(C(G)) can be denoted S ⊆ G.
+Definition 2.2. A subset S ⊆ G is closed under convolution if
+ϕ, ρ ∈ S =⇒ ϕ ⋆ ρ ∈ S.
+A subset S is closed under reverses if
+ϕ ∈ S =⇒ (ϕ ◦ S) ∈ S.
+A subset S contains the identity if C(G) admits a counit ε, and ε ∈ S.
+Proposition 2.3. Suppose that π : C(G) → C(H) gives a (closed) quantum subgroup
+H ⊆ G. Then the set:
+H⊆G := {ϕ ◦ π : ϕ ∈ H},
+is closed under convolution, and closed under reverses.
+There are subsets S ⊂ G that are closed under convolution, closed under reverses, and
+contain the identity that are not associated with quantum subgroups in this way.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+11
+Example 2.4. Let Γ be a finite group with a non-normal subgroup Λ ⊂ Γ. The state space
+of C(�Γ), denoted here �Γ, is the set of positive-definite functions on Γ. Define:
+(5)
+SΛ = {ϕ ∈ �Γ : ϕ(λ) = 1 for all λ ∈ Λ}.
+The convolution for states on C(�Γ) is pointwise multiplication, therefore SΛ is closed
+under convolution. The reverse of ϕ ∈ �Γ is:
+(ϕ ◦ S)(γ) = ϕ(γ−1),
+and Λ is a group so SΛ is closed under reverses. The identity, 1Γ ∈ SΛ.
+Example 2.5. Let G0 be the Kac–Paljutkin quantum group with algebra of functions
+C(G0) = Cf1 ⊕ Cf2 ⊕ Cf3 ⊕ Cf4 ⊕ M2(C).
+Where f i is dual to fi, and Eij is dual to the matrix unit Eij in the M2(C) factor, the
+convex hulls co({f 1, f 4, E11}) and co({f 1, f 4, E22}) are closed under convolution, under
+reverses, and contain the identity, ε = f 1.
+Example 2.6. Let G be a quantum permutation group with uii ∈ C(G) non-central. Define
+a subset Gi ⊂ G by:
+Gi := {ϕ ∈ G : ϕ(uii) = 1}.
+This set is closed under convolution, and closed under reverses because S(uii) = uii.
+Finally ε ∈ Gi as ε(uij) = δi,j. More in Section 3.
+Definition 2.7. A Pal set is a non-empty convex weak*-closed subset S ⊆ G that is closed
+under convolution.
+Theorem 2.8. A Pal set S ⊆ G contains a unique S-invariant state, φS ∈ S, such that
+for all ϕ ∈ S:
+φS ⋆ ϕ = φS = ϕ ⋆ φS.
+.
+Proof. This has exactly the same proof as Theorem 1.2, except rather than defining a
+Kω for each positive linear functional ω on C(G), they are defined only for each ω ∈
+cone(S).
+□
+The strength of the notion of a Pal set is that, as will be seen in Section 3, they can be
+easy to describe, and yield idempotent states with certain properties. The problem with
+Definition 2.7 is that Pal sets are not in general sub-objects, not state-spaces of algebras
+of continuous functions on a compact quantum group. It is possible to talk about compact
+quantum hypergroups in this setting [8, 9, 15], but this avenue will not be pursued here.
+Furthermore, the correspondence S → φS is not one-to-one. For example, the Pal set H⊆G
+yields the Haar idempotent hH ◦ π. The singleton {hH ◦ π} is a Pal set with the same
+idempotent hH ◦ π.
+
+12
+J.P. MCCARTHY
+Another such non-correspondence occurs for the Pal set of central states:
+Definition 2.9. Where:
+{uα
+ij : i, j = 1, . . . , dα, α ∈ Irr(G)}
+are matrix coefficients of mutually inequivalent irreducible unitary representations, a cen-
+tral state ϕ ∈ G is one such that for all α ∈ Irr(G) there exists ϕ(α) ∈ C such that:
+ϕ(uα
+ij) = ϕ(α)δi,j.
+Proposition 2.10. The set of central states is a Pal set with idempotent state h ∈ G.
+In [10], an S+
+N analogue of the measure on SN constant on transpositions, a central
+state ϕtr on C(S+
+N), is studied, and it is shown that the convolution powers (ϕ⋆k
+tr )k≥0 are
+a sequence of central states converging to the Haar state.
+2.2. Quasi-subgroups. To fix the non-injectivity of the association of a Pal set S with
+an idempotent φS, is to define a quasi-subgroup. This nomenclature of quasi-subgroup is
+inspired by Kasprzak and So�ltan [12].
+Proposition 2.11. Given an idempotent state φ ∈ G, the set:
+(6)
+Sφ := {ϕ ∈ G: ϕ ⋆ φ = φ = φ ⋆ ϕ}
+is a Pal set with idempotent state φ.
+Proof. By associativity, Sφ is closed under convolution. Convexity is straightforward. For
+weak*-closure, let (ϕλ) ⊆ Sφ converge to ϕ ∈ G, and take f ∈ O(G):
+(ϕ ⋆ φ)(f) =
+�
+ϕ(f(1))φ(f(2)) =
+� �
+lim
+λ ϕλ(f(1))
+�
+φ(f(2))
+= lim
+λ
+�
+ϕλ(f(1))φ(f(2)) = lim
+λ ((ϕλ ⋆ φ)(f)) = lim
+λ φ(f) = φ(f)
+□
+Definition 2.12. A quasi-subgroup is a subset of the state space of the form Sφ for an
+idempotent state φ on C(G); the quasi-subgroup generated by φ.
+The quasi-subgroup Sφ is the largest Pal set with idempotent φ, and there is a one-to-
+one correspondence between quasi-subgroups and idempotent states.
+2.3. Group-like projections. Group-like projections (and their link with idempotent
+states) were first noted by Lanstad and Van Daele [15]. This definition can be extended
+to the bidual:
+Definition 2.13. A group-like projection p ∈ C(G)∗∗ is a non-zero projection such that:
+∆∗∗(p)(1G ⊗ p) = p ⊗ p.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+13
+In the finite case, there is a bijective correspondence between idempotent states and
+group-like projections: every idempotent state has group-like density with respect to the
+Haar state [8] (and this group-like density coincides with the support projection [17]). In
+the compact case, continuous group-like projections p ∈ C(G) with h(p) > 0 give densities
+to idempotent states via the Fourier transform, p �→ h(·p)/h(p), but the converse does
+not hold (see Section 4 and Corollary 6.3). However it is shown here that every group-like
+projection in the bidual yields a Pal set, and thus an idempotent state, but as seen in
+Proposition 2.20 a converse statement does not hold. In general, it can only be said that
+idempotent states are associated with group-like projections in the multiplier algebra of
+the dual discrete quantum group [8].
+The language of wave-function collapse will be used talk about idempotent states with
+group-like density, and later illustrate the difference between Haar and non-Haar idem-
+potents:
+Definition 2.14. Let q ∈ C(G)∗∗ be a projection and ϕ ∈ G. If ωϕ(q) > 0, then ϕ
+conditioned by q = 1 is given by:
+�qϕ(g) := ωϕ(qgq)
+ωϕ(q)
+(g ∈ C(G)),
+and ϕ → �qϕ is referred to as wave-function collapse. Furthermore, say that a subset
+S ⊆ G is stable under wave-function collapse if for all projections q ∈ C(G)∗∗,
+(7)
+(ϕ ∈ S and ωϕ(q) > 0) =⇒ �qϕ ∈ S.
+The following is well known in the algebraic setting ([15], Prop. 1.8), and a similar
+proof is known to work in the finite quantum group setting ([8], Corollary 4.2). For the
+benefit of the reader, the proof is reproduced in the current setting:
+Proposition 2.15. If p ∈ C(G) is a continuous group-like projection such that h(p) > 0,
+then �ph ∈ G is an idempotent state.
+Proof. Let φ = �ph. The difference between ωh and h can be suppressed here as ωh |C(G) = h.
+Let f ∈ O(G):
+(φ ⋆ φ)(f) =
+1
+h(p)2
+�
+h(pf(1)p)h(pf(2)p) =
+1
+h(p)2
+�
+h(f(1)p)h(f(2)p)
+=
+1
+h(p)2(h ⊗ h) (∆(f)(p ⊗ p)) =
+1
+h(p)2(h ⊗ h) (∆(f)∆(p)(1G ⊗ p))
+=
+1
+h(p)2(h ⊗ h) (∆(fp)(1G ⊗ p)) =
+1
+h(p)2h(fp)h(p) = h(pfp)
+h(p)
+= φ(f),
+where the traciality of the Haar state, p2 = p, and (h⊗ϕ)(∆(f)(1G⊗g)) = h(f)ϕ(g) ([24],
+Remark 2.2.2 i.) were used. By norm-continuity this implies that �ph is idempotent.
+□
+
+14
+J.P. MCCARTHY
+Note that it is not claimed that the support projection of �ph ∈ G is p. In the below
+this is assumed, and a nice description of the quasi-subgroup follows
+Proposition 2.16. Let φ = �pφh be an idempotent with continuous group-like support
+projection pφ ∈ C(G). Then
+Sφ = {ϕ ∈ G : ϕ(pφ) = 1}.
+Proof. Suppose that ϕ(pφ) = 1. Similarly to the proof of Proposition 2.15, for f ∈ O(G):
+(8)
+(φ ⋆ ϕ)(f) =
+1
+h(pφ)(h ⊗ ϕ)(∆(fpφ)(1G ⊗ pφ)) = h(fpφ)
+h(pφ) ϕ(pφ) = φ(f),
+and by weak*-continuity, φ ⋆ ϕ = φ. On the other hand, suppose that φ ⋆ ϕ = φ so that
+ϕ ∈ Sφ, the quasi-subgroup generated by φ. Applying (8) at f = pφ, with the existence
+of �ph implying h(pφ) > 0:
+(φ ⋆ ϕ)(pφ) = h(pφ)
+h(pφ)ϕ(pφ) = φ(pφ) = 1 =⇒ ϕ(pφ) = 1.
+□
+Proposition 2.17. If states ϕ1, ϕ2 on C(G) are supported on a group-like projection
+p ∈ C(G)∗∗, then so is ϕ1 ⋆ ϕ2.
+Proof. The proof for the finite case ([16], Prop. 3.12) applies with some adjustments. Let
+(pλ) ⊂ O(G) converge σ-weakly to p ∈ C(G)∗∗. As the extension of ∆ to ∆∗∗ is σ-weakly
+continuous
+lim
+λ
+�
+∆(pλ)
+�
+(1 ⊗ p) = p ⊗ p
+The product is separately continuous, and ωϕ1 ⊗ ωϕ2 is σ-weakly continuous.
+=⇒ lim
+λ (ωϕ1 ⊗ ωϕ2)
+�
+pλ
+(0) ⊗ pλ
+(1)p = (ωϕ1 ⊗ ωϕ2)(p ⊗ p)
+=⇒ lim
+λ
+�
+ωϕ1(pλ
+(0))ωϕ2(pλ
+(1)p) = 1
+Note that as ϕ2 is supported on p:
+=⇒ lim
+λ
+�
+ϕ1(pλ
+(0))ϕ2(pλ
+(1)) = 1
+=⇒ lim
+λ (ϕ1 ⋆ ϕ2)(pλ) = 1
+=⇒ lim
+λ ωϕ1⋆ϕ2(pλ) = ωϕ1⋆ϕ2(p) = 1.
+□
+Proposition 2.18. Suppose p ∈ C(G)∗∗ is a group-like projection. Then:
+{ϕ ∈ G : ωϕ(p) = 1},
+is a Pal set, and so there is an idempotent φ supported on p such that pφ ≤ p.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+15
+Proof. First {ϕ ∈ G : ωϕ(p) = 1} is non-empty because p is normal and as ∥p∥C(G)∗∗ = 1,
+there exists a state ω on C(G)∗∗ such that ω(p) = 1 [19], whose restriction to C(G)
+is a state in Sp.
+Weak*-closure and convexity are straightforward, and closure under
+convolution follows from Proposition 2.17.
+□
+Note that p is not necessarily equal to the support projection of the idempotent state
+in {ϕ ∈ G : ωϕ(p) = 1}; and in the below the idempotent state in {ϕ ∈ G : ωϕ(p) = 1}
+is not necessarily equal to φ.
+Theorem 2.19. Suppose that an idempotent state φ ∈ G has group-like support projection
+p ∈ C(G)∗∗. Then the quasi-subgroup generated by φ:
+Sφ ⊆ {ϕ ∈ G : ωϕ(p) = 1}.
+Proof. Consider ϕ ∈ Sφ not supported on p. Then, where q = 1G−p, ωϕ(q) > 0. Consider
+ωϕ(q · q) ∈ C(G)∗ and note by Cauchy–Schwarz:
+0 ≤ ωϕ(q · q) ≤ ϕ.
+Then by Lemma 1.4:
+ωϕ(q · q) ⋆ φ = ωϕ(q1Gq)φ = ωϕ(q)φ,
+and it follows that �qϕ ∈ Sφ. Note �qϕ(p) = 0.
+Using similar notation and techniques to Proposition 2.17, apply the σ-weakly contin-
+uous ω�qϕ ⊗ ωφ to both sides of ∆∗∗(1G ⊗ p) = p ⊗ p, using the fact that p is the support
+of φ:
+=⇒ lim
+λ
+��
+ω�qϕ(pλ
+(0)) ⊗ ωφ(pλ
+(1)p)
+�
+= ω�qϕ(p) ⊗ ωφ(p)
+=⇒ lim
+λ
+��
+�qϕ(pλ
+(0)) ⊗ ωφ(pλ
+(1)p)
+�
+= 0
+=⇒ lim
+λ
+��
+�qϕ(pλ
+(0)) ⊗ φ(pλ
+(1))
+�
+= 0
+=⇒ lim
+λ
+�
+(�qϕ ⋆ φ)(pλ)
+�
+= 0
+=⇒ lim
+λ
+�
+φ(pλ)
+�
+= 0
+=⇒ ωφ(p) = 0,
+a nonsense, so ωϕ(q) = 0, and so ωϕ(p) = 1.
+□
+It is not the case that every idempotent state φ has group-like support projection
+pφ ∈ C(G)∗∗. Nor does Theorem 2.19 hold more generally:
+Corollary 2.20. Suppose G is non-coamenable. Then the support projection ph ∈ C(G)∗∗
+of the Haar state is not a group-like projection. Furthermore:
+{ϕ ∈ G : ωϕ(ph) = 1} ⊊ Sh.
+
+16
+J.P. MCCARTHY
+Proof. Assume that the support ph ∈ C(G)∗∗ is a group-like projection. As ωh(1G) = 1,
+1G − ph > 0 strictly as G is at the universal level and G is assumed non-coamenable.
+Therefore there exists a state ωϕ on C(G)∗∗ such that
+ωϕ(1G − ph) = 1 =⇒ ωϕ(ph) = 0.
+Restrict ωϕ to a state ϕ on C(G). By Theorem 2.19 it follows that ϕ is not invariant
+under the Haar state, which is absurd as Sh = G.
+□
+There is a group-like projection p such that
+{ϕ ∈ G : ωϕ(p) = 1} = Sh;
+the unit p = 1G.
+Note there is a relationship between quantum subgroups and wave-function collapse:
+Proposition 2.21. ([9], Th. 3.3) Let G be a compact quantum group and φ ∈ C(G)∗ an
+idempotent state. Then φ is a Haar idempotent if and only if the null-space
+Nφ = {f ∈ C(G) : φ(|f|2) = 0}
+is a two-sided ideal.
+Note in the below ωϕ0 is the extension of the state ϕ0 on C(H) to a state on C(H)∗∗.
+Lemma 2.22. Suppose that H ⊆ G via π : C(G) → C(H). Then the extension of ϕ0 ◦ π
+to a state on C(G)∗∗ is given by: ωϕ0 ◦ π∗∗.
+Proof. Consider f ∈ C(G). The result follows from the σ-continuity of the maps involved,
+and π∗∗ |C(G) = π.
+□
+Note that part (i) of the below is restricted to Haar idempotents coming from Haar
+states on universal versions.
+Theorem 2.23. Suppose that φ is an idempotent state on C(G).
+(i) If φ is a (universal) Haar idempotent, then Sφ is closed under wave-function col-
+lapse.
+(ii) If φ is a non-Haar idempotent with group-like projection support, then Sφ is not
+closed under wave-function collapse.
+Proof.
+(i) Suppose φ is a (universal) Haar idempotent via π : C(G) → C(H). By
+Vaes’s Remark 1.5, every element of Sφ is of the form ϕ0 ◦ π for a state ϕ0 on
+C(H). Suppose ϕ undergoes wave-function collapse to �qϕ. Then, using Lemma
+2.22
+ωϕ(q) > 0 =⇒ ωϕ0(π∗∗(q)) > 0
+(ωϕ0 ∈ S(C(H)∗∗)).
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+17
+Using Lemma 2.22 again, it can be shown that �qϕ = ψ ◦ π, where:
+ψ(g) = ωϕ0(π∗∗(q)gπ∗∗(q))
+ωϕ0(π∗∗(q))
+(g ∈ C(H), ωϕ0 ∈ S(C(H)∗∗)).
+Thus, again by Vaes’s remark, ψ ◦ π and thus �qϕ ∈ Sφ, that is Sφ is closed under
+wave-function collapse.
+(ii) Suppose φ is a non-Haar idempotent with group-like support projection. By The-
+orem 2.19
+Sφ ⊆ {ϕ ∈ G : ωϕ(pφ) = 1}.
+As φ is a non-Haar idempotent, N∗∗
+φ
+= C(G)∗∗qφ is only a left ideal, and qφ
+non-central. Suppose that for all uij ∈ C(G), uijqφuij ∈ N∗∗
+φ . Then uijqφuij =
+uijqφuijqφ
+=⇒
+uijqφuij = uijqφuijqφuij, so that uijqφuij is a projection. This
+implies, because [uij, qφ]3 = 0 and [uij, qφ] is skew adjoint, that uijqφ = qφuij.
+Therefore qφ is central and Nφ is an ideal. Therefore there exists uij such that
+uijqφuij ̸∈ N∗∗
+φ :
+ωφ(|uijqφuij|2) > 0.
+By Cauchy–Schwarz:
+0 < ωφ(|uijqφuij|2) ≤ ωφ(uijqφuij) ≤ ωφ(uij).
+=⇒ �
+uijφ(qφ) = ωφ(uijqφuij)
+ωφ(uij)
+> 0 =⇒ �
+uijφ(pφ) < 1 =⇒ �
+uijφ ̸∈ Sφ.
+□
+3. Stabiliser quasi-subgroups
+The analysis here is helped somewhat by defining the Birkhoff slice, a map Φ from the
+state space of the algebra of continuous functions C(G) on a quantum permutation group
+G to the doubly stochastic matrices:
+Φ(ϕ) := (ϕ(uij))N
+i,j=1.
+Given a finite group G ⊆ SN and a partition P = B1 ⊔ · · · ⊔ Bk of {1, . . . , N}, the
+P-stabiliser subgroup of G can be formed:
+GP = {σ ∈ G : σ(Bp) = Bp, 1 ≤ p ≤ k}.
+A P-stabiliser quasi-subgroup of G can also be defined. There are two, equivalent, defi-
+nitions. The first definition uses the equivalence relation ∼P associated to the partition:
+GP := {ϕ ∈ G: ϕ(uij) = 0 for all i ̸∼P j}.
+Alternatively, consider the Birkhoff slice S(C(G)) → MN(C). By relabelling if necessary,
+the blocks of a partition can be assumed to consist of consecutive labels. Define:
+GP := {ϕ ∈ G : Φ(ϕ) is block diagonal with pattern P},
+
+18
+J.P. MCCARTHY
+that is:
+ϕ ∈ GP ⇐⇒ Φ(ϕ) =
+
+
+ΦB1(ϕ)
+0
+· · ·
+0
+0
+ΦB2(ϕ)
+· · ·
+0
+...
+...
+...
+· · ·
+0
+0
+· · ·
+ΦBk(ϕ)
+
+ ,
+where ΦBp(ϕ) = [ϕ(uij)]i,j∈Bp.
+Theorem 3.1. For any partition P of {1, . . . , N}, GP is a quasi-subgroup.
+Proof. That GP is convex, weak*-closed, and closed under convolution is straightforward
+(using, for example that the Birkhoff slice is multiplicative Φ(ϕ1 ⋆ ϕ2) = Φ(ϕ1)Φ(ϕ2)).
+The universal version gives ε ∈ GP so that GP is non-empty, and so a Pal set.
+Suppose that φP is the associated idempotent. Therefore by Lemma 1.9:
+φP(uij) = (φP ◦ S)(uij) = φP(uji).
+For any fixed j ∈ {1, 2, . . . , N}, there exists i ∈ {1, 2, . . ., N} such that φP(uji) > 0. From
+here:
+φP(ujj) = (φP ⋆ φP)(ujj) = φP(uji)φP(uij) +
+�
+k̸=i
+φP(ujk)φP(ukj) > 0.
+To show that GP is equal to
+SφP = {ϕ ∈ G : ϕ ⋆ φP = φP = φP ⋆ ϕ},
+suppose ϕ ∈ SφP, but ϕ ̸∈ GP. That implies there exists uij such that ϕ(uij) ̸= 0 with
+i ̸∼P j. But this gives
+φP(uij) = (ϕ ⋆ φP)(uij) = ϕ(uij)φP(ujj) +
+�
+k̸=j
+ϕ(uik)φP(ukj) > 0,
+a contradiction.
+□
+For the partition j := {j} ⊔ ({1, 2, . . ., N}\{j}):
+Gj = {ϕ ∈ G : ϕ(ujj) = 1}.
+Note for any quantum permutation group G, and 1 ≤ j ≤ N, the diagonal element ujj is
+a polynomial group-like projection:
+∆(ujj)(1G ⊗ ujj) =
+� N
+�
+k=1
+ujk ⊗ ukj
+�
+(1G ⊗ ujj) = ujj ⊗ ujj.
+Using Proposition 2.16, it can be shown that the associated idempotent state is hj := �
+ujjh,
+that is:
+hj(f) = h(ujjfujj)
+h(ujj)
+(f ∈ C(G)).
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+19
+The below is (almost) a special case of Theorem 2.23, but included as it uses different
+proof techniques.
+Theorem 3.2. The following are equivalent:
+(i) hj is a Haar idempotent,
+(ii) ujj is central,
+(iii) Gj is stable under wave-function collapse.
+Proof. (i) =⇒ (ii): assume hj is a Haar idempotent, say equal to hH◦π where π : C(G) →
+C(H), uij �→ uH
+ij, is the quotient map. Note that because hj(ujj) = hH(π(ujj)) = 1, and
+hH is faithful on O(H),
+1H = π(1G) =
+N
+�
+m=1
+π(umj) = π(ujj),
+so that π(ujj) = 1H is central in C(H). Assume that ujj is non-central. Then there exists
+ukl ∈ C(G) such that |uklujj − ujjukl|2 > 0. Expanding:
+ujjuklujj − ujjuklujjukl − uklujjuklujj + uklujjukl > 0.
+Applying the Haar state, which is faithful on O(G), and using its traciality yields:
+h(ujjuklujj) > h(ujjuklujjuklujj)
+=⇒ hj(ukl) > hj(uklujjukl)
+=⇒ hH(π(ukl)) > hH(π(uklujjukl)) = hH(π(ukl)π(ujj)π(ukl)))
+=⇒ hH(π(ukl)) > hH(π(ukl)1Hπ(ukl))) = hH(π(ukl)),
+an absurdity, and so ujj is central.
+(ii) =⇒ (i): assume that ujj is central.
+Nj := {f ∈ C(G) : hj(|f|2) = 0}.
+If f ∈ Nj then h(ujjf ∗fujj) = 0 =⇒ fujj ∈ Nh, the null-space of the Haar state, so
+that:
+Nj = {f ∈ C(G) : fujj ∈ Nh}.
+The rest of the argument is the same as ([8], Th. 4.5).
+(ii) =⇒ (iii): assume that ujj is central. If ujj is central in C(G) then it is also central
+in C(G)∗∗. Let ϕ ∈ Gj and q ∈ C(G)∗∗ such that ωϕ(q) > 0. Let pϕ ∈ C(G)∗∗ be the
+support projection of ϕ. Note that
+ωϕ(ujj) = ϕ(ujj) = 1 =⇒ pϕ ≤ ujj =⇒ pϕ = pϕujj.
+Note
+ωϕ(qujjq) = ωϕ(pϕqujjqpϕ) = ωϕ(pϕujjqqpϕ) = ωϕ(pϕqpϕ) = ωϕ(q).
+
+20
+J.P. MCCARTHY
+It follows that:
+�qϕ(ujj) = ωϕ(qujjq)
+ωϕ(q)
+= 1 =⇒ �qϕ ∈ Gp.
+(iii)
+=⇒ (ii): assume now that ujj is non-central. Therefore there exists ukl ∈ C(G)
+such that:
+ujjukl ̸= uklujj.
+Represent C(G) with the universal GNS representation πGNS(C(G)) ⊆ B(H). Denote
+p := πGNS(ujj) and q := πGNS(ukl).
+As pq ̸= qp, using Halmos two projections theory there exists a unit vector x ∈ ran p that
+is orthogonal to both2 ran p ∩ ran q and ran p ∩ ker q. Define a state on C(G):
+ϕ0(f) = ⟨x, πGNS(f)x⟩
+(f ∈ C(G)).
+Note that:
+ϕ0(ujj) = ⟨x, px⟩ = ⟨x, x⟩ = 1 =⇒ ϕ0 ∈ Gj.
+Furthermore, together with x ∈ ran p
+ϕ0(ukl) = ⟨x, qx⟩ = 1 =⇒ x ∈ ran q
+ϕ0(ukl) = ⟨x, qx⟩ = 0 =⇒ x ∈ ker q
+but x is orthogonal to both ran p ∩ ran q and ran p ∩ ker q so
+0 < ⟨x, qx⟩ < 1 =⇒ 0 < ϕ0(ukl) < 1.
+Now consider ϕ = �
+uklϕ0:
+ϕ(f) := ϕ0(uklfukl)
+ϕ0(ukl)
+= ⟨qx, πGNS(f)qx⟩
+⟨qx, qx⟩
+(f ∈ C(G)).
+In particular
+ϕ(ujj) = ⟨qx, pqx⟩
+⟨qx, qx⟩
+Together with qx ∈ ran q:
+ϕ(ujj) = 1 =⇒ qx ∈ ran p
+ϕ(ujj) = 0 =⇒ qx ∈ ker p
+By ([7], (6)), qx is orthogonal to ran p ∩ ran q and ker p ∩ ran q, and it follows that:
+0 < ϕ(ujj) < 1,
+that is,
+ϕ0 ∈ Gj but �
+uklϕ0 ̸∈ Gj.
+□
+2in the notation of ([7],(1)), x ∈ M0
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+21
+Consider, at the universal level:
+(S+
+N)N := {ϕ ∈ S+
+N : ϕ(uNN) = 1}.
+If H given by π : C(S+
+N) → C(H) is an isotropy subgroup in the sense that H ⊆ (S+
+N)N
+and so π(uNN) = 1H, then H ⊆ S+
+N−1 by the universal property. In this way, where
+πN−1 : C(S+
+N) → C(S+
+N−1) is the quotient
+[u
+S+
+N
+ij ]N
+i,j=1 →
+�
+uS+
+N−1
+0
+0
+1S+
+N−1
+�
+,
+the following is a maximal (set of states on an algebra of continuous functions on a)
+quantum subgroup in the quasi-subgroup (S+
+N)N
+(S+
+N−1)⊂S+
+N = {ϕ ◦ πN−1 : ϕ ∈ S+
+N−1}.
+In the classical case, N ≤ 3, quasi-subgroups are subgroups, and so (S+
+N)N = (S+
+N−1)⊂S+
+N.
+However, for N ≥ 4, the inclusion is proper.
+Lemma 3.3. (Teo Banica) Consider a monomial of entries from the fundamental repre-
+sentation u ∈ M4(C(S+
+4 )):
+f = ui1j1 · · · uimjm.
+Then f can only be zero for trivial reasons; i.e. if and only if there exists 2 ≤ n ≤ m such
+that:
+δin−1,in + δjn−1,jn = 1,
+that is uin−1jn−1uinjn = 0.
+Proof. With the notation from [5], namely c1, . . . , c4 ∈ SU2 being the Pauli matrices, and
+x ∈ SU2 being a parameter, the Pauli representation of C(S+
+4 ) is:
+π(uij) = Pcixcj,
+the rank one projection on cixcj. Given unit norm ξ, Pξ(η) = ⟨η, ξ⟩ξ. By recurrence
+Pξ1 · · ·Pξm(η) = ⟨η, ξm⟩⟨ξm, ξm−1⟩ · · ·⟨ξ2, ξ1⟩ξ1.
+With η = ck, one of the Pauli matrices, therefore:
+ui1j1 · · ·uimjm(ck) = Pci1xcj1 · · · Pcimxcjm(ck)
+= ⟨ck, cimxcjm⟩⟨cimxcjm, cim−1xcjm−1⟩ · · ·⟨ci2xcj2, ci1xcj1⟩ci1xck1.
+Look at one of these inner products:
+⟨cinxcjn, cin−1xcjn−1⟩ = tr(cinxcjn(cin−1xcjn−1)∗)
+= ± tr(cinxcjncjn−1x∗cin−1)
+= ± tr(cin−1cinxcjncjn−1x∗).
+
+22
+J.P. MCCARTHY
+This vanishes for any x ∈ SU2 when one of cin−1cin or cjncjn−1 equals I2, and the other
+does not, and so when
+δin−1,in + δjn−1,jn = 1.
+□
+Proposition 3.4. Let S+
+N be the quantum permutation group on N symbols with Haar
+state h. Then, for any σ, τ ∈ SN:
+h(ui1j1 · · · uinjn) = h(uσ(i1)τ(j1) · · · uσ(in)τ(jn)).
+Proof. This is essentially ([17], Prop. 6.4), together with the fact that h is invariant.
+□
+Corollary 3.5. Let S+
+N be the quantum permutation group on N ≥ 4 symbols. Then
+|ui1j1ui2j2ui3j3|2 = 0.
+for trivial reasons only.
+Proof. Let 1 ≤ a, b, c, d, e, f ≤ 4 such that u
+S+
+4
+ab u
+S+
+4
+cd u
+S+
+4
+ef
+̸= 0. Using the quotient map
+π4 : C(S+
+N) → C(S+
+4 ), u → diag(uS+
+4 , 1S+
+4 , . . . , 1S+
+4 )
+π4(|uabucduef|2) ̸= 0 =⇒ |uabucduef|2 ̸= 0.
+Let σ(a) = i1, σ(c) = i2, σ(f) = i3 and similarly τ map b, d, f to j1, j2, j3. Proposition
+3.4 gives
+h(|ui1j1ui2j2ui3j3|2) = h(|uabucduef|2) ̸= 0 =⇒ |ui1j1ui2j2ui3j3|2 ̸= 0.
+□
+Proposition 3.6. The inclusion (S+
+N−1)⊂S+
+N ⊂ (S+
+N)N is proper for N ≥ 4.
+Proof. Note that for any (ϕ ◦ πN−1) ∈ (S+
+N−1)⊂S+
+N,
+(ϕ ◦ πN−1)(u11u2Nu11) = ϕ(πN−1(u11u2Nu11)) = ϕ(πN−1(u11)πN−1(u2N)πN−1(u11)) = 0,
+as πN−1(u2N) = 0. On the other hand, hN = �
+uNNh, the idempotent in the stabiliser
+quasi-subgroup (S+
+N)N, is not in (S+
+N−1)⊂S+
+N, because h faithful on O(S+
+N) implies
+hN(u11u2Nu11) = h(uNNu11u2Nu11uNN)
+h(uNN)
+= h(|u2Nu11uNN|2)
+h(uNN)
+> 0.
+□
+Trying to do something for more complicated partitions of {1, . . . , N}, with an (ex-
+plicit) idempotent state with a density with respect to ωh is in general more troublesome.
+Consider for example:
+Pi,j := ({1, . . . , N}\{i, j}) ⊔ {i} ⊔ {j}.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+23
+The obvious way to fix two points is to work with pi,j := uii ∧ ujj, an element of C(G)∗∗,
+and given a quantum permutation ϕ ∈ G, define a subset of G by:
+Gi,j := {ϕ ∈ G : ωϕ(pi,j) = 1}.
+Note that Gi,j = Gi ∩ Gj. However the following is not in general well defined because
+ωh(pi,j) is not necessarily strictly positive:
+φi,j := ωh(pi,j · pi,j)
+ωh(pi,j)
+,
+For example, consider the dual of the infinite dihedral group with the famous embedding
+�
+D∞ ⊂ S+
+4 . Working with alternating projection theory, and noting the Haar state on
+C(�
+D∞) is h(λ) = δλ,e,
+ωh(p1,3) = lim
+n→∞ h((u11u33)n) = lim
+n→∞
+1
+4n = 0.
+Proposition 3.7. The stabiliser quasi-subgroup (�
+D∞)1,3 is the trivial group.
+Proof. Let ϕ ∈ (�
+D∞)1,3 so that ϕ(u11) = ϕ(u33) = 1. Then Φ(ϕ) = I4 and, as will be seen
+later, by Proposition 4.1, ϕ is a character. There are four characters in �
+D∞ and only the
+counit has Birkhoff slice equal to the identity.
+□
+By Proposition 4.3, p1,3 = pε, the support projection of the counit.
+As C(�
+D∞) is
+coamenable, the Haar state is faithful and so ωh(pε) = 0 implies that pε ̸∈ C(�
+D∞) (and
+indeed p ∧ q ̸∈ C∗(p, q), the universal unital C∗-algebra generated by two projections).
+Note that in general {ε} is a quantum subgroup of any quantum permutation group in
+the sense that ε is a Haar idempotent via the quotient π : C(G) → C(e) to the trivial
+group {e} ⊆ G:
+[uij]N
+i,j=1 → diag(1C, . . . , 1C).
+4. Exotic quasi-subgroups of the quantum permutation group
+A second reason for studying Pal sets and their generated quasi-subgroups is to pos-
+tulate, or rather speculate, on, for some N ≥ 4, the existence of an exotic intermediate
+quasi-subgroup:
+SN ⊊ SN ⊊ S+
+N.
+It is currently unknown whether or not there is a Haar idempotent giving an exotic
+intermediate quantum subgroup SN ⊊ GN ⊊ S+
+N for some N ≥ 6. It is the case that
+SN = S+
+N for N ≤ 3, and for N = 4 [4] and N = 5 [1] there is no such Haar idempotent.
+Of course, if there is no exotic intermediate quasi-subgroup SN ⊊ SN ⊊ S+
+N then it is the
+case that SN is a maximal quantum subgroup of S+
+N for all N, but of course this is stronger
+than the non-existence of an exotic intermediate quantum subgroup. Indeed it is strictly
+
+24
+J.P. MCCARTHY
+stronger in the sense that given a quantum permutation group G and its classical version
+G ⊆ G (see below), the existence of a strictly intermediate quasi-subgroup G ⊊ S ⊊ G
+does not imply a strictly intermediate quantum subgroup. For example, the finite dual
+�
+A5 has trivial classical version, and for any non-trivial subgroup H ⊂ A5 the non-Haar
+idempotent 1H gives a strict intermediate quasi-subgroup:
+{ε} ⊊ SH ⊊ �
+A5.
+However �
+A5 has no non-trivial quantum subgroups because A5 is simple.
+The idea for an example of an exotic intermediate quasi-subgroup would be to find a
+Pal set given by some condition that is satisfied by the ‘elements of SN in S+
+N’ — and
+some states non-zero on a commutator [f, g] ∈ C(S+
+N) — but not by the Haar state on
+C(S+
+N). It will be seen that the ‘elements of SN in S+
+N’ correspond to the characters on
+C(S+
+N).
+4.1. The classical version of a quantum permutation group. The quotient of C(G)
+by the commutator ideal is the algebra of functions on the characters on C(G). The
+characters form a group G, with the group law given by the convolution:
+ϕ1 ⋆ ϕ2 = (ϕ1 ⊗ ϕ2)∆,
+the identity is the counit, and the inverse is the reverse ϕ−1 = ϕ ◦ S.
+This section contains some general analysis for the support projections of characters on
+algebras of continuous functions on quantum permutation groups. While passing to a von
+Neumann algebra to talk about support projections, it will not be the conventional choice
+of a von Neumann algebra associated to a compact quantum group. This conventional
+choice is the algebra:
+L∞(G) := Cr(G)′′.
+As discussed previously, the current work is at the universal level, so instead consider the
+bidual C(G)∗∗.
+As before the Birkhoff slice aids the analysis. See [17] for more, where the following
+proof is sketched.
+Proposition 4.1. A state ϕ on C(G) is a character if and only if Φ(ϕ) is a permutation
+matrix.
+Proof. If ϕ is a character,
+ϕ(uij) = ϕ(u2
+ij) = ϕ(uij)2 ⇒ ϕ(uij) = 0 or 1.
+As it is doubly stochastic, it follows that Φ(ϕ) is a permutation matrix. Suppose now that
+Φ(ϕ) = σ. Consider the GNS representation (Hσ, πσ, ξσ) associated to ϕ. By assumption
+(9)
+ϕ(uij) = ⟨ξσ, πσ(uij)(ξσ)⟩ = ⟨πσ(uij)(ξσ), πσ(uij)(ξσ)⟩ = ∥πσ(uij)(ξσ)∥2 = 0 or 1.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+25
+For f ∈ C(G), let (f (n))n≥1 ⊂ O(G) converge to f. For each f (n), (9) implies there exists
+an ∈ C such that
+πσ(f (n))(ξσ) = anξσ.
+The representation πσ is norm continuous, and so πσ(f (n)) → πσ(f), and (πσ(f (n)))n≥1 is
+Cauchy:
+∥πσ(f (m)) − πσ(f (n))∥ → 0
+=⇒ |am − an|∥ξσ∥ → 0,
+which implies that (an)n≥1 converges, to say af ∈ C. The norm convergence of f (n) → f
+implies the strong convergence of πσ(f (n)) to πσ(f):
+πσ(f)ξσ = lim
+n→∞
+�
+πσ(f (n))ξσ
+�
+= lim
+n→∞(anξσ) = afξσ.
+Therefore
+ϕ(gf) = ⟨ξσ, πσ(gf)ξσ⟩ = ⟨ξσ, πσ(g)πσ(f)(ξσ)⟩
+= ⟨ξσ, πσ(g)afξσ⟩ = af⟨ξσ, πσ(g)ξσ⟩ = ϕ(g)ϕ(f).
+□
+Define evσ : C(G) → C:
+evσ(f) := πab(f)(σ)
+(f ∈ C(G)).
+This is a *-homomorphism, but in general evσ need not be non-zero.
+Proposition 4.2. If ϕ is a state on C(G) such that Φ(ϕ) = σ, then ϕ = evσ.
+Proof. Suppose that Φ(ϕ) = σ. We know that evσ is a *-homomorphism, and by Propo-
+sition 4.1 so is ϕ.
+As C(G) admits a character, πab is non-zero.
+Furthermore, as *-
+homomorphisms they are determined by their values on the generators:
+ϕ(uij) = Φ(ϕ)ij = σij = δi,σ(j) = 1j→i(σ) = πab(uij)(σ) = evσ(uij).
+□
+The classical version of G is therefore the finite group G ⊆ SN given by:
+G := {evσ : σ ∈ SN, evσ ̸= 0}.
+References to uij in the below are in the embedding:
+C(G) ⊆ C(G)∗∗.
+Note that the proof of (i) doesn’t use minimality to show that pσ is central:
+Proposition 4.3. Associated to each character evσ on C(G) is a support projection
+pσ ∈ C(G)∗∗ such that:
+(i) pσ is a central projection in C(G)∗∗, and pσpτ = δσ,τpσ.
+(ii) pσ = uσ(1),1 ∧ uσ(2),2 ∧ . . . ∧ uσ(N),N.
+
+26
+J.P. MCCARTHY
+Proof.
+(i) Note that
+evσ(uσ(j),j) = 1 ⇒ ωσ(uσ(j),j) = 1 ⇒ pσ ≤ uσ(j),j,
+while pσuij = 0 for i ̸= σ(j). Therefore pσ commutes with all of C(G) ⊆ C(G)∗∗
+and thus, via the Sherman–Takeda Theorem, pσ is in the commutant of C(G).
+Everything in C(G)∗∗ commutes with the commutant of C(G). Any pair of per-
+mutations σ ̸= τ are distinguished by some σ(j) ̸= τ(j),
+pσpτ = pσuσ(j),juτ(j),jpτ = 0.
+(ii) Let
+qσ = uσ(1),1 ∧ uσ(2),2 ∧ . . . ∧ uσ(N),N.
+Define
+fσ := uσ(1),1 · · · uσ(N),N.
+The sequence (f n
+σ )n≥1 ⊂ C(G) converges σ-weakly to qσ. The extension ωσ of evσ
+is a character implying that:
+ωσ(qσ) = lim
+n→∞ ωσ(f n
+σ ) = 1 =⇒ pσ ≤ qσ.
+Suppose r := qσ − pσ is non-zero. Then there exists a state ωr on C(G)∗∗ such
+that ωr(r) = 1. Define a state ϕr on C(G) by:
+ϕr(f) = ωr(rfr)
+(f ∈ C(G)).
+Then ϕr(uσ(j),j) = 1 =⇒ ϕx = evσ, by Proposition 4.2, with equal extensions ωr
+and ωσ. However, in this case
+ωσ(pσ) = ωr(pσ) = 0,
+and this contradiction gives qσ = pσ.
+□
+In the following, whenever evσ = 0, then so is pσ. Properties of the bidual summarised
+in Section 1.4 are used.
+Theorem 4.4. Where G ⊆ G is the classical version, define
+pG :=
+�
+σ∈G
+pσ.
+Then pG is a group-like projection in C(G)∗∗. In addition, pG is the support projection of
+the Haar idempotent hC(G) ◦ πab.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+27
+Proof. Note pG is non-zero, as pεpG = pε. Consider pσ ̸= 0. Let (pλ
+σ) ⊂ O(G) converge
+σ-weakly to pσ ∈ C(G)∗∗. The extension of ∆ is σ-weakly continuous, and recall that pσ
+is a meet of projections in O(G):
+∆∗∗(pσ) = ∆∗∗(uσ(1),1 ∧ uσ(2),2 ∧ · · · ∧ uσ(N),N)
+= ∆(uσ(1),1) ∧ ∆(uσ(2),2) ∧ · · · ∧ ∆(uσ(N),N)
+= lim
+n→∞
+��
+∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N)
+�n�
+.
+Consider, for pτ ̸= 0
+∆(uσ(1),1)∆(uσ(2),2) · · · ∆(uσ(N),N)(1G ⊗ pτ)
+=
+�
+N
+�
+k1,...,kN=1
+uσ(1),k1uσ(2),k2 · · · uσ(N),kN ⊗ uk1,1uk2,2 · · ·ukN,N
+�
+(1G ⊗ pτ)
+Note pτ is central and
+pτukj =
+�
+pτukj,
+if k = τ(j)
+0,
+otherwise. ,
+and so
+∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)(1G ⊗ pτ)
+= uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),Npτ
+= (uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)(1G ⊗ pτ)
+Now
+∆∗∗(pσ)(1G ⊗ pτ) = lim
+n→∞
+�
+∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)
+�n (1G ⊗ pτ)
+= lim
+n→∞
+�
+∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)n(1G ⊗ pτ)
+�
+= lim
+n→∞
+�
+∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)n(1G ⊗ pτ)n�
+= lim
+n→∞
+�
+∆(uσ(1),1)∆(uσ(2),2) · · ·∆(uσ(N),N)(1G ⊗ pτ)
+�n
+= lim
+n→∞
+�
+(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)(1G ⊗ pτ)
+�n
+= lim
+n→∞
+�
+(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n(1G ⊗ pτ)n�
+= lim
+n→∞
+�
+(uσ(1),τ(1)uσ(2),τ(2) · · · uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n(1G ⊗ pτ)
+�
+= lim
+n→∞
+�
+uσ(1),τ(1)uσ(2),τ(2) · · ·uσ(N),τ(N) ⊗ uτ(1),1uτ(2),2 · · · uτ(N),N)n�
+(1G ⊗ pτ)
+= (pστ −1 ⊗ pτ)(1G ⊗ pτ) = pστ −1 ⊗ pτ.
+Finally, sum ∆∗∗(pσ)(1G ⊗ pτ) over σ, τ ∈ G.
+
+28
+J.P. MCCARTHY
+Note that C(G) = C(G)/Nab is finite dimensional, and so by (3):
+C(G)∗∗ ∼= C(G) ⊕ N∗∗
+ab.
+It follows that the support projection of hC(G) ◦ πab is pG.
+□
+4.2. The (classically) random and truly quantum parts of a quantum permu-
+tation. In the case of C(S+
+N), define pC := pSN and pQ := 1S+
+N − pC. In the rest of this
+section the Gelfand–Birkhoff picture will be used:
+ϕ ∈ S+
+N is a quantum permutation ⇐⇒ ϕ a state on C(S+
+N).
+Definition 4.5. Let ϕ ∈ S+
+N be a quantum permutation. Say that ϕ
+(i) is a (classically) random permutation if ωϕ(pQ) = 0,
+(ii) is a genuinely quantum permutation if ωϕ(pQ) > 0,
+(iii) is a mixed quantum permutation if 0 < ωϕ(pQ) < 1,
+(iv) is a truly quantum permutation if ωϕ(pQ) = 1.
+Random permutations are in bijection with probability measures ν ∈ Mp(SN):
+ϕ random
+⇐⇒ ϕ = ϕν where
+ϕν(f) :=
+�
+σ∈SN
+πab(f)(σ)ν({σ})
+(f ∈ C(S+
+N)).
+Theorem 4.6. Suppose hSN is the state on C(S+
+N) defined by hC(SN) ◦ πab. Then if
+ϕ ⋆ hSN = hSN = hSN ⋆ ϕ,
+ϕ is a random permutation.
+Proof. This follows from Theorem 2.19.
+□
+Lemma 4.7. Let ϕ, ρ be quantum permutations. The convolution operators ϕ → ρ ⋆ ϕ
+and ϕ → ϕ ⋆ ρ are weak*-continuous
+Proof. Follows from (ϕ ⋆ ρ)(f) = ϕ((IC(S+
+N) ⊗ ρ)∆(f)) = ρ((ϕ ⊗ IC(S+
+N))∆(f)).
+□
+4.3. Exotic quasi-subgroups.
+Theorem 4.8. Let ϕ ∈ S+
+N be genuinely quantum, ωϕ(pQ) > 0, and hSN ∈ S+
+N the Haar
+idempotent hC(SN) ◦ πab. Form the idempotent φϕ from the weak*-limit of Ces`aro means
+of ϕ, and then define an idempotent:
+(10)
+φ := w∗- lim
+n→∞
+1
+n
+n
+�
+k=1
+(hSN ⋆ φϕ)⋆k.
+Then the quasi-subgroup generated satisfies:
+SN ⊊ Sφ ⊆ S+
+N.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+29
+Proof. First let us show that SN ⊆ Sφ.
+For any σ ∈ SN, and φn a Ces`aro mean of
+(hSN ⋆ φϕ):
+evσ ⋆φn = φn =⇒ w∗- lim
+n→∞(evσ ⋆φn) = φ =⇒ evσ ⋆φ = φ =⇒ φ ⋆ evσ−1 = φ.
+by Proposition 1.8. Similarly evσ−1 ⋆φn → φ which implies that φ ⋆ evσ = φ, and so
+SN ⊆ Sφ
+Now suppose for the sake of contradiction that φ is random. Then
+φ ⋆ hSN = hSN = hSN ⋆ φ.
+However for all Ces`aro means φn:
+φn ⋆ ϕ = φn =⇒ φ ⋆ ϕ = φ =⇒ hSN ⋆ ϕ = hSN,
+by left convolving both sides of φ ⋆ ϕ = φ with hSN. But Theorem 4.6 says in this case
+that ϕ is random, a contradiction.
+□
+If in fact for all genuinely quantum ϕ ∈ S+
+N it is the case that Sφ = S+
+N for φ given
+by (10), then the maximality conjecture holds, and it is tenable to say that hSN and any
+genuinely quantum permutation ϕ ∈ S+
+N generates S+
+N.
+5. Convolution dynamics
+This section will explore, with respect to pQ ∈ C(S+
+N)∗∗, the qualitative dynamics of
+states on C(S+
+N) under convolution. Again, using the Gelfand–Birkhoff picture such states
+will be referred to as quantum permutations. The results of this section are illustrated
+qualitatively in a phase diagram, Figure 1.
+5.1. The convolution of random and truly quantum permutations.
+Lemma 5.1. Suppose p ∈ C(G)∗∗ is a group-like projection. Then, where q := 1G − p:
+∆∗∗(q)(1G ⊗ p) = q ⊗ p.
+Proof. Expand
+∆∗∗(p + q)(1G ⊗ p) = (1G ⊗ p),
+then multiply on the right with q ⊗ p.
+□
+Proposition 5.2. Consider quantum permutations in S+
+N:
+(i) The convolution of random permutations is random.
+(ii) The convolution of a truly quantum permutation and a random permutation is
+truly quantum.
+(iii) The convolution of a truly quantum permutations can be random, mixed, or truly
+quantum.
+Proof.
+(i) This is straightforward.
+
+30
+J.P. MCCARTHY
+(ii) Let ϕ be truly quantum, and ϕν random with extension ων. Let (pλ
+Q) ⊂ O(S+
+N)
+converge σ-weakly to pQ. Using Lemma 5.1, mimic the proof of Theorem 2.19,
+hitting both sides of
+∆∗∗(pQ)(1S+
+N ⊗ pC) = pQ ⊗ pC,
+with ωϕ ⊗ ων, to yield:
+ωϕ⋆ϕν(pQ) = 1,
+i.e. ϕ ⋆ ϕν is truly quantum.
+(iii) It will be seen in Corollary 6.3 that the Haar state is truly quantum. Note that
+for any N ≥ 4, the Kac–Paljutkin quantum group can be embedded G0 ⊂ S+
+N via
+πG0. It can be shown that E11 ◦ πG0 is truly quantum, and (E11 ◦ πG0)⋆2 = ϕν is a
+random permutation ([17], (4.6)). Let 0 ≤ c ≤ 1 and consider the truly quantum
+permutation:
+ϕ :=
+√
+1 − c (E11 ◦ πG0) + (1 −
+√
+1 − c) h.
+Then:
+ϕ⋆2 = (1 − c)ϕν + c h =⇒ ϕ⋆2(pQ) = c.
+□
+Corollary 5.3. If the convolution of two quantum permutations is a random permutation,
+then either both are random, or both are truly quantum.
+Proposition 5.4. A quantum permutation ϕ ∈ S+
+N can be written as a convex combination
+of a random permutation and a truly quantum permutation.
+Proof. If ϕ is random, or truly quantum, the result holds.
+Assume ϕ is mixed.
+The
+projections pC, pQ ∈ C(S+
+N)∗∗ are central, and thus
+ϕ = ωϕ(pC) �
+pCϕ + ωϕ(pQ) �
+pQϕ,
+and �
+pCϕ is random, while �
+pQϕ is truly quantum.
+□
+Definition 5.5. Let ϕ ∈ S+
+N be a quantum permutation. Define ϕC := �
+pCϕ, the (classi-
+cally) random part of ϕ, and ϕQ := �
+pQϕ, the truly quantum part of ϕ.
+Proposition 5.6. If ϕ ∈ S+
+N is a mixed quantum permutation with 0 < ωϕ(pQ) < 1, then
+no finite convolution power ϕ⋆k is random, or truly quantum.
+Proof. Let α := ωϕ(pQ) and write ϕ = (1 − α)ϕC + α ϕQ:
+ϕ⋆k > (1 − α)kϕ⋆k
+C
+=⇒ ωϕ⋆k(pQ) ≤ 1 − (1 − α)k,
+so no ϕ⋆k is truly quantum. In addition, ϕ⋆k = ϕ⋆ϕ⋆(k−1) cannot be random, by Corollary
+5.3, because ϕ is neither random nor truly quantum.
+□
+Definition 5.7. A quantum permutation ϕ ∈ S+
+N is called α-quantum if ωϕ(pQ) = α.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+31
+Proposition 5.8. If ϕ ∈ S+
+N is α-quantum and ρ ∈ S+
+N is β-quantum, then
+α + β − 2αβ ≤ ωϕ⋆ρ(pQ) ≤ α + β − αβ.
+Proof. Note that ϕ ⋆ ρ equals:
+(1 − α)(1 − β)(ϕC ⋆ ρC) + β(1 − α)(ϕC ⋆ ρQ) + α(1 − β)(ϕQ ⋆ ρC) + αβ(ϕQ ⋆ ρQ).
+Now apply Proposition 5.2.
+□
+Definition 5.9. Where (ϕ, ρ) = (ϕ + ρ)/2 is the mean of two quantum permutations, a
+quantum strictly 1-increasing pair of quantum permutations ϕ1, ϕ2 ∈ S+
+N are a pair such
+that:
+ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).
+A quantum strictly 2-increasing pair of quantum permutations are a pair such that:
+ω(ϕ1⋆ϕ2)⋆2(pQ) > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).
+Inductively, a quantum strictly (n + 1)-increasing pair of quantum permutations are a
+pair such that:
+ω(ϕ1⋆ϕ2)⋆(2n)(pQ) > ω(ϕ1⋆ϕ2)⋆(2n−1)(pQ) > · · · > ωϕ1⋆ϕ2(pQ) > ω(ϕ1,ϕ2)(pQ).
+Proposition 5.10. Let ϕ1 ∈ S+
+N be an α-quantum permutation, and ϕ2 ∈ S+
+N a β-
+quantum permutation.
+(i) If (α, β) ̸= (0, 0), then if α = 1/4 or β < α/(4α −1), the pair (ϕ1, ϕ2) is quantum
+strictly 1-increasing.
+(ii) If (α, β) ̸= (0, 0), and β = α/(4α − 1), then:
+ωϕ1⋆ϕ2(pQ) ≥ ω(ϕ1,ϕ2)(pQ).
+Equality is possible, with e.g. quantum permutations coming from the Kac–Paljutkin
+quantum group G0 ⊂ S+
+N.
+(iii) If β > α/(4α − 1) then ωϕ1⋆ϕ2(pQ) can be less than, equal to, or greater than
+ω(ϕ1,ϕ2)(pQ).
+(iv) Let
+(S+
+N × S+
+N)α,β := {(ϕ, ρ) : ωϕ(pQ) = α, ωρ(pQ) = β}.
+Then
+max{|ωϕ1⋆ϕ2(pQ) − ωϕ3⋆ϕ4(pQ)| : (ϕ1, ϕ2), (ϕ3, ϕ4) ∈ (S+
+N × S+
+N)α,β} = αβ.
+
+32
+J.P. MCCARTHY
+Proof. For (i)-(iii) apply Proposition 5.8. For (iv), the maximum in Proposition 5.8 is
+attained for
+ϕ1 = (1 − α) hSN + α h
+ϕ2 = (1 − β) hSN + β h
+ϕ3 = (1 − α) hSN + α (E11 ◦ πG0)
+ϕ4 = (1 − β) hSN + β (E11 ◦ πG0)
+□
+Suppose that ϕ1 is α-quantum, and ϕ2 is β-quantum. The subset of S+
+N × S+
+N given
+by condition (1) is called the QI-region. In this region the dynamics of the convolution
+(ϕ1, ϕ2) → ϕ with respect to pQ cannot be too wild:
+ωϕ1⋆ϕ2(pQ) ∈
+�
+ω(ϕ1,ϕ2)(pQ), ω(ϕ1,ϕ2)(pQ) + αβ
+�
+.
+Note that the width of this interval tends to zero for αβ → 0.
+On the other hand, the region of S+
+N ×S+
+N given by (3) is called the QW-region, and the
+dynamics can be more wild here. Given an arbitrary pair of quantum permutations in
+this region, the convolution can be more, equal, or less quantum than the mean, and, as
+αβ → 1, over the collection of (ϕ, ρ) ∈ QW the possible range of values of ωϕ⋆ρ(pQ) tends
+to one. Tracing from QI towards QW, on the boundary ∂W (given by (2)) ‘conservation
+of quantumness’,
+ωϕ1⋆ϕ2(pQ) = ω(ϕ1,ϕ2)(pQ),
+becomes possible for the first time.
+Similarly, higher order regions can be defined:
+(1) The region Q2I ⊆ QI given by β < (2α − 1)/(2α − 2) consists of quantum strictly
+2-increasing pairs;
+(2) The region Q3I ⊆ Q2I given by β < 1 −
+√
+2/(1 − 2α) consists of quantum strictly
+3-increasing pairs;
+(3) The region Q 1
+2 W ⊆ QW given by β > (1 − 1/
+√
+2)/α consists of pairs of quantum
+permutations (ϕ1, ϕ2) such that the pair (ϕ1 ⋆ ϕ2, ϕ1 ⋆ ϕ2) ̸∈ Q2I, etc.
+5.2. The truly quantum part of an idempotent state.
+Corollary 5.11. If φ ∈ S+
+N is an idempotent state, then
+ωφ(pQ) ∈ {0} ∪ [1/2, 1].
+Proof. If φ is an idempotent state,
+ωφ(pQ) = ωφ⋆φ(pQ).
+The rest follows from Proposition 5.10.
+□
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+33
+Figure 1. The phase diagram for the convolution of α-quantum and β-
+quantum permutations.
+The phases are quantum increasing, QI, in the
+bottom left, and quantum wild, QW, in the top right, with the bold line
+∂W the boundary.
+From the bottom left, Q3I ⊂ Q2I ⊂ QI, and then
+touching ∂W on the diagonal, Q 1
+2W ⊂ QW. The region Q 1
+2W is such that
+the convolution of states from this region cannot be too close to random:
+indeed the convolution cannot fall inside Q2I. The line α = β represents
+(ϕ, ϕ) → ϕ⋆2. The shading is proportional to αβ (see Proposition 5.10 (4)).
+An idempotent on the boundary ∂W is the Haar idempotent hG0 associated with the
+Kac–Paljutkin quantum group G0 ⊂ S+
+4 which satisfies ωhG0(pQ) = 1/2.
+Example 5.12. Let G be a finite quantum group given by π : C(S+
+N) → C(G). Where
+G ⊆ G is the classical version, the σ-weak extension π∗∗ to the biduals maps onto C(G),
+and in particular π∗∗(pσ) ∈ C(G) is the support projection of
+f �→ πab(π(f))(σ)
+(f ∈ C(S+
+N)).
+Let hG := hC(G) ◦ π with extension to the biduals ωG. From e.g. [13]:
+ωG(pσ) =
+1
+dim C(G)
+(σ ∈ G).
+
+0.8
+0.6
+β
+0.4-
+02
+0:
+0
+0.2
+0.4
+0.6
+0.834
+J.P. MCCARTHY
+This implies that
+(11)
+ωG(pQ) = 1 −
+|G|
+dim C(G).
+Let n ≥ 9, where Sn is generated by elements σ, τ of order two and three [18], and thus
+there is an embedding �
+Sn ⊂ S+
+5 given by Fourier type matrices uσ ∈ M2(C(�
+Sn)) and
+uτ ∈ M3(C(�
+Sn)) ([2], Chapter 13):
+u =
+�
+uσ
+0
+0
+uτ
+�
+.
+A finite dual �Γ ⊆ S+
+N has classical version with order equal to the number of one dimen-
+sional representations of Γ (see [17] for more). Therefore the classical version of �
+Sn is Z2
+and so, for n ≥ 9, the associated Haar idempotent:
+(12)
+ω�
+Sn(pQ) = 1 − 2
+n!,
+which tends to one for n → ∞.
+This suggests the following study: consider
+χN := {ωφ(pQ) : φ ∈ S+
+N, φ ⋆ φ = φ}.
+It is the case that χN = {0} for N ≤ 3, and otherwise a non-singleton. By (12), 1 is a
+limit point for χ5 ∩ [1/2, 1). Is there any other interesting behaviour: either at fixed N,
+or asymptotically N → ∞?
+It seems unlikely that there exists a finite exotic quantum permutation group SN ⊊
+GN ⊊ S+
+N for some N ≥ 6, but something can be said:
+Proposition 5.13. An exotic finite quantum permutation group at order N satisfies:
+dim C(G) ≥ 2N!
+In particular, there is no exotic finite quantum group with dim C(G) < 1440.
+Proof. This follows from (11) and Corollary 5.11, and the fact that any exotic quantum
+permutation group SN ⊊ G ⊊ S+
+N must satisfy N ≥ 6.
+□
+5.3. Periodicity. A periodicity in convolution powers of random permutations is possi-
+ble. For example, suppose that G ⊆ SN and N ⊳ G is a normal subgroup. Consider the
+probability ν uniform on the coset Ng. Then, where ϕν ∈ S+
+N is the associated state:
+ϕν(f) =
+�
+σ∈SN
+πab(f)(σ)ν({σ}) =
+1
+|Ng|
+�
+τ∈N
+πab(f)(τg)
+(f ∈ C(S+
+N)),
+the convolution powers (ϕ⋆k
+ν )k≥0 are periodic, with period equal to the order of g.
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+35
+There can also be periodicity with respect to pQ. For example, ϕ := E11 ◦ πG0 is such
+that
+ϕ⋆k(pQ) =
+�
+0,
+if k odd,
+1,
+if k odd.
+Proposition 5.14. Suppose that ϕ ∈ S+
+N is truly quantum. If ϕ⋆k is random, then ϕ⋆(k+1)
+is truly quantum.
+Proof. Follows from Corollary 5.3.
+□
+Corollary 5.15. Suppose that a truly quantum permutation ϕ has a random finite con-
+volution power. Let k0 be the smallest such power. Then:
+ωϕk(pQ) =
+�
+0,
+if k
+mod k0 = 0,
+1,
+otherwise.
+Is there a quantum permutation with k0 > 2? This phenomenon suggests looking at
+when the classical version of G is a normal quantum subgroup G⊳G. However, in general,
+the classical periodicity associated with probability measures constant on cosets of N ⊳G
+for G ⊆ SN does not extend to the quantum case. See [16], Section 4.3.1.
+6. Integer fixed points quantum permutations
+An example of an exotic intermediate quasi-subgroup would be nice: instead this section
+presents a non-example. For a quantum permutation group G, consider the observable:
+fix :=
+N
+�
+j=1
+ujj.
+Note that σ(fix) ⊆ [0, N]. Consider a finite partition P of the spectrum into Borel subsets,
+σ(fix) =
+m
+�
+i=1
+Ei.
+Borel functional calculus can be used to attach a (pairwise-distinct) label λi to each
+Ei ⊆ σ(fix), and the number of fixed points of a quantum permutation ϕ can be measured
+using fixP ∈ C(G)∗∗ given by:
+fixP :=
+m
+�
+i=1
+λi 1Ei(fix).
+Measurement is in the sense of algebraic quantum probability and the Gelfand–Birkhoff
+picture: when a quantum permutation ϕ ∈ G is measured with a finite spectrum observ-
+able f = �
+λ∈σ(f) λ pλ in the bidual C(G)∗∗, the result is an element of σ(f), with f = λ
+with probability ωϕ(pλ), and in that event there is wave-function collapse to �pλϕ.
+
+36
+J.P. MCCARTHY
+Definition 6.1. A quantum permutation ϕ ∈ S+
+N has integer fixed points only if for all
+Borel subsets E ⊆ σ(fix),
+E ∩ {0, 1, . . . , N} = ∅ =⇒ ωϕ(1E(fix)) = 0.
+Equivalently, if
+ωϕ(1{0,1,...,N}(fix)) = 1.
+Let F(G) ⊆ G be the set of quantum permutations with integer fixed points.
+In the quotient πab : C(G) → C(G) to the classical version G ⊆ G, the number of fixed
+points observable becomes a integer valued:
+πab(fix) = fixG =
+�
+λ=0,1...,N
+λ̸=N−1
+λ pλ,
+with
+pλ(σ) =
+�
+1,
+if σ has λ fixed points,
+0,
+otherwise.
+.
+Therefore, random permutations ϕν ∈ S+
+N are elements of F(S+
+N).
+There are plenty of concrete examples of genuinely quantum permutations with integer
+fixed points: e.g. the quantum permutation ϕ := E11 ◦ πG0 has zero fixed points. So,
+F(S+
+N) contains all the elements of SN in S+
+N, and also genuinely quantum permutations.
+Proposition 6.2. For N ≥ 4, the Haar state on C(S+
+N) is not an element of F(S+
+N). In
+fact:
+ωh(1{x}(fix)) = 0
+(x ∈ [0, N]).
+Proof. This follows from the fact that for N ≥ 4 the moments of fix with respect to
+the Haar state are the Catalan numbers [3], and thus the corresponding measure is the
+Marchenko-Pastur law of parameter one, which has no atoms:
+ωh(1{x}(fix)) =
+�
+{x}
+1
+2π
+�
+4
+t − 1 dt = 0.
+□
+Corollary 6.3. For N ≥ 4, the Haar state on C(S+
+N) is truly quantum.
+Proof. The Haar state h is genuinely quantum. Assume that h ∈ S+
+N is mixed:
+ωh(pC) > 0 =⇒ ωh(pσ) > 0
+for some σ ∈ SN. Let qσ := 1S+
+N − pσ. Recalling that pσ is central:
+ωh(f) = ωh(pσ) ( �pσh)(f) + ωh(qσ) ( �qσh)(f)
+(f ∈ C(S+
+N)∗∗).
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+37
+Note that �pσh has a central minimal projection for support, which implies it is a character.
+By Proposition 4.2, �pσh = evσ, which factors through the abelianisation πab:
+evσ(f) = πab(f)(σ)
+(f ∈ C(S+
+N)),
+while the extension ωσ factors through π∗∗
+ab. Suppose that σ has λ ∈ {0, 1, . . . , N} fixed
+points. Using Lemma 2.22, consider, where pλ = π∗∗
+ab(1{λ}(fix)),
+ωσ(1{λ}(fix)) = pλ(σ) = 1,
+=⇒ ωh(1{λ}(fix)) = ωh(pσ) ( �pσh)(1{λ}(fix)) + ωh(qσ) ( �qσh)(1{λ}(fix))
+≥ ωh(pσ) ωσ(1{λ}(fix)) = ωh(pσ) > 0,
+contradicting Proposition 6.2.
+□
+However, F(G) ⊆ G is in general not a Pal set:
+Example 6.4. Let �S4 ⊂ S+
+5 by:
+u =
+�
+u(12)
+0
+0
+u(234)
+�
+.
+Here u(12) ∈ M2(C( �S4)) and u(234) ∈ M3(C( �S4)) are Fourier-type magic unitaries associ-
+ated with (12) and (234) ([2], Chapter 13). Consider the regular representation:
+π : C( �S4) → B(C24).
+Consider:
+π(fix) = π(2e + (12) + (234) + (243)).
+The spectrum contains λ± := (5 ±
+√
+17)/2 (see [17]), but consider unit eigenvectors x2
+and x4 ∈ C24 of eigenvalues two and four that give quantum permutations:
+ϕ2 = ⟨x2, π(·)x2⟩ and ϕ4 = ⟨x4, π(·)x4⟩,
+with two and four fixed points. It can be shown that:
+ϕ := 1
+2ϕ2 + 1
+2ϕ4
+is strict, that is |ϕ(σ)| = 1 for σ = e only, and therefore as the convolution in �S4 is
+pointwise multiplication,
+ϕ⋆k → δe,
+which is the Haar state on C( �S4). The Haar state for finite quantum groups such as �S4
+is faithful, and so where pλ+ is the spectral projection associated with the eigenvalue λ+:
+h�
+S4(pλ+) > 0,
+which implies that (ϕ⋆k)k≥0 does not converge to an element with integer fixed points,
+and so F( �S4) is not a Pal set, and thus neither is F(S+
+N) for N ≥ 4.
+
+38
+J.P. MCCARTHY
+Example 6.5. In the case of C(S+
+N) (N ≥ 4), the central algebra C(S+
+N)0 generated by the
+characters of irreducible unitary representations is commutative [10], and generated by
+fix, and so the central algebra C(S+
+N)0 ∼= C([0, N]), and the central states are given by
+Radon probability measures.
+The quantum permutation ‘uniform on quantum transpositions’, ϕtr from [10], is a
+central state given by:
+ϕtr(f) = f(N − 2)
+(f ∈ C(S+
+N)0)
+It has N − 2 fixed points (see [17]) but its convolution powers converge to the Haar state
+h ∈ S+
+N, which is not in F(S+
+N) by Proposition 6.2.
+Acknowledgement. Some of this work goes back to discussions with Teo Banica. Indeed
+the proof of Lemma 3.3 is due to Teo. Thanks also to Matthew Daws for helping with
+Section 1.4, Stefaan Vaes with Remark 1.5, and Ruy Exel with the argument in Theorem
+2.23 (ii).
+References
+[1] T. Banica, Homogeneous quantum groups and their easiness level, Kyoto J. Math. 61, (2021) 1–30.
+[2] T. Banica, Introduction to quantum groups, Springer Nature Switzerland, (2023), doi:10.1007/978-
+3-031-23817-8.
+[3] T. Banica and J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math.
+Jussieu 6 (2007), 381–414.
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+74–114.
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+942–961.
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+Translated in Trans. Moscow Math. Soc. (1967), 251–284, (1966).
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+Soc. Japan, 3 (1940), 22:977-988. .
+
+IDEMPOTENT STATES ON QUANTUM PERMUTATION GROUPS
+39
+[15] M. B. Landstand and A. Van Daele, Compact and discrete subgroups of algebraic quantum groups,
+I (2007), available at arXiv:0702.458.
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+in Algebra, 49:9, (2021), 3850–3871, DOI:10.1080/00927872.2021.1908551
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+664.
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+Department of Mathematics, Munster Technological University, Cork, Ireland.
+[email protected]
+

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+arXiv:2301.03071v1  [math.DG]  8 Jan 2023
+Curves of Constant Breadth According to Darboux
+Frame in a Strict Walker 3-Manifold
+Ameth Ndiaye*
+D´epartement de Math´ematiques, FASTEF, UCAD, Dakar, Senegal.
+Abstract
+In this paper, we investigate the differential geometry properties of curves of constant
+breadth according to Darboux frame in a given strict Walker 3-manifold. The considered curves
+are lying on a timelike surface in the Walker 3-manifold.
+MSC: 53B25 ; 53C40.
+Keywords: Darboux frame, curvature, torsion, constant breadth curve, Walker 3-manifolds.
+1
+Introduction
+The study of curves of constant breadth were defined first in 1778 by Euler. Then, Solow [11]
+investigated the curves of constant breadth. Kose, Magden and Yilmaz in [9, 10] studied plane
+curves of constant breadth in Euclidean spaces E3 and E4. Fujiwara [7] defined constant breadth
+for space curves and obtained a problem to determine whether there exists space curve of con-
+stant breadth or not. Furthermore, Blaschke [3] defined the curves of constant breadth on a sphere.
+In [2], Altunkaya et al. defined null curves of constant breadth in Minkowski 4-space and obtain
+a characterization of these curves. Also Altunkaya et al. in [1] investigate constant breadth curves
+on a surface according to Darboux frame and give some characterizations of these curves.
+Motivated by the above papers, we investigate the geometries of curves of constant breadth accord-
+ing to Darboux frame in a Strict Walker 3-manifold which is a Lorentzian three-manifold admitting
+a parallel null vector field. It is known that Walker metrics have served as a powerful tool of con-
+structing interesting indefinite metrics which exhibit various aspects of geometric properties not
+given by any positive definite metrics. For more details about Walker 3-manifold see [5,6,8].
+2
+Preliminaries
+A Walker n-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r-
+planes, with r ≤ n
+2. The canonical forms of the metrics were investigated by A. G. Walker ( [4]).
+* E–mail: [email protected] (A. Ndiaye)
+1
+
+Walker has derived adapted coordinates to a parallel plan field. Hence, the metric of a three-
+dimensional Walker manifold (M, gǫ
+f) with coordinates (x, y, z) is expressed as
+gǫ
+f = dx ◦ dz + ǫdy2 + f(x, y, z)dz2
+(1)
+and its matrix form as
+gǫ
+f =
+
+
+0
+0
+1
+0
+ǫ
+0
+1
+0
+f
+
+
+with inverse (gǫ
+f)−1 =
+
+
+−f
+0
+1
+0
+ǫ
+0
+1
+0
+0
+
+
+for some function f(x, y, z), where ǫ = ±1 and thus D = Span∂x as the parallel degenerate line
+field. Notice that when ǫ = 1 and ǫ = −1 the Walker manifold has signature (2, 1) and (1, 2)
+respectively, and therefore is Lorentzian in both cases. In this study we take ǫ = 1.
+It follows after a straightforward calculation that the Levi-Civita connection of any metric (1)
+is given by:
+∇∂x∂z
+=
+1
+2fx∂x,
+∇∂y∂z = 1
+2fy∂x,
+∇∂z∂z
+=
+1
+2(ffx + fz)∂x + 1
+2fy∂y − 1
+2fx∂z
+(2)
+where ∂x, ∂y and ∂z are the coordinate vector fields
+∂
+∂x,
+∂
+∂y and
+∂
+∂z , respectively. Hence, if (M, gǫ
+f)
+is a strict Walker manifolds i.e., f(x, y, z) = f(y, z), then the associated Levi-Civita connection
+satisfies
+∇∂y∂z = 1
+2fy∂x,
+∇∂z∂z = 1
+2fz∂x − 1
+2fy∂y.
+(3)
+Note that the existence of a null parallel vector field (i.e f = f(y, z)) simplifies the non-zero
+components of the Christoffel symbols and the curvature tensor of the metric gǫ
+f as follows:
+Γ1
+23 = Γ1
+32 = 1
+2fy, Γ1
+33 = 1
+2fz, Γ2
+33 = −1
+2fy
+(4)
+Let now u and v be two vectors in M. Denoted by (⃗i,⃗j,⃗k) the canonical frame in R3.
+The vector product of u and v in (M, gǫ
+f) with respect to the metric gǫ
+f is the vector denoted by u×v
+in M defined by
+gǫ
+f(u × v, w) = det(u, v, w)
+(5)
+for all vector w in M, where det(u, v, w) is the determinant function associated to the canonical
+basis of R3. If u = (u1, u2, u3) and v = (v1, v2, v3) then by using (5), we have:
+u × v =
+�����
+u1
+v1
+u2
+v2
+���� − f
+����
+u2
+v2
+u3
+v3
+����
+�
+⃗i − ǫ
+����
+u1
+v1
+u3
+v3
+����⃗j +
+����
+u2
+v2
+u3
+v3
+����⃗k
+(6)
+2
+
+3
+Darboux equations in Walker 3-manifold
+Let α : I ⊂ R −→ (M, gǫ
+f) be a curve parametrized by its arc-length s. The Frenet frame of α is
+the vectors T, N and B along α where T is the tangent, N the principal normal and B the binormal
+vector. They satisfied the Frenet formulas
+
+
+
+∇TT(s)
+=
+ǫ2κ(s)N(s)
+∇TN(s)
+=
+−ǫ1κT(s) − ǫ3τB(s)
+∇TB(s)
+=
+ǫ2τ(s)N(s)
+(7)
+where κ and τ are respectively the curvature and the torsion of the curve α, with ǫ1 = gf(T; T); ǫ2 =
+gf(N; N) and ǫ3 = gf(B, B).
+Starting from local coordinates (x, y, z) for which (1) holds, it is easy to check that
+e1 = ∂y, e2 = 2 − f
+2
+√
+2 ∂x + 1
+√
+2∂z, e3 = 2 + f
+2
+√
+2 ∂x − 1
+√
+2∂z
+are local pseudo-orthonormal frame fields on (M, gǫ
+f), with gǫ
+f(e1, e1) = ǫ, gǫ
+f(e2, e2) = 1 and
+gǫ
+f(e3, e3) = −1. Thus the signature of the metric gǫ
+f is (1, ǫ, −1). If we choose ǫ = 1 then,
+pseudo-orthonormal frame is formed by two spacelike vectors and one timelike vector and If we
+choose ǫ = −1 then, pseudo-orthonormal frame is formed by one spacelike vector and two timelike
+vectors. For both cases we obtain Lorentzian manifold. In this work we assume that ǫ = 1
+Now we suppose that the curve α lies on a timelike surface S in M. Let U be the unit normal vector
+of S, then the Darboux frame is given by {T, Y, U}, where T is the tangent vector of the curve α(s)
+and Y = U × T.
+Case 1: Let α be timelike curve. Then the tangent vector T is timelike (ǫ1 = −1), the normal
+vector N and the binormal vector B are spacelike, that is (ǫ2 = ǫ3 = 1).
+Since S is timelike, the unit normal vector U is spacelike and so Y becomes spacelike. The usual
+transformations between the Walker Frenet frame and the Darboux takes the form
+Y = cos θN + sin θB
+(8)
+U = − sin θN + cos θB,
+(9)
+where θ is an angle between the vector Y and the vector N.
+Derivating Y along the curve alpha we get
+∇TY = cos θ∇TN − θ′ sin θN + sin θ∇TB + θ′ cos θB.
+Using the Frenet equation in (2.7) we have
+∇T Y = cos θ(κT − ǫ3τB) − θ′ sin θN + sin θ(ǫ2τN) + θ′ cos θB.
+Now we suppose that the principal normal and the binormal have the same sign. then we get
+∇TY = κ cos θT + (θ′ − τ)U
+(10)
+The same calculus gives
+∇TU = −κ sin θT − (θ′ − τ)Y.
+(11)
+3
+
+Then the Walker Darboux equation is expressed as
+
+
+
+∇TT = κgY + κnU
+∇TY = κgT + τgU
+∇TU = κnT − τgY,
+(12)
+where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on
+S, respectively. Also, (12) gives
+gǫ
+f (∇T Y, U) = τg = θ′ − τ,
+(13)
+gǫ
+f (∇TT, Y ) = κg = κ cos θ,
+(14)
+gǫ
+f (∇TT, U) = κn = −κ sin θ.
+(15)
+Case 2: Let α be spacelike curve. Then the tangent vector T is spacelike (ǫ1 = 1), the normal
+vector N is spacelike (ǫ2 = 1) and the binormal vector B is timelike (ǫ3 = −1) or normal vector N
+is timelike (ǫ2 = −1) and the binormal vector B is spacelike (ǫ3 = 1). So we have two following
+subcases:
+i): ǫ2 = 1 and ǫ3 = −1.
+Then the usual transformations between the Walker Frenet frame and the Darboux takes the form
+Y = cosh θN + sinh θB
+(16)
+U = sinh θN + cosh θB,
+(17)
+where θ is an angle between the vector Y and the vector N.
+Since ∇TT = κN, we have
+∇TT = −κ sinh θY + κ cosh θU.
+(18)
+Derivating Y along the curve alpha we get
+∇T Y = −κ sinh θT + (θ′ + τ)U
+(19)
+The same calculus gives
+∇TU = −κ cosh θT + (θ′ + τ)Y.
+(20)
+Then the Walker Darboux equation is expressed as
+
+
+
+∇TT = −κgY + κnU
+∇TY = −κgT + τgU
+∇TU = −κnT + τgY,
+(21)
+where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on
+S, respectively. Also, (21) gives
+gǫ
+f (∇TY, U) = τg = θ′ + τ,
+(22)
+gǫ
+f (∇TT, Y ) = κg = κ sinh θ,
+(23)
+gǫ
+f (∇TT, U) = κn = κ cosh θ.
+(24)
+4
+
+ii): ǫ2 = −1 and ǫ3 = 1.
+Then the usual transformations between the Walker Frenet frame and the Darboux takes the form
+Y = sinh θN + cosh θB
+(25)
+U = cosh θN + sinh θB,
+(26)
+where θ is an angle between the vector Y and the vector N.
+Since ∇TT = −κN, we have
+∇TT = −κ cosh θY + κ sinh θU.
+(27)
+Derivating Y with respect to s we get
+∇TY = −κ cosh θT + (θ′ − τ)U
+(28)
+Derivating Y with respect to s alpha we get
+∇TU = −κ sinh θT + (θ′ − τ)Y.
+(29)
+Then the Walker Darboux equation is expressed as
+
+
+
+∇TT = −κgY + κnU
+∇TY = −κgT + τgU
+∇TU = −κnT + τgY,
+(30)
+where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on
+S, respectively. Also, (30) gives
+gǫ
+f (∇T Y, U) = τg = θ′ − τ,
+(31)
+gǫ
+f (∇TT, Y ) = κg = κ cosh θ,
+(32)
+gǫ
+f (∇TT, U) = κn = κ sinh θ.
+(33)
+4
+Space curves of constant breadth According to Darboux Frame
+in Walker manifold
+In this section, we define space curves of constant breadth in the three dimensional Walker mani-
+fold.
+Definition 4.1. A curve α : I → (M, gǫ
+f) in the three-dimensional Walker manifold (M, gǫ
+f) is
+called a curve of constant breadth if there exists a curve β : I → Mf such that, at the corresponding
+points of curves, the parallel tangent vectors of α and β at α(s) and β(s⋆) at s; s⋆ ∈ I are opposite
+directions and the distance gǫ
+f(β − α, β − α) is constant. In this case, (α; β) is called a pair curve
+of constant breadth.
+Let now (α; β) be a pair of unit speed curves of constant breadth and s, s⋆ be arc-length of α
+and β, respectively.
+We suppose that the curve α lies on a timelike surface in Mf, then it has Darboux frame in addition
+to Frenet frame. Then we may write the following equation:
+β(s⋆) = α(s) + m1(s)T(s) + m2(s)Y (s) + m3(s)U(s);
+(34)
+where mi(i = 1, 2, 3) are smooth functions of s.
+5
+
+4.1
+Case where α is timelike.
+Differentiating (34) with respect to s and using (12) we obtain
+dβ
+ds
+=
+dβ
+ds⋆
+ds⋆
+ds
+=
+T ⋆(s⋆)ds⋆
+ds = (1 + m′
+1 + m2κg + m3κn)T(s)
++(m′
+2 + m1κg − m3τg)Y (s)
++(m′
+3 + m2τg + m1κn)U(s),
+(35)
+where T ⋆ denotes the unit tangent vector of β.
+Since T = −T ∗, from the equations in (35) we have
+
+
+
+m′
+1
+=
+−m2κg − m3κn − h(s)
+m′
+2
+=
+−m1κg + m3τg
+m′
+3
+=
+−m2τg − m1κn,
+(36)
+where h(s) =
+ds⋆
+ds + 1. We assume that (α, β) is a curve pair of constant breadth. Since α is a
+timelike curve and the vectors Y and U are spacelike vectors, we have
+∥β − α∥ = −m2
+1 + m2
+2 + m2
+3 = constant,
+(37)
+which imlplies that
+−m1
+dm1
+ds + m2
+dm2
+ds + m3
+dm3
+ds = 0.
+(38)
+If we combine (36) and (38), we get
+m1h(s) = 0.
+(39)
+If α and β are curves of constant breadth then m1 = 0 or h(s) = 0. If m1 ̸= 0 (that is h(s) = 0)
+then d = m1T(s) + m2Y (s) + m3U(s) becomes a constant vector. So β(s∗) is a translation of α
+along the constant vector d. Also h(s) = 0 gives s∗ = −s + c, where c is constant.
+Now, we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special case.
+4.1.1
+Case (For geodesic curves)
+Let α be non-straight line geodesic curve on a timelike surface. Then κg = κ cos θ = 0. As κ ̸= 0,
+we get cos θ = 0. So it implies that κn = −κ, τg = −τ. From (36), we have following differential
+equation system
+
+
+
+m′
+1
+=
+m3κ − h(s)
+m′
+2
+=
+−m3τ
+m′
+3
+=
+m1κ + m2τ.
+(40)
+By using (40), we obtain the following differential equation.
+1
+κ
+�1
+κ(m′
+1 + h)
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κ(m′
+1 + h)
+�′
++
+�τ
+κ
+�2
+(m′
+1+h)+
+�τ
+κ
+�′ κ
+τ m1−m′
+1 = 0.
+(41)
+6
+
+Subcase 1: m1 ̸= 0 (h(s) = 0).
+If we write h(s) = 0 in equation (41), we have.
+1
+κ
+�1
+κm′
+1
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κm′
+1
+�′
++
+��τ
+κ
+�2
+− 1
+�
+m′
+1 +
+�τ
+κ
+�′ κ
+τ m1 = 0.
+(42)
+Theorem 4.2. Let α be a timelike geodesic curve lying a timelike surface in M and let (α, β) be a
+pair of unit speed curves of constant breadth. If m1 is a non-zero constant then α is a general helix
+in the three dimensional Walker manifold (M, gǫ
+f). Also the curve β is given as:
+β(s⋆) = α(s) + m1T(s) + m2Y (s)
+(43)
+where m2 is a real constant and s∗ = −s + c.
+Proof. If m1 is non zero constant, then from (42) we obtain that
+� τ
+κ
+�′ = 0. So α is a general
+helix. Also from the first and second equations of (40) we get m3 = 0 and m2 is a real constant,
+respectively.
+Theorem 4.3. Let α be a timelike geodesic curve and a general helix lying a timelike surface in
+M. Let (α, β) be a pair of unit speed curves of constant breadth. If m1 is not zero, then the curve
+β can be expressed as one of the following cases:
+β(s∗) = α(s) + m1T(s) + 1
+c0
+( ¨m1 − m1)Y (s) + ˙m1U(s)
+(44)
+where
+i) m1 =
+1
+√
+c2
+0−1
+�
+a1 sin(
+�
+c2
+0 − 1z) − a2 cos(
+�
+c2
+0 − 1z)
+�
++ a3,
+c2
+0 − 1 > 0
+ii) m1 = a1
+2 z2 + a2z + a3,
+c2
+0 − 1 = 0
+iii) m1 =
+1
+√
+1−c2
+0
+�
+a1 sinh(
+�
+1 − c2
+0z) + a2 cosh(
+�
+1 − c2
+0z)
+�
++ a3,
+c2
+0 − 1 < 0
+where z =
+�
+κds and a1, a2, a3 are real constants.
+Proof. Let us consider that α is timelike geodesic curve and a general helix in Wlaker 3-manifold.
+Then we have τ
+κ = c0 = constant. From (42), we have
+�1
+κ
+�1
+κm′
+1
+�′�′
++
+�
+c2
+0 − 1
+�
+m′
+1 = 0.
+(45)
+By means of changing of the independant variable s with z =
+�
+κds, from (45) we obtain
+m′
+1 = dm1
+ds = dm1
+dz
+dz
+ds = ˙m1κ.
+...
+m1 + (c2
+0 − 1) ˙m1 = 0.
+(46)
+7
+
+If we solve this equation we get
+m1 =
+
+
+
+
+
+
+
+1
+√
+c2
+0−1
+�
+a1 sin(
+�
+c2
+0 − 1z) − a2 cos(
+�
+c2
+0 − 1z)
+�
++ a3, if c2
+0 − 1 > 0
+a1
+2 z2 + a2z + a2, if c2
+0 − 1 = 0
+1
+√
+1−c2
+0
+�
+a1 sinh(
+�
+1 − c2
+0z) + a2 cosh(
+�
+1 − c2
+0z)
+�
++ a3, if c2
+0 − 1 < 0.
+From (40) we obtain m3 = ˙m1 and m2 =
+1
+c0( ¨m1 − m1).
+Subcase 2: m1 = 0.
+If we take m1 = 0 in the equation (40), we get
+
+
+
+h(s)
+=
+m3κ
+m′
+2
+=
+−m3τ
+m′
+3
+=
+m2τ.
+(47)
+Since m3 = h
+κ, m2 = 1
+τ m′
+3 = 1
+τ
+�h
+κ
+�′, we get
+�1
+τ
+�h
+κ
+�′�′
++
+�h
+κ
+�
+τ = 0.
+(48)
+If we put y = h
+κ, the equation (48) becomes
+y′′ − τ ′
+τ y′ + τ 2y = 0.
+(49)
+For solving the equation (49), we put the new variable dw
+ds = τ. Then
+�
+y′ = dy
+dw
+dw
+ds = ˙yτ
+y′′ = d2y
+dw2τ 2 + dy
+dwτ ′
+(50)
+If we put the equation (50) in the equation (49) we obtain
+d2y
+dw2 + y = 0.
+(51)
+and the solution of (51) is y = b1 cos w + b2 sin w. Then we have
+h(s) = ��
+�
+b1 cos
+��
+τds
+�
++ b2 sin
+��
+τds
+��
+(52)
+m2 = h
+κ = b1 cos
+��
+τds
+�
++ b2 sin
+��
+τds
+�
+(53)
+m3 = 1
+τ
+�h
+κ
+�′
+= −b1 sin
+��
+τds
+�
++ b2 cos
+��
+τds
+�
+.
+(54)
+So we give the following theorem
+Theorem 4.4. Let (α, β) be a pair of constant breadth curve in (M, gf) where α is a timelike
+geodesic curve lying in a timelike surface in M. If m1 = 0, then the curve β is given by
+β(s∗) = α(s)+
+�
+b1 cos
+��
+τds
+�
++ b2 sin
+��
+τds
+��
+Y (s)+
+�
+−b1 sin
+��
+τds
+�
++ b2 cos
+��
+τds
+��
+U(s).
+8
+
+4.1.2
+Case (For asymptotic lines)
+Let α be non-straight line asymptotic line on a timelike surface. Then κn = −κ sin θ = 0. As
+κ ̸= 0, we get sin θ = 0. So it implies that κg = κ, τg = −τ. From (36), we have following
+differential equation system
+
+
+
+m′
+1
+=
+−m2κ − h(s)
+m′
+2
+=
+−m1κ − m3τ
+m′
+3
+=
+m2τ.
+(55)
+By using (55), we get
+1
+κ
+�1
+κ(m′
+1 + h)
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κ(m′
+1 + h)
+�′
++
+�τ
+κ
+�2
+(m′
+1+h)+
+�τ
+κ
+�′ κ
+τ m1−m′
+1 = 0.
+(56)
+Subcase 1: m1 ̸= 0 (h(s) = 0).
+If we take as h(s) = 0 in equation (56), we get following differential equation
+1
+κ
+�1
+κm′
+1
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κm′
+1
+�′
++
+��τ
+κ
+�2
+− 1
+�
+m′
+1 +
+�τ
+κ
+�′ κ
+τ m1 = 0.
+(57)
+Theorem 4.5. Let α be a timelike asymptotic line lying a timelike surface in M. Let (α, β) be a
+pair of unit speed curves of constant breadth. If m1 is non-zero constant then α is a general helix
+in the three dimensional Walker manifold (M, gǫ
+f). Also the curve β is given as:
+β(s⋆) = α(s) + m1T(s) + m3U(s)
+(58)
+where m3 is a real constant and s∗ = −s + c.
+Proof. If m1 is non zero constant, then from (57) we obtain that
+� τ
+κ
+�′ = 0. So α is a general
+helix. Also from the first and third equation of (55) we get m2 = 0 and m3 is a real constant,
+respectively.
+Theorem 4.6. Let α be a timelike asymptotic line lying in a timelike surface in M. Let (α, β) be a
+pair of unit speed curves of constant breadth. If m1 is not zero, then the curve β can be expressed
+as one of the following cases:
+β(s∗) = α(s) + m1T(s) − ˙m1Y (s) + 1
+c0
+( ¨m1 − m1)U(s),
+(59)
+where
+i) m1 =
+1
+√
+c2
+0−1
+�
+a1 sin(
+�
+c2
+0 − 1z) − a2 cos(
+�
+c2
+0 − 1z)
+�
++ a3, c2
+0 − 1 > 0
+ii) m1 = a1
+2 z2 + a2z + a3, c2
+0 − 1 = 0
+iii) m1 =
+1
+√
+1−c2
+0
+�
+a1 sinh(
+�
+1 − c2
+0z) + a2 cosh(
+�
+1 − c2
+0z)
+�
++ a3, c2
+0 − 1 < 0
+where z =
+�
+κds and a1, a2, a3 are constants.
+9
+
+Proof. The proof of Theorem (4.6) is done similarly to the proof of Theorem (4.3)
+Subcase 2: m1 = 0.
+If we take as m1 = 0 in (55) we get following differential equation system
+
+
+
+h(s)
+=
+−m2κ
+m′
+2
+=
+−m3τ
+m′
+3
+=
+m2τ.
+(60)
+Then we give the following theorem.
+Theorem 4.7. Let (α; β) be a curve pair of constant breadth in (M, gf) where α is a timelike
+asymptotic curve lying in a timelike surface in M. If m1 = 0, then the curve β is given by
+β(s∗) = α(s)+
+�
+−b1 cos
+��
+τds
+�
+− b2 sin
+��
+τds
+��
+Y (s)+
+�
+−b1 sin
+��
+τds
+�
++ b2 cos
+��
+τds
+��
+U(s).
+Proof. The proof of Theorem (4.7) is done similarly to the proof of Theorem (4.4).
+4.1.3
+Case (For Principal line)
+We suppose that α is a non-planar timelike principal line. Then we have τg = 0. Then it follows
+that τ = θ′. By using (36), we have the following differential equation system
+
+
+
+m′
+1
+=
+m3κ sin θ − m2κ cos θ − h(s)
+m′
+2
+=
+−m1κ cos θ
+m′
+3
+=
+m1κ sin θ.
+(61)
+By mean of changing of the independant variable s with θ =
+�
+τds, we get
+
+
+
+˙m1
+=
+φ(m3 sin θ − m2 cos θ) − g(θ)
+˙m2
+=
+−m1φ cos θ
+˙m3
+=
+m1φ sin θ.
+(62)
+where g(θ) = (− ds
+dθ − ds∗
+dθ ) and φ = κ
+τ . In here we denote the derivative with respect to θ with ”.”.
+From the equations in (62) we have
+...
+m1 + ¨g − d
+dθ
+� ˙φ
+φ( ˙m1 + g)
+�
+− d
+dθ(φ2m1) + ( ˙m1 + g)
+− ˙φ
+�
+− sin θ
+�
+m1φ cos θdθ + cos θ
+�
+m1φ sin θdθ
+�
+= 0.
+(63)
+Subcase 1: m1 ̸= 0 (h(s) = 0).
+In this case, we give the following theorem:
+Theorem 4.8. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a non-planar
+timelike principal line and a general helix then β is given by one of the following cases:
+β(s∗) = α(s) + m1T(s) − c
+�
+m1 cos θdθY (s) + c
+�
+m1 sin θdθU(s),
+(64)
+where
+10
+
+i) m1 =
+1
+√
+1−c2
+�
+a1 sin(
+√
+1 − c2θ) − a2 cos(
+√
+1 − c2θ)
+�
++ a3,
+1 − c2 > 0
+ii) m1 = a1
+2 θ2 + a2θ + a3,
+c2 − 1 = 0
+iii) m1 =
+1
+√
+c2−1
+�
+a1 sinh(
+√
+c2 − 1θ) + a2 cosh(
+√
+c2 − 1θ)
+�
++ a3,
+1 − c2 < 0
+Proof. If h(s) = 0 then g(θ) = 0 and from (63) we have
+...
+m1 − d
+dθ
+� ˙φ
+φ ˙m1
+�
+− d
+dθ(φ2m1) + ˙m1 − ˙φ
+�
+− sin θ
+�
+m1φ cos θdθ + cos θ
+�
+m1φ sin θdθ
+�
+= 0.(65)
+If α is helix curve then φ = κ
+τ = c = constant. From (65) we have
+...
+m1 + (1 − c2) ˙m1 = 0.
+(66)
+Then the solution is
+m1 =
+
+
+
+
+
+1
+√
+1−c2
+�
+a1 sin(
+√
+1 − c2θ) − a2 cos(
+√
+1 − c2θ)
+�
++ a3, if 1 − c2 > 0
+a1
+2 θ2 + a2θ + a3,
+if
+1 − c2 = 0
+1
+√
+c2−1
+�
+a1 sinh(
+√
+c2 − 1θ) + a2 cosh(
+√
+c2 − 1θ)
+�
++ a3, if 1 − c2 < 0,
+where θ =
+�
+τdθ.
+Subcase 2: m1 = 0.
+The case where m1 = 0, we have the following the following theorem:
+Theorem 4.9. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a non-planar
+timelike principal line. If m1 = 0 then α is general helix. The curve β is expressed as
+β(s∗) = α(s) + c2Y (s) + c3U(s),
+(67)
+where c2 and c3 are constants.
+Proof. From (63) we have
+¨g − d
+dθ
+� ˙φ
+φg
+�
++ g = 0.
+(68)
+On the other hand, from (61) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0 and
+from (62)
+g = φ(−c2 cos θ + c3 sin θ).
+(69)
+By considering (68) and (69) with together, we get
+˙φ(c2 sin θ + c3 cos θ) = 0.
+(70)
+Then we have ˙φ = 0 or c2 sin θ + c3 cos θ = 0. If c2 sin θ + c3 cos θ = 0 then we have that θ is a
+constant. So α becomes a planar curve. It is a contridiction. So ˙φ = 0. Then we obtain that φ = κ
+τ
+is a constant. Thus α is a general helix.
+11
+
+4.2
+Case where α is spacelike and ǫ2 = 1 and ǫ3 = −1.
+Here we suppose that the curve α is spacelike and lying on a timelike surface in Mf.
+Differentiating (34) with respect to s and using (21) we obtain
+dβ
+ds
+=
+dβ
+ds⋆
+ds⋆
+ds
+=
+T ⋆ds⋆
+ds = (1 + m′
+1 − m2κg − m3κn)T
++(m′
+2 − m1κg + m3τg)Y
++(m′
+3 + m2τg + m1κn)U,
+(71)
+where T ⋆ denotes the tangent vector of β.
+Since T = −T ∗, from the equation in (35) we have
+
+
+
+m′
+1
+=
+m2κg + m3κn − h(s)
+m′
+2
+=
+m1κg − m3τg
+m′
+3
+=
+−m2τg − m1κn,
+(72)
+where h(s) = ds∗
+ds + 1.
+Since α is spacelike and ǫ2 = 1 andǫ3 = −1, then, if we assume that (α, β) is a curve pair of
+constant breadth, we have
+∥β − α∥ = m2
+1 + m2
+2 − m2
+3 = constant,
+(73)
+which imlplies that
+m1
+dm1
+ds + m2
+dm2
+ds − m3
+dm3
+ds = 0.
+(74)
+If we combine (72) and (74) we get
+m1(2m′
+1 + h(s)) = 0.
+(75)
+If α and β are curves of constant breadth then m1 = 0 or 2m′
+1 − h(s) = 0.
+Now we investigate the case where α is geodesic curve or principal line curve because κn ̸= 0.
+4.2.1
+Case (For geodesic curves)
+Let α be non-straight line geodesic curve on a timelike surface. Then κg = κ sinh θ = 0. As κ ̸= 0,
+we get sinh θ = 0. So it implies that κn = κ, τg = τ. From (72), we have the following differential
+equation system
+
+
+
+m′
+1
+=
+m3κ − h(s)
+m′
+2
+=
+−m3τ
+m′
+3
+=
+−m1κ − m2τ.
+(76)
+From (76) we have
+
+
+
+m3
+=
+1
+κ(m′
+1 + h)
+m′
+2
+=
+− τ
+κ(m′
+1 + h)
+m2
+=
+− 1
+τ
+�
+( 1
+κ(m′
+1 + h))′ + m1κ
+�
+.
+(77)
+12
+
+Differentiating the third equation of (76) with respect to s and using the first, the second and the
+third equations of (77), we obtain the following equation:
+1
+κ
+�1
+κ(m′
+1 + h)
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κ(m′
+1 + h)
+�′
+−
+�τ
+κ
+�2
+(m′
+1+h)−
+�τ
+κ
+�′ κ
+τ m1+m′
+1 = 0.
+(78)
+Subcase 1: m1 ̸= 0 (h(s) = −2m′
+1).
+The equation (78) becomes
+1
+κ
+�1
+κm′
+1
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κm′
+1
+�′
+−
+��τ
+κ
+�2
++ 1
+�
+m′
+1 +
+�τ
+κ
+�′ κ
+τ m1 = 0.
+(79)
+Theorem 4.10. Let α be a geodesic curve. Let (α; β) be a pair of unit speed curves of constant
+breadth where α is spacelike (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface in Mf. If m1 is
+non-zero constant then m3 = 0 and α is a general helix in the three dimensional Walker manifold
+(M, gǫ
+f). Also the curve β is given as:
+β(s⋆) = α(s) + m1T + cY
+(80)
+where c is a real constant and s∗ = −s + c.
+Proof. If m1 is non zero constant, then from (79) we obtain that
+� τ
+κ
+�′ = 0. So α is a general helix.
+Also from the second and third equation of (76) we get m3 = 0 because h = 0 and m2 is a real
+constant.
+Theorem 4.11. Let α be a geodesic curve. Let (α, β) be a pair of unit speed curves of constant
+breadth where α is spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface Mf. If m1 is
+not zero, then the curve β can be expressed as one of the following cases:
+β(s∗) = α(s) + m1T + 1
+c0
+( ¨m1 − m1)Y + ˙m1U,
+(81)
+where m1 =
+1
+√
+1+c2
+0
+�
+a1e
+√
+1+c2
+0θ − a2e−√
+1+c2
+0θ�
+, m3 = − ˙m1 and m2 = 1
+c0( ¨m1 − m1).
+Proof. Let us consider that α is a general helix in Wlaker 3-manifold. Then we have τ
+κ = c0 =
+constant. From (79), we have
+�1
+κ
+�1
+κm′
+1
+�′�′
+−
+�
+c2
+0 + 1
+�
+m′
+1 = 0.
+(82)
+By means of changing of the independant variable s with z =
+�
+κds, we obtain
+m′
+1 = dm1
+ds = dm1
+dz
+dz
+ds = ˙m1κ.
+From (82), we get
+...
+m1 − (c2
+0 + 1) ˙m1 = 0.
+(83)
+If we solve this equation we get
+m1 =
+1
+�
+1 + c2
+0
+�
+a1e
+√
+1+c2
+0θ − a2e−√
+1+c2
+0θ�
+.
+(84)
+From (77) we have m3 = − ˙m1 and m2 =
+1
+c0( ¨m1 − m1).
+13
+
+Subcase 2: m1 = 0.
+Theorem 4.12. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be a geodesic
+spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface on Mf. If m1 = 0 then the curve
+β is expressed as
+β(s∗) = α(s) + cY,
+(85)
+where c is a constant real.
+Proof. If m′
+1 = 0 then h = 0 and from (76) we have m3 = 0 and m2 = constant.
+4.2.2
+Case (For Principal line)
+If α is principal line, then τg = 0 and τ = −θ′. From (72)
+
+
+
+m′
+1
+=
+m2κ sinh θ + m3κ cosh θ − h(s)
+m′
+2
+=
+m1κ sinh θ
+m′
+3
+=
+−m1κ cosh θ,
+(86)
+By mean of changing of the independant variable s with θ =
+�
+τds, we get
+
+
+
+˙m1
+=
+m3
+κ
+τ cosh θ + m2
+κ
+τ sinh θ − h(s)
+τ(s)
+˙m2
+=
+m1
+κ
+τ sinh θ
+˙m3
+=
+−m1
+κ
+τ cosh θ.
+(87)
+Denoted by h(s)
+τ(s) = g(θ) and κ
+τ = φ, we have
+
+
+
+˙m1
+=
+φ(m3 cosh θ + m2 sinh θ) − g(θ)
+˙m2
+=
+m1φ sinh θ
+˙m3
+=
+−m1φ cosh θ.
+(88)
+From the equations in (88) we have
+
+
+
+1
+φ( ˙m1 + g)
+=
+m3 cosh θ + m2 sinh θ
+˙m2 sinh θ + ˙m3 cosh θ
+=
+−m1φ
+˙m2 cosh θ
+=
+−m3 sinh θ.
+(89)
+Differentiating the first equation in (88), we get
+...
+m1 + ¨g − d
+dθ
+� ˙φ
+φ( ˙m1 + g)
+�
++ d
+dθ(φ2m1) − ( ˙m1 + g)
+− ˙φ
+�
+cosh θ
+�
+m1φ sinh θdθ − sinh θ
+�
+m1φ cosh θdθ
+�
+= 0.
+(90)
+Subcase 1: m1 ̸= 0 (m′
+1 = −h
+2).
+If m′
+1 = −h
+2 then ˙m1 = −g
+2. From (90) we obtain
+−...
+m1 + d
+dθ
+� ˙φ
+φ ˙m1
+�
++ d
+dθ(φ2m1) + ˙m1 − ˙φ
+�
+cosh θ
+�
+m1φ sinh θdθ − sinh θ
+�
+m1φ cosh θdθ
+�
+= 0.(91)
+14
+
+Theorem 4.13. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line
+and a general helix then β is given by
+β(s∗) = α(s) + m1T + m2Y + m3U,
+(92)
+where
+m1 =
+1
+√
+1 + c2
+�
+a1e
+√
+1+c2θ − a2e−
+√
+1+c2θ�
+,
+m2 = c
+�
+m1 sinh θdθ and m3 = −c
+�
+m1 cosh θdθ.
+Proof. If α is helix curve then φ = κ
+τ = c = constant. From (91) we have
+...
+m1 − (1 + c2) ˙m1 = 0.
+(93)
+m1 =
+1
+√
+1 + c2
+�
+a1e
+√
+1+c2θ − a2e−
+√
+1+c2θ�
+.
+(94)
+Subcase 2: m1 = 0.
+From the equations in (72) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0. The first
+equation in (72) gives
+tanh θ = −c2
+c3
+.
+(95)
+Then θ is a constant and we have τ = 0.
+Theorem 4.14. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line.
+If m1 = 0 then α is planar curve. The curve β is expressed as
+β(s∗) = α(s) + c2Y + c3U,
+(96)
+where c2 and c3 are constants.
+4.3
+Case where α is spacelike and ǫ2 = −1 and ǫ3 = 1.
+Let α be a spacelike with ǫ2 = −1 and ǫ3 = 1 lying on a timelike surface in Mf.
+Differentiating (34) with respect to s and using (30) we obtain
+
+
+
+m′
+1
+=
+m2κg + m3κn − h(s)
+m′
+2
+=
+m1κg − m3τg
+m′
+3
+=
+−m2τg − m1κn,
+(97)
+where h(s) = ds∗
+ds + 1.
+Since α is spacelike and ǫ2 = −1 andǫ3 = 1, then, if we assume that (α, β) is a curve pair of
+constant breadth, we have
+∥β − α∥ = m2
+1 − m2
+2 + m2
+3 = constant,
+(98)
+15
+
+which imlplies that
+m1
+dm1
+ds + m2
+dm2
+ds − m3
+dm3
+ds = 0.
+(99)
+If we combine (97) and (99) we get
+m1h(s) = 0.
+(100)
+If α and β are curves of constant breadth then m1 = 0 or h(s) = 0. If m1 ̸= 0 (that is h(s) = 0)
+then d = m1T + m2Y + m3U becomes a constant vector because d′ = 0. So β(s∗) is a translation
+of α along the constant vector d. Also h(s) = 0 gives s∗ = −s + c, where c is constant.
+Since κg ̸= 0, here we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special
+case (asymptotic line or principal line).
+4.3.1
+Case (For Asymptotic line)
+Let α be non-straight line asymptotic line on a timelike surface. Then κn = κ sinh θ = 0. As
+κ ̸= 0, we get cosh θ = 0. So it implies that κg = κ, τg = −τ. From (97), we have following
+differential equation system
+
+
+
+m′
+1
+=
+m2κ − h(s)
+m′
+2
+=
+m1κ + m3τ
+m′
+3
+=
+−m2τ.
+(101)
+By differentiating the second equation in (101) with respect to s and using the first and third equa-
+tions in (101), we get
+1
+κ
+�1
+κ(m′
+1 + h)
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κ(m′
+1 + h)
+�′
+−
+�τ
+κ
+�2
+(m′
+1+h)+
+�τ
+κ
+�′ κ
+τ m1 −m′
+1 = 0.
+(102)
+Subcase 1: m1 ̸= 0 (h(s) = 0).
+The equation (102) is given by
+1
+κ
+�1
+κm′
+1
+�′′
++
+��1
+κ
+�′
+− 1
+τ
+�τ
+κ
+�′� �1
+κm′
+1
+�′
+−
+��τ
+κ
+�2
++ 1
+�
+m′
+1 +
+�τ
+κ
+�′ κ
+τ m1 = 0.
+(103)
+Theorem 4.15. Let α be a asymptotic curve. Let (α; β) be a pair of unit speed curves of constant
+breadth where α is spacelike (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 is
+non-zero constant then m2 = 0 and α is a general helix in the three dimensional Walker manifold
+(M, gǫ
+f). Also the curve β is given as:
+β(s⋆) = α(s) + m1T + m3U
+(104)
+where m3 is a real constant and s∗ = −s + c.
+Proof. If m1 is non zero constant, then from (103) we obtain that
+� τ
+κ
+�′ = 0. So α is a general helix.
+Also from the first and third equation of (101) we get m2 = 0 and m3 is a real constant.
+16
+
+Theorem 4.16. Let α be a asymptotic line. Let (α, β) be a pair of unit speed curves of constant
+breadth where α is timelike curve and lying in a timelike surface Mf. If m1 is not zero, then the
+curve β can be expressed as one of the following cases:
+β(s∗) = α(s) + m1T + ˙m1Y + 1
+c0
+( ¨m1 + m1)U,
+(105)
+where
+m1 =
+1
+�
+c2
+0 + 1
+�
+a1e
+√
+c2
+0+1z − a2e
+√
+c2
+0+1z�
+.
+Proof. Let us consider that α is a general helix in Walker 3-manifold. Then we have τ
+κ = c0 =
+constant. From (103), we have
+�1
+κ
+�1
+κm′
+1
+�′�′
+−
+�
+c2
+0 + 1
+�
+m′
+1 = 0.
+(106)
+By means of changing of the independant variable s with z =
+�
+κds, we obtain
+...
+m1 − (c2
+0 + 1) ˙m1 = 0.
+(107)
+If we solve this equation we get
+m1 =
+1
+�
+c2
+0 + 1
+�
+a1e
+√
+c2
+0+1z − a2e
+√
+c2
+0+1z�
+(108)
+From (101) we obtain m2 = ˙m1 and m3 = 1
+c0( ¨m1 + m1).
+Subcase 2: m1 = 0
+With the same computation as above, we have the following theorem:
+Theorem 4.17. Let (α; β) be a curve pair of constant breadth in (M, gf). If α is a spacelike
+asymptotic curve (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 = 0, then the
+curve β is given by
+β(s∗) = α(s)+
+�
+b1 cos
+��
+τds
+�
++ b2 sin
+��
+τds
+��
+Y (s)+
+�
+−b1 sin
+��
+τds
+�
++ b2 cos
+��
+τds
+��
+U(s).
+4.3.2
+Case (For Principal line)
+In this case we have the two following theorems:
+Theorem 4.18. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be spacelike
+principal line (with ǫ2 = −1 and ǫ3 = 1) and a general helix then β is given by
+β(s∗) = α(s) + m1T + m2Y + m3U,
+(109)
+where
+m1 =
+1
+√
+1 + c2
+�
+a1e
+√
+1+c2θ − a2e−
+√
+1+c2θ�
+,
+m2 = c
+�
+m1 cosh θdθ and m3 = −c
+�
+m1 sinh θdθ.
+17
+
+Theorem 4.19. Let (α, β) be a pair curves of constant breadth in (M, gfǫ). Let α be principal line
+(with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf. If m1 = 0 then α is general helix or
+α is planar curve and the curve β is expressed as
+β(s∗) = α(s) + c2Y + c3U,
+(110)
+where c2 and c3 are constants.
+Acknowledgments
+The author would like to thank the anonymous Referees for their comments and suggestions. All
+many thanks to professor Ferdag Kahraman from Ahi Evran University (Turkish) for their remarks
+and suggestions.
+References
+[1] B. Altunkaya, F. Kahraman, Curves of constant breadth according to Darboux frame. Com-
+mun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66, (2), 44–52 (2017).
+[2] B. Altunkaya, F. Kahraman, Null curves of constant breadth in Minkowski 4-space. Commun.
+Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68, (1), 451–456 (2019).
+[3] Blaschke W., Einige Bemerkungen uber Kurven und Flachen konstanter Breite, Ber. Verh.
+sachs. Akad. Leipzig 67, 290–297 (1915).
+[4] M. Brozos-V´azquez, E. Garc´ıa-Rio, P. Gilkey, S. Nikevi´c and R. V´azquez-Lorenzo, The Ge-
+ometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, 5. Morgan
+and Claypool Publishers, Williston, VT, (2009).
+[5] G. Calvaruso and J. Van der Veken, Parallel surfaces in Lorentzian three-manifolds admitting
+a parallel null vector field, J. Phys. A 43, no. 32, 325207 (2010).
+[6] A. S. Diallo, A. Ndiaye and A. Niang, Minimal graphs on three-dimensional Walker man-
+ifolds, Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, Trends Math.,
+Birkh¨a user/Springer, Cham, 425-438 (2020).
+[7] M. Fujivara, On space curve of constant breadth, Tohoku Math. J. 5, 179–184 (1914).
+[8] M. Gningue, A. Ndiaye, R. Nkunzimana, Biharmonic Curves in a Strict Walker 3-Manifold.
+Int. J. Math. Math. Sci., 1–6 (2022).
+[9] O. Kose, Some properties of ovals and curves of constant width in a plane, Doga Sci. J. Serial
+B (8), 2, 119-126 (1984).
+[10] A. Magden and O. Kose, On the curves of constant breadth in E4 space, Turkish J. Math., 21,
+277-284 (1997).
+[11] R. M. Solow, Quarterly Journal of Economics, 70, 6594 (1956).
+18
+

5NE1T4oBgHgl3EQfSwPe/content/tmp_files/load_file.txt

@@ -0,0 +1,462 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf,len=461
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='03071v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='DG]  8 Jan 2023  Curves of Constant Breadth According to Darboux  Frame in a Strict Walker 3-Manifold  Ameth Ndiaye*  D´epartement de Math´ematiques, FASTEF, UCAD, Dakar, Senegal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Abstract  In this paper, we investigate the differential geometry properties of curves of constant  breadth according to Darboux frame in a given strict Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The considered curves  are lying on a timelike surface in the Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  MSC: 53B25 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' 53C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Keywords: Darboux frame, curvature, torsion, constant breadth curve, Walker 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  1  Introduction  The study of curves of constant breadth were defined first in 1778 by Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then, Solow [11]  investigated the curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Kose, Magden and Yilmaz in [9, 10] studied plane  curves of constant breadth in Euclidean spaces E3 and E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Fujiwara [7] defined constant breadth  for space curves and obtained a problem to determine whether there exists space curve of con-  stant breadth or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Furthermore, Blaschke [3] defined the curves of constant breadth on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  In [2], Altunkaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' defined null curves of constant breadth in Minkowski 4-space and obtain  a characterization of these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also Altunkaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' in [1] investigate constant breadth curves  on a surface according to Darboux frame and give some characterizations of these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Motivated by the above papers, we investigate the geometries of curves of constant breadth accord-  ing to Darboux frame in a Strict Walker 3-manifold which is a Lorentzian three-manifold admitting  a parallel null vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' It is known that Walker metrics have served as a powerful tool of con-  structing interesting indefinite metrics which exhibit various aspects of geometric properties not  given by any positive definite metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' For more details about Walker 3-manifold see [5,6,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  2  Preliminaries  A Walker n-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r-  planes, with r ≤ n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The canonical forms of the metrics were investigated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Walker ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  E–mail: ameth1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='ndiaye@ucad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='sn (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Ndiaye)  1  Walker has derived adapted coordinates to a parallel plan field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Hence, the metric of a three-  dimensional Walker manifold (M, gǫ  f) with coordinates (x, y, z) is expressed as  gǫ  f = dx ◦ dz + ǫdy2 + f(x, y, z)dz2  (1)  and its matrix form as  gǫ  f =  \uf8eb  \uf8ed  0  0  1  0  ǫ  0  1  0  f  \uf8f6  \uf8f8  with inverse (gǫ  f)−1 =  \uf8eb  \uf8ed  −f  0  1  0  ǫ  0  1  0  0  \uf8f6  \uf8f8  for some function f(x, y, z), where ǫ = ±1 and thus D = Span∂x as the parallel degenerate line  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Notice that when ǫ = 1 and ǫ = −1 the Walker manifold has signature (2, 1) and (1, 2)  respectively, and therefore is Lorentzian in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' In this study we take ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  It follows after a straightforward calculation that the Levi-Civita connection of any metric (1)  is given by:  ∇∂x∂z  =  1  2fx∂x,  ∇∂y∂z = 1  2fy∂x,  ∇∂z∂z  =  1  2(ffx + fz)∂x + 1  2fy∂y − 1  2fx∂z  (2)  where ∂x, ∂y and ∂z are the coordinate vector fields  ∂  ∂x,  ∂  ∂y and  ∂  ∂z , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Hence, if (M, gǫ  f)  is a strict Walker manifolds i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=', f(x, y, z) = f(y, z), then the associated Levi-Civita connection  satisfies  ∇∂y∂z = 1  2fy∂x,  ∇∂z∂z = 1  2fz∂x − 1  2fy∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (3)  Note that the existence of a null parallel vector field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='e f = f(y, z)) simplifies the non-zero  components of the Christoffel symbols and the curvature tensor of the metric gǫ  f as follows:  Γ1  23 = Γ1  32 = 1  2fy, Γ1  33 = 1  2fz, Γ2  33 = −1  2fy  (4)  Let now u and v be two vectors in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Denoted by (⃗i,⃗j,⃗k) the canonical frame in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  The vector product of u and v in (M, gǫ  f) with respect to the metric gǫ  f is the vector denoted by u×v  in M defined by  gǫ  f(u × v, w) = det(u, v, w)  (5)  for all vector w in M, where det(u, v, w) is the determinant function associated to the canonical  basis of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If u = (u1, u2, u3) and v = (v1, v2, v3) then by using (5), we have:  u × v =  �����  u1  v1  u2  v2  ���� − f  ����  u2  v2  u3  v3  ����  �  ⃗i − ǫ  ����  u1  v1  u3  v3  ����⃗j +  ����  u2  v2  u3  v3  ����⃗k  (6)  2  3  Darboux equations in Walker 3-manifold  Let α : I ⊂ R −→ (M, gǫ  f) be a curve parametrized by its arc-length s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The Frenet frame of α is  the vectors T, N and B along α where T is the tangent, N the principal normal and B the binormal  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' They satisfied the Frenet formulas  \uf8f1  \uf8f2  \uf8f3  ∇TT(s)  =  ǫ2κ(s)N(s)  ∇TN(s)  =  −ǫ1κT(s) − ǫ3τB(s)  ∇TB(s)  =  ǫ2τ(s)N(s)  (7)  where κ and τ are respectively the curvature and the torsion of the curve α, with ǫ1 = gf(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' ǫ2 =  gf(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' N) and ǫ3 = gf(B, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Starting from local coordinates (x, y, z) for which (1) holds, it is easy to check that  e1 = ∂y, e2 = 2 − f  2  √  2 ∂x + 1  √  2∂z, e3 = 2 + f  2  √  2 ∂x − 1  √  2∂z  are local pseudo-orthonormal frame fields on (M, gǫ  f), with gǫ  f(e1, e1) = ǫ, gǫ  f(e2, e2) = 1 and  gǫ  f(e3, e3) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Thus the signature of the metric gǫ  f is (1, ǫ, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If we choose ǫ = 1 then,  pseudo-orthonormal frame is formed by two spacelike vectors and one timelike vector and If we  choose ǫ = −1 then, pseudo-orthonormal frame is formed by one spacelike vector and two timelike  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' For both cases we obtain Lorentzian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' In this work we assume that ǫ = 1  Now we suppose that the curve α lies on a timelike surface S in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let U be the unit normal vector  of S, then the Darboux frame is given by {T, Y, U}, where T is the tangent vector of the curve α(s)  and Y = U × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Case 1: Let α be timelike curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then the tangent vector T is timelike (ǫ1 = −1), the normal  vector N and the binormal vector B are spacelike, that is (ǫ2 = ǫ3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since S is timelike, the unit normal vector U is spacelike and so Y becomes spacelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The usual  transformations between the Walker Frenet frame and the Darboux takes the form  Y = cos θN + sin θB  (8)  U = − sin θN + cos θB,  (9)  where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Derivating Y along the curve alpha we get  ∇TY = cos θ∇TN − θ′ sin θN + sin θ∇TB + θ′ cos θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Using the Frenet equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='7) we have  ∇T Y = cos θ(κT − ǫ3τB) − θ′ sin θN + sin θ(ǫ2τN) + θ′ cos θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Now we suppose that the principal normal and the binormal have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' then we get  ∇TY = κ cos θT + (θ′ − τ)U  (10)  The same calculus gives  ∇TU = −κ sin θT − (θ′ − τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (11)  3  Then the Walker Darboux equation is expressed as  \uf8f1  \uf8f2  \uf8f3  ∇TT = κgY + κnU  ∇TY = κgT + τgU  ∇TU = κnT − τgY,  (12)  where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on  S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also, (12) gives  gǫ  f (∇T Y, U) = τg = θ′ − τ,  (13)  gǫ  f (∇TT, Y ) = κg = κ cos θ,  (14)  gǫ  f (∇TT, U) = κn = −κ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (15)  Case 2: Let α be spacelike curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then the tangent vector T is spacelike (ǫ1 = 1), the normal  vector N is spacelike (ǫ2 = 1) and the binormal vector B is timelike (ǫ3 = −1) or normal vector N  is timelike (ǫ2 = −1) and the binormal vector B is spacelike (ǫ3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So we have two following  subcases:  i): ǫ2 = 1 and ǫ3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Then the usual transformations between the Walker Frenet frame and the Darboux takes the form  Y = cosh θN + sinh θB  (16)  U = sinh θN + cosh θB,  (17)  where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since ∇TT = κN, we have  ∇TT = −κ sinh θY + κ cosh θU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (18)  Derivating Y along the curve alpha we get  ∇T Y = −κ sinh θT + (θ′ + τ)U  (19)  The same calculus gives  ∇TU = −κ cosh θT + (θ′ + τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (20)  Then the Walker Darboux equation is expressed as  \uf8f1  \uf8f2  \uf8f3  ∇TT = −κgY + κnU  ∇TY = −κgT + τgU  ∇TU = −κnT + τgY,  (21)  where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on  S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also, (21) gives  gǫ  f (∇TY, U) = τg = θ′ + τ,  (22)  gǫ  f (∇TT, Y ) = κg = κ sinh θ,  (23)  gǫ  f (∇TT, U) = κn = κ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (24)  4  ii): ǫ2 = −1 and ǫ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Then the usual transformations between the Walker Frenet frame and the Darboux takes the form  Y = sinh θN + cosh θB  (25)  U = cosh θN + sinh θB,  (26)  where θ is an angle between the vector Y and the vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since ∇TT = −κN, we have  ∇TT = −κ cosh θY + κ sinh θU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (27)  Derivating Y with respect to s we get  ∇TY = −κ cosh θT + (θ′ − τ)U  (28)  Derivating Y with respect to s alpha we get  ∇TU = −κ sinh θT + (θ′ − τ)Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (29)  Then the Walker Darboux equation is expressed as  \uf8f1  \uf8f2  \uf8f3  ∇TT = −κgY + κnU  ∇TY = −κgT + τgU  ∇TU = −κnT + τgY,  (30)  where κg, κn and τg are the geodesic curvature, normal curvature and geodesic torsion of α(s) on  S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also, (30) gives  gǫ  f (∇T Y, U) = τg = θ′ − τ,  (31)  gǫ  f (∇TT, Y ) = κg = κ cosh θ,  (32)  gǫ  f (∇TT, U) = κn = κ sinh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (33)  4  Space curves of constant breadth According to Darboux Frame  in Walker manifold  In this section, we define space curves of constant breadth in the three dimensional Walker mani-  fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' A curve α : I → (M, gǫ  f) in the three-dimensional Walker manifold (M, gǫ  f) is  called a curve of constant breadth if there exists a curve β : I → Mf such that, at the corresponding  points of curves, the parallel tangent vectors of α and β at α(s) and β(s⋆) at s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' s⋆ ∈ I are opposite  directions and the distance gǫ  f(β − α, β − α) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' In this case, (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) is called a pair curve  of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Let now (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) be a pair of unit speed curves of constant breadth and s, s⋆ be arc-length of α  and β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  We suppose that the curve α lies on a timelike surface in Mf, then it has Darboux frame in addition  to Frenet frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we may write the following equation:  β(s⋆) = α(s) + m1(s)T(s) + m2(s)Y (s) + m3(s)U(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (34)  where mi(i = 1, 2, 3) are smooth functions of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1  Case where α is timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Differentiating (34) with respect to s and using (12) we obtain  dβ  ds  =  dβ  ds⋆  ds⋆  ds  =  T ⋆(s⋆)ds⋆  ds = (1 + m′  1 + m2κg + m3κn)T(s)  +(m′  2 + m1κg − m3τg)Y (s)  +(m′  3 + m2τg + m1κn)U(s),  (35)  where T ⋆ denotes the unit tangent vector of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since T = −T ∗, from the equations in (35) we have  \uf8f1  \uf8f2  \uf8f3  m′  1  =  −m2κg − m3κn − h(s)  m′  2  =  −m1κg + m3τg  m′  3  =  −m2τg − m1κn,  (36)  where h(s) =  ds⋆  ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' We assume that (α, β) is a curve pair of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Since α is a  timelike curve and the vectors Y and U are spacelike vectors, we have  ∥β − α∥ = −m2  1 + m2  2 + m2  3 = constant,  (37)  which imlplies that  −m1  dm1  ds + m2  dm2  ds + m3  dm3  ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (38)  If we combine (36) and (38), we get  m1h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (39)  If α and β are curves of constant breadth then m1 = 0 or h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 ̸= 0 (that is h(s) = 0)  then d = m1T(s) + m2Y (s) + m3U(s) becomes a constant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So β(s∗) is a translation of α  along the constant vector d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also h(s) = 0 gives s∗ = −s + c, where c is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Now, we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1  Case (For geodesic curves)  Let α be non-straight line geodesic curve on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then κg = κ cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' As κ ̸= 0,  we get cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So it implies that κn = −κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (36), we have following differential  equation system  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m3κ − h(s)  m′  2  =  −m3τ  m′  3  =  m1κ + m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (40)  By using (40), we obtain the following differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  1  κ  �1  κ(m′  1 + h)  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κ(m′  1 + h)  �′  +  �τ  κ  �2  (m′  1+h)+  �τ  κ  �′ κ  τ m1−m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (41)  6  Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If we write h(s) = 0 in equation (41), we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  1  κ  �1  κm′  1  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κm′  1  �′  +  ��τ  κ  �2  − 1  �  m′  1 +  �τ  κ  �′ κ  τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (42)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a timelike geodesic curve lying a timelike surface in M and let (α, β) be a  pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is a non-zero constant then α is a general helix  in the three dimensional Walker manifold (M, gǫ  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also the curve β is given as:  β(s⋆) = α(s) + m1T(s) + m2Y (s)  (43)  where m2 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is non zero constant, then from (42) we obtain that  � τ  κ  �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So α is a general  helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also from the first and second equations of (40) we get m3 = 0 and m2 is a real constant,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a timelike geodesic curve and a general helix lying a timelike surface in  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is not zero, then the curve  β can be expressed as one of the following cases:  β(s∗) = α(s) + m1T(s) + 1  c0  ( ¨m1 − m1)Y (s) + ˙m1U(s)  (44)  where  i) m1 =  1  √  c2  0−1  �  a1 sin(  �  c2  0 − 1z) − a2 cos(  �  c2  0 − 1z)  �  + a3,  c2  0 − 1 > 0  ii) m1 = a1  2 z2 + a2z + a3,  c2  0 − 1 = 0  iii) m1 =  1  √  1−c2  0  �  a1 sinh(  �  1 − c2  0z) + a2 cosh(  �  1 − c2  0z)  �  + a3,  c2  0 − 1 < 0  where z =  �  κds and a1, a2, a3 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let us consider that α is timelike geodesic curve and a general helix in Wlaker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Then we have τ  κ = c0 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (42), we have  �1  κ  �1  κm′  1  �′�′  +  �  c2  0 − 1  �  m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (45)  By means of changing of the independant variable s with z =  �  κds, from (45) we obtain  m′  1 = dm1  ds = dm1  dz  dz  ds = ˙m1κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 + (c2  0 − 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (46)  7  If we solve this equation we get  m1 =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  1  √  c2  0−1  �  a1 sin(  �  c2  0 − 1z) − a2 cos(  �  c2  0 − 1z)  �  + a3, if c2  0 − 1 > 0  a1  2 z2 + a2z + a2, if c2  0 − 1 = 0  1  √  1−c2  0  �  a1 sinh(  �  1 − c2  0z) + a2 cosh(  �  1 − c2  0z)  �  + a3, if c2  0 − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  From (40) we obtain m3 = ˙m1 and m2 =  1  c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If we take m1 = 0 in the equation (40), we get  \uf8f1  \uf8f2  \uf8f3  h(s)  =  m3κ  m′  2  =  −m3τ  m′  3  =  m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (47)  Since m3 = h  κ, m2 = 1  τ m′  3 = 1  τ  �h  κ  �′, we get  �1  τ  �h  κ  �′�′  +  �h  κ  �  τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (48)  If we put y = h  κ, the equation (48) becomes  y′′ − τ ′  τ y′ + τ 2y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (49)  For solving the equation (49), we put the new variable dw  ds = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then  �  y′ = dy  dw  dw  ds = ˙yτ  y′′ = d2y  dw2τ 2 + dy  dwτ ′  (50)  If we put the equation (50) in the equation (49) we obtain  d2y  dw2 + y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (51)  and the solution of (51) is y = b1 cos w + b2 sin w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we have  h(s) = κ  �  b1 cos  ��  τds  �  + b2 sin  ��  τds  ��  (52)  m2 = h  κ = b1 cos  ��  τds  �  + b2 sin  ��  τds  �  (53)  m3 = 1  τ  �h  κ  �′  = −b1 sin  ��  τds  �  + b2 cos  ��  τds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (54)  So we give the following theorem  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair of constant breadth curve in (M, gf) where α is a timelike  geodesic curve lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0, then the curve β is given by  β(s∗) = α(s)+  �  b1 cos  ��  τds  �  + b2 sin  ��  τds  ��  Y (s)+  �  −b1 sin  ��  τds  �  + b2 cos  ��  τds  ��  U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2  Case (For asymptotic lines)  Let α be non-straight line asymptotic line on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then κn = −κ sin θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' As  κ ̸= 0, we get sin θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So it implies that κg = κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (36), we have following  differential equation system  \uf8f1  \uf8f2  \uf8f3  m′  1  =  −m2κ − h(s)  m′  2  =  −m1κ − m3τ  m′  3  =  m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (55)  By using (55), we get  1  κ  �1  κ(m′  1 + h)  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κ(m′  1 + h)  �′  +  �τ  κ  �2  (m′  1+h)+  �τ  κ  �′ κ  τ m1−m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (56)  Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If we take as h(s) = 0 in equation (56), we get following differential equation  1  κ  �1  κm′  1  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κm′  1  �′  +  ��τ  κ  �2  − 1  �  m′  1 +  �τ  κ  �′ κ  τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (57)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a timelike asymptotic line lying a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a  pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is non-zero constant then α is a general helix  in the three dimensional Walker manifold (M, gǫ  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also the curve β is given as:  β(s⋆) = α(s) + m1T(s) + m3U(s)  (58)  where m3 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is non zero constant, then from (57) we obtain that  � τ  κ  �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So α is a general  helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also from the first and third equation of (55) we get m2 = 0 and m3 is a real constant,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a timelike asymptotic line lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a  pair of unit speed curves of constant breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is not zero, then the curve β can be expressed  as one of the following cases:  β(s∗) = α(s) + m1T(s) − ˙m1Y (s) + 1  c0  ( ¨m1 − m1)U(s),  (59)  where  i) m1 =  1  √  c2  0−1  �  a1 sin(  �  c2  0 − 1z) − a2 cos(  �  c2  0 − 1z)  �  + a3, c2  0 − 1 > 0  ii) m1 = a1  2 z2 + a2z + a3, c2  0 − 1 = 0  iii) m1 =  1  √  1−c2  0  �  a1 sinh(  �  1 − c2  0z) + a2 cosh(  �  1 − c2  0z)  �  + a3, c2  0 − 1 < 0  where z =  �  κds and a1, a2, a3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='6) is done similarly to the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3)  Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If we take as m1 = 0 in (55) we get following differential equation system  \uf8f1  \uf8f2  \uf8f3  h(s)  =  −m2κ  m′  2  =  −m3τ  m′  3  =  m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (60)  Then we give the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) be a curve pair of constant breadth in (M, gf) where α is a timelike  asymptotic curve lying in a timelike surface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0, then the curve β is given by  β(s∗) = α(s)+  �  −b1 cos  ��  τds  �  − b2 sin  ��  τds  ��  Y (s)+  �  −b1 sin  ��  τds  �  + b2 cos  ��  τds  ��  U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='7) is done similarly to the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3  Case (For Principal line)  We suppose that α is a non-planar timelike principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we have τg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then it follows  that τ = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' By using (36), we have the following differential equation system  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m3κ sin θ − m2κ cos θ − h(s)  m′  2  =  −m1κ cos θ  m′  3  =  m1κ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (61)  By mean of changing of the independant variable s with θ =  �  τds, we get  \uf8f1  \uf8f2  \uf8f3  ˙m1  =  φ(m3 sin θ − m2 cos θ) − g(θ)  ˙m2  =  −m1φ cos θ  ˙m3  =  m1φ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (62)  where g(θ) = (− ds  dθ − ds∗  dθ ) and φ = κ  τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' In here we denote the derivative with respect to θ with ”.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  From the equations in (62) we have  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 + ¨g − d  dθ  � ˙φ  φ( ˙m1 + g)  �  − d  dθ(φ2m1) + ( ˙m1 + g)  − ˙φ  �  − sin θ  �  m1φ cos θdθ + cos θ  �  m1φ sin θdθ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (63)  Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  In this case, we give the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a non-planar  timelike principal line and a general helix then β is given by one of the following cases:  β(s∗) = α(s) + m1T(s) − c  �  m1 cos θdθY (s) + c  �  m1 sin θdθU(s),  (64)  where  10  i) m1 =  1  √  1−c2  �  a1 sin(  √  1 − c2θ) − a2 cos(  √  1 − c2θ)  �  + a3,  1 − c2 > 0  ii) m1 = a1  2 θ2 + a2θ + a3,  c2 − 1 = 0  iii) m1 =  1  √  c2−1  �  a1 sinh(  √  c2 − 1θ) + a2 cosh(  √  c2 − 1θ)  �  + a3,  1 − c2 < 0  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If h(s) = 0 then g(θ) = 0 and from (63) we have  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 − d  dθ  � ˙φ  φ ˙m1  �  − d  dθ(φ2m1) + ˙m1 − ˙φ  �  − sin θ  �  m1φ cos θdθ + cos θ  �  m1φ sin θdθ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' (65)  If α is helix curve then φ = κ  τ = c = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (65) we have  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 + (1 − c2) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (66)  Then the solution is  m1 =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  √  1−c2  �  a1 sin(  √  1 − c2θ) − a2 cos(  √  1 − c2θ)  �  + a3, if 1 − c2 > 0  a1  2 θ2 + a2θ + a3,  if  1 − c2 = 0  1  √  c2−1  �  a1 sinh(  √  c2 − 1θ) + a2 cosh(  √  c2 − 1θ)  �  + a3, if 1 − c2 < 0,  where θ =  �  τdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  The case where m1 = 0, we have the following the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a non-planar  timelike principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0 then α is general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The curve β is expressed as  β(s∗) = α(s) + c2Y (s) + c3U(s),  (67)  where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (63) we have  ¨g − d  dθ  � ˙φ  φg  �  + g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (68)  On the other hand, from (61) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0 and  from (62)  g = φ(−c2 cos θ + c3 sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (69)  By considering (68) and (69) with together, we get  ˙φ(c2 sin θ + c3 cos θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (70)  Then we have ˙φ = 0 or c2 sin θ + c3 cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If c2 sin θ + c3 cos θ = 0 then we have that θ is a  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So α becomes a planar curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' It is a contridiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So ˙φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we obtain that φ = κ  τ  is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Thus α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2  Case where α is spacelike and ǫ2 = 1 and ǫ3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Here we suppose that the curve α is spacelike and lying on a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Differentiating (34) with respect to s and using (21) we obtain  dβ  ds  =  dβ  ds⋆  ds⋆  ds  =  T ⋆ds⋆  ds = (1 + m′  1 − m2κg − m3κn)T  +(m′  2 − m1κg + m3τg)Y  +(m′  3 + m2τg + m1κn)U,  (71)  where T ⋆ denotes the tangent vector of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since T = −T ∗, from the equation in (35) we have  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m2κg + m3κn − h(s)  m′  2  =  m1κg − m3τg  m′  3  =  −m2τg − m1κn,  (72)  where h(s) = ds∗  ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since α is spacelike and ǫ2 = 1 andǫ3 = −1, then, if we assume that (α, β) is a curve pair of  constant breadth, we have  ∥β − α∥ = m2  1 + m2  2 − m2  3 = constant,  (73)  which imlplies that  m1  dm1  ds + m2  dm2  ds − m3  dm3  ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (74)  If we combine (72) and (74) we get  m1(2m′  1 + h(s)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (75)  If α and β are curves of constant breadth then m1 = 0 or 2m′  1 − h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Now we investigate the case where α is geodesic curve or principal line curve because κn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1  Case (For geodesic curves)  Let α be non-straight line geodesic curve on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then κg = κ sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' As κ ̸= 0,  we get sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So it implies that κn = κ, τg = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (72), we have the following differential  equation system  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m3κ − h(s)  m′  2  =  −m3τ  m′  3  =  −m1κ − m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (76)  From (76) we have  \uf8f1  \uf8f2  \uf8f3  m3  =  1  κ(m′  1 + h)  m′  2  =  − τ  κ(m′  1 + h)  m2  =  − 1  τ  �  ( 1  κ(m′  1 + h))′ + m1κ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (77)  12  Differentiating the third equation of (76) with respect to s and using the first, the second and the  third equations of (77), we obtain the following equation:  1  κ  �1  κ(m′  1 + h)  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κ(m′  1 + h)  �′  −  �τ  κ  �2  (m′  1+h)−  �τ  κ  �′ κ  τ m1+m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (78)  Subcase 1: m1 ̸= 0 (h(s) = −2m′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  The equation (78) becomes  1  κ  �1  κm′  1  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κm′  1  �′  −  ��τ  κ  �2  + 1  �  m′  1 +  �τ  κ  �′ κ  τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (79)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a geodesic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) be a pair of unit speed curves of constant  breadth where α is spacelike (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is  non-zero constant then m3 = 0 and α is a general helix in the three dimensional Walker manifold  (M, gǫ  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also the curve β is given as:  β(s⋆) = α(s) + m1T + cY  (80)  where c is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is non zero constant, then from (79) we obtain that  � τ  κ  �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Also from the second and third equation of (76) we get m3 = 0 because h = 0 and m2 is a real  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a geodesic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair of unit speed curves of constant  breadth where α is spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is  not zero, then the curve β can be expressed as one of the following cases:  β(s∗) = α(s) + m1T + 1  c0  ( ¨m1 − m1)Y + ˙m1U,  (81)  where m1 =  1  √  1+c2  0  �  a1e  √  1+c2  0θ − a2e−√  1+c2  0θ�  , m3 = − ˙m1 and m2 = 1  c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let us consider that α is a general helix in Wlaker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we have τ  κ = c0 =  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (79), we have  �1  κ  �1  κm′  1  �′�′  −  �  c2  0 + 1  �  m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (82)  By means of changing of the independant variable s with z =  �  κds, we obtain  m′  1 = dm1  ds = dm1  dz  dz  ds = ˙m1κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  From (82), we get  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 − (c2  0 + 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (83)  If we solve this equation we get  m1 =  1  �  1 + c2  0  �  a1e  √  1+c2  0θ − a2e−√  1+c2  0θ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (84)  From (77) we have m3 = − ˙m1 and m2 =  1  c0( ¨m1 − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  13  Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a geodesic  spacelike curve (ǫ2 = 1, ǫ3 = −1) and lying in a timelike surface on Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0 then the curve  β is expressed as  β(s∗) = α(s) + cY,  (85)  where c is a constant real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m′  1 = 0 then h = 0 and from (76) we have m3 = 0 and m2 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2  Case (For Principal line)  If α is principal line, then τg = 0 and τ = −θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (72)  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m2κ sinh θ + m3κ cosh θ − h(s)  m′  2  =  m1κ sinh θ  m′  3  =  −m1κ cosh θ,  (86)  By mean of changing of the independant variable s with θ =  �  τds, we get  \uf8f1  \uf8f2  \uf8f3  ˙m1  =  m3  κ  τ cosh θ + m2  κ  τ sinh θ − h(s)  τ(s)  ˙m2  =  m1  κ  τ sinh θ  ˙m3  =  −m1  κ  τ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (87)  Denoted by h(s)  τ(s) = g(θ) and κ  τ = φ, we have  \uf8f1  \uf8f2  \uf8f3  ˙m1  =  φ(m3 cosh θ + m2 sinh θ) − g(θ)  ˙m2  =  m1φ sinh θ  ˙m3  =  −m1φ cosh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (88)  From the equations in (88) we have  \uf8f1  \uf8f2  \uf8f3  1  φ( ˙m1 + g)  =  m3 cosh θ + m2 sinh θ  ˙m2 sinh θ + ˙m3 cosh θ  =  −m1φ  ˙m2 cosh θ  =  −m3 sinh θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (89)  Differentiating the first equation in (88), we get  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 + ¨g − d  dθ  � ˙φ  φ( ˙m1 + g)  �  + d  dθ(φ2m1) − ( ˙m1 + g)  − ˙φ  �  cosh θ  �  m1φ sinh θdθ − sinh θ  �  m1φ cosh θdθ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (90)  Subcase 1: m1 ̸= 0 (m′  1 = −h  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If m′  1 = −h  2 then ˙m1 = −g  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (90) we obtain  −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 + d  dθ  � ˙φ  φ ˙m1  �  + d  dθ(φ2m1) + ˙m1 − ˙φ  �  cosh θ  �  m1φ sinh θdθ − sinh θ  �  m1φ cosh θdθ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' (91)  14  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be principal line  and a general helix then β is given by  β(s∗) = α(s) + m1T + m2Y + m3U,  (92)  where  m1 =  1  √  1 + c2  �  a1e  √  1+c2θ − a2e−  √  1+c2θ�  ,  m2 = c  �  m1 sinh θdθ and m3 = −c  �  m1 cosh θdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If α is helix curve then φ = κ  τ = c = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (91) we have  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 − (1 + c2) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (93)  m1 =  1  √  1 + c2  �  a1e  √  1+c2θ − a2e−  √  1+c2θ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (94)  Subcase 2: m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  From the equations in (72) we have m2 = c2 = constant ̸= 0, m3 = c3 = constant ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The first  equation in (72) gives  tanh θ = −c2  c3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (95)  Then θ is a constant and we have τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be principal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  If m1 = 0 then α is planar curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' The curve β is expressed as  β(s∗) = α(s) + c2Y + c3U,  (96)  where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3  Case where α is spacelike and ǫ2 = −1 and ǫ3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Let α be a spacelike with ǫ2 = −1 and ǫ3 = 1 lying on a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Differentiating (34) with respect to s and using (30) we obtain  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m2κg + m3κn − h(s)  m′  2  =  m1κg − m3τg  m′  3  =  −m2τg − m1κn,  (97)  where h(s) = ds∗  ds + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since α is spacelike and ǫ2 = −1 andǫ3 = 1, then, if we assume that (α, β) is a curve pair of  constant breadth, we have  ∥β − α∥ = m2  1 − m2  2 + m2  3 = constant,  (98)  15  which imlplies that  m1  dm1  ds + m2  dm2  ds − m3  dm3  ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (99)  If we combine (97) and (99) we get  m1h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (100)  If α and β are curves of constant breadth then m1 = 0 or h(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 ̸= 0 (that is h(s) = 0)  then d = m1T + m2Y + m3U becomes a constant vector because d′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So β(s∗) is a translation  of α along the constant vector d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also h(s) = 0 gives s∗ = −s + c, where c is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Since κg ̸= 0, here we investigate curves of constant breadth for m1 ̸= 0 or m1 = 0 in some special  case (asymptotic line or principal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='1  Case (For Asymptotic line)  Let α be non-straight line asymptotic line on a timelike surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then κn = κ sinh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' As  κ ̸= 0, we get cosh θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So it implies that κg = κ, τg = −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (97), we have following  differential equation system  \uf8f1  \uf8f2  \uf8f3  m′  1  =  m2κ − h(s)  m′  2  =  m1κ + m3τ  m′  3  =  −m2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (101)  By differentiating the second equation in (101) with respect to s and using the first and third equa-  tions in (101), we get  1  κ  �1  κ(m′  1 + h)  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κ(m′  1 + h)  �′  −  �τ  κ  �2  (m′  1+h)+  �τ  κ  �′ κ  τ m1 −m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (102)  Subcase 1: m1 ̸= 0 (h(s) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  The equation (102) is given by  1  κ  �1  κm′  1  �′′  +  ��1  κ  �′  − 1  τ  �τ  κ  �′� �1  κm′  1  �′  −  ��τ  κ  �2  + 1  �  m′  1 +  �τ  κ  �′ κ  τ m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (103)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a asymptotic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) be a pair of unit speed curves of constant  breadth where α is spacelike (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is  non-zero constant then m2 = 0 and α is a general helix in the three dimensional Walker manifold  (M, gǫ  f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Also the curve β is given as:  β(s⋆) = α(s) + m1T + m3U  (104)  where m3 is a real constant and s∗ = −s + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is non zero constant, then from (103) we obtain that  � τ  κ  �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' So α is a general helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Also from the first and third equation of (101) we get m2 = 0 and m3 is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  16  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be a asymptotic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair of unit speed curves of constant  breadth where α is timelike curve and lying in a timelike surface Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 is not zero, then the  curve β can be expressed as one of the following cases:  β(s∗) = α(s) + m1T + ˙m1Y + 1  c0  ( ¨m1 + m1)U,  (105)  where  m1 =  1  �  c2  0 + 1  �  a1e  √  c2  0+1z − a2e  √  c2  0+1z�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let us consider that α is a general helix in Walker 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Then we have τ  κ = c0 =  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' From (103), we have  �1  κ  �1  κm′  1  �′�′  −  �  c2  0 + 1  �  m′  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (106)  By means of changing of the independant variable s with z =  �  κds, we obtain  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  m1 − (c2  0 + 1) ˙m1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  (107)  If we solve this equation we get  m1 =  1  �  c2  0 + 1  �  a1e  √  c2  0+1z − a2e  √  c2  0+1z�  (108)  From (101) we obtain m2 = ˙m1 and m3 = 1  c0( ¨m1 + m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Subcase 2: m1 = 0  With the same computation as above, we have the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' β) be a curve pair of constant breadth in (M, gf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If α is a spacelike  asymptotic curve (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0, then the  curve β is given by  β(s∗) = α(s)+  �  b1 cos  ��  τds  �  + b2 sin  ��  τds  ��  Y (s)+  �  −b1 sin  ��  τds  �  + b2 cos  ��  τds  ��  U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='2  Case (For Principal line)  In this case we have the two following theorems:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be spacelike  principal line (with ǫ2 = −1 and ǫ3 = 1) and a general helix then β is given by  β(s∗) = α(s) + m1T + m2Y + m3U,  (109)  where  m1 =  1  √  1 + c2  �  a1e  √  1+c2θ − a2e−  √  1+c2θ�  ,  m2 = c  �  m1 cosh θdθ and m3 = −c  �  m1 sinh θdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  17  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let (α, β) be a pair curves of constant breadth in (M, gfǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Let α be principal line  (with ǫ2 = −1 and ǫ3 = 1) lying in a timelike surface in Mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' If m1 = 0 then α is general helix or  α is planar curve and the curve β is expressed as  β(s∗) = α(s) + c2Y + c3U,  (110)  where c2 and c3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Acknowledgments  The author would like to thank the anonymous Referees for their comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' All  many thanks to professor Ferdag Kahraman from Ahi Evran University (Turkish) for their remarks  and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Kahraman, Curves of constant breadth according to Darboux frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Com-  mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=', Einige Bemerkungen uber Kurven und Flachen konstanter Breite, Ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Verh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  sachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Leipzig 67, 290–297 (1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Brozos-V´azquez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Gilkey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Nikevi´c and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Ndiaye and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Niang, Minimal graphs on three-dimensional Walker man-  ifolds, Proceedings of the First NLAGA-BIRS Symposium, Dakar, Senegal, Trends Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' Fujivara, On space curve of constant breadth, Tohoku Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' 5, 179–184 (1914).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Gningue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Ndiaye, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Nkunzimana, Biharmonic Curves in a Strict Walker 3-Manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=', 1–6 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  [9] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Kose, Some properties of ovals and curves of constant width in a plane, Doga Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Serial  B (8), 2, 119-126 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Magden and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Kose, On the curves of constant breadth in E4 space, Turkish J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}
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+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfSwPe/content/2301.03071v1.pdf'}

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+arXiv:2301.13448v1  [nlin.AO]  31 Jan 2023
+Delay, resonance and the Lambert W function
+Kenta Ohira1 and Toru Ohira2
+1Future Value Creation Research Center,
+Graduate School of Informatics, Nagoya University, Japan
+2Graduate School of Mathematics, Nagoya University, Japan
+February 1, 2023
+Abstract
+We present here a connection between the solutions of the transcenden-
+tal trigonometric equation and the Lambert W function. This connection
+emerged through an analysis of resonant conditions with a recently pro-
+posed simple delay differential equation that shows transient oscillatory
+behaviors. We investigate and present the connection both analytically
+and numerically.
+1
+Introduction
+In the various fields including mathematics, biology, physics, engineering, eco-
+nomics, and so on, there have been interests in investigating the effect of delays
+in the system.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). Typically, delays intro-
+duce oscillations and complex behaviors to otherwise simple and well-behaved
+systems. Longer delays are known to induce an increase in the complexity of
+dynamics. The representative example is the Mackey–Glass equation[8] which
+shows the sequence of the monotonic convergence, transient oscillations, persis-
+tent oscillations, and chaotic dynamics as the delay parameter in the feedback
+function becomes longer.
+The main mathematical approaches and modeling tools are “Delay Differen-
+tial Equations”. Mathematical analysis of such delay systems has been posing
+challenges. Though understanding of the delay systems have been gradually
+gained (e.g.[15]), there is more to be investigated and explored. One recent an-
+alytical approach to simple delay differential equations is an application of the
+Lambert W function. It has been shown that the solution of the simple delay
+differential equation can be expressed using the W function.
+In this paper, we follow this path of analyzing a simple delay differential
+equation using the W function. Specifically, we analyze the transient oscillatory
+behaviors of a delay differential equation that shows a resonant behavior[16]. In
+this resonance, the optimal height of the power spectrum of the dynamical
+trajectory is observed with the suitably tuned value of the delay parameter.
+1
+
+We focus on the appearance of the power spectrum peaks and found that it
+relates to the transcendental trigonometric equation. That condition, on the
+other hand, can be also expressed in terms of the W function. We connect these
+two approaches both analytically and numerically to provide a new way that
+the W function can be useful.
+2
+Delay Differential Equation
+Recently, we have proposed and studied the following delay differential equation:
+dX(t)
+dt
++ atX(t) = bX(t − τ)
+(1)
+where a ≥ 0, b ≥ 0, τ ≥ 0 are real parameters with τ interpreted as the delay.
+This equation is a slight extension of much-studied Hayes’s equation[4],
+dX(t)
+dt
++ αX(t) = βX(t − τ)
+(2)
+where α, β are real constants.
+Even though the apparent change from (2) to (1) is small with only the
+second term changed to a linear function of time, their behaviors are quite
+different. Particularly for (1), we have shown oscillatory transient dynamics
+appear and disappear as the value of delay increases without losing asymptotic
+stability of X = 0.
+2.1
+Analysis
+Let us first review some properties of (1).
+When b = 0.
+With the initial
+condition X(t = 0) = X0, the solution to the equation is given as
+X(t) = X0e− 1
+2 at2
+(3)
+Thus, this solution is a gaussian shape. We also note in this case that (1) is the
+equation for the ground state of the quantum simple harmonic oscillator with
+the interpretation of t as a position rather than time (e.g.[18]).
+The case that a = 0 is a special case of (2). In this case, the origin X = 0 is
+asymptotically stable only in the range of
+− π/2τ < b < 0.
+(4)
+For the general case with a > 0, b > 0 with the delay τ = 0, the solution
+X(t = 0) = X0 is obtained as
+X(t) = X0e− 1
+2 at2+bt
+(5)
+This is again a Gaussian with its peak at b/a.
+2
+
+For the case with a > 0, b > 0 with the delay τ → ∞, the dynamics is
+influenced by the initial function for all 0 ≤ t. Thus for the initial function
+X(t) = X0, (−τ ≤ t ≤ 0), we can replace the right hand side of equation (1) as
+bX(t − τ) → X0. The solution can be obtained as
+X(t) = X0e− 1
+2 at2(1 + b
+� t
+0
+e
+1
+2 as2ds) = X0e− 1
+2 at2(1 + b
+� π
+2aerfi(
+� a
+2 t))
+(6)
+where erfi(x) is the imaginary error function defined as
+erfi(x) =
+2
+√π
+� x
+0
+es2 ds
+(7)
+The shape of this function is also a single peaked function approaching to the
+origin X = 0.
+We now see one of the major differences between the equations (1) and (2).
+In the latter, the asymptotic stability of X = 0 is lost for the larger delay with
+0 < α < β, while in the former, it is kept even with the large delay for all
+a > 0, b > 0. Also, even though both exhibit transient oscillations, (1) shows
+coherent oscillations with the tuned value of the delay τ. We now turn our
+attention to these resonating phenomena.
+2.2
+Power Spectrum and Resonance
+The transient dynamics of equation (1) are investigated through numerical sim-
+ulations. Some examples are shown in Fig. 1. With zero delays, the shape of
+the dynamics is the gaussian as derived in the previous subsection. The oscil-
+latory behaviors arise on top of the gaussian trajectory with increasing delay.
+Further increase of delay changes the oscillatory shape into trains of pulses with
+decreasing height at the delay interval, and asymptotically the pulses disappear
+leading to the gaussian shape (5). As mentioned, the asymptotic stability of
+X = 0 does not change by the increasing delay in this parameter set.
+This property is in contrast to that of equation (2) where the onset of the
+oscillation by the increasing delay leads to the loss of stability. Thus, it is differ-
+ent from stability switching phenomena (e.g.[17]) with the delay as the bifurca-
+tion parameter. It is also different from the delay induced transient oscillation
+(DITO)[19, 20]. The phenomena arise in coupled delay differential equations
+exhibiting the prolonged duration of oscillatory behaviors with increasing delay.
+We investigate these oscillatory behaviors for the case a > 0, b > 0, and
+finite τ ̸= 0 by taking the Fourier transform of the equation (1).
+iω ˆX(ω) + iad ˆX(ω)
+dω
+= −b ˆX(ω)eiωτ
+(8)
+where
+ˆX(ω) =
+� ∞
+−∞
+eiωtX(t)dt
+(9)
+3
+
+(B)
+X(t)
+t
+(A)
+X(t)
+t
+(E)
+X(t)
+t
+(F)
+X(t)
+t
+(C)
+X(t)
+t
+(D)
+X(t)
+t
+50
+100
+150
+200
+0
+20000
+40000
+60000
+80000
+100000
+120000
+50
+100
+150
+200
+0
+50
+100
+150
+200
+250
+300
+350
+50
+100
+150
+200
+0
+5
+10
+15
+20
+50
+100
+150
+200
+0
+2
+4
+6
+50
+100
+150
+200
+0.0
+0.5
+1.0
+1.5
+2.0
+50
+100
+150
+200
+0.0
+0.5
+1.0
+1.5
+Figure 1: Representative dynamics of the main equation (1) with different values
+of the delays, τ. The parameters are set at a = 0.15, b = 6.0 with the initial
+interval condition as X(t) = 0.1(−τ ≤ t ≤ 0). The values of the delays τ are
+(A)2, (B)4, (C)7, (D)10, (E)20, (F)25.
+4
+
+The solution is given as
+ˆX(ω) = CExp[− 1
+2aω2 + b
+τaeiωτ]
+(10)
+with C as the integration constant. We can calculate the power spectrum from
+equation(9).
+S(ω) = | ˆX(ω)|2 = ˆX(ω) ˆ
+X∗(ω) = C2Exp[−1
+aω2 + 2b
+τa cos ωτ]
+(11)
+We have plotted this equation for the power spectrum for the various delays.
+Results with the same parameter setting as in Fig. 1 are shown in Fig. 2.
+In the previous work, we noted that the peak of the power spectrum shows
+a maximum height with the tuned value of the delay. The higher peak indicates
+a more coherent oscillation. It is in this sense that the resonance exists with the
+delay as a tuning parameter.
+2.3
+Peaks of the Power Spectrum
+We now focus on the analysis of these power spectrum peaks. The appearance
+and disappearance of the peaks in the power spectrum correspond to those of
+oscillatory behavior. By taking the derivative of (11), we see the maximum and
+minimum points of the power spectrum function occur at ω satisfying,
+ω = −b sin ωτ
+(12)
+They are given by the intersection points of the two functions from both
+sides of this condition. The position of the first peak corresponds to the second
+non-zero smallest intersection point (the first one corresponds to the minimum
+before the peak).
+We can also infer the condition for the appearance of the series of power
+spectrum peaks. Each peak appears when the intersection point is the tangent
+point (Fig.3). This gives the following conditions for the n-th peak.
+ω = −b sin ωτ,
+1 = −bτ cos ωτ,
+(2n − 1)π
+τ
+< ω < (2n − 1
+2)π
+τ
+, (n = 1, 2, . . .).
+(13)
+If we set λ = bτ,θ = ωτ, the above condition leads to
+θ = −λ sin θ,
+1 = −λ cosθ,
+(2n − 1)π < θ < (2n − 1
+2)π, (n = 1, 2, . . . ).
+(14)
+From this set of equations, we can numerically estimate the solutions (θn, λn)
+for each n. First, we can derive that θn and λn are related by
+λ2
+n = θ2
+n + 1
+(15)
+Then, the followings are obtained:
+θ = −
+�
+θ2 + 1 sin θ,
+(2n − 1)π < θ < (2n − 1
+2)π, (n = 1, 2, . . .),
+(16)
+5
+
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+5
+10
+15
+20
+25
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+5.0×1016
+1.0×1017
+1.5×1017
+2.0×1017
+(A)
+ω
+S(ω)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1×108
+2×108
+3×108
+4×108
+5×108
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+200
+400
+600
+800
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+10
+20
+30
+40
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+200
+400
+600
+800
+(B)
+ω
+S(ω)
+(C)
+ω
+S(ω)
+(F)
+ω
+S(ω)
+(E)
+ω
+S(ω)
+(D)
+ω
+S(ω)
+Figure 2: Representative power spectrums given by the equation (1) with dif-
+ferent values of the delays τ.
+The parameters are set as the same as Fig.1;
+a = 0.15, b = 6.0, C = 1 with the initial interval condition as X(t) = 0.1(−τ ≤
+t ≤ 0). The values of the delays τ are (A)2, (B)4, (C)7, (D)10, (E)20, (F)25.
+6
+
+5
+10
+15
+20
+- 20
+- 10
+10
+20
+5
+10
+15
+20
+- 20
+- 10
+10
+20
+30
+(A)
+(B)
+θ1
+θ2
+θ3
+θ1
+θ2
+θ3
+Figure 3: The plots of equations (A) (16) and (B) (17).
+or
+θ = tan θ,
+(2n − 1)π < θ < (2n − 1
+2)π, (n = 1, 2, . . . ),
+(17)
+or
+− 1
+λ = cos
+�
+λ2 − 1,
+�
+((2n − 1)π)2 + 1 < λ <
+�
+((2n − 1
+2)π)2 + 1.
+(18)
+The values of the solutions (θn, λn) are listed in the Table 1.
+n
+θn
+λn
+tan θn
+1
+4.49341
+4.60334
+4.49341
+2
+10.9041
+10.9499
+10.9041
+3
+17.2208
+17.2498
+17.2208
+4
+23.5195
+23.5407
+23.5195
+5
+29.8116
+29.8284
+29.8116
+6
+36.1006
+36.1145
+36.1006
+7
+42.3879
+42.3997
+42.3879
+8
+48.6741
+48.6844
+48.6741
+9
+54.9597
+54.9688
+54.9597
+10
+61.2447
+61.2529
+61.2447
+Table 1: Numerically estimated values of θn, λn and tan θn
+2.4
+Lambert W function
+We present here that we can alternatively obtain the solutions (θn, λn) discussed
+in the previous subsection using the Lambert W function.
+W function is defined as a multivalued complex function with a complex
+variable z satisfying
+z = Wk(z)eWk(z),
+(k = 0, 1, 2, . . .),
+(19)
+7
+
+where k is the branch number. It has been pointed out that the W function
+can be used to express the solution of simple delay differential equations[21, 22].
+What we present here is another way the W function can be utilized.
+We start with (14) and use e−iθ = cos θ − i sin θ to obtain
+1 − iθ = −λe−iθ.
+(20)
+By defining Q ≡ −1 + iθ we can rewrite (20) as
+QeQ = λ
+e .
+(21)
+We can now use the W function on (21), and Q can be expressed as
+Q = Wk(λ
+e ).
+(22)
+By the constraints that θ is real, the real part of Q must be equal to −1, or
+Re[Q] = Re[Wk(λ
+e )] = −1.
+(23)
+Also by the definition of Q, we have θ from the imaginary part,
+θ = Im[Q] = Im[Wk(λ
+e )].
+(24)
+Further, we can prove the following.
+Lemma
+Re[Wk(λ
+e )] = −1 ⇐⇒ |Wk(λ
+e )| = λ
+(25)
+Proof
+Necessary Part:
+By the definition of the W function,
+Wk(λ
+e )Exp[Wk(λ
+e )] = λ
+e
+(26)
+Also, by the assumption of
+Re[Wk(λ
+e )] = −1,
+(27)
+we can write
+Wk(λ
+e ) = −1 + iµ,
+(µ ∈ R).
+(28)
+8
+
+Then, (26) leads to
+λ
+e = Wk(λ
+e )Exp[Wk(λ
+e )] = Wk(λ
+e )Exp[−1 + iµ].
+(29)
+Thus,
+Wk(λ
+e )Exp[iµ] = λ,
+(30)
+which is equivalent to
+|Wk(λ
+e )| = λ.
+(31)
+Sufficient Part:
+By (26) and λ > 0 we have
+λ
+e = |Wk(λ
+e )Exp[Wk(λ
+e )]| = |Wk(λ
+e )||Exp[Wk(λ
+e )]|.
+(32)
+Also, by the assumption
+|Wk(λ
+e )| = λ,
+(33)
+this leads to
+λ
+e = λ|Exp[Wk(λ
+e )]|
+(34)
+If we set
+Wk(λ
+e ) = η + iµ,
+(η, µ ∈ R)
+(35)
+(34) can be re-writen as
+1
+e = |Exp[Wk(λ
+e )]| = eη,
+(36)
+leading to
+η = Re[Wk(λ
+e )] = −1
+(37)
+We are now in a position to put together pieces obtained through the analysis
+of resonant peaks. They can be summarized as follows.
+9
+
+Theorem
+Let (θ, λ) satisfy the following,
+θ = −λ sin θ,
+1 = −λ cosθ,
+(2n − 1)π < θ < (2n − 1
+2)π, (n = 1, 2, . . . ),
+(38)
+then they also satisfy the following for some k
+Re[Wk(λ
+e )] = −1,
+(39)
+and
+θ = Im[Wk(λ
+e )],
+λ = |Wk(λ
+e )|
+(40)
+Based on the above, we further want to investigate between the n-th root and
+the n-th branch of the W function. With numerical estimations, we conjecture
+the following.
+Conjecture
+The n-th root θn of the following,
+θn = tan θn,
+(2n − 1)π < θn < (2n − 1
+2)π, (n = 1, 2, . . .),
+(41)
+is given by the n-th branch of the W function
+θn = Im[Wn(λn
+e )],
+(42)
+where λn satisfies
+Re[Wn(λn
+e )] = −1.
+(43)
+In Table 2, we show the results of estimated related numerical values. Com-
+paring Tables 1 and 2 supports the above theorems and conjecture.
+Thus,
+through the analysis of resonant peaks, we have connected the solutions of the
+trigonometric transcendental function with a specific value of the n-th branch of
+the W function. To the author’s knowledge, this relation has not been explicitly
+pointed out.
+3
+Discussion
+In this paper, we presented some properties of the Lambert W function through
+the analysis of resonant behaviors of a simple delay differential equation. The
+connection between the solutions of trigonometric transcendental equation and
+that of the W function is revealed. It remains to be explored if these properties
+of the W function can be utilized in more broader context.
+10
+
+n
+λn
+Wn( λn
+e )
+|Wn( λn
+e )|
+1
+4.60334
+-1.0 + i 4.49341
+4.60334
+2
+10.9499
+-1.0 + i 10.9041
+10.9499
+3
+17.2498
+-1.0 + i 17.2208
+17.2498
+4
+23.5407
+-1.0 + i 23.5195
+23.5407
+5
+29.8284
+-1.0 + i 29.8116
+29.8284
+6
+36.1145
+-1.0 + i 36.1006
+36.1145
+7
+42.3997
+-1.0 + i 42.3879
+42.3997
+8
+48.6844
+-1.0 + i 48.6741
+48.6844
+9
+54.9688
+-1.0 + i 54.9597
+54.9688
+10
+61.2529
+-1.0 + i 61.2447
+61.2529
+Table 2: Numerically estimated values of λn, Wn( λn
+e ) and |Wn( λn
+e )|
+Acknowledgments
+The authors would like to thank useful discussions with Prof. Hideki Ohira
+and members of his research group at Nagoya University. This work was sup-
+ported by ”Yocho-gaku” Project sponsored by Toyota Motor Corporation, JSPS
+Topic-Setting Program to Advance Cutting-Edge Humanities and Social Sci-
+ences Research Grant Number JPJS00122674991, JSPS KAKENHI Grant Num-
+ber 19H01201, and the Research Institute for Mathematical Sciences, an Inter-
+national Joint Usage/Research Center located in Kyoto University.
+References
+[1] U. an der Heiden. Delays in physiological systems. J. Math. Biol., 8:345–364,
+1979.
+[2] R. Bellman and K. Cooke. Differential-Difference Equations. Academic
+Press, New York, 1963.
+[3] J. L. Cabrera and J. G. Milton. On–off intermittency in a human balancing
+task. Phys. Rev. Lett., 89:158702, 2002.
+[4] N. D. Hayes. Roots of the transcendental equation associated with a certain
+difference–differential equation. J. Lond. Math. Soc., 25:226–232, 1950.
+[5] T. Insperger. Act-and-wait concept for continuous-time control systems with
+feedback delay. IEEE Trans. Control Sys. Technol., 14:974–977, 2007.
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+tended by a time-delayed term. Stoch. Stoch. Rep., 40:23–42, 1992.
+[7] A. Longtin and J. G. Milton. Insight into the transfer function, gain and
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+Biol. Cybern., 61:51–58, 1989.
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+and S. A. Campbell. The time–delayed inverted pendulum: Implications for
+human balance control. Chaos, 19:026110, 2009.
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+to the life sciences. Springer, New York, 2010.
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+tions. Wiley & Sons, New York, 1989.
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+a route to act-and-wait control. Ann. Rev. Control, 30:159–168, 2006.
+[14] M. Szydlowski and A. Krawiec. The Kaldor–Kalecki model of business cycle
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+[15] S. R.Taylor and S. A. Campbell. Approximating chaotic saddles for delay
+differential equations. Phys. Rev. E, 75: 046215, 2007.
+[16] K. Ohira. Resonating Delay Equation. EPL, 137: 23001, 2022.
+[17] X. Yan, F. Liu and C. Zhang. Multiple stability switches and Hopf bifur-
+cation in a damped harmonic oscillator with delayed feedback. Nonlinear
+Dynamics, 99:2011, 2020.
+[18] J. J. Sakurai. Modern Quantum Mechanics. Benjamin/Cummings, Menlo
+Park, California, 1985.
+[19] J. Milton, P. Naik, C. Chan, and S. A. Campbell. Indecision in neural
+decision making models. Math. Model. Nat. Phenom., 5:125–145, 2010.
+[20] K. Pakdaman, C. Grotta-Ragazzo, and C. P. Malta. Transient regime dura-
+tion in continuous-time neural networks with delay. Phys. Rev. E, 58:3623–
+3627, 1998.
+[21] R. Pusenjak. Application of Lambert function in the control of production
+systems with delay. Int. J. Eng. Sci, 6:28?–38, 2017.
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+system by Lambert W function. Automatica, 42: 1791–1799, 2006.
+12
+

AdFQT4oBgHgl3EQf8zdx/content/tmp_files/load_file.txt

@@ -0,0 +1,437 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf,len=436
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='13448v1  [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='AO]  31 Jan 2023  Delay, resonance and the Lambert W function  Kenta Ohira1 and Toru Ohira2  1Future Value Creation Research Center,  Graduate School of Informatics, Nagoya University, Japan  2Graduate School of Mathematics, Nagoya University, Japan  February 1, 2023  Abstract  We present here a connection between the solutions of the transcenden-  tal trigonometric equation and the Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' This connection  emerged through an analysis of resonant conditions with a recently pro-  posed simple delay differential equation that shows transient oscillatory  behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' We investigate and present the connection both analytically  and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  1  Introduction  In the various fields including mathematics, biology, physics, engineering, eco-  nomics, and so on, there have been interests in investigating the effect of delays  in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Typically, delays intro-  duce oscillations and complex behaviors to otherwise simple and well-behaved  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Longer delays are known to induce an increase in the complexity of  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The representative example is the Mackey–Glass equation[8] which  shows the sequence of the monotonic convergence, transient oscillations, persis-  tent oscillations, and chaotic dynamics as the delay parameter in the feedback  function becomes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  The main mathematical approaches and modeling tools are “Delay Differen-  tial Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Mathematical analysis of such delay systems has been posing  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Though understanding of the delay systems have been gradually  gained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' [15]), there is more to be investigated and explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' One recent an-  alytical approach to simple delay differential equations is an application of the  Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' It has been shown that the solution of the simple delay  differential equation can be expressed using the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  In this paper, we follow this path of analyzing a simple delay differential  equation using the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Specifically, we analyze the transient oscillatory  behaviors of a delay differential equation that shows a resonant behavior[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' In  this resonance, the optimal height of the power spectrum of the dynamical  trajectory is observed with the suitably tuned value of the delay parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  1  We focus on the appearance of the power spectrum peaks and found that it  relates to the transcendental trigonometric equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' That condition, on the  other hand, can be also expressed in terms of the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' We connect these  two approaches both analytically and numerically to provide a new way that  the W function can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  2  Delay Differential Equation  Recently, we have proposed and studied the following delay differential equation:  dX(t)  dt  + atX(t) = bX(t − τ)  (1)  where a ≥ 0, b ≥ 0, τ ≥ 0 are real parameters with τ interpreted as the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  This equation is a slight extension of much-studied Hayes’s equation[4],  dX(t)  dt  + αX(t) = βX(t − τ)  (2)  where α, β are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Even though the apparent change from (2) to (1) is small with only the  second term changed to a linear function of time, their behaviors are quite  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Particularly for (1), we have shown oscillatory transient dynamics  appear and disappear as the value of delay increases without losing asymptotic  stability of X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1  Analysis  Let us first review some properties of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  When b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  With the initial  condition X(t = 0) = X0, the solution to the equation is given as  X(t) = X0e− 1  2 at2  (3)  Thus, this solution is a gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' We also note in this case that (1) is the  equation for the ground state of the quantum simple harmonic oscillator with  the interpretation of t as a position rather than time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  The case that a = 0 is a special case of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' In this case, the origin X = 0 is  asymptotically stable only in the range of  − π/2τ < b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (4)  For the general case with a > 0, b > 0 with the delay τ = 0, the solution  X(t = 0) = X0 is obtained as  X(t) = X0e− 1  2 at2+bt  (5)  This is again a Gaussian with its peak at b/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  2  For the case with a > 0, b > 0 with the delay τ → ∞, the dynamics is  influenced by the initial function for all 0 ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Thus for the initial function  X(t) = X0, (−τ ≤ t ≤ 0), we can replace the right hand side of equation (1) as  bX(t − τ) → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The solution can be obtained as  X(t) = X0e− 1  2 at2(1 + b  � t  0  e  1  2 as2ds) = X0e− 1  2 at2(1 + b  � π  2aerfi(  � a  2 t))  (6)  where erfi(x) is the imaginary error function defined as  erfi(x) =  2  √π  � x  0  es2 ds  (7)  The shape of this function is also a single peaked function approaching to the  origin X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  We now see one of the major differences between the equations (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  In the latter, the asymptotic stability of X = 0 is lost for the larger delay with  0 < α < β, while in the former, it is kept even with the large delay for all  a > 0, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Also, even though both exhibit transient oscillations, (1) shows  coherent oscillations with the tuned value of the delay τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' We now turn our  attention to these resonating phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  Power Spectrum and Resonance  The transient dynamics of equation (1) are investigated through numerical sim-  ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Some examples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' With zero delays, the shape of  the dynamics is the gaussian as derived in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The oscil-  latory behaviors arise on top of the gaussian trajectory with increasing delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Further increase of delay changes the oscillatory shape into trains of pulses with  decreasing height at the delay interval, and asymptotically the pulses disappear  leading to the gaussian shape (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' As mentioned, the asymptotic stability of  X = 0 does not change by the increasing delay in this parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  This property is in contrast to that of equation (2) where the onset of the  oscillation by the increasing delay leads to the loss of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Thus, it is differ-  ent from stability switching phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' [17]) with the delay as the bifurca-  tion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' It is also different from the delay induced transient oscillation  (DITO)[19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The phenomena arise in coupled delay differential equations  exhibiting the prolonged duration of oscillatory behaviors with increasing delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  We investigate these oscillatory behaviors for the case a > 0, b > 0, and  finite τ ̸= 0 by taking the Fourier transform of the equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  iω ˆX(ω) + iad ˆX(ω)  dω  = −b ˆX(ω)eiωτ  (8)  where  ˆX(ω) =  � ∞  −∞  eiωtX(t)dt  (9)  3  (B)  X(t)  t  (A)  X(t)  t  (E)  X(t)  t  (F)  X(t)  t  (C)  X(t)  t  (D)  X(t)  t  50  100  150  200  0  20000  40000  60000  80000  100000  120000  50  100  150  200  0  50  100  150  200  250  300  350  50  100  150  200  0  5  10  15  20  50  100  150  200  0  2  4  6  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  50  100  150  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5  Figure 1: Representative dynamics of the main equation (1) with different values  of the delays, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The parameters are set at a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='15, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0 with the initial  interval condition as X(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1(−τ ≤ t ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The values of the delays τ are  (A)2, (B)4, (C)7, (D)10, (E)20, (F)25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  4  The solution is given as  ˆX(ω) = CExp[− 1  2aω2 + b  τaeiωτ]  (10)  with C as the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' We can calculate the power spectrum from  equation(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  S(ω) = | ˆX(ω)|2 = ˆX(ω) ˆ  X∗(ω) = C2Exp[−1  aω2 + 2b  τa cos ωτ]  (11)  We have plotted this equation for the power spectrum for the various delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Results with the same parameter setting as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' 1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  In the previous work, we noted that the peak of the power spectrum shows  a maximum height with the tuned value of the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The higher peak indicates  a more coherent oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' It is in this sense that the resonance exists with the  delay as a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='3  Peaks of the Power Spectrum  We now focus on the analysis of these power spectrum peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The appearance  and disappearance of the peaks in the power spectrum correspond to those of  oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' By taking the derivative of (11), we see the maximum and  minimum points of the power spectrum function occur at ω satisfying,  ω = −b sin ωτ  (12)  They are given by the intersection points of the two functions from both  sides of this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The position of the first peak corresponds to the second  non-zero smallest intersection point (the first one corresponds to the minimum  before the peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  We can also infer the condition for the appearance of the series of power  spectrum peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Each peak appears when the intersection point is the tangent  point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' This gives the following conditions for the n-th peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  ω = −b sin ωτ,  1 = −bτ cos ωτ,  (2n − 1)π  τ  < ω < (2n − 1  2)π  τ  , (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (13)  If we set λ = bτ,θ = ωτ, the above condition leads to  θ = −λ sin θ,  1 = −λ cosθ,  (2n − 1)π < θ < (2n − 1  2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (14)  From this set of equations, we can numerically estimate the solutions (θn, λn)  for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' First, we can derive that θn and λn are related by  λ2  n = θ2  n + 1  (15)  Then, the followings are obtained:  θ = −  �  θ2 + 1 sin θ,  (2n − 1)π < θ < (2n − 1  2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ),  (16)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  5  10  15  20  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0×1016  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0×1017  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5×1017  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0×1017  (A)  ω  S(ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  1×108  2×108  3×108  4×108  5×108  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  200  400  600  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  200  400  600  800  (B)  ω  S(ω)  (C)  ω  S(ω)  (F)  ω  S(ω)  (E)  ω  S(ω)  (D)  ω  S(ω)  Figure 2: Representative power spectrums given by the equation (1) with dif-  ferent values of the delays τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  The parameters are set as the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='15, b = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0, C = 1 with the initial interval condition as X(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1(−τ ≤  t ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The values of the delays τ are (A)2, (B)4, (C)7, (D)10, (E)20, (F)25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  6  5  10  15  20  20  10  10  20  5  10  15  20  20  10  10  20  30  (A)  (B)  θ1  θ2  θ3  θ1  θ2  θ3  Figure 3: The plots of equations (A) (16) and (B) (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  or  θ = tan θ,  (2n − 1)π < θ < (2n − 1  2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ),  (17)  or  − 1  λ = cos  �  λ2 − 1,  �  ((2n − 1)π)2 + 1 < λ <  �  ((2n − 1  2)π)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (18)  The values of the solutions (θn, λn) are listed in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  n  θn  λn  tan θn  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='49341  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='60334  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='49341  2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9041  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9499  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9041  3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2208  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2498  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2208  4  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5195  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5407  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5195  5  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8116  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8284  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='8116  6  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1006  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1145  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1006  7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='3879  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='3997  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='3879  8  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6741  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6844  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='6741  9  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9597  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9688  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9597  10  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2447  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2529  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2447  Table 1: Numerically estimated values of θn, λn and tan θn  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='4  Lambert W function  We present here that we can alternatively obtain the solutions (θn, λn) discussed  in the previous subsection using the Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  W function is defined as a multivalued complex function with a complex  variable z satisfying  z = Wk(z)eWk(z),  (k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ),  (19)  7  where k is the branch number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' It has been pointed out that the W function  can be used to express the solution of simple delay differential equations[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  What we present here is another way the W function can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  We start with (14) and use e−iθ = cos θ − i sin θ to obtain  1 − iθ = −λe−iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (20)  By defining Q ≡ −1 + iθ we can rewrite (20) as  QeQ = λ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (21)  We can now use the W function on (21), and Q can be expressed as  Q = Wk(λ  e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (22)  By the constraints that θ is real, the real part of Q must be equal to −1, or  Re[Q] = Re[Wk(λ  e )] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (23)  Also by the definition of Q, we have θ from the imaginary part,  θ = Im[Q] = Im[Wk(λ  e )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (24)  Further, we can prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Lemma  Re[Wk(λ  e )] = −1 ⇐⇒ |Wk(λ  e )| = λ  (25)  Proof  Necessary Part:  By the definition of the W function,  Wk(λ  e )Exp[Wk(λ  e )] = λ  e  (26)  Also, by the assumption of  Re[Wk(λ  e )] = −1,  (27)  we can write  Wk(λ  e ) = −1 + iµ,  (µ ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (28)  8  Then, (26) leads to  λ  e = Wk(λ  e )Exp[Wk(λ  e )] = Wk(λ  e )Exp[−1 + iµ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (29)  Thus,  Wk(λ  e )Exp[iµ] = λ,  (30)  which is equivalent to  |Wk(λ  e )| = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (31)  Sufficient Part:  By (26) and λ > 0 we have  λ  e = |Wk(λ  e )Exp[Wk(λ  e )]| = |Wk(λ  e )||Exp[Wk(λ  e )]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (32)  Also, by the assumption  |Wk(λ  e )| = λ,  (33)  this leads to  λ  e = λ|Exp[Wk(λ  e )]|  (34)  If we set  Wk(λ  e ) = η + iµ,  (η, µ ∈ R)  (35)  (34) can be re-writen as  1  e = |Exp[Wk(λ  e )]| = eη,  (36)  leading to  η = Re[Wk(λ  e )] = −1  (37)  We are now in a position to put together pieces obtained through the analysis  of resonant peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' They can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  9  Theorem  Let (θ, λ) satisfy the following,  θ = −λ sin θ,  1 = −λ cosθ,  (2n − 1)π < θ < (2n − 1  2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ),  (38)  then they also satisfy the following for some k  Re[Wk(λ  e )] = −1,  (39)  and  θ = Im[Wk(λ  e )],  λ = |Wk(λ  e )|  (40)  Based on the above, we further want to investigate between the n-th root and  the n-th branch of the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' With numerical estimations, we conjecture  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Conjecture  The n-th root θn of the following,  θn = tan θn,  (2n − 1)π < θn < (2n − 1  2)π, (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' ),  (41)  is given by the n-th branch of the W function  θn = Im[Wn(λn  e )],  (42)  where λn satisfies  Re[Wn(λn  e )] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  (43)  In Table 2, we show the results of estimated related numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Com-  paring Tables 1 and 2 supports the above theorems and conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  Thus,  through the analysis of resonant peaks, we have connected the solutions of the  trigonometric transcendental function with a specific value of the n-th branch of  the W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' To the author’s knowledge, this relation has not been explicitly  pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  3  Discussion  In this paper, we presented some properties of the Lambert W function through  the analysis of resonant behaviors of a simple delay differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' The  connection between the solutions of trigonometric transcendental equation and  that of the W function is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' It remains to be explored if these properties  of the W function can be utilized in more broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  10  n  λn  Wn( λn  e )  |Wn( λn  e )|  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='60334  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0 + i 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='49341  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='60334  2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9499  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='2498  4  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='0 + i 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5195  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='5407  5  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='0 + i 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='8284  6  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1145  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0 + i 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='1006  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='0 + i 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='3997  8  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='0 + i 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content='6844  9  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9688  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0 + i 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9597  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='9688  10  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2529  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='0 + i 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2447  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='2529  Table 2: Numerically estimated values of λn, Wn( λn  e ) and |Wn( λn  e )|  Acknowledgments  The authors would like to thank useful discussions with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Hideki Ohira  and members of his research group at Nagoya University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' This work was sup-  ported by ”Yocho-gaku” Project sponsored by Toyota Motor Corporation, JSPS  Topic-Setting Program to Advance Cutting-Edge Humanities and Social Sci-  ences Research Grant Number JPJS00122674991, JSPS KAKENHI Grant Num-  ber 19H01201, and the Research Institute for Mathematical Sciences, an Inter-  national Joint Usage/Research Center located in Kyoto University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  References  [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' an der Heiden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content=' Bellman and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Cooke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Differential-Difference Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Academic  Press, New York, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Cabrera and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
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+page_content=' Grotta-Ragazzo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Malta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Transient regime dura-  tion in continuous-time neural networks with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' E, 58:3623–  3627, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Pusenjak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Application of Lambert function in the control of production  systems with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Sci, 6:28?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='–38, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Shinozaki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Mori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Robust stability analysis of linear time delay  system by Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content=' Automatica, 42: 1791–1799, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFQT4oBgHgl3EQf8zdx/content/2301.13448v1.pdf'}

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+David Maurice Brink
+20 July 1930 - 8 March 2021
+Elected FRS 1981
+C. V. Sukumar1∗, A. Bonaccorso2 †
+1Wadham College, Oxford OX1 3PN, U.K, 2INFN, Sezione di Pisa, 56127 Pisa, Italy.
+January 10, 2023
+Abstract
+David Brink was one of the leading theoretical nuclear physicists of his generation. He
+made major contributions to the study of all aspects of nuclear physics embracing nuclear
+structure, nuclear scattering, and nuclear instability. His wide ranging interests and inter-
+actions with theorists and experimentalists alike helped him in providing both theoretical
+analysis and interpretations and suggesting experiments.
+He had the gift of visualising
+complex problems in simple terms and provided clear analysis of the underlying processes.
+He was an expert on the use of semi-classical methods which provided an intuitively clear
+picture of complex phenomena. His research work and books are characterised by scientific
+clarity, transparency, and depth. David possessed outstanding skills in mathematical com-
+putation, and he was an expert on special functions, group theory, and the Feynman path
+integral method. David had many research students and collaborated with a large number
+of scientists from across the world, for whom he was a source of scientific and human in-
+spiration and admiration. His most fundamental belief was that research was a means of
+trying to discover and understand the beauties of Nature and explain them in simple terms
+to others. His absolute belief in the value of truth and his unselfish and generous attitude
+in sharing knowledge makes him an outstanding figure in contemporary Nuclear Physics.
+1
+Early years and family memories
+David Maurice Brink was born on 20 July 1930 in Hobart, Tasmania. His father, Maurice Brink
+had been born in the village of Bjuv in Sweden in 1900. David’s grandparents emigrated to
+Australia in July 1900. At the age of 14 David’s father moved to Sydney where he trained to
+become an accountant. After this he went to Tasmania and joined an accountancy firm Wise,
+Lord and Ferguson, where he eventually became a partner. In 1929 he married Victoria Finlayson,
+(born in 1900). Her father David had emigrated with his parents from Scotland in 1884. They had
+an engineering firm in Devonport, Tasmania whose main activity was maintaining and repairing
+machinery for mining, shipping, and timber companies. David’s grandfather and his colleagues
+built the first steam car in Tasmania and between 1900 and 1904 built nine vehicles including
+three passenger cars and one 12-passenger bus.
+David visited his grandparents often during
+vacations. He saw the casting floor and other parts of the factory and enjoyed playing amongst
+the remains of old steam traction engines.
+∗[email protected]
+†[email protected]
+1
+arXiv:2301.02907v1  [physics.hist-ph]  7 Jan 2023
+
+Figure 1: David M. Brink
+2
+
+David was the eldest of three brothers. The Brink brothers went to a Quaker school in Hobart,
+Australia 1936 to 1948. David attended the University of Tasmania during 1948-51 studying
+Physics, Mathematics, and Chemistry, graduating with a BSc in December 1950 and was elected
+as a Rhodes Scholar at Magdalen College, Oxford, from October 1951. From February 1951 to
+September 1951 he studied for BSc Honours in Hobart but did not complete the course because
+he moved to Oxford in September 1951.
+As a student at the University of Tasmania David joined the Hobart Walking Club. With this
+club he went on many trips to the interior of the island. When he arrived in Oxford he became
+a member of the Oxford University Alpine Club. Its activities took him to the Alps where he
+climbed in the Valais and the Engadine in Switzerland. It was in Switzerland that he met his
+future wife Verena. Verena and David married in 1958 and had three children together. His love
+for walking was transmitted to his three children who continue to enjoy walking in urban, rural,
+and mountainous settings. While always very committed and absorbed with his Physics he was
+also a devoted husband and father, transmitting his joy for walking and travel to his family. He
+often helped his children with their homework and was very patient with them, even when they
+were not! Together David and his family travelled to, and lived in many countries across the
+world, where their horizons were broadened and they were introduced to the idea that there are
+many different ways of living and being. When his children had left home and travelled to other
+countries he would often be found in front of an atlas studying their exact whereabouts.
+David was very open minded and curious, always accepting other people’s opinions and points of
+view. David and Verena were very close, shared everything and had full respect for each other.
+Verena was a wonderful host and the Brinks often organised tea and dinner parties for students,
+visitors, and their families. Verena also helped visitors find accommodation, and with other
+issues related to living in Oxford. They were also very generous in offering accommodation at
+their place whenever possible.
+In Oxford David developed an interest in birds, initially just birds he saw in Oxford, but when
+he travelled he always liked to look for birds and made lists of species he saw. This curiosity
+in nature extended to other species as well, including trees. When in 1993 he moved to Trento,
+Italy, he became a member of the SOSAT, a branch of the alpine club, and went regularly with
+them on Sunday trekking trips.
+2
+Graduate studies and Oxford beginnings
+David started his studies at Oxford in October 1951. When he arrived at Magdalen College there
+was no tutor in Theoretical Physics at the college. His maths tutor was David Kendal who sent
+him for tutorials to Jack De Wet at Balliol College. Jack asked David to read Von Neumann’s
+book on the foundations of Quantum Mechanics in German. He also encouraged David to change
+his studies from a BA in Mathematics to a D. Phil in Theoretical Physics. Maurice H. L. Pryce
+(FRS 1951) was the Wykeham Professor and head of the Theoretical Physics Department in
+Oxford from 1946 to 1954. He was David’s supervisor. Pryce was also the part-time leader of the
+Theoretical Physics Division of the Atomic Energy Research Establishment (AERE) at Harwell,
+not far from Oxford, where nuclear theory was very much in the forefront and Rudolf E. Peierls
+(FRS 1945) was a consultant. At Harwell there was a very productive theory group including
+Tony Skyrme and J. P. (Phil) Elliott (FRS 1980). Skyrme organized regular informal meetings
+known as ’Skyrmishes’. Important papers in the latest journals were presented and discussed.
+Members of the group attended Oxford seminars and while the local group including Roger Blin-
+Stoyle (FRS 1976), David Brink, and Pryce attended the Harwell meetings. Elliott gave some
+3
+
+Figure 2: David (right) in Tasmania 1950.
+4
+
+EVENT
+RESlectures at Oxford on Racah algebra. Later on his best-known work brought together the shell
+and collective models to explain rotational bands in deformed nuclei using the unitary group
+SU(3). During this time he wrote a long article in Handbuch der Physik with A. M.(Tony)Lane
+(FRS 1975) [1] on the shell model.
+The foundations of David Brink’s lifelong research, can all be found in his thesis ”Some Aspects
+of the Interactions of Fields with Matter” [2] which was submitted in May 1955. It is a remark-
+able document for its breadth and early contributions to the field of nuclear physics. M. Pryce,
+his thesis adviser was interested largely in atomic spectroscopy but also studied the spectroscopy
+of nuclear energy levels. The advent of the shell model around 1950 opened the door to new
+theoretical approaches for understanding the properties of nuclei and applying quantum mechan-
+ical tools to calculate them. There was also a great interest in reactions involving heavy nuclei
+and which could only be treated by statistical methods that had been developed much earlier.
+Brink’s two-part thesis contained contributions to both areas, reflecting the interactions between
+the Harwell and Oxford groups. The first part was inspired by the shell model and the second
+contains important contributions to the statistical theory of nuclear reactions.
+In the first part of his thesis, dealing with Nuclear Structure, David analyzed the spectroscopic
+consequences of the nucleon-nucleon interaction acting on the valence nucleons in nuclei close to
+the doubly-magic 208Pb. David was able to estimate the order of magnitude of the interaction
+matrix elements from the properties of the deuteron. He also proposed treating the interac-
+tion through a density matrix expansion. This would figure prominently in later work in the
+field.
+The second part of his thesis dealt with reactions involving heavy nuclei. It was probably inspired
+by the work of experimental group at Harwell. There, a Van de Graaff accelerator was used to
+measure energy levels, moments and transition rates in nuclei.
+David was also fortunate to
+have contact with the strong experimental group working on neutron resonances. While David
+was working on gamma widths of neutron resonances he benefited from contacts with Prof.
+Hughes [3] and Prof. Weisskopf who were visiting Oxford. Weisskopf was very much interested
+in applying the detailed balance theory to nuclear reaction and interactions with him must
+have influenced David because at the end of the thesis he acknowledges discussions with Victor
+Weisskopf. The first subject in this part was the theory of inelastic scattering on deformed nuclei.
+David constructed a theory for the excitation of rotational bands in deformed nuclei based on
+two new ideas, namely Bohr’s model of deformed nuclei and the optical model of Weisskopf et al.
+[4] published the previous year. David was able to carry out the calculations to a point where
+the relative importance of this mechanism in the total cross section could be estimated. This
+was an impressive achievement at a time before computers were available to carry out the full
+calculations.
+The final section of his thesis deals with the decays of the compound-nucleus resonances produced
+in reactions on heavy nuclei. The formulas he presented here are still in use for modeling the
+spectra and reactions in heavy nuclei [5]. The best known is the formula for gamma decay rates
+in compound-nucleus resonances. This formula is based on a treatment widely known as the
+”Brink-Axel” hypothesis. At a fundamental level, the theory was derived from the principle of
+detailed balance which Weisskopf had used very successfully in other contexts. The principle
+gives a formula to relate decay rates to absorption cross sections in the inverse reaction. The
+Brink-Axel hypothesis simply states that the absorption cross sections for gamma radiation on
+excited states of heavy nuclei can be estimated by the corresponding cross sections on the ground
+states. Axel and Brink worked independently. Peter Axel’s paper appeared in 1962 [6]. The
+important statement is made on page 101 of David’s thesis and is expressed in equation (11) of
+5
+
+Figure 3: David and his children (left to right), Barbara, Thomas, and Anne-Katherine 1969.
+Axel’s paper. The prediction of the statistics of the widths of nuclear resonances, based on the
+generalization of the central limit theorem which David had learned about in his statistics course
+in Tasmania. David published the results in [7] where he showed the close connection between
+the shell-model description of the giant dipole resonance and the collective model of Goldhaber
+and Teller [8] and Steinwedel and Jensen [9]. After his paper, theory of the giant resonances
+used the shell model as a starting point. Confirmation of the Brink-Axel hypothesis first came
+from the Berkeley experiments in 1981 [10].
+The last part of thesis has formulas related to another important topic in compound-nucleus
+theory, the fluctuations in decay widths of individual resonances. Here, David speculated that
+the fluctuations would follow a chi-squared distribution with one or two degrees of freedom.
+This is borne out experimentally and is now considered one of the hallmark properties of the
+compound nucleus. It also became a part of random matrix theory in mathematical physics.
+Unlike the early parts of the thesis, David never published the parts on compound-nucleus decay
+widths. However, physicists at the Harwell Laboratory knew about David’s results and J.E.
+Lynn explained them in his book [11]. Unfortunately, David’s treatment of fluctuations was not
+recognized until very recently [12] and the distributions are known today under other author’s
+names [13].
+6
+
+3
+Research areas
+David’s interactions with the physicists mentioned earlier were reflected not only in David’s thesis
+but also in his early publications. One paper [14], which dealt with angular momentum couplings
+and angular distributions of γ-rays and other particles, is still the ”Bible” most experimentalist
+use when they analyse their data, as we have been told by Peter Butler (FRS 2019) (Liverpool)
+and Yorick Blumenfeld (Orsay), and others. Early in his research career David wrote the textbook
+Angular Momentum [15] with Ray Satchler. This textbook was prominent among several texts
+published in this time period.
+It was widely used by graduate students and post-graduates
+working in nuclear theory. David also published a book on Nuclear Forces [16].
+3.1
+Effective interactions and calculations tools
+In his thesis David had laid the basis for the use of effective interactions in the calculations
+of matrix elements for nuclear structure studies. The idea was greatly advanced in three later
+papers. The first proposes a gaussian form for the effective nucleon-nucleon interaction known
+as the ”Brink-Boeker” interaction [17] that all nuclear physicists have used at least once in
+their lives. This paper was very influential at the time and was later developed by Gogny and
+collaborators in the interaction that is widely used even today [18, 19].
+In 1959 Tony Skyrme proposed modelling the effective interaction between nucleons in nuclei
+by a short-range potential, an idea which is useful in nuclear structure and the equation of
+state of neutron stars [20]. The Skyrme force is an effective interaction depending on a small
+number of parameters whose strength could be fitted to reproduce various bulk properties of
+nuclei as well as selected properties of some nuclei, especially the doubly magic nuclei. At the
+beginning of the 1970s David was a frequent visitor to the Theoretical Division at the Institut
+de Physique Nucl´eaire, Orsay where his sixty-fifth birthday was celebrated (figure 4). The work
+done there produced two papers with Dominique Vautherin [21, 22] which were the basis for the
+intense use of the so-called Skyrme interactions, in all their many present variants. The papers
+revived a general interest in using Skyrme’s parametrization of the nucleon-nucleon interaction
+to calculate nuclear binding energies, and later to other aspects of nuclear structure. In effect,
+the interaction is treated as an energy-density functional theory in the spirit of the Kohn-Sham
+theory in condensed matter physics.
+The Hartree-Fock calculations in [21] for spherical nuclei used Skyrme’s density dependent effec-
+tive interaction. This seminal paper showed how the Skyrme force could be used to make accurate
+calculations of certain nuclear properties and Vautherin and Brink developed these ideas further
+in a series of papers which had a strong impact on nuclear structure calculations. T. Otsuka
+comments: “The paper [21] has had a huge impact, as verified by the number of citations >2000.
+In nuclear theory, papers having the citation index >1000 are rather few, which implies how
+important the Vautherin-Brink paper is. This year is the 50 year anniversary of this paper, and
+it is amazing that the basic formulation within the mean-field approach has not changed too
+much, implying that the scheme presented in this paper is so solid”.
+The calculations of Vautherin and Brink were extended by many other physicists during the
+subsequent period.
+In particular at Oxford, Micky Engel, Klaus Goeke and Steve Krieger,
+together with Dominique Vautherin derived the energy density using a Slater determinant where
+the single particle states were no longer invariant under time reversal, as it is in the Hartree-Fock
+method. With the Skyrme interaction the TDHF approach leads to an equation of continuity
+for the single particle density [22]. This paper showed how Dirac’s time-dependent Hartree-Fock
+theory could be applied to nuclear dynamics in a light nucleus. In the year immediately following
+7
+
+the publication, the theory was applied to collisions involving a large number of nucleons [23],
+showing that the method would be a powerful one for heavy nuclei as well.
+The method is
+justified as a time-dependent density-functional theory, and it remains in widespread use.
+In 1973 Ica Stancu came to Oxford as a post doctoral fellow and worked with David on heavy
+ion reactions in deriving the interaction potential of two 16O nuclei starting from the Skyrme
+energy density formalism [24]. They included the previously ignored tensor part of the Skyrme
+interaction.
+Along with an additional effort from Hubert Flocard at Orsay, the Skyrme HF
+calculations yielded single particle levels of spherical closed nuclei [25]. The role of the tensor
+force is to contribute to the spin-orbit splitting of the single-particle levels. For spherical closed
+shell nuclei the effect turned out to be small. Later it was found that in spherical spin unsaturated
+nuclei it makes a dramatic difference, giving the correct order of single particle levels, as, for
+example, in the Sn isotopes [26]. Many experiments on neutron-rich nuclei since 2006 have shown
+that the Skyrme formalism including the tensor force was the simplest way to describe the shell
+evolution of neutron-rich or proton-rich nuclei and indicated new magic numbers.
+3.2
+Heavy-ions and Semi-classical methods in Nuclear Physics
+As tandem accelerators and cyclotrons were built to study heavy-ion Physics, David started an
+intense collaboration with the experimentalists at the Department of Nuclear Physics in Oxford.
+The accelerators were used to study heavy-ion elastic scattering and direct reactions such as
+transfer and measure masses and perform spectroscopy of neutron-rich matter. In those years
+semiclassical methods were widely used in the Nuclear Physics community to analyse data. They
+were particularly appropriate for heavy ions because of the high incident energies and the large
+impact parameters involved. Thus David started the Oxford school on the subject, more or
+less parallel in time to the Copenhagen school of Broglia and Winter and collaborators.
+At
+that time, these heavy-ion reactions were analyzed through the partial wave expansions of the
+colliding partners, a methodology that was computationally demanding and giving little insight
+to the underlying dynamics. David’s semi-classical treatment of the collision was much simpler.
+Some of the early papers on the theory of peripheral reactions were based on his student’s thesis,
+including Hashima Hasan and Luigi Lo Monaco [27, 28].
+David’s investigation of the kinematical effects in such reactions, for which there was concrete
+experimental evidence from the work of Peter Twin (FRS 1993) and his collaborators at Liver-
+pool, became a key element for experimentalists. In the paper by the title ”Kinematical effects in
+heavy-ion reactions” [29] David introduced a ”semi-classical amplitude” [30] that could be used
+in DWBA-like calculations of transfer [31] and proposed a matching condition to predict a large
+reaction cross-sections, a condition that was beautifully adapted to understand spin-polarization
+experiments. He showed that energy and angular momentum couplings in heavy-ion reactions led
+to very selective matching rules by which high angular momentum single-particle states could
+be populated.
+High angular momentum single-particle states sometimes appear as low-lying
+continuum resonances. They have been studied by the method of transfer-to-the-continuum [32]
+which has helped disentangle single-particle from collective degrees of freedom and has also been
+applied in the so called ”surrogate reactions” as a substitute for free neutron beams.
+Semi-classical ideas have been helpful in studying breakup and dissociation of weakly bound
+radioactive ions including halo nuclei and other such unstable nuclei whose dynamics is rather
+involved and difficult to study experimentally due to the very low intensity of beams. David,
+Angela Bonaccorso and her students got heavily involved in this new physics from the ’90s on,
+with a long series of papers (see [33] and references therein), conference contributions, meeting
+organization, some of them at the ECT* in Trento, spanning the last forty years of David’s
+8
+
+career. Finally it has recently been shown [34] that the semi-classical treatment of breakup by
+David and his collaborators is fully consistent with a quantum mechanical treatment.
+David studied microscopic models for the real and imaginary parts of the ion-ion optical potential
+to be used in elastic scattering calculations with Ica Stancu. He also studied fusion with Neil
+Rowley and N. Takigawa. David and Takigawa developed a semi-classical reaction theory with
+three classical turning points which explained the anomalous large angle scattering of α particles
+as a quantum-mechanical interference between the barrier wave and the internal wave, thereby
+providing an intuitively clear picture of a complex phenomenon underlying nuclear reactions in
+terms of classical and quantum ideas. David, Vautherin, and M.C. Nemes studied the effect of
+intrinsic degrees of freedom on the quantum tunnelling of a collective variable. This work was
+further developed by other theorists including Kouichi Hagino who studied the deviation from
+adiabaticity in quantum tunnelling with many degrees of freedom.
+David met Uzi Smilanski in Munich when they were both there on sabbatical. Both had worked
+on semi-classical approximations and gave a joint series of lectures on this topic. David was con-
+cerned that the standard WKB method was insufficient to explain tunnelling through a barrier
+and was particularly bad near the barrier top. David and Uzi applied the uniform semi-classical
+method evolved by Michael Berry (FRS 1982) to successfully address the problem [35].
+Uzi
+remembers David as a physicist with excellent intuition and an ability to grasp the essence
+of a problem before cracking the problem with rigorous mathematics and complex computa-
+tion.
+David, Massimo di Toro, and Alberto Dellafiore developed a semi-classical description of col-
+lective responses with a mean field approach paving the way for a study of the dynamics of a
+nuclear Hartree-Fock fluid. When the national heavy-ion laboratory started in Catania (LNS-
+INFN) around an advanced superconducting cyclotron, David was a reference point for simple
+physics suggestions.
+3.3
+Path integral methods in Nuclear Physics
+David’s expertise with semi-classical methods for tackling quantum problems naturally led him
+towards the Feynman path integral approach to quantum mechanics which was based on a
+Lagrangian approach. Hans Weidenm¨uller had met David at various conferences in the 1950s
+and 1960s and spent 1977-78 on a sabbatical in Oxford. During this period David and Hans
+worked on the application of the Feynman path integral method to the study of the heavy-ion
+reactions and developed the Influence Functional approach to this problem which David and his
+collaborators later used to establish master equations. Hans remembers that at a summer school
+a few years later David delivered a series of lectures on nuclear reactions. In the first lecture
+he developed the topic using a dozen transparencies and in subsequent lectures used the same
+transparencies in a different order to display and illuminate aspects of the topic that had gone
+unnoticed before. Hans remembers it as a display of the combination of simplicity and depth
+that were hallmarks of David’s approach to Physics.
+The path integral method was particularly well suited for studying problems with many degrees
+of freedom in which classical description in terms of trajectories was good for some degrees of
+freedom but not for all. Coulomb excitation in heavy-ion collisions is an example where the
+relative motion of the ions could be described in terms of coulomb trajectories but the excitation
+of the quantum states of the ions had to be treated using quantum mechanics.
+David and
+Sukumar [36] used the Feynman path integral method to evolve a systematic way of arranging
+the correction terms for the quantum amplitudes for processes involving coupled degrees of
+9
+
+Figure 4: David and his wife Verena, May 2018.
+freedom where the description in terms of classical trajectories was good for some degrees of
+freedom. David, Sukumar, and Fernando Dos Aidos used this method to provide corrections
+to the primitive semi-classical amplitude for Coulomb excitation of heavy-ions. Sukumar and
+David used the path integral method to describe spin-orbit coupling effects and together with Ron
+Johnson at Surrey and his group successfully explained the experimental data on polarization
+effects.
+4
+Other topics
+David was very quick at grasping the core of a Physics problem and putting it in simple, calculable
+terms.
+Often the problem required somewhat involved analytical calculations, but he was a
+master of that. Thus anytime a visitor went to Oxford with a new problem, David would start
+a very successful line of research which he often followed up with his graduate students.
+10
+
+4.1
+Cluster models
+It happened for example with the cluster model physics, starting with the seminal paper [37].
+This paper developed the generator coordinate method of Hill and Wheeler [38] to produce
+a practical tool to reduce the many-particle Hamiltonian to an ordinary Schr¨odinger equation
+for a collective variable. Thus the nuclear cluster model was related to the shell model. To
+treat nuclear states in such different circumstances, a formulation which includes clustering at
+one extreme and shell structure at the other extreme was needed. David proposed microscopic
+multi-α-clusters treating four nucleons with different spin-isospin states as a single particle orbit.
+Under anti-symmetrisation of nucleons the cluster model wave-functions approximate shell model
+functions and enabled the description of both cluster and shell model structures in a unified way.
+Their approach was adopted and is in widespread use even in present-day nuclear theory. The
+main applications up to now are on spectroscopy and large-amplitude collective motion.
+Y. Suzuki’s work on the cluster model was largely inspired by David’s paper on ”Do alpha
+clusters exist in nuclei?” [39] presented at a meeting in Tokyo in 1975. This paper contained
+all the essential components needed in the alpha particle model, the microscopic theory beyond
+the shell model description based on many-particle many-hole excitations, the relation between
+the resonating group method GCM, the equilibrium arrangement of clusters, extension of the
+Hill-Wheeler method, the angular momentum projection, and the Slater determinant technique
+for evaluating matrix elements. Suzuki remembers that David never forgot to mention that the
+original model was proposed by H. Margenau and C.Bloch [40, 41, 42].
+At the Varenna School in 1955 David met S. Yoshida from Japan and they discussed inelastic
+scattering of protons and neutrons by deformed nuclei. By chance David had a chapter in his
+thesis on this topic and Yoshida had been studying the same subject. This interaction with
+Yoshida helped David to develop strong connections with nuclear theory groups in Japan over
+many years.
+4.2
+Bose-Einstein condensation of atoms
+During his period as Deputy Director of ECT* in Trento, 1993-1998, David interacted with many
+members of the Physics Department in Trento. One such interaction with Sandro Stringari led to
+David’s interest in Bose-Einstein condensation of alkali atoms in magnetic traps [43]. Sukumar
+and David [44] developed an approximate method for calculating the rate of escape from the
+magnetic trap thereby enabling an estimation of the duration for which the condensate atoms
+can be held in the trap as a function of the ultra-cold temperature and the strength of the
+magnetic field.
+4.3
+Miscellaneous
+David was interested in the role of pairing interaction in finite nuclei and this led to the study
+of nuclear superfluids. His book with R. Broglia [45] is considered to be a wonderful exposition
+of this subject. David’s knack for explaining detailed Physics in a simple and clear manner is
+abundantly evident in this book. In the 1990s Ica Stancu raised David’s interest in the quark
+structure of exotic hadrons named tetraquarks, a system of two quarks and two antiquarks, and
+studied the stability of such systems containing heavy quarks/antiquarks in a QCD inspired
+quark model. Even though David had not worked on the Interacting Boson Model (IBM) he
+nevertheless provided supervision for doctoral students such as Martin Zirnbauer who chose
+topics in this field. He also supervised Hans Peter Pavel’s thesis on Schwinger pair production
+in a flux tube model containing a chromomagnetic field.
+11
+
+5
+Teaching and administrative roles
+David’s doctoral students remember him for the gentle way he corrected them when they had
+made errors. Many of the students learned from him how to take a critical approach to their
+results and how it is possible to look at a complex problem from several different viewpoints and
+find the one that gives the best physical insight. They also remember the immense support he
+gave to their research and pastoral care. Many graduate students also remember how much they
+had learned from the courses he taught at Oxford and at Summer schools. His book with Satchler
+[15] and paper with Rose [14] on angular momentum algebra were found to be of immense value
+in formulating and tackling problems in Nuclear Physics. Many researchers and students who
+met David were astonished that someone with such towering achievements could be so humble,
+nice and honest. David was very open-minded and we report a number of episodes to illustrate
+this aspect of his character.
+Future Nobel laureate Prof. Tony Leggett remembers: ” My undergraduate major at Balliol was
+in Greats (classical languages, ancient history and philosophy) and I was set to graduate (and
+eventually did so) in the summer of 1959. Towards the end of the academic year 1957-1958,
+partly encouraged by the post-Sputnik cultural swing towards science in the UK, I conceived
+the ambition of taking a second undergraduate degree in physics and perhaps eventually making
+my career in academia in that subject. Given that I had essentially no meaningful exposure to
+physics at the high-school level and only a brief and informal exposure to any kind of mathematics
+beyond simple differential calculus (I’m not sure that I had even had that), such a drastic change
+of academic direction was extremely unusual, indeed at the time almost unheard-of. My first
+concern was to find a higher education institution which would accept me for it and I rapidly
+concluded that my only hope was to apply to my existing Oxford college, Balliol. David had
+just recently become the college’s first tutor in theoretical physics (most Oxford colleges did
+not have such a thing in 1958), so it fell on him to take the decision on my application. To
+this end he asked me to read over the summer vacation a few chapters from the book ”What
+is Mathematics?” by Courant and Robbins [46], perhaps the most beautiful presentation I have
+ever seen of mathematical topics for the layperson. When I returned to Oxford in the Fall of
+1958 he gave me an informal mini-exam on that material, and on the basis of my performance
+decided to recommend to Balliol to accept me. In the event I did my physics degree at Merton,
+who offered me a scholarship, but since they did not at the time have a tutor in theoretical
+physics David played that role for me for much of the two years which it took me to complete
+the degree. I think it is virtually certain that had he made the opposite decision, I would never
+have had a career in physics, and I am profoundly grateful to him for the imagination he showed
+in going beyond my formal academic qualifications.”
+Another story comes from Paul Stevenson: ”I was called up for interview at Balliol in December
+1991. The office I was in for that interview was David Brink’s office, above the Senior Common
+Room. In the interview were me, David Brink, David Wark, Jonathan Hodby (those three there
+for physics) and Bill Newton-Smith (for philosophy). I don’t remember all the questions. I do
+remember that David Brink showed me a postcard and asked me what, physically, was wrong
+with the picture. It was a Japanese style print with a mountain in the background and a lake in
+the foreground. There was a reflection of the mountain in the lake, but it was off to one side. I
+saw what was wrong, and struggled to articulate it in the language of a physicist, and in the end
+David prompted me by asking what is particular about an incident light ray, a reflected light
+ray, and the normal to the surface at which it is reflected and I said the right thing - that they
+are all in the same plane. I was duly accepted to Balliol and spent three years there studying
+physics”. Danny Chapman remembers: ”I don’t think I’ll ever forget the ”sense” of David Brink’s
+12
+
+tutorials, and of being in the presence of such a sharp and insightful mind. I remember being
+quite inspired once when my fellow student had tried to answer a question in what I thought was
+an odd and probably wrong way, ending up with a sum, which he then attempted to turn into
+an integral, which didn’t work out. Rather than saying ”don’t do it like that, do it like this”,
+David was able to continue from there and make it work, which was a really positive experience
+and encouragement to follow every path to its end. I feel lucky to have been at Balliol when he
+was there.”
+Angela Bonaccorso remembers daily life as one of David’s students: At the Department of The-
+oretical Physics there was a coffee room where coffee was served between 11:00 and 11:30. We
+would try to be there on time to sit around David who would be chatting with other senior
+members of the department or some visitor. There would always be someone bringing up some
+interesting and challenging new problem. Everyone gave an opinion, the atmosphere was com-
+petitive. Most of the time David would win the argument and his students felt very proud.Not
+all supervisors were so nice, helpful, and respectful of us as David was. But it was not at all
+easy to be David’s student. First of all we needed to have detective skills. David was very busy
+and very elusive. In those days there was no email or SMS. The only way to be sure that he was
+inside was to look for his bicycle. If the bicycle was outside we would knock at the door of his
+office and if we were lucky he would answer and let us in. In spite of all his many commitments
+we always managed to have at least one chat per week with him. Another reason why it was
+not easy to be his student was that David had a very original way of understanding things and
+finding the way out of problems. During our conversations often he would stop talking and be
+silent for five to ten minutes, rubbing his hand on his forehead. Then he came up with some
+equation, or a drawing or something like that and he would tell us: I think it is like this...I think
+we should get something like that...etc. I (we) would stare at him speechless and in wonder.
+Where did the ’oracle’ come from? Most of the time this was the end of the meeting. I (we)
+left his office rather puzzled, worked desperately hard for one week and if we had managed to
+understand his line of thought, after pages and pages of calculations, we would find exactly what
+he had predicted. We all knew it was like that, we all passed this information on to each other,
+generation after generation: listen to David, he is always right, just try to reproduce the miracle
+of his craftsmanship in physics.
+A further proof of how much busy David was and how precious was for everyone the time spent in
+conversation with him can be found in the comment Gerry Brown made in his review for Science
+[47] of the Proceedings of the Varenna summer school [41] : ’Let me draw special attention also
+to the article of David Brink, ”The alpha-particle model of light nuclei,” which is one of the most
+beautiful developments in this subject. Brink likes to sit on his work for years and, on the whole,
+doesn’t even answer letters inquiring about it, so that one must either adopt the expedient of
+traveling to Oxford to talk with him, or invite him to lecture at summer schools. Both are worth
+while.’
+David was a pillar of Balliol college and Department of Theoretical Physics for decades, an
+immensely popular tutor and supervisor, a cheerful and always helpful colleague, and a wonderful
+guide to younger colleagues and administrative staff who happened to be working with him.
+David had another long and distinguished career in Italy after he left Oxford. Following an
+invitation from Renzo Leonardi he moved to Trento as full professor of History of Physics and
+helped in establishing the ECT*, European Center for Theoretical Studies in Nuclear Physics
+and Related Areas. The Nobel laureate Ben Mottelson was the founding director and David
+the vice-director, while Renzo Leonardi was the Scientific Secretary. In the five years David
+spent at Trento he took care of organising various technical aspects of the secretarial offices,
+13
+
+Figure 5: David’s sixty-fifth birthday celebration. Orsay, 1995.
+library, computer center and visitor hospitality.
+At the same time he gave very productive
+contributions to workshops with his constant presence, his huge knowledge of nuclear physics
+and stimulating discussions. The superb reputation and international standing of this extremely
+important European initiative is undoubtedly due in large part to David’s wisdom in its crucial,
+formative years.
+6
+Career, Honours and Awards
+1954-55 Royal Society Rutherford Scholarship.
+1957-1958 Instructor at the Massachusetts Institute of Technology (MIT).
+1958 Fellow of Balliol College and Lecturer in Theoretical Physics, Oxford.
+1976-1978 Vice-Master of Balliol College.
+1981 Fellow of the Royal Society.
+1982 Rutherford Medal of the Institute of Physics.
+1988 H. J. G. Mosley Reader at Oxford.
+1990-1993 Senior Tutor, Balliol College, academic planning and administration, Oxford.
+1992 Foreign member of the Royal Society of Sciences, Uppsala.
+1993-1998 ECT*, Trento, Vice-Director .
+1993-1998 Full professor of History of Physics, University of Trento.
+14
+
+2006 Varenna Conference on Nuclear Reactions dedicated to him.
+2006 Lise Meitner prize of the European Physical Society shared with H. J. Kluge.
+Visiting scientist at :
+• Niels Bohr Institute 1964,
+• University of British Columbia 1975,
+• Institut de Physique Nucl´eaire d’Orsay 1969 and 1981-1982,
+• The Technical University of Munich 1982,
+• University of Trento 1988,
+• University of Catania 1988,
+• Michigan State University 1988-1989.
+7
+Acknowledgements
+The authors are greatly indebted to the Brink family for sharing with them private memories
+and photographs and for a critical reading of the manuscript. A large number of friends and
+colleagues, too many to be individually mentioned, contributed with their appreciation of David’s
+life and scientific career. Ica Stancu and Sharon McGrayne Bertsch read and commented the
+manuscript. One of us (AB) gratefully acknowledges George F. Bertsch for his help in digging
+out from David’s thesis and early work the roots of several founding pillars of modern Nuclear
+Physics.
+References
+[1] Elliott J.P., Lane A.M. 1957 The Nuclear Shell-Model. Structure of Atomic Nuclei / Bau
+der Atomkerne. Encyclopedia of Physics / Handbuch der Physik vol 8 / 39. Springer, Berlin,
+Heidelberg. https://doi.org/10.1007/978-3-642-45872-9 4
+[2] Brink, D. M., 1955
+https://ora.ox.ac.uk/objects/uuid:334ec4a3-8a89-42aa-93f4-2e54d070ee09.
+[3] Hughes, D. J., and Harvey, J. A. 1954 Radiation-widths of nuclear energy-levels Nature 173,
+942 - 943. DOI: https://doi.org/10.1038/173942a0
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+15
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+Nucl. Phys. 4 215 - 220. DOI: 10.1016/0029-5582(87)90021-6
+[8] Goldhaber, M. and Teller, E. 1948 On Nuclear dipole vibrations Phys. Rev. 74 1046 - 49.
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+[9] Steinwedel, H., Jensen, J. H. D. and Jensen, P. 1950 Nuclear dipole vibrations Phys. Rev.
+79 1019. DOI:https://doi.org/10.1103/PhysRev.79.1019
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+1981
+Observation
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+built
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+Rev.
+Lett.
+46
+,
+1383-1386.
+DOI:https://doi.org/10.1103/PhysRevLett.46.1383
+[11] Lynn, J.E., 1968 Theory of neutron resonance reactions, (OUP) 321.
+[12] Hagino, K. and Bertsch, G. F., 2021, Porter-Thomas fluctuations in complex quantum sys-
+tems Phys. Rev. E104 L052104. DOI:https://doi.org/10.1103/PhysRevE.104.L052104 and
+references therein.
+[13] Porter, C.E. and Thomas, R.G. 1956 Fluctuations of Nuclear Reaction widths Phys. Rev.
+104, 483-491. DOI:https://doi.org/10.1103/PhysRev.104.483
+[14] Rose, H.J. and Brink, D.M. 1967 Angular Distributions of Gamma Rays in Terms of Phase-
+Defined Reduced Matrix Elements Rev. Mod. Phys. 39 , 306-347. DOI: 10.1103/RevMod-
+Phys.39.306
+[15] Brink, D. M. and Satchler, G. R., 1962 Angular Momentum, (OUP).
+[16] Brink, D. M. 1965 Nuclear Forces, (Pergamon).
+[17] Brink, D.M. and Boeker, E. 1967 Effective interactions for Hartree-Fock calculations Nucl.
+Phys. A 91, 1-26. DOI: 10.1016/0375-9474(67)90446-0
+[18] Gogny, D., Pires, D. P. and De Tourreil, R. 1970 A smooth realistic nucleon-nucleon
+force suitable for nuclear Hartree-Fock calculations Phys. Lett. B32 591-595. DOI:
+https://doi.org/10.1016/0370-2693(70)90552-6
+[19] Decharge,
+J.
+and
+Gogny,
+D.1980
+Hartree-Fock-Bogolyubov
+calculations
+with
+the
+D1effective
+interaction
+on
+spherical
+nuclei
+Phys.
+Rev.
+C21
+1568-1593.
+DOI:https://doi.org/10.1103/PhysRevC.21.1568
+[20] Skyrme,
+A.
+1959
+The
+effective
+nuclear
+potential
+Nucl.
+Phys.
+9
+615-634.
+DOI:https://doi.org/10.1016/0029-5582(58)90345-6
+[21] Vautherin , D. and Brink, D. M.1972 Hartree-Fock calculations with Skyrme’s interaction.
+1. Spherical nuclei Phys. Rev. C5 626-647. DOI: 10.1103/PhysRevC.5.626
+[22] Engel, Y.M., Brink, D. M., Goeke, K., Kriege, S.J. and Vautherin, D. 1975 Time de-
+pendent Hartree-Fock theory with Skyrme’s interaction Nucl. Phys. A249, 215-238. DOI:
+10.1016/0375-9474(75)90184-0
+[23] Bonche, P., Koonin S., and Negele, J. W., 1976 One-dimensional nuclear dynam-
+ics in the time-dependent Hartree-Fock approximation Phys. Rev. C13 1226-1258.
+DOI:https://doi.org/10.1103/PhysRevC.13.1226
+[24] Brink, D.M.and Stancu, Fl.1975 Interaction potential between two O-16 nuclei derived from
+the Skyrme interaction Nucl. Phys. A 243 175-188.
+16
+
+[25] Stancu, Fl., Brink, D.M. and Flocard, H. 1977 The tensor part of Skyrme’s interaction Phys.
+Lett. B68 108-112.
+[26] Brink, D.M. and Stancu, Fl., 2007 Evolution of nuclear shells with the Skyrme density
+dependent interaction Phys. Rev. C 75 064311.
+[27] Hasan, H. and Brink, D. M. 1979 The transfer amplitude and angular distributions in
+heavy-ion reactions J. Phys. G: Nucl. Phys. 5 771.
+[28] Lo Monaco, L. and Brink, D. M. 1985 Perturbation approach to nucleon transfer in heavy-ion
+reactions J. Phys. G: Nucl. Phys. 11 935-952.
+[29] Brink, D.M. 1972 Kinematical effects in heavy-ion reactions Phys .Lett. B40 37-40. DOI:
+10.1016/0370-2693(72)90274-2
+[30] Brink, D. M., 1985. Semi-Classical Methods in Nucleus-Nucleus Scattering, (Cambridge
+University Press).
+[31] Anyas-Weiss, N., Cornell, J.C., Fisher, P.S. ,Hudson, P.N., Menchaca-Rocha, A., Millener,
+D.J., Panagiotou, A.D., Scott, D.K., Strottman, D., Brink, D.M., Buck, B., Ellis, T.P.
+and Engeland, J., 1974 Nuclear structure of light nuclei using the selectivity of high en-
+ergy transfer reactions with heavy ions Physics Reports 12 201-272. DOI: 10.1016/0370-
+1573(74)90045-3
+[32] Bonaccorso, A. and Brink, D. M., 1988 Nuclear transfer to continuum state Phys. Rev. C38
+1776-1786; 1991 Stripping to the continuum of 208Pb Phys. Rev. C44 1559-1568.
+[33] Bonaccorso, A. and Brink, D. M., 2021 Models of breakup: a final state interaction problem
+Eur. Phys. J.A57 171.
+[34] Jin Lei, Bonaccorso, A. 2021 Comparison of semiclassical transfer to continuum model with
+Ichimura-Austern-Vincent model in medium energy knockout reactions Phys. Lett. B813
+136032. DOI:https://doi.org/10.1016/j.physletb.2020.136032
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+479-490.
+[36] Sukumar, C. V. and Brink, D.M. 1983 Path integral methods for inelastic scattering Nucl.
+Phys. A404 121-141.
+[37] Brink, D.M. and Weiguny, A. 1968 The generator coordinate theory of collective motion
+Nucl. Phys. A120 59-93. DOI: 10.1016/0375-9474(68)90059-6
+[38] Hill, D. L. and Wheeler, J. A. 1953 Phys. Rev. 89, 1102.
+[39] Brink, D. M. 1975 Do alpha-clusters exist in nuclei. Proceedings of the INS-IPCR Symposium
+on Cluster Structure of Nuclei and Transfer Reactions Induced by Heavy-Ions, Tokyo, March
+17-22, 1975.
+[40] Margenau, H. 1940 Phys. Rev. 5, 37.
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+’Enrico FERMI’, XXXVI Corso, Ed. C. Bloch, 247.
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+[43] Brink, D.M., Stringari, S., 1990 Density of states and evaporation rate of helium clusters.
+Z Phys D-Atoms, Molecules and Clusters 15 257-263 .
+17
+
+[44] Sukumar, C. V. and Brink, D.M. 1997 Spin flip transitions in a magnetic trap Phys. Rev.
+A56 2451-2454.
+[45] Brink, D. M. and Broglia, R. A., 2005 Nuclear Superfluidity Pairing in Finite Systems,
+(CUP).
+[46] Courant, R. and Robbins, H. 1941 What is mathematics?
+Oxford, UK: Oxford University
+Press.
+[47] Brown, G. E., 1967 Nuclear Physics: Many-Body Description of Nuclear Structure and Re-
+actions. Course 36, International School of Physics ’Enrico Fermi.’ C. Bloch, Ed. Academic
+Press, New York, 1966. Science, 158 (3807), DOI: 10.1126/science.158.3807.1440
+18
+

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+ 
+ 
+High-fidelity ptychographic imaging of highly periodic structures 
+enabled by vortex high harmonic beams 
+Bin Wang1*†, Nathan J. Brooks1*, Peter C. Johnsen1, Nicholas W. Jenkins1, Yuka Esashi1, Iona 
+Binnie1, Michael Tanksalvala1, Henry C. Kapteyn1,2, Margaret M. Murnane1 
+1STROBE Science and Technology Center, JILA, University of Colorado, Boulder, CO 80309, USA 
+2KMLabs Inc., 4775 Walnut St. #102, Boulder, CO 80301, USA 
+*These authors contributed equally 
+†[email protected] 
+ 
+Abstract 
+Ptychographic Coherent Diffractive Imaging enables diffraction-limited imaging of nanoscale structures at extreme 
+ultraviolet and x-ray wavelengths, where high-quality image-forming optics are not available. However, its reliance 
+on a set of diverse diffraction patterns makes it challenging to use ptychography to image highly periodic samples, 
+limiting its application to defect inspection for electronic and photonic devices. Here, we use a vortex high harmonic 
+light beam driven by a laser carrying orbital angular momentum to implement extreme ultraviolet ptychographic 
+imaging of highly periodic samples with high fidelity and reliability. We also demonstrate, for the first time to our 
+knowledge, ptychographic imaging of an isolated, near-diffraction-limited defect in an otherwise periodic sample 
+using vortex high harmonic beams. This enhanced metrology technique can enable high-fidelity imaging and 
+inspection of highly periodic structures for next-generation nano, energy, photonic and quantum devices. 
+Introduction 
+In recent decades, a powerful coherent diffractive imaging (CDI) technique known as ptychography has enabled robust, 
+diffraction-limited, phase-contrast imaging of nanoscale structures [1-5]. Although ptychography has been 
+implemented using a range of illumination wavelengths, its use in the extreme-ultraviolet (EUV) and x-ray regions is 
+particularly attractive for achieving high spatial resolution with inherent elemental and chemical contrast [6-10]. In 
+ptychography, a coherent illumination (the probe) is focused and scanned across an extended sample. A series of far-
+field diffraction patterns are recorded, while maintaining a large overlap between adjacent scan positions. Iterative 
+phase-retrieval algorithms [11-15] can then be used to robustly and uniquely reconstruct the complex-valued probe 
+field and sample transmission or reflection functions. However, successful reconstruction relies heavily on diversity 
+in the data provided by the lateral scanning of the probe relative to the sample, i.e., interferences at the detector plane 
+mix amplitude and phase, allowing the reconstruction algorithms to unravel both. this means that ptychographic 
+imaging of highly periodic samples with a sufficiently small period is extremely challenging due to the lack of 
+diversity in a series of diffraction patterns, leading to poor convergence of the reconstructed sample images. This 
+significantly limits ptychography’s application to a wide variety of nanoscale periodic structures such as photonic 
+crystals [16-17], semiconductor devices [18], and EUV photomasks [19-25]. Consequently, it is critical to fill this 
+characterization gap to aid the advancement of a host of next-generation nano-devices. 
+High harmonic upconversion of femtosecond lasers can produce bright coherent beams from the EUV to the soft x-
+ray regions of the spectrum, in a tabletop-scale setup [26-28]. When combined with ptychography, high harmonic 
+generation (HHG) enables phase-sensitive lensless imaging with diffraction-limited nanoscale spatial resolution and 
+excellent elemental specificity [9,15,29-31]. Moreover, because of the quantum-coherent nature of the HHG 
+upconversion process, polarization and phase structure present in the driving laser beam can be transferred to the 
+generated harmonics, provided energy, spin and orbital angular momentum are conserved [32,33]. This makes it 
+possible to create designer short-wavelength structured light for a variety of applications in advanced spectro-
+microscopies [34,35]. 
+Light beams carrying orbital angular momentum (OAM) have attracted considerable interest for super-resolution 
+imaging [36] and enhanced optical sensing, communication and inspection [37-39]. Recently, by considering 
+
+ 
+ 
+conservation of OAM in HHG upconversion, additional routes for controlling the OAM, polarization, as well as the 
+spectral and temporal properties of HHG have been revealed [40-47]. A property particularly interesting for 
+ptychography is the relationship between OAM and the HHG beam divergence/propagation: the spiral phase structure 
+characteristic of OAM-HHG beams forces them to diverge more quickly from the focal point [46]. This means that 
+by using one or more OAM beams to drive the HHG process (referred to as OAM-HHG), one can control the 
+divergence of the emitted HHG probe without changing the focusing optics of the HHG driving laser. 
+In this article, we demonstrate a solution to a decade-long challenge by showing that high harmonic beams carrying 
+orbital angular momentum can be used to advantage in high-resolution, high-fidelity and fast-convergence 
+ptychographic imaging of highly periodic two-dimensional (2D) grating structures, using the standard extended 
+ptychographic iterative engine (ePIE) algorithm [13]. The key to this technique is that the increased divergence of the 
+OAM-HHG source, combined with the ring-shaped intensity distribution, introduces strong interference fringes 
+between adjacent diffraction orders in the far-field. These encode the non-measurable phase information into the 
+measurable intensity modulation in diffraction fields, significantly enhancing data diversity so that the phase of the 
+diffracted field can be reliably retrieved. We further show that using OAM-HHG beams for illumination provides 
+three significant advantages compared to standard Gaussian-HHG beams, all of which lead to enhanced signal-to-
+noise-ratio (SNR) for imaging periodic structures: First, due to the conservation of OAM in the HHG process and the 
+resulting strong spiral phase structure in the generated EUV beams, OAM-HHG beams naturally have a significantly 
+increased divergence compared to that of Gaussian-HHG beams. This enhanced illumination NA makes it possible to 
+achieve overlap between different diffraction orders for small pitch periodic samples, beyond what is possible with a 
+Gaussian-HHG probe, and without making any changes to the focusing optics of the ptychographic EUV microscope. 
+Second, the unique ring-shaped OAM beam intensity distribution, which is determined by the strong spiral phase 
+structure in the EUV beams, leads to overlap between different diffraction orders in the high-intensity regions of the 
+beam. And third, OAM-HHG also allows a higher total number of photons to be collected by the detector given a 
+fixed detector dynamic range. Therefore, by leveraging OAM-HHG beams for ptychography, we successfully imaged 
+highly periodic samples with substantially reduced gridding artifacts, and reliably detected defects near the diffraction 
+limit. This new structured-EUV HHG metrology technique can support the advancement of next-generation EUV 
+lithography, nanoelectronics, photonic and quantum devices. 
+Methodology 
+ 
+To date, imaging highly periodic structures has been extremely challenging for ptychography. In a conventional 
+implementation of ptychography using a Gaussian EUV beam to illuminate highly periodic 2D structures (see Fig. 
+1(a)), the far-field diffraction patterns consist of many isolated diffraction orders, each of which is a copy of the far-
+field beam profile, and is modulated by an envelope in both amplitude and phase. The zoomed-in green circle in Fig. 
+1(a) shows this characteristic behavior, with the white circles indicating the edge of each diffraction order. In the 
+resulting ptychographic dataset, diffraction patterns taken at different positions on the highly periodic sample are 
+almost identical to each other. This is because, in contrast to diffraction from non-periodic structures, changes in the 
+far-field diffraction field happen almost entirely in the relative phase between the diffraction orders, but not in the 
+intensity (i.e., diffraction efficiency) of the diffracted orders. The phase information is thus totally lost in this case. 
+Ptychography, as a phase retrieval algorithm, tries to retrieve the phase distributions of diffraction patterns from their 
+intensity measurements. The fact that the phase information is totally lost for highly periodic samples with a 
+sufficiently short period, as opposed to being encoded in the intensity measurements as would be the case for non-
+periodic structures, makes it extremely challenging to achieve successful ptychographic imaging of such highly 
+periodic structures. As expected, the reconstruction fails for ptychography using a Gaussian-HHG beam, as shown in 
+Fig. 1(b), in which the amplitude and phase of the reconstruction are plotted in brightness and hue, respectively. This 
+phase problem can also be understood through the convolution theorem, as discussed in detail in Supplementary 
+Section 1. 
+In 1969, Hoppe proposed to achieve electron diffraction imaging of periodic atomic lattices by encoding the non-
+measurable phase information into the measurable intensity modulation in diffraction patterns, through interference 
+between neighboring diffraction orders [48]. As schematically shown in Fig. 1(c), OAM-HHG beams are able to 
+achieve overlap and interference between neighboring diffraction orders due to their intrinsically larger beam 
+
+ 
+ 
+divergence and ring-shaped intensity distribution. The zoomed-in blue circle in Fig. 1(c) shows the interference fringes, 
+with the yellow circles indicating the edge of each diffraction order. As one scans the probe relative to the periodic 
+structures, the relative phase of each diffraction order changes accordingly, which then causes the interference fringes 
+to shift. In other words, the phase information in the diffraction patterns is now encoded in the intensity measurements 
+through interference. These interference fringes contain the missing phase information, and increase the diversity in 
+diffraction patterns, thereby enabling robust and reliable ptychographic reconstructions (see Supplementary Section 
+2). Figure 1(d) shows a high-fidelity ptychographic reconstruction of a 2D periodic structure under an OAM-HHG 
+illumination. 
+ 
+Figure 1. Robust and reliable ptychographic imaging of highly periodic structures. (a) Schematic showing HHG 
+ptychographic imaging of a periodic structure using conventional Gaussian-HHG illumination. The resulting diffraction orders are 
+isolated (see zoomed-in green circle), where the white circles indicate the edges of each diffraction order. This leads to a complete 
+loss of the relative phase information between the orders in the far-field diffraction, which subsequently leads to the failure of the 
+ptychographic reconstruction in (b). (c) OAM-HHG illumination intrinsically has a larger source divergence and a ring-shaped 
+intensity profile, to support overlap and interference between diffraction orders (see zoomed-in blue circle), in which the yellow 
+circles indicate the edges of each diffraction order. This interference converts the relative phase between the diffraction orders into 
+measurable intensity modulation, enabling fast and robust ptychographic reconstruction of the 2D periodic structure in (d). In (b, 
+d), the complex-valued amplitude and phase are plotted as brightness and hue, respectively. 
+
+5um 
+ 
+The required NA for high fidelity imaging of periodic samples can be understood as follows. When a periodic structure 
+is illuminated by a focused coherent beam, the angular separation between two neighboring diffraction orders is given 
+by 𝛥𝜃 =  𝜆/𝛬, where 𝜆 is the illumination wavelength, and 𝛬 is the period of the structure. The illumination NA for 
+the microscope is defined as the half-cone angle of the focusing beam. Geometrically, for fixed 𝜆 and 𝛬, there exists 
+a critical value for illumination NA: 
+𝑁𝐴𝑐  = 
+1
+2 𝛥𝜃 = 
+𝜆
+2𝛬.                                                                   (1) 
+Only for illumination NAs larger than 𝑁𝐴𝑐 will neighboring diffraction orders have sufficiently large footprints on 
+the detector to overlap with each other and generate interference fringes, thus enabling successful ptychographic 
+reconstructions. 
+Experimental configuration 
+We designed and built an EUV ptychographic microscope in a transmission geometry, as shown in Fig. 2. The driving 
+laser for the HHG process is a frequency-doubled Ti:sapphire laser amplifier system (𝜆~395 nm), with an intrinsic 
+near-Gaussian mode (vortex charge of ℓ = 0), that can be converted to an OAM beam of vortex charge ℓ = 1 using a 
+spiral phase plate. The 7th harmonic of the driving laser (𝜆~56 nm) exhibits either a Gaussian mode or an OAM of 
+vortex charge ℓ = 7 depending on whether a spiral phase plate is used. The EUV beam is then focused by a double-
+toroid focusing system onto the periodic samples, with a spot size of ~13 × 18 μm (1/𝑒2 intensity) for Gaussian-mode 
+HHG, or ~27 × 32 μm (donut intensity peak-to-peak) for OAM-HHG. The reconstructed Gaussian- and OAM-HHG 
+beam profiles are shown in a complex representation in Fig. S4, with the beam amplitude and phase indicated by 
+brightness and hue. The test samples are three Quantifoil holey carbon films (~20 nm thick) which have various hole 
+sizes and shapes arranged in a periodic rectangular grid. The three Quantifoil holey carbon films have a pitch of 9 μm, 
+4.5 μm and 3 μm, respectively. These Quantifoil holey carbon films are mounted on standard Ted Pella Ø3mm Cu 
+200 mesh TEM grids. (See the Methods section for more information.) 
+ 
+Figure 2. EUV ptychographic microscope using OAM-HHG EUV beams for imaging highly periodic structures. A spiral 
+phase plate (ℓ = 1 at 395 nm wavelength) converts the driving laser at 395 nm wavelength to an OAM beam, which is focused into 
+a semi-infinite gas cell to produce a nearly monochromatic 7th harmonic beam with a wavelength of 56 nm and an OAM charge 
+of ℓ = 7. A double-toroidal mirror focusing system focuses this OAM-HHG beam onto a 2D periodic sample, and an EUV-CCD 
+camera is used to record the far-field diffraction patterns. 
+
+e=7
+入=56nm
+Detector
+Periodic
+Sample
+0=3.5mrad
+To
+l= 1
+入=395nm
+do= 100 μm 
+ 
+During the ptychography scans, the test samples are translated in the plane perpendicular to the beam path in 7 × 7 
+rectangular grids (49 scan positions) with nominally 3.3 μm distance between adjacent scan positions. A random offset 
+of ±20% of the scan step size was added to each scan position to avoid gridding artifacts in the reconstructions [49]. 
+The far-field diffraction patterns are recorded by an EUV-CCD detector (Andor iKon-L, 2048 × 2048, 13.5 μm pixel 
+size) positioned 50 mm after the sample. To obtain the best ptychographic reconstructions possible for each 
+illumination case, we carefully pre-characterized each probe function in the sample plane by taking ptychographic 
+scans on a non-periodic sample and reconstructing both the sample and the probe functions through blind 
+deconvolution. These pre-characterized probe functions were used as initial guesses in the ptychographic 
+reconstructions of highly periodic structures. Other than using the pre-characterized probe as the initial guess, we used 
+the standard ePIE algorithm [13] for all reconstructions in this study, without the need for additional constraints such 
+as modulus enforced probe [15] or total variation regularization [25,50]. 
+ 
+Results 
+OAM-HHG enables robust and reliable ptychographic imaging of periodic structures 
+We experimentally demonstrate that OAM-HHG beams enable robust and reliable ptychographic imaging of highly 
+periodic structures because of three intrinsic advantages compared to Gaussian-HHG illumination. First, due to the 
+conservation of OAM in the HHG process and the resulting strong spiral phase structure in the generated EUV beams, 
+OAM-HHG beams naturally exhibit a significantly increased divergence (i.e., increased illumination NA for the 
+microscope given the same focusing optics) compared with Gaussian-HHG beams. This enhanced illumination NA 
+allows us to achieve overlap between diffraction orders for smaller pitch periodic samples beyond what is possible 
+with a Gaussian-mode probe, without making any changes to the EUV microscope end-station. Second, the 
+characteristic ring-shaped intensity distribution of OAM-HHG beams ensures that the majority of photons fall in the 
+overlap area (in contrast to the Gaussian-HHG beams, for which the overlap between diffraction orders occurs at the 
+tails of the intensity distributions), which increases the SNR for the interference fringes. Third, the ring-shaped 
+intensity distribution of OAM-HHG beams allows one to collect a higher total number of photons by the detector 
+given a fixed dynamic range, which also leads to higher SNR without the need for high dynamic range (HDR). 
+We performed ptychographic imaging on three highly periodic structures with 9 μm, 4.5 μm and 3 μm pitches using 
+Gaussian- and OAM-HHG beams at a wavelength of 56 nm. Example diffraction patterns and reconstructed images 
+from each ptychography scan can be found in Fig. 3. Furthermore, each ptychography scan collected 49 far-field 
+diffraction intensity patterns, as shown in a log scale in Supplementary Video 1. Note that while there is a clear change 
+in the diffraction patterns from frame to frame for the OAM-HHG case, particularly in the interference fringes between 
+the adjacent diffraction orders, the diffraction patterns in the Gaussian-HHG case do not change very much for small 
+period samples. All ptychography datasets were taken without making any changes to the EUV microscope — the 
+difference in divergence between Gaussian- and OAM-HHG beams is intrinsic to the HHG upconversion process, 
+which conserves energy and OAM. 
+For the 9 μm pitch sample, the small diffraction angle means that successive orders largely overlap even for the 
+Gaussian-HHG beam, as shown in Fig. 1(a). This results in a reasonably good image, apart from some gridding 
+artifacts as shown in Fig. 1(d). In comparison, the ptychography scan using OAM-HHG illumination sees more overlap 
+resulting in improved SNR in the interference fringes and a much higher-fidelity image with greatly reduced gridding 
+artifacts, as shown in Fig. 1(g,j). 
+For the smaller 4.5 μm pitch sample, the diffraction orders are further apart, causing the Gaussian-HHG beams to lose 
+most of the interference in the diffraction patterns, as shown in Fig. 3(b). The low SNR in the interference fringes 
+results in reduced quality image reconstruction, as shown in Fig. 3(e). However, due to their higher intrinsic divergence 
+and the unique ring-shaped intensity distribution, OAM-HHG maintains a large area of overlap between neighboring 
+diffraction orders with more photons, as shown in Fig. 3(h). This results in higher-quality images of the periodic 
+structure with a 4.5 μm period, as shown in Fig. 3(k). Thus, simply by inserting a spiral phase plate to convert the 
+
+ 
+ 
+driving laser to an OAM beam, while keeping everything else in the microscope the same, a greatly improved 
+reconstruction quality is obtained. 
+Lastly, for the smallest 3 μm period sample, Gaussian-HHG illumination totally fails due to the lack of interference 
+between diffraction orders, as shown in Fig. 3(c,f). In this case, OAM beams can reconstruct a reasonable image, 
+although the quality of the unit cell is degraded, as shown in Fig. 3(i,l). 
+We also evaluated the quality of these ptychographic reconstructions using complex histogram analysis and the results 
+can be found in Supplementary Section 3. We note that all reconstructions in Fig. 3 have the correct sample periodicity 
+because this information is directly available from the measured intensity of the diffracted fields — the success or 
+failure of the reconstructions of the unit cells depends on whether the relative phase between the diffraction orders 
+can be successfully retrieved or not. The ptychography reconstructed images in Fig. 3(d–f, j–l) are complex-valued 
+and are shown in a complex representation where the amplitude and phase information are represented by the 
+brightness and hue, respectively. The color wheel is shown in the bottom left corner of each panel. 
+
+ 
+ 
+ 
+Figure 3. High divergence OAM-HHG beams are able to produce higher-quality ptychography images of periodic 
+structures than low divergence Gaussian-HHG beams. Three test samples with different periods and shapes, i.e., 9 μm period 
+with square holes, 4.5 μm period with circular holes and 3 μm period with circular holes, are investigated. For Gaussian-HHG 
+beams, example diffraction patterns from the three test samples are shown in (a–c) and the corresponding ptychography 
+reconstructed images are shown in (d–f). The example diffraction patterns and ptychography images from OAM-HHG beams are 
+shown in (g–i) and (j–l). The complex-valued image in (d–f, j–l) are plotted in a complex representation where amplitude and 
+phase are shown in brightness and hue, respectively. 
+
+9 μm pitch
+4.5 μm pitch
+3 μm pitch
+(a)
+1.0
+(b)
+C
+Intensity (a.u.)
+Gaussian HHG
+0.0
+1.0
+(g)
+h
+1
+Intensity (a.u.)
+OAM HHG
+0.0
+(k)
+um 
+ 
+OAM-HHG beams reveal nanoscale defects in otherwise periodic samples 
+A major motivation for imaging periodic structures is to reliably detect and pinpoint small areas where the periodicity 
+is broken, i.e., to locate defects. However, when the diffraction orders are insufficiently overlapped, the artifacts in 
+the ptychographic reconstructions make it difficult or even impossible to locate defects. In contrast, the increase in 
+reconstruction quality enabled by OAM-HHG beams, especially the suppression of periodic artifacts in the 
+reconstructions (inherent to ptychographic imaging of periodic structures), enables reliable location of nanoscale 
+defects in otherwise highly periodic structures. This can potentially find its application in metrology for micro- and 
+nano-fabrication and manufacturing, including in advanced metrologies in support of EUV lithography. 
+In the 9 μm pitch sample, a damaged carbon bar (~300 nm wide) can be seen in the scanning electron microscopy 
+(SEM) image in Fig. 4(e), indicated by the red arrow. We first performed ptychographic imaging of the corresponding 
+area of the sample using an OAM-HHG beam. During data acquisition at each scan position, we acquired two 
+diffraction patterns with exposure times of 0.1 and 1 second, and combined them to form a composite high dynamic 
+range (HDR) image to increase SNR. The reconstructed image of the transmitted amplitude is shown in Fig. 4(a), in 
+which the defect is clearly resolved and is indicated by the red arrow. Given that the pixel size in the ptychography 
+reconstruction images is 200 nm, the fact that our EUV microscope using OAM-HHG illuminations can clearly image 
+a defect with size of about 300 nm (i.e., 1.5× the pixel size in the reconstruction images) makes it very promising to 
+detect or image smaller defects down to 10’s of nanometers using shorter EUV wavelengths and increased imaging 
+NA. 
+Next, a similar experiment is performed using a Gaussian-HHG beam under the same conditions, resulting in the same 
+approximate maximum detector count in the diffraction patterns as for the OAM case. The reconstructed image of the 
+transmitted amplitude is shown in Fig. 4(b), where reconstruction artifacts heavily corrupt the image details and render 
+the defect unidentifiable. Furthermore, due to the different intensity distributions of the Gaussian- and OAM-HHG 
+beams, even though the two datasets have the same maximum detector count, the OAM dataset has 3 times more total 
+detector counts than the Gaussian one. 
+To confirm that the difference in reconstructed image quality is not simply due to this different in the total number of 
+photons collected, but is due to how those photons are distributed in the diffraction plane (i.e., in the area of overlap 
+between diffraction orders), we performed a third experiment using the Gaussian-HHG beam and triple HDR exposure 
+(0.1-, 1- and 3-second exposure time), which leads to longer data acquisition time by a factor of 1.67, to have 
+approximately equal total counts in the combined diffraction data compared to the OAM-HHG case. The resulting 
+amplitude image is shown in Fig. 4(c). There is significant improvement over the reconstructed image from Gaussian-
+HHG beams with double HDR exposure, but the reconstruction artifacts still make it difficult to identify the nanoscale 
+defect. We further quantitatively analyzed the SNR of the defect in these three reconstruction amplitude images using 
+the transmission profiles of the thin carbon bar in the boxes in Fig. 4(a-c). These transmission profiles are obtained by 
+averaging the transmission images in the vertical direction, and are plotted along the horizontal direction, as shown in 
+Fig. 4(d). The SNRs of the defect in Figs. 4(a–c) are calculated (see Methods) and summarized in Table 1. The SNR 
+for the defect image from OAM-HHG illumination is improved by a factor of >135 compared to that from Gaussian-
+HHG illumination with equal exposure time. Furthermore, we evaluated the quality of these ptychographic 
+reconstructions using complex histogram analysis and verified that OAM-HHG illuminations result in higher fidelity 
+images, as discussed in detail in Supplementary Section 3. It is worth emphasizing that when taking the SEM image 
+in Fig. 4(e), the high-energy electron beam at 300 keV severely damaged the thin carbon bar, causing shrinkage and 
+the appearance of the bright areas on the top and bottom edges. In contrast, the EUV HHG beam can non-destructively 
+image both the periodic sample and the defect. 
+
+ 
+ 
+ 
+Figure 4. Enhanced sensitivity to nanoscale defects in periodic structures using OAM-HHG beams. (a–c) Amplitude images 
+of ptychographic reconstruction of a 2D square periodic structure (9 µm period, with a nanoscale defect of ~300 nm in size) under 
+various conditions: (a) OAM-HHG beams with double HDR (0.1- and 1- second exposure times), (b) Gaussian-HHG beams with 
+equal exposure time as the OAM-HHG case using double HDR (0.1- and 1- second exposure times), and (c) Gaussian-HHG beams 
+with roughly equal number of photons as the OAM-HHG case using triple HDR (0.1-, 1- and 3-second exposure times). The red 
+arrows indicate the nano-defect in the thin carbon bar. (d) Ptychography reconstructed transmission profile of the thin carbon bar 
+containing a nano-defect (indicated by the red arrow) in the boxes in (a–c). The transmission profiles are averaged in the vertical 
+direction. The red arrow indicates the nano-defect. (e) An SEM image of the same sample area shows a 300-nm-wide defect. Bright 
+areas on the top and bottom edges are due to sample damage from the high energy electron (300 keV) beams. 
+ 
+Table 1. Signal-to-noise ratio analysis for the three ptychographic reconstructions shown in Fig. 4(a–c). 
+Ptychographic reconstructions 
+signal 
+background 
+noise 
+SNR 
+Improvement 
+factor 
+OAM-HHG in Fig. 4(a) 
+1.46e-1 
+1.52e-2 
+5.62e-3 
+23.25 
+135.8 
+Gaussian-HHG, equal 
+exposure time in Fig. 4(b) 
+7.34e-2 
+6.94e-2 
+2.35e-2 
+0.17 
+Benchmark 
+Gaussian-HHG, equal number 
+of photons in Fig. 4(c) 
+7.40e-2 
+1.72e-2 
+1.31e-2 
+4.34 
+25.53 
+
+(a)
+(b)
+C
+5um
+(p)
+0.2
+(e)
+OAM-HHG
+Gaussian-HHG, equal exposure time
+Gaussian-HHG, equal number of photons
+300 nm
+0.05
+6
+8
+10
+12
+14
+16
+Sample position (μm) 
+ 
+Conclusion 
+In conclusion, we demonstrated that by incorporating illumination engineering via OAM-HHG beams into an EUV 
+ptychography microscope, we can address the long-standing challenge of high-fidelity coherent diffractive imaging 
+of periodic structures. The intrinsic large divergence and ring-shaped intensity distribution of OAM-HHG beams leads 
+to the formation of higher SNR interference fringes in the diffraction patterns — thus enabling faster and higher 
+fidelity image reconstructions using the basic ePIE algorithm, without extra algorithmic effort. Furthermore, the 
+improvement in image fidelity allowed sensitive detection of a 300 nm wide defect, which is 1.5× the pixel size of 
+the reconstructed images, in an otherwise periodic thin carbon mesh with 9 μm period. 
+Ptychographic imaging of highly periodic structures has been widely recognized to be challenging, which has 
+precluded its application in critical science and technology fields such as semiconductor metrology and EUV 
+photomask inspection. Future studies can employ coherent EUV and X-ray vortex beams to enable nanometer- or 
+even sub-nanometer-scale spatial resolution in a broad range of next-generation nanoelectronics, photonics and 
+quantum devices. A particularly interesting direction would be to use coherent EUV light at a wavelength of 13.5 nm 
+for actinic imaging and inspection of EUV photomasks [19-25]. Finally, this work can provide inspiration for the 
+electron ptychography community (e.g., cryo-EM and 4D-STEM), where recent work has explored the potential 
+benefits of engineered vortex electron beams for enhanced imaging fidelity and lower dose [51,52]. 
+ 
+Methods 
+Experimental setup 
+A Ti:sapphire amplifier system (KMLabs Wyvern HE) with a 𝜆 = 790 nm central wavelength, 45 fs pulse duration, 8 
+mJ pulse energy, and 1 kHz repetition rate was used for this demonstration. Part of the laser output is used for second 
+harmonic generation (SHG) in a 𝛽-barium borate crystal (𝛽-BBO), yielding a frequency doubled beam at 395 nm 
+central wavelength for driving the HHG process. This SHG beam is focused into a semi-infinite gas cell, which 
+consists of a Brewster-cut entrance window, a 20 cm length filled with 50 torr of argon gas, and a copper gasket placed 
+in the focal plane of the driving laser where a coherent HHG beam is generated [53,54]. The driving laser at 395 nm 
+central wavelength is separated from the high-harmonic beam by using a 200 nm aluminum filter. This filter also 
+blocks any harmonics with 𝜆 > 77 nm, while harmonics with 𝜆 < 39 nm exceed the HHG cutoff energy, and so are 
+not generated. Furthermore, due to the centrosymmetry of the medium, only odd-numbered harmonic orders are 
+generated. The resulting EUV beam after the aluminum filter thus consists of narrow peaks at the 7th (𝜆 = 56 nm) and 
+9th (𝜆 = 44 nm) harmonics. The intensity ratio of the two harmonics in our experimental setup is estimated to be 
+Iλ=56nm/Iλ=44nm ~30:1, which can be well-approximated as a monochromatic illumination suitable for ptychographic 
+imaging. For generating HHG beams with a Gaussian spatial profile, we used an SHG beam with pulse energy of 
+~500 µJ. For generating HHG beams carrying OAM, we increased the pulse energy of the SHG beam to ~1.5 mJ, and 
+inserted a spiral phase plate (Holo-Or, VL-214-395-Y-A, OAM charge number ℓ = 1 at 395 nm wavelength) right 
+after the focusing optics into the semi-infinite gas cell to generate a driving beam with OAM charge number ℓ = 1, 
+and 𝜆 = 395 nm. The increased pulse energy is necessary in order to make the peak intensity (located at a central point 
+for the Gaussian beam, but distributed in a ring for the OAM beam) equal for the two cases, thus matching HHG cutoff 
+energies and conversion efficiency. Due to the conservation of OAM in HHG, the resulting quasi-monochromatic 7th 
+harmonic beam (𝜆 = 56 nm) carries an OAM charge number of ℓ = 7. 
+The HHG beam at 56 nm wavelength is focused sequentially by two toroidal mirrors (1: B4C-coated, feff = 27 cm, θ = 
+15°; 2: Au-coated, feff = 50 cm, θ = 10°) in a Wolter configuration to create an imaging system with higher NA (feff = 
+17 cm) while managing coma aberration [55]. The resulting focusing beam is redirected towards the sample at normal 
+incidence using a glancing incidence mirror (B4C coating, fused silica substrate, 3° incidence angle from grazing, 
+nominal reflectivity 95%). The testing samples are three Quantifoil holey carbon films which have various hole sizes 
+and shapes arranged in a rectangular grid, and are mounted on standard Ted Pella Ø3mm Cu TEM grids with 200 
+mesh (125 um pitch, 90 um hole width and 35 um bar width). More specifically, the three Quantifoil holey carbon 
+
+ 
+ 
+films have a pitch of 9 μm (7 μm square hole and 2 μm bar, product number 656-200-CU), 4.5 μm (3.5 μm diameter 
+circular holes and 1 μm separation, product number 660-200-CU) and 3 μm (2 um diameter circular holes and 1 um 
+separation, product number 661-200-CU), respectively. The samples are positioned close to the beam focus, and are 
+mounted on a precision translation stage ensemble (SmarAct XYZ-SLC17:30). They are translated in the plane 
+perpendicular to the beam path to perform ptychographic scans in 7 × 7 rectangular grids (49 positions) with nominally 
+3.3 μm between adjacent positions. A random offset of ±20% of the scan step size was added to each scan position 
+to avoid artifacts originating from the scan grid itself. The far-field diffraction patterns are recorded by an EUV-CCD 
+detector (Andor iKon-L, 2048 × 2048, 13.5 μm pixel size) positioned 50 mm after the sample. In order to obtain the 
+best ptychographic reconstructions possible for each illumination case, we carefully characterized each probe function 
+in the sample planes by taking ptychographic scans on a non-periodic sample and reconstructing both the sample and 
+the probe function through blind deconvolution. The reconstructed probe functions were used in the ptychographic 
+reconstructions of highly periodic samples as initial guesses. 
+ 
+Ptychographic data processing and image reconstructions 
+The diffraction patterns were recorded by an EUV-CCD detector with 2048 × 2048 pixels and 13.5 μm detector pixel 
+size. During data processing, we cropped them to 1024 × 1024 because very few photons were detected outside this 
+area. The resulting pixel size of the reconstructed images is 
+𝑑𝑟 =
+𝜆∙𝑧
+𝑁∙𝑑𝑥 ≈ 200 nm, 
+ 
+ 
+ 
+ 
+ 
+(2) 
+where λ = 56 nm is the wavelength, z = 50 cm is the distance from the sample to the CCD detector, N = 1024 is the 
+number of pixels in one direction and dx = 13.5 μm is the detector pixel size. 
+The ptychographic reconstructions were performed in two steps using only the ePIE algorithm [13]. In the first step, 
+the complex-valued probe functions (both Gaussian- and OAM-HHG beams) were characterized by performing 
+ptychography on a non-periodic sample and using the ePIE algorithm for reconstruction. In the second step, the pre-
+characterized probe functions were used as initial guesses for reconstructing the periodic structures. For the first 100 
+iterations, only the sample images were updated while the probe functions were kept fixed. Then, both the sample 
+images and probe functions were updated by the ePIE algorithm for another 900 iterations. The total number of 
+iterations for ptychographic reconstructions of periodic structures was 1000. This procedure was kept constant for all 
+Gaussian-HHG and OAM-HHG reconstructions in this paper. We also want to emphasize the fast convergence speed 
+of our technique compared to that in the work by Gardner et al. [15], which takes more than 10,000 iterations. 
+ 
+SNR analysis of imaging of the nano-defect 
+The SNR of the defect detection in Table 1 is calculated as follows: We start from the three curves in Fig. 4(d). For 
+each curve, corresponding to an experimental condition shown in the legend, the signal level is the transmission value 
+in the defect, the background level and noise level are calculated as the average and the standard deviation, respectively, 
+of the transmission values excluding the defect. The SNR is then calculated using the following formula: 
+𝑆𝑁𝑅 = 
+𝑠𝑖𝑔𝑛𝑎𝑙 − 𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑
+𝑛𝑜𝑖𝑠𝑒
+. 
+ 
+ 
+ 
+ 
+(3) 
+ 
+Data availability 
+The data that supports the plots and other findings within this paper are available from the corresponding author upon 
+reasonable request. 
+ 
+ 
+
+ 
+ 
+References 
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+ 
+Acknowledgements 
+This research was primarily supported by STROBE: a National Science Foundation (NSF) Science and Technology 
+Center (STC) under award DMR-1548924 for the setup and new illumination engineering and algorithms, and also 
+by a DARPA STTR grant 140D0419C0094 for imaging periodic samples. A Moore Foundation Grant No. 10784 
+supported the low-dose imaging research. The authors thank Guan Gui, Drew Morrill, Yunzhe Shao, Chen-Ting Liao, 
+Emma Cating-Subramanian for comments on the text. 
+ 
+Author contributions 
+B. W., N. J. B., M. M. M. and H. C. K. conceived the experiment. B. W., N. J. B. and P. J. built and maintained the 
+EUV source. B. W. and N. J. B. collected the data sets and performed the reconstructions and data analysis. N. J., Y. 
+E. and B. W. performed the SEM imaging of the test samples. M.T., Y.E. and N.J. advised on the phase retrieval 
+
+ 
+ 
+algorithms and setup, while I.B. helped to develop the laser setup. All authors designed aspects of the experiment, 
+performed the research and wrote the paper. 
+ 
+Competing financial interests 
+B. W., N. J. B., M. M. M. and H. C. K. have submitted a patent disclosure based on this work. M.M M. and H. C. K. 
+are partial owners of KMLabs Inc. who manufactured the ultrafast laser used in this study. 
+ 
+ 
+ 
+
+ 
+ 
+Supplementary information: 
+S1. Convolution theorem perspective on ptychographic imaging of highly periodic structures 
+In ptychography, the far-field diffraction fields are approximated as the Fourier transform of the product of the 
+complex probe and object functions, p(x,y) and o(x,y), i.e., 
+Ψ (u,v) = ℱ [p(x,y) × o(x,y)],                                                                    (S1) 
+where ℱ is the Fourier transform operation, (x,y) are the real space coordinates and (u,v) are the reciprocal space 
+coordinates. According to the convolution theorem, this can also be represented as a convolution of the Fourier 
+transform of the probe function, P(u,v) = ℱ [p(x,y)], and that of the object function, O(u,v) = ℱ [o(x,y)], i.e., 
+Ψ(u,v) = P(u,v) ⁕ O(u,v),                                                                     (S2) 
+where ⁕ is the convolution operation. Often, P(u,v), which is the complex beam in the detector plane when no sample 
+is in the way, has an edge resulting from apertures in the system, as indicated by the circle in the close-ups in Fig. 
+1(a,c). In the case of 2D highly periodic structures, O(u,v) consists of a 2D comb of 𝛿 functions (diffraction peaks) 
+arranged in a 2D periodic grid, the amplitudes and phases of which are modulated by the Fourier transform of the unit 
+cell of the periodic structure. This is a sparse function in the reciprocal space. The convolution operation in Eq. (S2) 
+puts a copy of P(u,v) at the location of each 𝛿 function with modulated amplitude and phase. 
+In cases where P(u,v) is small in size such that all diffraction orders are isolated, the modulated phase of each 
+diffraction order is totally lost when we collect intensity measurements, thus causing ptychography to fail. However, 
+in cases where P(u,v) is sufficiently large in size, the interference fringes in the overlapped regions between 
+neighboring diffraction orders encoded the relative phase of each diffraction orders into the intensity modulations that 
+are directly measurable with the EUV-CCD camera, thus enabling fast and robust ptychographic reconstructions of 
+the highly periodic structures. 
+ 
+S2. A phase-change-like behavior in ptychography demonstrated by Gaussian-HHG 
+illuminations with controlled divergence 
+Since the key to successfully achieving ptychographic imaging of highly periodic structures is to obtain overlap and 
+interference between neighboring diffraction orders, an abrupt, phase-change-like behavior in reconstruction quality 
+is expected as one smoothly changes the illumination NA. We experimentally demonstrated this behavior in 
+ptychographic imaging of highly periodic structures, as shown in Fig. S1, using Gaussian-HHG illuminations with a 
+controlled illumination NA. This is achieved by installing an in-vacuum iris ~0.5 m after the semi-infinite gas cell, 
+which allows direct control of the divergence of the HHG beams, and thus of the illumination NA on the sample and 
+the overlap between neighboring diffraction orders given the same focusing optics. 
+We performed four ptychography scans on the same 2D square periodic structure with a 9 μm period under various 
+illumination NAs controlled by the in-vacuum iris. Fig. S1(a–d) shows example diffraction patterns from each scan 
+from small illumination NA in (a) to large illumination NA in (d). The close-ups in the blue circles show the effect of 
+illumination NA on the resulting diffraction patterns. We then reconstructed these four datasets using the standard 
+ePIE algorithm [13] and the corresponding results are shown in Fig. S1(e–h). It is clear that for ptychography scans 
+where diffraction orders are isolated, the periodic structure cannot be reliably reconstructed due to the lost phase 
+information, as shown in Fig. S1(e–f), while for ptychography scans where the illumination NA is large enough to 
+support overlap between diffraction orders, the ePIE algorithm can quickly and reliably reconstruct the periodic 
+structures. 
+
+ 
+ 
+ 
+ 
+Figure S1. Experimental demonstrations of a phase-change-like behavior in ptychographic imaging of 2D square periodic 
+structures with 9 um period using Gaussian-HHG beams with controlled divergence. (a–d) Example diffractions from a 2D square 
+periodic structure using Gaussian-HHG beams with various divergences. The inserts show close-ups of the center of the diffraction 
+patterns. (e–h) The corresponding ptychographic reconstructions of the 2D square periodic structure under various illumination 
+conditions. The reconstructions are successful only when the diffraction orders have overlap, showing a phase-change-like behavior. 
+ 
+S3. Image quality assessment using complex histogram analysis 
+We use a complex histogram analysis to evaluate the quality of the ptychographic reconstructions in Fig. 3(d–f), 3(j–
+l) and 4(a–c). A 2D complex histogram is an extension of a normal histogram showing how many data points of a 
+complex field lie within a certain range of real and imaginary parts. For the approximately binary test samples used in 
+this study, ideally, the complex histograms consist of only two 𝛿-function peaks corresponding to the transmissive 
+and opaque areas. In reality, the two 𝛿-function peaks are broadened due to limited SNR and spatial resolution. The 
+quality of the ptychographic reconstructions can thus be assessed by examining the degree of broadening of these 
+peaks, where reconstructions with higher quality have narrower peaks. 
+We first evaluate the quality of the ptychographic reconstructions in Fig. 3. In the complex histograms shown in Fig. 
+S2, the two parts of the sample (free space and the carbon bars) are indicated by the red and yellow circles respectively. 
+The complex histograms for OAM-HHG images (d–f) have narrower peaks than those for Gaussian-HHG images (a–
+c), which indicates that OAM-HHG images have better quality. The reconstruction in (c) (Gaussian-HHG 
+illuminations on a 3-um-pitch structure) failed, thus not showing the double-peak feature. 
+We then evaluate the quality of the ptychographic reconstructions in Fig. 4. As shown in Fig. S3(a–c), the 
+ptychographic reconstructions are shown in the complex representation with amplitude and phase indicated by 
+brightness and hue. Visually, the image from OAM-HHG illumination in a has the best quality in terms of a sharp 
+transition from free space area to thin carbon bar area and smoothness within free space or carbon bar areas. The 
+complex histograms in (d–f) confirmed this: the primary peaks (indicated by the red and yellow circles) in the complex 
+histogram from OAM-HHG illuminations (as shown in d) are the narrowest. 
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+10um 
+ 
+ 
+ 
+Figure S2. Quality assessment of ptychographic reconstructions in Fig. 3 using complex histogram analysis. (a–c) Complex 
+histograms of ptychographic reconstructions of 9 μm, 4.5 μm and 3 μm pitch periodic structures using Gaussian-HHG illuminations. 
+The ptychographic reconstructions are shown in each bottom left corner and correspond to Fig. 3(d–f). (d–f) Complex histograms 
+of ptychographic reconstructions of 9 μm, 4.5 μm and 3 μm pitch periodic structures using OAM-HHG illuminations. The 
+ptychographic reconstructions are shown in each bottom left corner and correspond to Fig. 3(j–l). These complex histograms consist 
+of two primary peaks (except panel c because the reconstruction failed), which correspond to the open space area (indicated by the 
+red circles) and the thin carbon bar area (indicated by the yellow circles). The complex histograms from OAM-HHG illuminations 
+(the bottom row) have narrower primary peaks than those from Gaussian-HHG illuminations (the top row), which shows superior 
+image quality for OAM-HHG reconstructions. The ‘Re’ and ‘Im’ axes in (a) show the complex coordinate. 
+
+Im 
+ 
+ 
+Figure S3. Quality assessment of ptychographic reconstructions in Fig. 4 using complex histogram analysis. (a–c) Complex 
+representations of ptychographic reconstructions of 2D square periodic structures with 9 μm period under three different 
+experimental conditions: (a) an OAM-HHG illumination, (b) a Gaussian-HHG illumination with equal exposure time, and (c) a 
+Gaussian-HHG illumination with equal number of photons. The amplitude and phase of these images are presented in brightness 
+and hue, respectively. The color wheel is shown in the bottom left corner of panel a. (d–f) Complex histograms of ptychographic 
+reconstructions are shown in (a–c). These complex histograms all consist of two primary peaks, which correspond to the open 
+space area, indicated by the red circles, and the thin carbon bar area, indicated by the yellow circles. The complex histogram in 
+panel d has the narrowest primary peaks, which indicates its superior image quality provided by the intrinsic advantages of OAM-
+HHG illumination. 
+ 
+
+(a)
+(b)
+(c)
+Im
+Re 
+ 
+ 
+Figure S4. Complex representations of the ptychography reconstructed Gaussian-HHG and OAM-HHG beams in the 
+sample plane (a–b) and in the detector plane (c–d). The amplitude and phase of the beams are shown in brightness and hue, 
+respectively. The scale bars in (a–b) indicate beam size in the sample plane, and those in (c–d) indicate beam divergence angle in 
+the detector plane. The OAM-HHG beam in the detector plane in (d) shows a characteristic donut intensity profile, while the OAM-
+HHG beam in the sample plane does not show a donut intensity profile due to aberrations introduced by the focusing optics. 
+
+Gaussian-HHG
+OAM-HHG
+(a)
+(b)
+Sample plane
+0
+元
+50 um
+50 μm
+(c)
+(d)
+Detectorplane
+10mrad
+10mrad

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+Anomalous conductivities in the holographic
+Stückelberg model
+Nishal Rai1,2 and Eugenio Megías1,3
+1 Departamento de Física Atómica, Molecular y Nuclear,
+Universidad de Granada, Avenida de Fuente Nueva s/n, E-18071 Granada, Spain
+2 Department of Physics, SRM University Sikkim, Upper Tadong, Sikkim, India
+3 Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
+E-18071 Granada, Spain
+January 3, 2023
+Abstract
+We have studied a massive U(1) gauge holographic model with pure
+gauge and mixed gauge-gravitational Chern-Simons terms. The full
+backreaction of the gauge field on the metric tensor has been consid-
+ered in order to explore the vortical and energy transport sector. The
+background solution has been computed numerically. On this back-
+ground, we have considered the fluctuation of the fields and evaluated
+the different correlators. We have found that all the correlators depend
+on the mass of the gauge field. Correlators such as the current-current
+one, ⟨JxJx⟩, which were completely absent in the massless case, in the
+presence of a finite gauge boson mass start picking up some finite value
+even at zero chemical potential. Similarly, the energy-current corre-
+lator, ⟨T0xJx⟩, which was also absent in the massless theory, has now
+a non-vanishing value but for finite values of the chemical potential.
+Using Kubo formulae we have evaluated the chiral magnetic and chiral
+vortical conductivities and studied their behaviour with the variation
+of the mass of the gauge field. Our findings for the chiral vortical con-
+ductivity, σV , and the chiral magnetic/vortical conductivity of energy
+current, σε
+B = σε
+V , are completely new results. In addition to this, we
+have found that these anomalous transport coefficients depend linearly
+both on the pure Chern-Simon coupling, κ, and on the mixed gauge-
+gravity Chern-Simon coupling, λ. One of the results which we would
+like to highlight is the contribution to σV induced by λ in the massive
+theory, which was not present in the massless case.
+0
+arXiv:2301.00361v1  [hep-th]  1 Jan 2023
+
+1
+Introduction
+The AdS/CFT correspondence [1, 2] has been one of the most prominent
+theoretical handles for studying systems which were very hard to tackle pre-
+viously. It states that, in the low energy limit, the large-Nc, N = 4 super
+Yang-Mills field theory in four-dimensional space is equivalent to the type
+IIB string theory in AdS5 × S5 space. It has been widely applied for the
+study of strongly coupled systems such as condensed matter systems, QCD
+and hydrodynamics. Our current objective is to study the hydrodynamical
+approach using this correspondence.
+Quantum chiral anomalies are very fascinating properties which arise in
+the context of relativistic field theories of chiral fermions beyond perturba-
+tion theory [3–5]. Chiral anomalies have played a very crucial role in the
+formulation of relativistic hydrodynamics [6]. Anomaly-induced transport
+mechanisms have appeared on many occasions since the 80’s [7]. The ax-
+ial current was the main topic in [8], and AdS/CFT correspondence was
+first used to anomalous hydrodynamics in [9]. Recently a lot of attention is
+gained by the effect of quantum anomalies on the hydrodynamics of otherwise
+conserved currents. The chiral magnetic effect [10] and the chiral vortical ef-
+fect [11–14] are two of such effects. In the former, the axial anomaly induces
+a current parallel to the external magnetic field, while in the latter a current
+is generated due to the presence of a vortex in the charged relativistic fluid.
+It has been argued that these and other anomaly-induced effects may be pro-
+duced in non-central heavy ion collisions at RHIC and LHC [15], inducing in
+particular an event-by-event parity violation. These effects can also lead to
+anomalous transport properties in some condensed matter systems, such as
+the Weyl semi-metals [16,17].
+In the past few years, these anomalous effects has been implemented
+in holography giving a lot of insights.
+One of such works is [18], where
+they considered a holographic model with a pure Chern-Simon term, and
+they computed the chiral magnetic conductivity which exactly matches with
+the results of the weakly coupled system. This is due to the fact that the
+anomalous conductivities have non-renormalization properties so that they
+are independent of the coupling constant. Later on, this model was extended
+to incorporate the effect of the energy-momentum tensor related to the energy
+current as well, and the mixed gauge-gravitational Chern-Simon term was
+added in the gravitational action [19–21]. In these references the gauge fields
+were considered to be massless.
+In a similar line of work, the authors of [22] have studied the depen-
+dence of the anomalous transport properties with the mass of the gauge field
+which is introduced via the Stückelberg mechanism. In their case, they have
+1
+
+considered the probe limit. As a consequence the sectors comprising of the
+correlators related to the energy-momentum tensor were not accessible, in
+particular: i) the chiral vortical conductivity, ii) the chiral vortical conduc-
+tivity of energy current, and iii) the chiral magnetic conductivity of energy
+current. In a sense, this model only comprises a pure gauge Chern-Simon
+term. Our goal in the present work is to access those sectors and to study
+the chiral vortical effects as well. To this end, we have considered the full
+backreaction of the gauge field onto the metric tensor, and included in the
+action of the model a mixed gauge-gravitational Chern-Simons term.
+The paper has been organized as follows. In Section 2, we will discuss the
+model under consideration and get the full backreacted numerical solution
+for the background. Next, we will discuss in Section 3 the Kubo formulae
+and their relation with the retarded Green’s functions, i.e. the correlators.
+Using the AdS/CFT dictionary we will define these correlators in terms of
+the boundary terms. In Section 4 we will start presenting our results; first,
+we will compare the results with the known results for the massless case [19],
+and after that, we will present our main results regarding the behaviour of the
+two-point correlators including the mass term for the gauge boson. We will
+discuss in the same section the effect of the mixed gauge-gravitational Chern-
+Simons term in these correlators, and finally we will show how the gauge
+boson mass affects the anomalous conductivities, namely the chiral vortical
+conductivity, σV , the chiral vortical conductivity of energy current, σε
+V , the
+chiral magnetic conductivity, σB, and the chiral magnetic conductivity for
+energy current, σε
+B. Finally, we end with a discussion in Section 5.
+2
+Holographic massive U(1) gauge theory
+We consider a holographic model with a massive U(1) gauge boson that
+includes both a pure gauge and a mixed gauge-gravitational Chern-Simon
+term in the action [19,22]. The action of the model is
+S
+=
+1
+16πG
+�
+d5x√−g
+�
+R + 2Λ − 1
+4FMNF MN
+−m2
+2 (AM − ∂Mθ)(AM − ∂Mθ)
++ϵMNPQR(AM − ∂Nθ)
+�κ
+3FNPFQR + λRA
+BNPRB
+AQR
+� �
++SGH + SCSK ,
+(2.1)
+2
+
+where
+SGH
+=
+1
+8πG
+�
+∂
+d4x
+√
+hK ,
+(2.2)
+SCSK
+=
+− 1
+2πG
+�
+∂
+d4x
+√
+hKλnMϵMNPQRANKPLDQKL
+R ,
+(2.3)
+are the Gibbons-Hawking boundary term, and a boundary term induced by
+the mixed gauge-gravitational anomaly, respectively, which have been well
+discussed in [19]. θ is a field which ensures gauge invariance (up to gauge
+anomalies), and thus the mass term enters in a consistent way. As it men-
+tioned in [23–25], the Stückelberg term arises as the holographic realization
+of dynamical anomalies. A comparison of the consistent form of the anomaly
+for chiral fermions [3] with the variation of the action under axial gauge
+transformations, allows to fix the anomaly coefficients to
+κ = κp ≡ − G
+2π ,
+λ = λp ≡ − G
+48π .
+(2.4)
+See e.g. Ref. [19] for a discussion. In the following we will refer to the values
+of Eq. (2.4) as the physical values of the anomaly coefficients.
+The bulk equations of motion for the action of Eq. (2.1) turn out to be
+GMN − ΛgMN
+=
+1
+2FMLFN
+L − 1
+8gMNF 2 + m2
+2 BMBN − m2
+4 gMNBPBP
++2λϵLPQR(M▽B
+�
+F PLRB
+N)
+QR�
+,
+(2.5)
+▽NF NM
+=
+−ϵMNPQR �
+κFNPFQR + λRA
+BNPRB
+AQR
+�
++ m2BM ,(2.6)
+where we have defined a new field BM ≡ AM − ∂Mθ, so that in the following
+θ will not appear explicitly anywhere. We have used the notation X(MN) ≡
+1
+2(XMN + XNM).
+The ansatz for the background metric is a black hole solution in Fefferman-
+Graham coordinates, which is given by [26,27]
+ds2 = −ℓ2
+ρ gττ(ρ)dτ 2 + ℓ2
+ρ gxx(ρ)d⃗x2 + ℓ2
+4ρ2dρ2,
+(2.7)
+where the boundary lies at ρ = 0 and the horizon at ρ = ρh, while ℓ is the
+radius of AdS. The horizon ρh is chosen in such a way that gττ(ρh) = 0, and
+the temperature of the black hole turns out to be
+T = 1
+2π
+�
+2ρhg′′
+ττ(ρh) .
+(2.8)
+3
+
+The asymptotic expansion (ρ → 0) of the solution of Eq. (2.6) shows that
+the gauge field near the boundary behaves as
+BM(ρ) = a0ρ− ∆
+2 + a1ρ
+∆
+2 +1 + · · · ,
+(2.9)
+where m2ℓ2 = ∆(∆ + 2), with ∆ the anomalous dimension of the dual
+current [22].
+The first(second) term in Eq. (2.9) corresponds to a non-
+normalizable(normalizable) mode. The scaling dimension of the normaliz-
+able mode is (3 + ∆), and this puts an upper bound on the value ∆ = 1. For
+∆ > 1 the dual operators become irrelevant (in the IR), and so we will be
+working in the range of values of ∆ below this bound.
+2.1
+Numerical solution for the background
+In order to account for the chiral vortical effects within the present model, we
+will be considering the full backreaction of the gauge field onto the metric.
+Plugging Eq. (2.7) into Eqs. (2.5) and (2.6), the equations of motion for the
+background metric and gauge field turn out to be
+g′′
+xx(ρ) − g′
+xx(ρ)
+ρ
++
+1
+6ℓ2ρ
+gxx(ρ)
+gττ(ρ)
+�m2ℓ2
+4
+Bt(ρ)2 + ρ2B′
+t(ρ)2
+�
+= 0 ,(2.10)
+g′
+ττ(ρ)
+�
+1 − ρg′
+xx(ρ)
+gxx(ρ)
+�
++ gττ(ρ)g′
+xx(ρ)
+gxx(ρ)
+�
+3 − ρg′
+xx(ρ)
+gxx(ρ)
+�
+−1
+3
+ρ2
+ℓ2 B′
+t(ρ)2 + 1
+12m2Bt(ρ)2 = 0 ,
+(2.11)
+B′′
+t (ρ) + 1
+2
+�
+3g′
+xx(ρ)
+gxx(ρ) − g′
+ττ(ρ)
+gττ(ρ)
+�
+B′
+t(ρ) − ℓ2m2
+4ρ2 Bt(ρ) = 0 ,
+(2.12)
+where the gauge field has been chosen in the following way
+BMdxM = Bt(ρ)dt ,
+(2.13)
+so that Br = 0. We will solve numerically the above coupled differential
+equations with the following boundary conditions
+Bt(ρh) = 0 ,
+lim
+ρ→0
+�
+ρ∆/2Bt(ρ)
+�
+= µ5 ,
+(2.14)
+with µ5 being the source. As it is discussed in [22], in the presence of a finite
+gauge boson mass µ5 does not correspond to a thermodynamic parameter,
+but it is instead a coupling in the Hamiltonian. As a result, different values of
+chemical potential correspond to different theories. For completeness, we will
+4
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+ρ
+gxx
+gττ
+ρΔ/2·Bt
+Figure 2.1: (color) Dependence of the background metric and gauge field
+with ρ.
+We display the results for gxx(ρ) (blue), gττ(ρ) (orange), and
+ρ∆/2Bt(ρ) (green). We have chosen ∆ = 0.1 and µ5 = 0.5.
+provide the analytical solution of the background equations of motion (2.10)-
+(2.12) for vanishing µ5. These are
+gττ(ρ) = 1
+ρ2
+h
+(ρ2
+h − ρ2)2
+ρ2
+h + ρ2
+,
+gxx(ρ) = 1 + ρ2
+ρ2
+h
+,
+Bt(ρ) = 0 ,
+(2.15)
+while the temperature turns out to be T = 1
+π
+�
+2
+ρh. For the metric tensor, we
+demand that the solution is regular at the horizon, while at the boundary
+it reaches some constant value which we can always scale to set it to 1.
+Hereafter we will set the values of ℓ = 1 and ρh = 1 for our numerical
+calculations, which one can fix as such by using the scaling symmetry of the
+metric tensor. This will set the units of all the quantities, i.e. µ5, Q, M,
+etc. We have plotted in Fig. 2.1 the numerical solution of all the background
+fields, i.e. gxx(ρ), gττ(ρ) and Bt(ρ). One may note from this figure that
+limρ→0
+�
+ρ∆/2Bt(ρ)
+�
+= µ5.
+3
+Kubo formulae and correlators
+In this section, we will discuss the Kubo formulae needed to compute the
+anomalous transport coefficients in our model, and set up the equations to
+evaluate these transport properties. The Kubo formulae for the anomalous
+conductivities have been well studied [28]. The authors of this reference have
+shown that the chiral vortical conductivity for charge and energy transport
+5
+
+can be obtained from the following two-point functions
+σV = lim
+kc→0
+i
+2kc
+�
+a,b
+ϵabc⟨JaT 0b⟩|w=0 ,
+σε
+V = lim
+kc→0
+i
+2kc
+�
+a,b
+ϵabc⟨T 0aT 0b⟩|w=0 ,
+(3.1)
+where σV is the chiral vortical conductivity and σε
+V the chiral vortical con-
+ductivity of energy current, respectively. The chiral magnetic conductivities
+for charge, σB, and energy, σε
+B, current are given by
+σB = lim
+kc→0
+i
+2kc
+�
+a,b
+ϵabc⟨JaJb⟩|w=0 ,
+σε
+B = lim
+kc→0
+i
+2kc
+�
+a,b
+ϵabc⟨T 0aJb⟩|w=0 .
+(3.2)
+To compute these correlators one can use the AdS/CFT dictionary [19,
+29,30]. Keeping this in mind, we proceed with the perturbation of the fields,
+where the background is set by the numerical solution as shown in Fig. 2.1.
+We will study the linear response of the fluctuation, so that we split the
+metric and gauge field into a background and a linear perturbation part, i.e.
+gMN = g(0)
+MN + ϵhMN,
+BM = B(0)
+M + ϵbM .
+(3.3)
+Then, we will follow the general procedure of Fourier mode decomposition [28]
+hMN(ρ, xµ)
+=
+�
+ddk
+(2π)dhMN(ρ)e−iωt+i⃗k.⃗x ,
+(3.4)
+bM(ρ, xµ)
+=
+�
+ddk
+(2π)dbM(ρ)e−iωt+i⃗k.⃗x .
+(3.5)
+Without the loss of generality, one can consider perturbations of frequency ω
+and momentum k in the z-direction. In order to study the anomalous effect
+we will switch on the fluctuations Bi, hi
+t and hi
+z, where i = x, y. Following
+this, we will substitute (3.3) in the equations of motion (2.5) and (2.6), and
+consider the resulting expressions at order O(ϵ).
+Since we are interested in computing correlators at zero frequency, we
+can set the frequency-dependent parts as zero in the equations, and solve
+the system up to first order in k. In this limit, the fields hi
+z decouple from
+the system and take a constant value. Finally, we can write the system of
+6
+
+differential equations for the shear sector as
+b′′
+i (ρ) + 1
+2
+�g′
+xx(ρ)
+gxx(ρ) + g′
+ττ(ρ)
+gττ(ρ)
+�
+b′
+i(ρ) − ∆(∆ + 2)
+4ρ2
+bi(ρ)
+(3.6)
++
+�
+4iκkϵijbj(ρ)
+�
+gxx(ρ)gττ(ρ)
++ gxx(ρ)hi′
+t(ρ)
+gττ(ρ)
+�
+B′
+t(ρ) + iλkϵijhj′
+t(ρ)Ω(ρ) = 0 ,
+hi′′
+t (ρ) −
+� g′
+ττ(ρ)
+2gττ(ρ) − 5g′
+xx(ρ)
+2gxx(ρ) + 1
+ρ
+�
+hi′
+t(ρ) + ρB′
+t(ρ)
+gxx(ρ) b′
+i(ρ)
++∆(∆ + 2)Bt(ρ)
+4ρgxx(ρ)
+bi(ρ) + iλkϵijΦj(ρ) = 0 ,
+(3.7)
+where i, j = x, y. The explicit expressions of the functions Ω(ρ) and Φj(ρ)
+are given in Appendix A.
+Asymptotic analysis of the fluctuations near the boundary (ρ → 0) up to
+the first subleading order shows
+bi(ρ)
+=
+b(0)
+i ρ− ∆
+2 + b(1)
+i ρ
+∆
+2 +1 + · · · ,
+(3.8)
+hi
+t(ρ)
+=
+hi
+t
+(0) + hi
+t
+(1)ρ2 + · · · ,
+(3.9)
+where the leading order terms b(0)
+i
+and hi
+t
+(0) are the sources. From the holo-
+graphic description of the correlation functions, one can evaluate the one-
+point functions as
+⟨Ja⟩
+=
+δSren
+δb(0)
+a
+= −
+2
+16πG(∆ + 1)b(1)
+a ,
+(a = x, y) ,
+(3.10)
+⟨T0a⟩
+=
+δSren
+δha
+t (0) =
+1
+16πG
+�
+2ha
+t
+(0) + ha
+t
+(1)�
+,
+(a = x, y) ,
+(3.11)
+where Sren = S + Sct is the renormalized action, with S the action given
+in Eq. (2.1) and Sct the counterterm. The procedure to evaluate this coun-
+terterm is given in [19] and [22]. We find that the counterterm needed to
+renormalize this theory is the same as the one given in [22], i.e. the mixed
+gauge-gravitational Chern-Simons term does not introduce new divergences,
+and so the renormalization is not modified by it (see e.g. Ref. [19] for a discus-
+sion in the massless case). In this regards, we are not writing the counterterm
+Sct explicitly. ⟨Ja⟩ and ⟨T0a⟩ correspond to current and energy-momentum
+tensor one-point functions, respectively 1. Similarly, the two-point functions
+can be obtained by taking the variation of one-point function with respect
+1Ji and T0i are related with the fluctuations bi and hi
+t, respectively, with i = x, y.
+7
+
+to the corresponding source term, i.e.
+⟨JaJb⟩
+=
+δ⟨Ja⟩
+δb(0)
+b
+,
+(a, b = x, y) ,
+(3.12)
+⟨JaT0b⟩
+=
+δ⟨Ja⟩
+δhb
+t(0) ,
+(a, b = x, y) ,
+(3.13)
+⟨T0aJb⟩
+=
+δ⟨T0a⟩
+δb(0)
+b
+,
+(a, b = x, y) ,
+(3.14)
+⟨T0aT0b⟩
+=
+δ⟨T0a⟩
+δhb
+t(0) ,
+(a, b = x, y) .
+(3.15)
+From the above expressions, it is clear that it is required the leading and
+subleading parts of the asymptotic expansion of the fluctuations to evaluate
+the two-point functions we are interested in. To do so we have solved numer-
+ically the coupled differential equations of the fluctuations (3.6) and (3.7)
+and imposed suitable boundary conditions, i.e. i) regularity at the horizon,
+and ii) sourceless condition at the asymptotic boundary.
+4
+Results
+In this section, we will start presenting our results. Firstly, we will start
+with the massless case (∆ = 0) and compare the results with the previous
+work done in [19]. In the second part of this section, we will consider the
+massive case ∆ ̸= 0, and study the dependence of the two-point functions
+with ∆ for different values of µ5. In both cases, we will set G = 1/(16π) so
+that the physical values of the anomalous couplings are κ = −1/(32π2) and
+λ = −1/(768π2), cf. Eq. (2.4). Later on, we will study the dependence of the
+two-point functions with the parameters κ and λ. This is done to show that
+the parametric dependence of the correlators is linear in these parameters,
+but values of κ and λ different from κ/λ = 24 are non-physical. In addition
+to this, to make a direct comparison with the previous work in [22], all the
+anomalous correlators have been displayed normalized by |κ|−1.
+8
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+μ5
+-Im  < Jx Jy >
+k κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+μ5
+<JyT0 y>
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+2.5
+3.0
+3.5
+4.0
+μ5
+-Im  < Jx T0 y >
+k κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+1.00
+1.05
+1.10
+1.15
+μ5
+<T0 yT0 y>
+Figure 4.1: Upper panel: Plots of the correlators ⟨JxJy⟩ (left) and ⟨JyT0y⟩
+(right) vs µ5. Lower panel: Plots of the correlators ⟨JxT0y⟩(left) and ⟨T0yT0y⟩
+(right) vs µ5. These plots are obtained in the massless case (∆ = 0).
+4.1
+Massless case
+In the absence of mass, the correlators have been evaluated in [19,22], leading
+to
+⟨JxT0x⟩
+=
+⟨JyT0y⟩ =
+√
+3Q
+4πGℓ3 ,
+⟨JxJy⟩
+=
+−⟨JyJx⟩ = κi
+√
+3kQ
+2πGr2
+h
+− κ ikα
+6πG = −ik(3µ5 − α)
+12π2
+,
+⟨JxT0y⟩
+=
+−⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ = κ 3ikQ2
+4πGr4
+h
++ λ2ikπT 2
+G
+,
+= −ik
+� µ2
+5
+8π2 + T 2
+24
+�
+,
+(4.1)
+⟨T0xT0x⟩
+=
+⟨T0yT0y⟩ =
+M
+16πGℓ3 ,
+⟨T0xT0y⟩
+=
+−⟨T0yT0x⟩ = κi
+√
+3kQ3
+2πGr6
+h
++ λ4πi
+√
+3kQT 2
+Gr2
+h
+= −ik
+� µ3
+5
+12π2 + µ5T 2
+12
+�
+,
+9
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+1
+2
+3
+4
+μ5
+-Im  < T0 x T0 y >
+k κ
+Figure 4.2: Plot of the correlator ⟨T0xT0y⟩ vs µ5 in the massless case (∆ = 0).
+with M = r4
+h
+ℓ2 + Q2
+r2
+h
+and Q = µ5r2
+h
+√
+3 the mass and charge of the black hole
+solution computed in Poincaré coordinates, with blackening factor
+f(r) = 1 − Mℓ2
+r4
++ Q2ℓ2
+r6
+.
+(4.2)
+The Hawking temperature is given in terms of these black hole parameters
+as
+T =
+r2
+h
+4πℓ2f ′(rh) = (2r2
+hM − 3Q2)
+2πr5
+h
+.
+(4.3)
+The parameter α in Eq. (4.1) corresponds to the asymptotic value of the
+gauge field At for ρ → 0. In our case, we are assuming α = µ5 for ∆ = 0, cf.
+Eq. (2.14). The other correlators are vanishing in the massless case, i.e.
+⟨JxJx⟩ = ⟨JyJy⟩ = 0 ,
+⟨T0xJx⟩ = ⟨T0yJy⟩ = 0 .
+(4.4)
+While the correlators with the same indices are not induced by quantum
+anomalies (i.e. they are non-anomalous) and they become real, the correla-
+tors with different indexes are anomalous and they become imaginary. We
+will be comparing the numerical results with the analytical expressions given
+in the above equations, Eq. (4.1). We plot in Figs. 4.1 and 4.2 five inde-
+pendent non-vanishing correlators, while the other correlators are related to
+them through the expressions given in Eq. (4.1). In these and subsequent
+plots, it is understood that it has been taken the limit k → 0 with k ≡ kz.
+In these figures the dots stand for the numerical results, and the solid lines
+correspond to the analytic results of Eq. (4.1). One may observe that the
+numerical results are in good agreement with the analytic expression.
+10
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+1
+2
+3
+4
+5
+6
+7
+Δ
+-<JxJx>
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.0
+0.5
+1.0
+1.5
+Δ
+<JxT0 x>
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+2
+4
+6
+8
+10
+Δ
+-<T0 xJx>
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+1.000
+1.005
+1.010
+1.015
+1.020
+1.025
+1.030
+Δ
+<T0 xT0 x>
+Figure 4.3: (color) Plots for non-anomalous correlators vs ∆. Upper panel:
+Plot of the correlators ⟨JxJx⟩ (left) and ⟨JxT0x⟩ (right) vs ∆. Lower panel:
+Plot of the correlators ⟨T0xJx⟩ (left) and ⟨T0xT0x⟩ (right) vs ∆. We have
+considered in all the panels µ5 = {0, 0.1, 0.2} (blue, orange and green).
+4.2
+Massive case
+We will split our discussion into anomalous and non-anomalous correlators.
+We have found that the above mentioned relations between different corre-
+lators still hold in the massive case, i.e.
+⟨JxT0x⟩ = ⟨JyT0y⟩ ,
+⟨JxJy⟩ = −⟨JyJx⟩ ,
+⟨JxT0y⟩ = −⟨JyT0x⟩ = ⟨T0xJy⟩ = −⟨T0yJx⟩ ,
+⟨T0xT0x⟩ = ⟨T0yT0y⟩ ,
+⟨T0xT0y⟩ = −⟨T0yT0x⟩ .
+(4.5)
+In addition to this, there are two more independent correlators, i.e ⟨T0xJx⟩ =
+⟨T0yJy⟩ and ⟨JxJx⟩ = ⟨JyJy⟩. In this regard, we will be plotting only seven
+independent correlators.
+11
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+1
+2
+3
+4
+5
+Δ
+Im  < Jy Jx >
+k κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Δ
+Im  < T0 y T0 x >
+k κ
+Figure 4.4: (color) Plot of the correlators ⟨JyJx⟩ (left) and ⟨T0yT0x⟩ (right)
+vs ∆ with µ5 = {0, 0.1, 0.2} (blue, orange and green).
+4.2.1
+Non-anomalous correlators
+While the correlator ⟨JxJx⟩ is vanishing for ∆ = 0 (cf. Section 4.1), we can
+see from the Fig. 4.3 (upper-left panel) that this correlator starts picking
+up some finite value in the massive case (∆ ̸= 0). With the increase of ∆
+the absolute value of this correlator increases quite sharply, and gets even
+shaper with the increase in µ5. This property, i.e. an increasing value of
+the (absolute value of the) correlator for increasing ∆ and for finite µ5, is a
+general feature for all the non-anomalous coefficients as we will discuss below.
+We can see from Fig. 4.3 (upper-right panel) that for µ5 = 0 the correlator
+⟨JxT0x⟩ is zero for all values of ∆.
+As the value of µ5 increases, ⟨JxT0x⟩
+becomes finite and its value increases with ∆ in a somewhat linear fashion.
+The slope of ⟨JxT0x⟩ vs ∆ also increases with the increase of µ5.
+In Fig. 4.3 (lower panel-left) we can see that even though the correlator
+⟨T0xJx⟩ is vanishing for ∆ = 0, for finite values of ∆ and µ5 this corre-
+lator is non-vanishing. More in details, for a given finite value of µ5, the
+absolute value |⟨T0xJx⟩| increases quite sharply with ∆. Notice that ⟨T0xJx⟩
+was completely absent in the previous work [19], but we find now that it is
+non-vanishing at finite µ5 in the massive theory.
+Finally, we can see in Fig. 4.3 (lower-right panel) that ⟨T0xT0x⟩ is inde-
+pendent of ∆ for µ5 = 0, i.e. it has a constant value corresponding to the
+pressure term, a feature that has been well discussed in [18–20, 22]. At fi-
+nite chemical potential, this correlator increases with ∆, a behavior which is
+sharper for larger values of µ5.
+4.2.2
+Anomalous correlators
+We display in Fig. 4.4 (left) the behaviour of ⟨JyJx⟩ vs ∆. One can see that
+the absolute value of this correlator increases with ∆, and the change is quite
+12
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+2.5
+3.0
+3.5
+4.0
+4.5
+Δ
+-Im  < Jx T0 y >
+k κ
+Figure 4.5: (color) Plot of the correlator ⟨JxT0y⟩ vs ∆ with µ5 = {0, 0.15, 0.3}
+(blue, orange and green).
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.0002
+-0.0010
+-0.0005
+0.0000
+0.0005
+0.0010
+-0.2
+-0.1
+0.0
+0.1
+0.2
+-2
+-1
+0
+1
+2
+λ
+κp
+Im  < T0 y T0 x >
+k κp
+-0.0010
+-0.0005
+0.0000
+0.0005
+0.0010
+-0.03
+-0.02
+-0.01
+0.00
+0.01
+0.02
+0.03
+-0.2
+-0.1
+0.0
+0.1
+0.2
+-15
+-10
+-5
+0
+5
+10
+15
+λ
+κp
+Im  < Jy T0 x >
+k κp
+Figure 4.6: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs λ/|κp|.
+The inset figures correspond to zooms of the main figures in the small λ
+regime. We have considered in both panels, µ5 = 0.1, ∆ = 0.1 and κp =
+−1/(32π2).
+subtle. It is plotted in Fig. 4.4 (right) the correlator ⟨T0yT0x⟩ vs ∆, and unlike
+the other correlator, its absolute value decreases with the increase of ∆.
+In Fig. 4.5 we have plotted ⟨JxT0y⟩ vs ∆. We find that the absolute value
+of this correlator increases with the increase in ∆ and µ5. We have taken a
+different value of µ5 as compared to the other correlators, because for those
+values of µ5 the correlator did not have any substantial changes. The new
+values of µ5 = {0, 0.15, 0.3} are chosen to make these changes distinct in
+the figure. We can see from the figure that even in the absence of µ5 this
+correlator is non-zero. This can be traced back to the temperature term, as
+the temperature does not vanish for µ5 = 0. Finally, one may notice that
+in all the cases the values of two point correlators tend toward the analytic
+values as given in (4.1) when considering the limit ∆ → 0. This is also shown
+in the figures for the massless case.
+13
+
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.674
+0.676
+0.678
+0.680
+0.682
+λ
+κp
+Im  < Jy Jx >
+k κp
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+κ
+κp
+Im  < Jy Jx >
+k κp
+Figure 4.7: Left: Plot of the correlator ⟨JyJx⟩ vs λ/|κp|. Right: Plot of the
+correlator ⟨JyJx⟩ vs κ/|κp| for λ = −1/(768π2). We have considered in both
+panels µ5 = 0.1 and ∆ = 0.1, while κp = −1/(32π2).
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.492
+0.493
+0.494
+0.495
+κ
+κp
+Im  < T0 y T0 x >
+k κp
+-1.0
+-0.5
+0.0
+0.5
+1.0
+2.82
+2.84
+2.86
+2.88
+2.90
+κ
+κp
+Im  < Jy T0 x >
+k κp
+Figure 4.8: Plot of the correlator ⟨T0yT0x⟩ (left) and ⟨JyT0x⟩ (right) vs κ/|κp|.
+In both cases, µ5 = 0.1, ∆ = 0.1, |κp| = 1/(32π2) and λ = −1/(768π2).
+λ and κ dependence:
+To study the dependence of the two-point functions
+with the parameter λ, we will consider the case where we fix the values as
+µ5 = ∆ = 0.1 and κ = −1/(32π2), and vary λ. Here we will only present the
+correlators that have a dependence on λ, while the λ independent correlators
+are given in Fig. B.1 of Appendix B. We have plotted in Figs. 4.6 and 4.7
+(left) the dependence of the anomalous correlators with λ. One can see that
+the behaviour is linear with λ in all the cases. The inset figures are given
+to show that the corresponding correlators do not vanish at λ = 0. This is
+in fact true, as the non-vanishing values arise due to the κ coupling, which
+leads to ⟨T0xT0y⟩ ∼ µ3
+5κ and ⟨J0yT0x⟩ ∼ µ2
+5κ at λ = 0, with some contribution
+from ∆. In the case of ⟨JyJx⟩ the λ dependence only arises in the massive
+case.
+Setting the values of ∆ = µ5 = 0.1, λ = −1/(768π2) and varying κ, we
+see a similar kind of linear behaviour with κ. The effect of κ is only seen in
+⟨T0xT0y⟩, ⟨J0yT0x⟩ and ⟨JyJx⟩ as shown in Fig. 4.7 (right) and Fig. 4.8. The
+non-anomalous correlators are independent of κ, and they are displayed in
+14
+
+Fig. B.2 of Appendix B. These correlators are in fact independent of both the
+parameters κ and λ, and hence they are non-anomalous in nature even in the
+massive theory. This means that they do not contribute to anomalous trans-
+port, unlike the correlators studied above which are associated to anoma-
+lous conductivities. This can be seen in the Kubo formulae for anomalous
+conductivities, Eqs.
+(3.1) and (3.2), as these formulae involve Levi-Civita
+(ϵijz) symbols which runs over i = j = {x, y}. Hence, the correlators with
+i = j = x and i = j = y do not lead to anomalous transport effects.
+Anomalous conductivities:
+Finally, as a summary of the previous nu-
+merical results, we now present the anomalous conductivities which are com-
+puted with the Kubo formulas (3.1) and (3.2), i.e.
+σV = − lim
+k→0
+1
+kIm⟨JxT0y⟩ ,
+σε
+V = − lim
+k→0
+1
+kIm⟨T0xT0y⟩ ,
+(4.6)
+σB = − lim
+k→0
+1
+kIm⟨JxJy⟩ ,
+σε
+B = − lim
+k→0
+1
+kIm⟨T0xJy⟩ .
+(4.7)
+The results are displayed in Fig. 4.9. We can see from this figure that the
+chiral vortical conductivity and the chiral magnetic conductivity for energy
+current are the same either at zero or finite mass, i.e. σV = σε
+B, and these
+quantities increase with ∆. We also see in this figure that the chiral vortical
+conductivity of energy current, σε
+V , decreases with ∆ but the rate decreases
+rapidly. In the case of the chiral magnetic conductivity, σB, it increases with
+∆ as shown in Fig. 4.9.
+Regarding the other dependences of the anomalous conductivities, for
+instance the dependence in the parameters κ and λ, it would be sufficient to
+study them from Fig. 4.6 and Fig. 4.7, as the two-point functions and the
+anomalous conductivities are related through Kubo formulae. We conclude
+that for a given value of µ5 and ∆, the anomalous transport coefficients:
+σV , σε
+B, σB and σε
+B; change linearly with the pure (κ) and mixed (λ) gauge-
+gravitational Chern-Simon couplings. At the limit of vanishing mass, our
+results lead to
+σB
+µ5|κ| ≃ 16/3, which exactly coincides with the results in [18,
+19,22] where α has been set to µ5 in both references 2. In order to reproduce
+the results of [18] where they have set α = 0, our κ needs to be rescaled by
+a factor 3/2. Finally, let us emphasize that all the correlators involving the
+energy-momentum tensor are completely new results at finite mass (∆ ̸= 0),
+i.e. σV , σε
+V and σε
+B.
+2α corresponds to the asymptotic value of the gauge field At for ρ → 0. In our case,
+we assume α = µ5 for ∆ = 0.
+15
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+Δ
+σV
+κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Δ
+σε
+V
+κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Δ
+σB
+κ
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+Δ
+σε
+B
+κ
+Figure 4.9: Upper panel: Plot of σV (left) and σε
+V (right) vs ∆. Lower panel:
+Plot of σB (left) and σε
+B (right) vs ∆. We have considered µ5 = 0.15 in all
+the panels.
+5
+Discussion
+We have studied the anomalous and non-anomalous conductivities in the
+holographic Stückelberg model including both pure gauge and mixed gauge-
+gravitational anomaly terms. To access the sectors concerning the energy-
+momentum tensor we have to consider the full backreaction of the massive
+gauge field onto the metric tensor. We have evaluated the numerical back-
+ground solution and on this background, we have considered the fluctuations
+of the fields. From these fluctuations, we have calculated the different corre-
+lators and studied their behaviors with the relevant parameters of the model
+(µ5, ∆, κ and λ).
+We have found that the correlators in the massless case match with previ-
+ous results in the literature [28]. Later on, we have studied the dependence of
+these correlators with the mass of the gauge field, m2 = ∆(∆+2), and found
+that all the correlators explicitly depend on the mass for a given non-zero
+value of µ5. One of the results that it is important to emphasize here is that
+the non-anomalous correlators such as ⟨JxJx⟩ and ⟨T0xJx⟩ are non-vanishing
+in the massive theory for finite values of µ5. Moreover ⟨JxJx⟩ is non-zero in
+this theory even for µ5 = 0, while ⟨T0xJx⟩ is vanishing for µ5 = 0 indepen-
+dently of the mass. These correlators are vanishing in the massless theory,
+independently of µ5. The mass of the gauge field highly enhances the abso-
+16
+
+lute value of the correlators, and this gets translated into an enhancement
+of the anomalous conductivities. The behaviours of the correlators on the
+pure gauge and mixed gauge-gravitational Chern-Simon couplings, κ and λ,
+were also studied. We found that the correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and
+⟨T0xT0x⟩ are independent of κ and λ, and hence they are non-anomalous in
+nature. They do not contribute to the anomalous conductivities, as it can
+be seen from the Kubo formulae (3.1) and (3.2) as well.
+Finally, we have computed the anomalous conductivities and studied their
+dependence with the mass of the gauge field (m). We have found that the
+chiral vortical conductivity, σV , and the chiral magnetic conductivity for
+energy current, σε
+B, are equal and increase with ∆. One interesting result is
+that there are contributions to σB coming from λ in the massive theory, which
+was completely absent in the massless case. The conductivities σB, σV and
+σε
+B increase with ∆, while the chiral vortical conductivity of energy current,
+σε
+V , decreases with ∆. We have explicitly checked that all our numerical
+results for the conductivities at finite mass tend to the known results at zero
+mass in the limit ∆ → 0. For instance, it is known that at zero mass, the
+chiral magnetic conductivity is σB = − 16
+3 κµ5 when α = µ5, which implies
+that the ratio − σB
+κµ5 = 16/3, independently of κ and µ5. As one can see
+from Fig. 4.9 (left), our numerics produces in this limit σB/|κ| = 0.8 for
+µ5 = 0.15, in agreement with the expected result. We have also checked the
+ratio − σB
+κµ5 = 16/3 for other values of κ and µ5.
+This work can be extended in several ways. One possible extension could
+be to consider the U(1)V × U(1)A gauge group. There are some studies in
+holography with this gauge group, see e.g. Refs. [22,31,32]. However, in these
+works: i) either the probe limit has been considered so that the chiral vortical
+effect and the transport conductivities in the energy-momentum tensor are
+not accessible, or ii) they correspond to studies for massless gauge bosons.
+In particular, it would be interesting to study the interplay between the
+anomalous and non-anomalous currents in the set-up of the full backreacted
+background of Ref. [32], both for massless and massive gauge bosons. We
+will explore these and other issues in future works.
+Acknowledgments
+We would like to thank Karl Landsteiner for enlightening discussions. E.M.
+is grateful to Manuel Valle for collaboration in the early stages of this work.
+N.R. thanks the Instituto de Física Teórica UAM/CSIC, Spain, for its hos-
+pitality and partial support during his research visits in the final stages
+of this work.
+The works of N.R. and E.M. are supported by the project
+17
+
+PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/501100011033, by
+the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento
+2014-2020 Operational Program under Grant A-FQM-178-UGR18, and by
+the Ramón y Cajal Program of the Spanish MCIN under Grant RYC-2016-
+20678. The work of E.M. is also supported by Junta de Andalucía under
+Grant FQM-225.
+Appendix A
+Explicit expressions for the func-
+tions Ω(ρ) and Φj(ρ)
+These functions have been introduced in the equations of motion of the fluc-
+tuations (3.6)-(3.7). Their explicit expressions are given by
+Ω(ρ) =
+4
+�
+gττ(ρ) (g′
+xx(ρ) + 2ρg′′
+xx(ρ)) + ρg′
+xx(ρ)g′
+ττ(ρ)
+�
+�
+gxx(ρ)gττ(ρ)3/2
+− 8ρ g′
+xx(ρ)2
+gxx(ρ)3/2
+�
+gττ(ρ)
++
+�
+gxx(ρ)
+gττ(ρ)5/2
+�
+4ρg′
+ττ(ρ)2 − 4gττ(ρ) (g′
+ττ(ρ) + 2ρg′′
+ττ(ρ))
+�
+,
+(A.1)
+and
+Φj(ρ) = b′
+j(ρ)
+�
+− 8ρ2�
+gττ(ρ)g′
+xx(ρ)2
+gxx(ρ)7/2
++
+4ρ
+�
+gττ(ρ)
+�
+g′
+xx(ρ) + 2ρg′′
+xx(ρ)
+�
++ ρg′
+xx(ρ)g′
+ττ(ρ)
+�
+gxx(ρ)5/2�
+gττ(ρ)
+−
+4ρ
+�
+gττ(ρ)
+�
+g′
+ττ(ρ) + 2ρg′′
+ττ(ρ)
+�
+− ρg′
+ττ(ρ)2�
+gxx(ρ)3/2gττ(ρ)3/2
+�
++ bj(ρ)
+�
+8ρ2�
+gττ(ρ)g′
+xx(ρ)3
+gxx(ρ)9/2
+− 8ρ
+�
+gττ(ρ)g′
+xx(ρ)
+gxx(ρ)7/2
+�
+g′
+xx(ρ) + 2ρg′′
+xx(ρ)
+�
++
+4ρ
+�
+ρg′′
+xx(ρ)g′
+ττ(ρ) − ρg′
+xx(ρ)g′′
+ττ(ρ) + 3gττ(ρ)g′′
+xx(ρ) + 2ρgxx(3)(ρ)gττ(ρ)
+�
+gxx(ρ)5/2�
+gττ(ρ)
+−
+4ρ
+�
+− 2gττ(ρ)g′
+ττ(ρ)
+�
+g′
+ττ(ρ) + 2ρg′′
+ττ(ρ)
+�
++ 2ρg′
+ττ(ρ)3 + gττ(ρ)2�
+3g′′
+ττ(ρ) + 2ρgττ (3)(ρ)
+��
+gxx(ρ)3/2gττ(ρ)5/2
+�
+(A.2)
+18
+
++ hj′
+t(ρ)
+�
+B′
+t(ρ)
+�
+�
+16ρ2g′
+xx(ρ)
+gxx(ρ)3/2�
+gττ(ρ)
++
+8ρ
+�
+gττ(ρ) − ρg′
+ττ(ρ)
+�
+�
+gxx(ρ)gττ(ρ)3/2
+�
+� +
+8ρ2B′′
+t (ρ)
+�
+gxx(ρ)
+�
+gττ(ρ)
+�
++ hj′′
+t (ρ)
+8ρ2B′
+t(ρ)
+�
+gxx(ρ)
+�
+gττ(ρ)
+.
+Appendix B
+Some additional results for the non-
+anomalous correlators
+We show in this Appendix the numerical results for the non-anomalous cor-
+relators as a function of the anomalous parameters κ and λ, in the massive
+case (∆ ̸= 0). The correlators ⟨JxJx⟩, ⟨JxT0x⟩, ⟨T0xJx⟩ and ⟨T0xT0x⟩, are
+displayed in Figs. B.2 and B.1. These correlators turn out to be constant in
+both κ and λ. The lack of dependence in these parameters implies that they
+lead to non-anomalous transport effects.
+19
+
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+λ
+κp
+-<JxJx>
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+λ
+κp
+-<JxT0 x>
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.00
+0.05
+0.10
+0.15
+λ
+κp
+-<T0 xJx>
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.0
+0.5
+1.0
+1.5
+2.0
+λ
+κp
+<T0 xT0 x>
+Figure B.1: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩
+(right) vs λ/|κp|.
+Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and
+⟨T0xT0x⟩ (right) vs λ/|κp|.
+We have considered µ5 = 0.1, ∆ = 0.1 and
+κp = −1/(32π2) in all the panels.
+20
+
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.1
+0.2
+0.3
+0.4
+κ
+κp
+-<JxJx>
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+κ
+κp
+<JxT0 x>
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.00
+0.05
+0.10
+0.15
+κ
+κp
+-<T0 xJx>
+-2
+-1
+0
+1
+2
+0.0
+0.5
+1.0
+1.5
+2.0
+κ
+κp
+<T0 xT0 x>
+Figure B.2: Upper panel: plot of the correlator ⟨JxJx⟩ (left) and ⟨JxT0x⟩
+(right) vs κ/|κp|.
+Lower panel: plot of the correlator ⟨T0xJx⟩ (left) and
+⟨T0xT0x⟩ (right) vs κ/|κp|. We have considered µ5 = 0.1, ∆ = 0.1, |κp| =
+1/(32π2) and λ = −1/(768π2) in all the panels.
+21
+
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+ 
+ 
+Deep Learning For Classification Of Chest X-Ray Images (Covid 19) 
+ 
+Benbakreti Samir1, Said Mwanahija1, Benbakreti Soumia2, Umut Özkaya3 
+1 Specialty Department, Ecole Nationale des Télécommunications et des Technologies de l'Information et de la Communication 
+(ENSTTIC), Oran, Algeria. 
+2 Laboratoire des Mathématiques, University of Djillali Liabes, Sidi Bel Abbes, Algeria. 
+3Electrical ans Electronic Engineering, Konya Technical University, Turkey 
+ 
+ABSTRACT
+ 
+ 
+In medical practice, the contribution of information technology can be considerable. Most of these practices include the 
+images that medical assistance uses to identify different pathologies of the human body. One of them is X-ray images 
+which cover much of our work in this paper. Chest x-rays have played an important role in Covid 19 identification and 
+diagnosis. The Covid 19 virus has been declared a global pandemic since 2020 after the first case found in Wuhan 
+China in December 2019. Our goal in this project is to be able to classify different chest X-ray images containing Covid 
+19, viral pneumonia, lung opacity and normal images. We used CNN architecture and different pre-trained models. The 
+best result is obtained by the use of the ResNet 18 architecture with 94.1% accuracy. We also note that The GPU 
+execution time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge 
+much faster than the CNN. The time saving is very considerable. 
+With these results not only will solve the diagnosis time for patients, but will provide an interesting tool for practitioners, 
+thus helping them in times of strong pandemic in particular. 
+ 
+Keywords: Deep learning, Image classification, CNN, Covid-19, Chest Xray, Pre-trained models. 
+
+ 
+ 
+ 
+1. Introduction 
+ 
+Computerized Tomography (CT) and X-ray scans are frequently used for chest imaging. An X-ray is a scan of 
+the body that looks for pneumonia, tumors, fractures, and lung infections. An upgraded X-ray machine called a 
+CT scan can produce sharper images of bones, tissue, and organs. Compared to CT, the X-ray approach is 
+simpler, faster, and more affordable, but it is also more dangerous. Doctors can visually diagnose viral 
+bacterial infections, viruses like covid 19 [1], and other infections by examining chest X-ray images. The 
+technique of visual diagnosis is typically unappealing, time-consuming, and inaccurate, because it can result 
+in low accuracy and requires specialized human resources. 
+Coronavirus disease 2019 (COVID-19) is an infectious disease brought on by the coronavirus strain known as 
+severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) [2]. It is a lung infection that is respiratory in 
+nature. The root of the coronavirus word is Greek (κορώνη) which means "crown or halo." It relates to the 
+virus's appearance under an electron microscope, which resembles a royal crown. Because of this, 
+coronavirus is also known as the crowned virus. The purpose of this paper is therefore to provide a decision 
+making tool that will lighten the burden on medical staff, especially during pandemic peaks.  
+ 
+2. Related work 
+ 
+Since Corona was announced as a pandemic, different projects were carried out since 2020 to 2022 that it 
+became among an interesting subject to learn from, some of the works related to image classification for 
+different reasons are discussed in this paragraph. 
+The COVID-CT dataset of 2560 images was the database used in [3], 2214 of which were used for training 
+and the remaining 246 for testing. By employing WOA to optimize the network's hyperparameters, the model 
+used to train ResNet-50 became the WOANet model. This last experiment looked at the accuracy of the 
+classification using the suggested method on 246 CT scans, and found that 98.37% of them were categorized 
+as COVID-19, while 99.18% were identified as non-COVID-19. The radiologists will be greatly assisted by this 
+proposed WOANet in reducing the burden on the healthcare system and hospitals. 
+In this study [4], patients' X-ray images are used to classify patients using CNN deep learning. One of the 
+most powerful algorithms with generative and deterministic capabilities is the capsule network (CapsNet). 
+However, compared to the basic CNN structures, this network has been relatively more responsive to images. 
+The dataset utilized was the NIH complete Chest X-rays [5] collection. VDSNet has a validation accuracy 
+value of 73%, which is higher than the sample dataset's score of 70.8%. 
+Using a dataset of 6432 images, the DLH COVID [6] model is distinct, trustworthy, and independently created 
+without any input from the transfer learning approach. The experimental findings from the prospective 
+validation phase suggest that the DLH MODEL outperformed the majority of the pre-trained models since it 
+distinguished COVID-19, pneumonia, and healthy/unhealthy patients from the image dataset with a promising 
+accuracy of 96%. 
+ 
+3. Dataset 
+ 
+As seen in figure 1, a database of chest X-ray images for COVID-19 positive cases as well as images of 
+normal and viral pneumonia was created in collaboration with medical professionals by a group of researchers 
+from Qatar University, Doha, Qatar, and the University of Dhaka, Bangladesh, as well as their collaborators 
+from Pakistan and Malaysia. This dataset contains 3616 COVID-19 positive cases, 10,192 Normal, 6012 Lung 
+Opacity (Non-COVID lung infection), and 1345 Viral Pneumonia images. The COVID-19 x-ray image database 
+was created using different sources [7, 8, 9].  
+ 
+Fig. 1 CXR scans with four categories of pathology. 
+
+COVID
+Lung_Opacity
+Normal
+ViralPneumonia 
+ 
+ 
+ 
+For training of datasets, we employed Matlab 2021 installed on computer with 64-bit operating system, 
+windows 10 Pro, 24 GB of Random Access Memory (RAM), with an Intel(R) Xeon(R) CPU E5-2620 v3 @ 
+2.40GHz and Graphical Processing Unit (GPU). Eighty percent of the datasets are used for training and 20% 
+for testing (evaluating the model performance). 
+ 
+4. Proposed Model 
+ 
+In this study, we analyzed the different techniques for image classification of COVID-19 using X-Ray 
+radiographic images of the chest, then examined CNN’s architecture that is based on research on the visual 
+cortex of the cat by Hubel and Wisiel [10], and different pre-trained models: AlexNet, ResNet18 and 
+GoogleNet in order to see the variation of answers in our work. 
+ 
+ 
+Fig. 2 The proposed model with different algorithms. 
+ 
+The use of deep learning methods not only allows us to process a very large number of images but at the 
+same time allows us to skip the feature extraction step, such a cumbersome step because it is done by hand 
+crucially. 
+Due to their capacity to extract features (see figure 2) and learn to distinguish between various classes, 
+convolutional neural networks (CNNs) are the top DL tool that are widely employed in several fields of the 
+healthcare system (i.e., positive and negative, infected). Transfer learning (TL) has made it simpler to quickly 
+and accurately retrain neural networks on chosen datasets. 
+ 
+5. Experiments and Results  
+5.1 Experiment 1: Application of the CNN model 
+ 
+The structure of our CNN includes a number of layers, as shown in table I. CNN receives a CXR image with a 
+size of 299 by 299 pixels as its input, and the rest of the architecture is mentioned in the table I. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Model: CNN/AlexNet/GoogleNet/ResNet 18
+Fully
+Convolution
+Connected
+Pooling
+Output
+Input
+O
+O
+O
+299*299*1
+4 classes
+FeatureExtraction
+Classification 
+ 
+ 
+Table 1 The architecture of the CNN model. 
+Name 
+Type 
+Description of output size 
+Input layer 
+Input data 
+299*299 
+Conv 1 
+Convolution +ReLU 
+32*32*8  
+S1 
+Max pooling 
+3,2  
+Conv 2 
+Convolution +ReLU 
+64*64*3  
+S2 
+Max pooling 
+3,2  
+Conv 3 
+Convolution +ReLU 
+128*128*5  
+S3 
+Max pooling 
+3,2  
+Conv 4 
+Convolution +ReLU 
+256*256*5  
+S4 
+Max pooling 
+3,2  
+Conv 5 
+Convolution +ReLU 
+512*512*5  
+S5 
+Max pooling 
+3,2  
+Conv 6 
+Convolution +ReLU 
+1024*1024*5  
+S6 
+Max pooling 
+3,2  
+Fc 
+Fully connected 
+1 Fc (4) 
+ 
+The trained parameters used in this model are in the options side where all the hyperparameters used were 
+defined including the number of epochs used (1 or 5), the mini batch (64), the learning rate is 0.001 and 
+frequency validation is 20. The given CNN was trained using different parameters to test the accuracy for this 
+model. We utilized the accuracy parameter to evaluate how well the trained models performed. The 
+percentage of correctly classified images over all the images is what is referred to as accuracy. The following 
+formula is utilized: 
+𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 
+𝑇𝑃 + 𝑇𝑁
+𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁 
+ 
+Table 2 The results of the CNN model. 
+The CNN model 
+ 
+1 epoch 
+5 epochs 
+Accuracy 
+75.61% 
+89.13% 
+Time GPU 
+execution 
+146 min 58s 
+703 min 16s 
+ 
+The first top accuracy after training the model using one epoch provided us with 75.61% accuracy for our 4 
+classes classification. As training the model using only one epoch did not provide the best result, we had to 
+increase the number of epochs and see the performance of our model and the results for our model gave us 
+89.13%, which is a lot better compared to our first experiment with one epoch. 
+ 
+5.2 Experiment 2: Application of the pretrained models 
+ 
+AlexNet: With 5 convolutional layers and convolutional filter sizes of 3*3 and 2*2 for max pooling operation, 
+AlexNet is an 8-layer convolutional neural network [11]. Fully connected layers are the final three layers. The 
+AlexNet model's standard input size is 227*227*3. 
+GoogleNet (Inception v3): A convolutional neural network with 50 layers in depth is called GoogleNet [12]. 
+The program, titled "Going deeper with convolutions," was developed and taught by Google. Up to 1000 
+objects can be classified using the pre-trained Inceptionv3 model with the ImageNet dataset [13] weights. This 
+network's image input size was 299x299 pixels. 
+ResNet18: A convolutional neural network with 18 layers in depth is called ResNet18. Deep Residual 
+Learning for Image Recognition, as it is known, was developed and trained by Microsoft in 2015 [14]. To 
+address the issue of vanishing gradient that may affect the weightage change in neural networks, ResNet 
+architectures introduced the use of residual layers and skip connections. This made training easier and 
+allowed neural networks to get much deeper with greater performance. The network was trained on colored 
+images with a resolution of 224x224 pixels. 
+ 
+
+ 
+ 
+ 
+In addition to the accuracy parameters, we estimated the time GPU execution for each model. The results 
+obtained are shown in Table 3. 
+Table 3 The results of the pretrained models. 
+ 
+Pretrained models 
+ 
+AlexNet 
+GoogleNet 
+ResNet18 
+Accuracy 
+89.93% 
+91.87% 
+94.1% 
+Time 
+GPU 
+execution 
+14 min 
+58s 
+41 min 
+34s  
+33 min 
+13s 
+ 
+Confusion matrix is the common approach used for evaluation of model performance based on true positive 
+(TP), true negative (TN), false positive (FP), and false negative (FN).  
+ 
+The figure 3 represents the confusion matrix of the Resnet 18 model which gave the best result in terms 
+of accuracy.  
+ 
+ 
+Fig. 3 The confusion matrix for the RestNet 18 model with the best result. 
+When the same dataset was used using the same hyper parameters the accuracy found was 89.93%, 91.87% 
+and 94.1 % for AlexNet, GoogleNet and ResNet18 respectively. Note that the pretrained models used only 
+one epoch. We see that the results can be improved by using pretrained architectures, attaining an accuracy 
+of 94.1%. The increased classification rate attained by Resnet 18 can be attributed to the network's use of 
+novel techniques to lessen over-fitting in its model. 
+The first method involved artificially enlarging the dataset with the aid of a label-preserving transformation. 
+This involved extracting random patches (224x224 for ResNet 18) and training the network on them while 
+varying the intensities of the RGB channels in the training images. The result was the generation of image 
+translations and horizontal reflections. The second strategy was "dropout," which involves removing neurons 
+that do not participate in the forward pass or the backward propagation. As a result, the model is forced to 
+learn more robust characteristics and decreases the complex co-adaptations of neurons. The GPU execution 
+time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge 
+much faster than the CNN. The time saving is very considerable. 
+ 
+
+707
+COVID
+6
+6
+0
+97.9%
+16.7%
+0.1%
+0.2%
+0.0%
+2.1%
+Lungopacity
+S
+1060
+LL
+1
+92.7%
+0.1%
+25.1%
+1.8%
+0.0%
+7.3%
+Output Class
+Normal
+8
+136
+1949
+5
+92.9%
+0.2%
+3.2%
+46.1%
+0.1%
+7.1%
+1
+0
+3
+263
+ViralPneumonia
+98.5%
+0.0%
+%00
+0.1%
+6.2%
+1.5%
+981%
+88.2%
+95.6%
+97.8%
+94.1%
+1.9%
+118%
+4.4%
+2.2%
+5.9%
+alPpr
+Target Class 
+ 
+ 
+6. Conclusion 
+This work aimed at developing a convolutional neural network (CNN) model that will help classify COVID-19 
+and non-COVID 19 disease such as viral pneumonia cases using chest X-ray images in the period caused by 
+the pandemic. The model used in this work was CNN as well as pre-trained models including AlexNet, 
+GoogleNet, and ResNet18. The CNN gave a result with 89.13% accuracy for classifying the four classes after 
+training 80% of the dataset and testing on 20%. This motivated us not only to keep changing settings, but also 
+to work on pretrained model. In the latter, the pre-trained models were used on the same dataset but with just 
+one epoch for each model. And the results were 89.93%, 91.87% and 94.1 % for AlexNet, GoogleNet and 
+ResNet18 respectively.  
+ 
+7. References 
+[1] Kutlu, Yakup, and Yunus Camgözlü. "Detection of coronavirus disease (COVID-19) from X-ray images 
+using deep convolutional neural networks." Natural and Engineering Sciences 6, no. 1 (2021): 60-74. 
+[2] Alakus, T.B. and Turkoglu, I., 2020. Comparison of deep learning approaches to predict COVID-19 
+infection. Chaos, Solitons & Fractals, 140, p.110120 
+[3] Murugan, R., Goel, T., Mirjalili, S., & Chakrabartty, D. K. (2021). WOANet: Whale optimized deep neural 
+network for the classification of COVID-19 from radiography images. Biocybernetics and Biomedical 
+Engineering, 41(4), 1702-1718. 
+[4] Apostolopoulos, I.D., Mpesiana, T.A. Covid-19: automatic detection from X-ray images utilizing transfer 
+learning with convolutional neural networks. Phys Eng Sci Med 43, 635–640 (2020).  
+[5] Patel 
+P. 
+Chest 
+X-ray 
+(COVID-19 
+& 
+Pneumonia). 
+Kaggle. 
+(2020); 
+https://www.kaggle.com/prashant268/chest-xray-covid19- pneumonia 
+[6] CDey, S., Bacellar, G. C., Chandrappa, M. B., & Kulkarni, R. (2021). COVID-19 Chest X-Ray Image 
+Classification 
+Using 
+Deep 
+Learning. 
+medRxiv 
+2021.07.15.21260605; 
+doi: 
+https://doi.org/10.1101/2021.07.15.21260605 
+[7] https://bimcv.cipf.es/bimcv-projects/bimcv covid19/#1590858128006-9e640421-6711 
+[8] https://github.com/ml-workgroup/covid-19-image-repository/tree/master/png 
+[9] https://sirm.org/category/senza-categoria/covid-19/ 
+[10] D. H. Hubel, T. N. Wiesel, Receptive fields and functional architecture of monkey striate cortex, J. Physiol, 
+vol. 195, pp. 215-243, 1968. 
+[11] Li, Shaojuan, Lizhi Wang, Jia Li, and Yuan Yao. "Image classification algorithm based on improved 
+AlexNet." In Journal of Physics: Conference Series, vol. 1813, no. 1, p. 012051. IOP Publishing, 2021. 
+[12] Hu, J., Shen, L. and Sun, G., 2018. Squeeze-and-excitation networks. In Proceedings of the IEEE 
+conference on computer vision and pattern recognition (pp. 7132-7141). 
+[13] Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. "Imagenet classification with deep convolutional 
+neural networks." Advances in neural information processing systems 25 (2012). 
+[14] Huang L, Ruan S, Denoeux T. Covid-19 classification with deep neural network and belief functions. 
+InThe Fifth International Conference on Biological Information and Biomedical Engineering 2021 Jul 20 
+(pp. 1-4). 
+

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+page_content="Deep Learning For Classification Of Chest X-Ray Images (Covid 19)  Benbakreti Samir1, Said Mwanahija1, Benbakreti Soumia2, Umut Özkaya3 1 Specialty Department, Ecole Nationale des Télécommunications et des Technologies de l'Information et de la Communication (ENSTTIC), Oran, Algeria." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' 3Electrical ans Electronic Engineering, Konya Technical University, Turkey  ABSTRACT  In medical practice, the contribution of information technology can be considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Most of these practices include the images that medical assistance uses to identify different pathologies of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' One of them is X-ray images which cover much of our work in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Chest x-rays have played an important role in Covid 19 identification and diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The Covid 19 virus has been declared a global pandemic since 2020 after the first case found in Wuhan China in December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Our goal in this project is to be able to classify different chest X-ray images containing Covid 19, viral pneumonia, lung opacity and normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' We used CNN architecture and different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The best result is obtained by the use of the ResNet 18 architecture with 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='1% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' We also note that The GPU execution time is optimal in the case of AlexNet but what requires our attention is that the pretrained models converge much faster than the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The time saving is very considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' With these results not only will solve the diagnosis time for patients, but will provide an interesting tool for practitioners, thus helping them in times of strong pandemic in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Keywords: Deep learning, Image classification, CNN, Covid-19, Chest Xray, Pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Introduction  Computerized Tomography (CT) and X-ray scans are frequently used for chest imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' An X-ray is a scan of the body that looks for pneumonia, tumors, fractures, and lung infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' An upgraded X-ray machine called a CT scan can produce sharper images of bones, tissue, and organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Compared to CT, the X-ray approach is simpler, faster, and more affordable, but it is also more dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Doctors can visually diagnose viral bacterial infections, viruses like covid 19 [1], and other infections by examining chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The technique of visual diagnosis is typically unappealing, time-consuming, and inaccurate, because it can result in low accuracy and requires specialized human resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Coronavirus disease 2019 (COVID-19) is an infectious disease brought on by the coronavirus strain known as severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' It is a lung infection that is respiratory in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The root of the coronavirus word is Greek (κορώνη) which means "crown or halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='" It relates to the virus\'s appearance under an electron microscope, which resembles a royal crown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Because of this, coronavirus is also known as the crowned virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The purpose of this paper is therefore to provide a decision making tool that will lighten the burden on medical staff, especially during pandemic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Related work  Since Corona was announced as a pandemic, different projects were carried out since 2020 to 2022 that it became among an interesting subject to learn from, some of the works related to image classification for different reasons are discussed in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The COVID-CT dataset of 2560 images was the database used in [3], 2214 of which were used for training and the remaining 246 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=" By employing WOA to optimize the network's hyperparameters, the model used to train ResNet-50 became the WOANet model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' This last experiment looked at the accuracy of the classification using the suggested method on 246 CT scans, and found that 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='37% of them were categorized as COVID-19, while 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='18% were identified as non-COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The radiologists will be greatly assisted by this proposed WOANet in reducing the burden on the healthcare system and hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=" In this study [4], patients' X-ray images are used to classify patients using CNN deep learning." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' One of the most powerful algorithms with generative and deterministic capabilities is the capsule network (CapsNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' However, compared to the basic CNN structures, this network has been relatively more responsive to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The dataset utilized was the NIH complete Chest X-rays [5] collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=" VDSNet has a validation accuracy value of 73%, which is higher than the sample dataset's score of 70." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Using a dataset of 6432 images, the DLH COVID [6] model is distinct, trustworthy, and independently created without any input from the transfer learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The experimental findings from the prospective validation phase suggest that the DLH MODEL outperformed the majority of the pre-trained models since it distinguished COVID-19, pneumonia, and healthy/unhealthy patients from the image dataset with a promising accuracy of 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Dataset  As seen in figure 1, a database of chest X-ray images for COVID-19 positive cases as well as images of normal and viral pneumonia was created in collaboration with medical professionals by a group of researchers from Qatar University, Doha, Qatar, and the University of Dhaka, Bangladesh, as well as their collaborators from Pakistan and Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' This dataset contains 3616 COVID-19 positive cases, 10,192 Normal, 6012 Lung Opacity (Non-COVID lung infection), and 1345 Viral Pneumonia images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The COVID-19 x-ray image database was created using different sources [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 1 CXR scans with four categories of pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  COVID  Lung_Opacity  Normal  ViralPneumonia  For training of datasets, we employed Matlab 2021 installed on computer with 64-bit operating system, windows 10 Pro, 24 GB of Random Access Memory (RAM), with an Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='40GHz and Graphical Processing Unit (GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Eighty percent of the datasets are used for training and 20% for testing (evaluating the model performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Proposed Model  In this study, we analyzed the different techniques for image classification of COVID-19 using X-Ray radiographic images of the chest, then examined CNN’s architecture that is based on research on the visual cortex of the cat by Hubel and Wisiel [10], and different pre-trained models: AlexNet, ResNet18 and GoogleNet in order to see the variation of answers in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 2 The proposed model with different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  The use of deep learning methods not only allows us to process a very large number of images but at the same time allows us to skip the feature extraction step, such a cumbersome step because it is done by hand crucially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Due to their capacity to extract features (see figure 2) and learn to distinguish between various classes, convolutional neural networks (CNNs) are the top DL tool that are widely employed in several fields of the healthcare system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', positive and negative, infected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Transfer learning (TL) has made it simpler to quickly and accurately retrain neural networks on chosen datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Experiments and Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='1 Experiment 1: Application of the CNN model  The structure of our CNN includes a number of layers, as shown in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' CNN receives a CXR image with a size of 299 by 299 pixels as its input, and the rest of the architecture is mentioned in the table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Model: CNN/AlexNet/GoogleNet/ResNet 18  Fully  Convolution  Connected  Pooling  Output  Input  O  O  O  299 299 1  4 classes  FeatureExtraction  Classification  Table 1 The architecture of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Name Type Description of output size Input layer Input data 299*299 Conv 1 Convolution +ReLU 32*32*8 S1 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Conv 2 Convolution +ReLU 64*64*3 S2 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Conv 3 Convolution +ReLU 128*128*5 S3 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Conv 4 Convolution +ReLU 256*256*5 S4 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Conv 5 Convolution +ReLU 512*512*5 S5 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Conv 6 Convolution +ReLU 1024*1024*5 S6 Max pooling 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Fc Fully connected 1 Fc (4)  The trained parameters used in this model are in the options side where all the hyperparameters used were defined including the number of epochs used (1 or 5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' the mini batch (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='001 and frequency validation is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The given CNN was trained using different parameters to test the accuracy for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' We utilized the accuracy parameter to evaluate how well the trained models performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The percentage of correctly classified images over all the images is what is referred to as accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The following formula is utilized: 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃 + 𝑇𝑁 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃 + 𝐹𝑁  Table 2 The results of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The CNN model  1 epoch  5 epochs  Accuracy  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='61%  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='13%  Time GPU  execution  146 min 58s  703 min 16s  The first top accuracy after training the model using one epoch provided us with 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='61% accuracy for our 4 classes classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' As training the model using only one epoch did not provide the best result, we had to increase the number of epochs and see the performance of our model and the results for our model gave us 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='13%, which is a lot better compared to our first experiment with one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='2 Experiment 2: Application of the pretrained models  AlexNet: With 5 convolutional layers and convolutional filter sizes of 3*3 and 2*2 for max pooling operation, AlexNet is an 8-layer convolutional neural network [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Fully connected layers are the final three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=" The AlexNet model's standard input size is 227*227*3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' GoogleNet (Inception v3): A convolutional neural network with 50 layers in depth is called GoogleNet [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The program, titled "Going deeper with convolutions," was developed and taught by Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Up to 1000 objects can be classified using the pre-trained Inceptionv3 model with the ImageNet dataset [13] weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=" This network's image input size was 299x299 pixels." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' ResNet18: A convolutional neural network with 18 layers in depth is called ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Deep Residual Learning for Image Recognition, as it is known, was developed and trained by Microsoft in 2015 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' To address the issue of vanishing gradient that may affect the weightage change in neural networks, ResNet architectures introduced the use of residual layers and skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' This made training easier and allowed neural networks to get much deeper with greater performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The network was trained on colored images with a resolution of 224x224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  In addition to the accuracy parameters, we estimated the time GPU execution for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The results obtained are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Table 3 The results of the pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Pretrained models  AlexNet  GoogleNet  ResNet18  Accuracy  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='93%  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='87%  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='1%  Time  GPU  execution  14 min  58s  41 min  34s  33 min  13s  Confusion matrix is the common approach used for evaluation of model performance based on true positive (TP), true negative (TN), false positive (FP), and false negative (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  The figure 3 represents the confusion matrix of the Resnet 18 model which gave the best result in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 3 The confusion matrix for the RestNet 18 model with the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' When the same dataset was used using the same hyper parameters the accuracy found was 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' Note that the pretrained models used only one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' The first method involved artificially enlarging the dataset with the aid of a label-preserving transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' The result was the generation of image translations and horizontal reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The second strategy was "dropout," which involves removing neurons that do not participate in the forward pass or the backward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' Conclusion This work aimed at developing a convolutional neural network (CNN) model that will help classify COVID-19 and non-COVID 19 disease such as viral pneumonia cases using chest X-ray images in the period caused by the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The model used in this work was CNN as well as pre-trained models including AlexNet, GoogleNet, and ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' The CNN gave a result with 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='13% accuracy for classifying the four classes after training 80% of the dataset and testing on 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' This motivated us not only to keep changing settings, but also to work on pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' In the latter, the pre-trained models were used on the same dataset but with just one epoch for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' And the results were 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='93%, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='87% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='1 % for AlexNet, GoogleNet and ResNet18 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' References [1] Kutlu, Yakup, and Yunus Camgözlü.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' "Detection of coronavirus disease (COVID-19) from X-ray images using deep convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='" Natural and Engineering Sciences 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' Chaos, Solitons & Fractals, 140, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='110120 [3] Murugan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' WOANet: Whale optimized deep neural network for the classification of COVID-19 from radiography images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Biocybernetics and Biomedical Engineering, 41(4), 1702-1718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' [4] Apostolopoulos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Covid-19: automatic detection from X-ray images utilizing transfer learning with convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Phys Eng Sci Med 43, 635–640 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' [5] Patel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Chest X-ray (COVID-19 & Pneumonia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content='com/prashant268/chest-xray-covid19- pneumonia [6] CDey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', Bacellar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', Chandrappa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', & Kulkarni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' COVID-19 Chest X-Ray Image Classification Using Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' medRxiv 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content='21260605 [7] https://bimcv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='cipf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
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+page_content='com/ml-workgroup/covid-19-image-repository/tree/master/png [9] https://sirm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='org/category/senza-categoria/covid-19/ [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Hubel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Wiesel, Receptive fields and functional architecture of monkey striate cortex, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Physiol, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 195, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 215-243, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='  [11] Li, Shaojuan, Lizhi Wang, Jia Li, and Yuan Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' "Image classification algorithm based on improved AlexNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='" In Journal of Physics: Conference Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 1813, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 012051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' IOP Publishing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' [12] Hu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', Shen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' and Sun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Squeeze-and-excitation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 7132-7141).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' [13] Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' "Imagenet classification with deep convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content='" Advances in neural information processing systems 25 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' [14] Huang L, Ruan S, Denoeux T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' Covid-19 classification with deep neural network and belief functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' InThe Fifth International Conference on Biological Information and Biomedical Engineering 2021 Jul 20 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}
+page_content=' 1-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfkAHC/content/2301.02468v1.pdf'}

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+Heterogeneous Beliefs and Multi-Population Learning in
+Network Games
+Shuyue Hu1, Harold Soh2, and Georgios Piliouras3
+1Shanghai Artificial Intelligence Laboratory
+2National University of Singapore
+3Singapore University of Technology and Design
+Abstract
+The effect of population heterogeneity in multi-agent learning is practically relevant but remains
+far from being well-understood. Motivated by this, we introduce a model of multi-population learning
+that allows for heterogeneous beliefs within each population and where agents respond to their
+beliefs via smooth fictitious play (SFP). We show that the system state — a probability distribution
+over beliefs — evolves according to a system of partial differential equations. We establish the
+convergence of SFP to Quantal Response Equilibria in different classes of games capturing both
+network competition as well as network coordination. We also prove that the beliefs will eventually
+homogenize in all network games. Although the initial belief heterogeneity disappears in the limit,
+we show that it plays a crucial role for equilibrium selection in the case of coordination games as it
+helps select highly desirable equilibria. Contrary, in the case of network competition, the resulting
+limit behavior is independent of the initialization of beliefs, even when the underlying game has many
+distinct Nash equilibria.
+1
+Introduction
+Smooth Fictitious play (SFP) and variants thereof are arguably amongst the most well-studied learning
+models in AI and game theory [2, 3, 21, 22, 9, 19, 36, 37, 42, 18, 17]. SFP describes a belief-based learning
+process: agents form beliefs about the play of opponents and update their beliefs based on observations.
+Informally, an agent’s belief can be thought as reflecting how likely its opponents will play each strategy.
+During game plays, each agent plays smoothed best responses to its beliefs. Much of the literature of
+SFP is framed in the context of homogeneous beliefs models where all agents in a given role have the
+same beliefs. This includes models with one agent in each player role [3, 2, 39] as well as models with a
+single population but in which all agents have the same beliefs [21, 22]. SFP are known to converge in
+large classes of homogeneous beliefs models (e.g., most 2-player games [9, 19, 3]). However, in the context
+of heterogeneous beliefs, where agents in a population have different beliefs, SFP has been explored to a
+less extent.
+The study of heterogeneous beliefs (or more broadly speaking, population heterogeneity) is important
+and practically relevant. From multi-agent system perspective, heterogeneous beliefs widely exist in
+many applications, such as traffic management, online trading and video game playing. For example,
+it is natural to expect that public opinions generally diverge on autonomous vehicles and that people
+have different beliefs about the behaviors of taxi drivers vs non-professional drivers. From machine
+learning perspective, recent empirical advances hint that injecting heterogeneity potentially accelerates
+population-based training of neural networks and improves learning performance [25, 29, 44]. From game
+theory perspective, considering heterogeneity of beliefs better explains results of some human experiments
+[10, 11].
+Heterogeneous beliefs models of SFP are not entirely new. In the pioneering work [12], Fudenberg
+and Takahashi examine the heterogeneity issue in 2-population settings by appealing to techniques from
+the stochastic approximation theory. This approach, which is typical in the SFP literature, relates the
+limit behavior of each individual to an ordinary differential equation (ODE) and has yielded significant
+insights for many homogeneous beliefs models [3, 2, 19, 39]. However, this approach, as also noted by
+Fudenberg and Takahashi, “does not provide very precise estimates of the effect of the initial condition of
+the system.” Consider an example of a population of agents each can choose between two pure strategies
+1
+arXiv:2301.04929v1  [cs.MA]  12 Jan 2023
+
+s1 and s2. Let us imagine two cases: (i) every agents in the population share the same belief that their
+opponents play a mixed strategy choosing s1 and s2 with equal probability 0.5, and (ii) half of the agents
+believe that their opponents determinedly play the pure strategy s1 and the other half believe that
+their opponents determinedly play the pure strategy s2. The stochastic approximation approach would
+generally treat these two cases equally, providing little information about the heterogeneity in beliefs as
+well as its consequential effects on the system evolution. This drives our motivating questions:
+How does heterogeneous populations evolve under SFP? How much and under what conditions does
+the heterogeneity in beliefs affect their long-term behaviors?
+Model and Solutions. In this paper, we study the dynamics of SFP in general classes of multi-
+population network games that allow for heterogeneous beliefs. In a multi-population network game, each
+vertex of the network represents a population (continuum) of agents, and each edge represents a series
+of 2-player subgames between two neighboring populations. Note that multi-population network games
+include all the 2-population games considered in [12] and are representation of subclasses of real-world
+systems where the graph structure is evident [?]. We consider that for a certain population, individual
+agents form separate beliefs about each neighbor population and observe the mean strategy play of
+that population. Taking a approach different from stochastic approximation, we define the system state
+as a probability measure over the space of beliefs, which allows us to precisely examine the impact of
+heterogeneous beliefs on system evolution. This probability measure changes over time in response to
+agents’ learning. Thus, the main challenge is to analyze the evolution of the measure, which in general
+requires the development of new techniques.
+As a starting point, we establish a system of partial differential equations (PDEs) to track the
+evolution of the measure in continuous time limit (Proposition 1). The PDEs that we derive are akin
+to the continuity equations1 commonly encountered in physics and do not allow for a general solution.
+Appealing to moment closure approximation [13], we circumvent the need of solving the PDEs and
+directly analyze the dynamics of the mean and variance (Proposition 2 and Theorem 1). As one of our
+key results, we prove that the variance of beliefs always decays quadratically fast with time in all network
+games (Theorem 1). Put differently, eventually, beliefs will homogenize and the distribution of beliefs
+will collapse to a single point, regardless of initial distributions of beliefs, 2-player subgames that agents
+play, and the number of populations and strategies. This result is non-trivial and perhaps somewhat
+counterintuitive. Afterall, one may find it more natural to expect that the distribution of beliefs would
+converge to some distribution rather than a single point, as evidenced by recent studies on Q-learning
+and Cross learning [23, 24, 27].
+Technically, the eventual belief homogenization has a significant implication — it informally hints
+that the asymptotic system state of initially heterogeneous systems are likely to be the same as in
+homogeneous systems. We show that the fixed point of SFP correspond to Quantual Response Equilibria
+(QRE)2 in network games for both homogeneous and initially heterogeneous systems (Theorem 2). As
+our main result, we establish the convergence of SFP to QRE in different classes of games capturing both
+network competition as well as network coordination, independent of belief initialization. Specifically,
+for competitive network games, we first prove via a Lyapunov argument that the SFP converges to a
+unique QRE in homogeneous systems, even when the underlying game has many distinct Nash equilibria
+(Theorem 3). Then, we show that this convergence result can be carried over to initially heterogeneous
+systems (Theorem 4), by leveraging that the mean belief dynamics of initially heterogeneous systems is
+asymptotically autonomous [31] with its limit dynamics being the belief dynamics of a homogeneous system
+(Lemma 7). For coordination network games, we also prove the convergence to QRE for homogeneous
+and initially heterogeneous systems, in which the underlying network has star structure (Theorem 5).
+On the other hand, the eventual belief homogenization may lead to a misconception that belief
+heterogeneity has little effect on system evolution. Using an example of 2-population stag hunt games,
+we show that belief heterogeneity actually plays a crucial role in equilibrium selection, even though it
+eventually vanishes. As shown in Figure 1, changing the variance of initial beliefs results in different limit
+behaviors, even when the mean of initial beliefs remains unchanged; in particular, while a small variance
+leads to the less desirable equilibrium pH, Hq, a large variance leads to the payoff dominant equilibrium
+pS, Sq. Thus, in the case of network coordination, initial belief heterogeneity can help select the highly
+desirable equilibrium and provides interesting insights to the seminal thorny problem of equilibrium
+selection [26]. On the contrary, in the case of network competition, we prove (Theorems 3 and 4 on the
+1The continuity equation is a PDE that describes the transport phenomena of some quantity (e.g., mass, energy,
+momentum and other conserved quantities) in a physical system.
+2QRE is a game theoretic solution concept under bounded rationality. By QRE, in this paper we refer to their canonical
+form also referred to as logit equilibria or logit QRE in the literature [14].
+2
+
+Figure 1: The system dynamics under the effects of different variances of initial beliefs (thin lines:
+predictions of our PDE model, shaded wide lines: simulation results). ¯µ2S represents the mean belief
+about population 2 and ¯x1S represents the mean probability of playing strategy S in population 1. Initially,
+we set the mean beliefs ¯µ2S “ ¯µ1S “ 0.3 (details of the setup are summarized in the supplementary).
+Given the same initial mean belief, different initial variances σ2pµ2Sq lead to the convergence to different
+beliefs (the left panel) and even to different strategy choices (the right panel). In particular, a large initial
+variance helps select the payoff dominant equilibrium pS, Sq in stag hunt games.
+convergence to a unique QRE in competitive network games) as well as showcase experimentally that the
+resulting limit behavior is independent of initialization of beliefs, even if the underlying game has many
+distinct Nash equilibria.
+Related Works.
+SFP and its variants have recently attracted a lot of attention in AI research
+[36, 37, 42, 18, 17]. There is a significant literature that analyze SFP in different models [3, 7, 21, 19], and
+the paper that is most closely related to our work is [12]. Fudenberg and Takahashi [12] also examines the
+heterogeneity issue and anticipate belief homogenization in the limit under 2-population settings. In this
+paper, we consider multi-population network games, which is a generalization of their setting.3 Moreover,
+our approach is more fundamental, as the PDEs that we derive can provide much richer information
+about the system evolution and thus precisely estimates the temporal effects of heterogeneity, which is
+generally intractable in [12]. Therefore, using our approach, we are able to show an interesting finding —
+the initial heterogeneity plays a crucial role in equilibrium selection (Figure 1) — which unfortunately
+cannot be shown using the approach in [12]. Last but not least, to our knowledge, our paper is the first
+work that presents a systematic study of smooth fictitious play in general classes of network games.
+On the other hand, networked multi-agent learning constitutes one of the current frontiers in AI and
+ML research [43, 30, 16]. Recent theoretical advances on network games provide conditions for learning
+behaviors to be not chaotic [6, 34], and investigate the convergence of Q-learning and continuous-time FP
+in the case of network competitions [7, 28]. However, [7, 28] consider that there is only one agent on each
+vertex, and hence their models are essentially for homogeneous systems.
+Lahkar and Seymour [27] and Hu et al. [23, 24] also use the continuity equations as a tool to study
+population heterogeneity in multi-agent systems where a single population of agents applies Cross learning
+or Q-learning to play symmetric games. They either prove or numerically showcase that heterogeneity
+generally persists. Our results complement these advances by showing that heterogeneity vanishes under
+SFP and that heterogeneity helps select highly desirable equilibria. Moreover, methodologically, we
+establish new proof techniques for the convergence of learning dynamics in heterogeneous systems by
+leveraging seminal results (Lemmas 1 and 2) from the asymptotically autonomous dynamical system
+literature, which may be of independent interest.
+3The analysis presented in this paper covers all generic 2-population network games, all generic bipartite network games
+where the game played on each edge is the same along all edges, and all weighted zero-sum games which do not require the
+graph to be bipartite nor to have the same game played on each edge.
+3
+
+Small vs. Large Variance of Initial Population Beliefs
+Mean Belief about the Others' Playing S
+Mean Probability of Playing S
+1S
+0.8
+0.8
+0.6
+Mean Prob.
+0.6
+)~ 0.023
+(μ2s) ~ 0.023
+0.A
+α(μ2s)
+= 0.1
+0.4
+g(μ2s)
+= 0.1
+0.2
+0.2
+50
+100
+150
+200
+50
+100
+150
+200
+Time t
+Time t2
+Preliminaries
+Population Network Games.
+A population network game (PNG) Γ “ pN, pV, Eq, pSi, ωiq@iPV , pAijqpi,jqPEq
+consists of a multi-agent system N distributed over a graph pV, Eq, where V “ t1, ..., nu is the set of
+vertices each represents a population (continuum) of agents, and E is the set of pairs, pi, jq, of population
+i ‰ j P V . For each population i P V , agents of this population has a finite set Si of pure strategies (or
+actions) with generic elements si P Si. Agents may also use mixed strategies (or choice distributions).
+For an arbitrary agent k in population i, its mixed strategy is a vector xipkq P ∆i, where ∆i is the
+simplex in R|Si| such that ř
+siPSi xisipkq “ 1 and xisipkq ě 0, @si P Si. Each edge pi, jq P E defines
+a series of two-player subgames between populations i and j, such that for a given time step, each
+agent in population i is randomly paired up with another agent in population j to play a two-player
+subgame. We denote the payoff matrices for agents of population i and j in these two-player subgames by
+Aij P R|Si|ˆ|Sj| and Aji P R|Sj|ˆ|Si|, respectively. Note that at a given time step, each agent chooses a
+(mixed or pure) strategy and plays that strategy in all two-player subgames. Let x “ pxi, txjupi,jqPEq be
+a mixed strategy profile, where xi (or xj) denotes a generic mixed strategy in population i (or j). Given
+the mixed strategy profile x, the expected payoff of using xi in the game Γ is
+ripxq “ ripxi, txjupi,jqPEq :“
+ÿ
+pi,jqPE
+xJ
+i Aijxj.
+(1)
+The game Γ is competitive (or weighted zero-sum), if there exist positive constants ω1, . . . , ωn such that
+ÿ
+iPV
+ωiripxq “
+ÿ
+pi,jqPE
+`
+ωixJ
+i Aijxj ` ωjxJ
+j Ajixi
+˘
+“ 0,
+@x P
+ź
+iPV
+∆i.
+(2)
+On the other hand, Γ is a coordination network game, if for each edge pi, jq P E, the payoff matrices of
+the two-player subgame satisfy Aij “ AJ
+ji.
+Smooth Fictitious Play.
+SFP is a belief-based model for learning in games. In SFP, agents form
+beliefs about the play of opponents and respond to the beliefs via smooth best responses. Given a
+game Γ, consider an arbitrary agent k in a population i P V . Let Vi “ tj P V : pi, jq P Eu be the set
+of neighbor populations. Agent k maintains a weight κi
+jsjpkq for each opponent strategy sj P Sj of a
+neighbor population j P Vi. Based on the weights, agent k forms a belief about the neighbor population
+j, such that each opponent strategy sj is played with probability
+µi
+jsjpkq “
+κi
+jsjpkq
+ř
+s1
+jPSj κi
+js1
+jpkq.
+(3)
+Let µi
+jpkq be the vector of beliefs with the sj-th element equals µi
+jsjpkq. Agent k forms separate beliefs
+for each neighbor population, and plays a smooth best response to the set of beliefs tµi
+jpkqujPVi. Given a
+game Γ, agent k’s expected payoff for using a pure strategy si P Si is
+uisipkq “ ripesi, tµi
+jpk, tqujPViq “
+ÿ
+jPVi
+eJ
+siAijµi
+jpkq
+(4)
+where esi is a unit vector where the si-th element is 1. The probability of playing strategy si is then
+given by
+xisipkq “
+exppβuisipkqq
+ř
+s1
+iPSi exppβuis1
+ipkqq
+(5)
+where β is a temperature (or the degree of rationality). We consider that agents observe the mean mixed
+strategy of each neighbor population. As such, at a given time step t, agent k updates the weights for
+each opponent strategy sj P Sj, j P Vi as follows:
+κi
+jsjpk, t ` 1q “ κi
+jsjpk, tq ` ¯xjsjptq
+(6)
+where ¯xjsj is the mean probability of playing strategy sj in population j, i.e., ¯xjsj “
+1
+nj
+ř
+lPpopulation j xjsjplq
+with the number of agents denoted by nj.
+For simplicity, we assume the initial sum of weights
+ř
+sjPSj κi
+jsjpk, 0q to be the same for every agent in the system N and denote this initial sum by λ.
+Observe that Equation 6 can be rewritten as
+pλ ` t ` 1qµi
+jsjpk, t ` 1q “ pλ ` tqµi
+jsjpk, tq ` ¯xjsjptq.
+(7)
+4
+
+Hence, even though agent k directly updates the weights, its individual state can be characterized by the
+set of beliefs tµi
+jpkqujPVi. In the following, we usually drop the time index t and agent index k in the
+bracket (depending on the context) for notational convenience.
+3
+Belief Dynamics in Population Network Games
+Observe that for an arbitrary agent k, its belief µi
+jpkq is in the simplex ∆j “ tµi
+jpkq P R|Sj|| ř
+sjPSj µi
+jsjpkq “
+1, µi
+jsjpkq ě 0, @sj P Sju. We assume that the system state is characterized by a Borel probability measure
+P defined on the state space ∆ “ ś
+iPV ∆i. Given µi P ∆i, we write the marginal probability density func-
+tion as ppµi, tq. Note that ppµi, tq is the density of agents having the belief µi about population i throughout
+the system. Define µ “ tµiuiPV P ∆. Since agents maintain separate beliefs about different neighbor
+populations, the joint probability density function ppµ, tq can be factorized, i.e., ppµ, tq “ ś
+iPV ppµi, tq.
+We make the following assumption for the initial marginal density functions.
+Assumption 1. At time t “ 0, for each population i P V , the marginal density function ppµi, tq is
+continuously differentiable and has zero mass at the boundary of the simplex ∆i.
+This assumption is standard and common for a “nice” probability distribution. Under this mild
+condition, we determine the evolution of the system state P with the following proposition, using the
+techniques similar to those in [27, 23].
+Proposition 1 (Population Belief Dynamics). The continuous-time dynamics of the marginal density
+function ppµi, tq for each population i P V is governed by a partial differential equation
+´Bppµi, tq
+Bt
+“ ∇ ¨
+ˆ
+ppµi, tq ¯xi ´ µi
+λ ` t ` 1
+˙
+(8)
+where ∇¨ is the divergence operator and ¯xi is the mean mixed strategy with each si-th element
+¯xisi “
+ż
+ś
+jPVi ∆j
+exp pβuisiq
+ř
+s1
+iPSi exp pβuis1
+iq
+ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+(9)
+where uisi “ ř
+jPVi eJ
+siAijµj.
+For every marginal density function ppµi, tq, the total mass is always conserved (Corollary 1 of the
+supplementary); moreover, the mass at the boundary of the simplex ∆i always remains zero, indicating
+that agents’ beliefs will never go to extremes (Corollary 2 of the supplementary).
+Generalizing the notion of a system state to a distribution over beliefs allows us to address a very
+specific question — the impact of belief heterogeneity on system evolution. That said, partial differential
+equations (Equation 8) are notoriously difficult to solve. Here we resort to the evolution of moments
+based on the evolution of the distribution (Equation 8). In the following proposition, we show that the
+characterization of belief heterogeneity is important, as the dynamics of the mean system state (or the
+mean belief dynamics) is indeed affected by belief heterogeneity.
+Proposition 2 (Mean Belief Dynamics). The dynamics of the mean belief ¯µi about each population
+i P V is governed by a system of differential equations such that for each strategy si,
+d¯µisi
+dt
+« fsiptµjujPViq ´ ¯µisi
+λ ` t ` 1
+`
+ř
+jPVi
+ř
+sjPSj
+B2fsiptµjujPViq
+pBµjsj q2
+Varpµjsjq
+2pλ ` t ` 1q
+.
+(10)
+where fsiptµujPViq is the logit choice function (Equation 5) applied to strategy si P Si, and Varpµjsjq is
+the variance of belief µjsj in the entire system.
+In general, the mean belief dynamics is under the joint effects of the mean, variance, and infinitely
+many higher moments of the belief distribution. To allow for more conclusive results, we apply the
+moment closure approximation4 and assume the effects of the third and higher moments to be negligible.
+Now, just for a moment, suppose that the system beliefs are homogeneous —- the beliefs of every
+individuals are the same. Hence, the mean belief dynamics are effectively the belief dynamics of individuals.
+The following proposition follows from Equation 7.
+4Moment closure is a typical approximation method used to estimate moments of population models [13, 15, 32]. To use
+moment closure, a level is chosen past which all cumulants are set to zero. The conventional choice of the level is 2, i.e.,
+setting the third and higher cumulants to be zero.
+5
+
+Proposition 3 (Belief Dynamics for Homogeneous Populations). For a homogeneous system, the
+dynamics of the belief µi about each population i P V is governed by a system of differential equations
+such that for each strategy si,
+dµisi
+dt
+“ xisi ´ µisi
+λ ` t ` 1 “ fsiptµjujPViq ´ µisi
+λ ` t ` 1
+(11)
+where µisi is the same for all agents in each neighbor population j P Vi.
+Intuitively, the mean belief dynamics indicates the trend of beliefs in a system, and the variance of
+beliefs indicates belief heterogeneity. Contrasting Propositions 2 and 3, it is clear that the variance of
+belief (belief heterogeneity) plays a role in determining the mean belief dynamics (the trend of beliefs) for
+heterogeneous systems. It is then natural to ask: how does the belief heterogeneity evolve over time?
+How much does the belief heterogeneity affect the trend of beliefs? Our investigation to these questions
+reveals an interesting finding — the variance of beliefs asymptotically tends to zero.
+Theorem 1 (Quadratic Decay of the Variance of Population Beliefs). The dynamics of the
+variance of beliefs µi about each population i P V is governed by a system of differential equations such
+that for each strategy si,
+dVarpµisiq
+dt
+“ ´2Varpµisiq
+λ ` t ` 1 .
+(12)
+At given time t, Varpµisiq “
+´
+λ`1
+λ`t`1
+¯2
+σ2pµisiq, where σ2pµisiq is the initial variance. Thus, the variance
+Varpµisiq decays to zero quadratically fast with time.
+Such quadratic decay of the variance stands no matter what 2-player subgames agents play and what
+initial conditions are. Put differently, the beliefs will eventually homogenize for all population network
+games. This fact immediately implies the system state in the limit.
+Corollary 1. As time t Ñ 8, the density function ppµi, tq for each population i P V evolves into a Dirac
+delta function, and the variance of the choice distributions within each population i P V also goes to zero.
+Note that while the choice distributions will homogenize within each population, they are not necessarily
+the same across different populations. This is because the strategy choice of each population is in response
+to its own set of neighbor populations (which are generally different).
+4
+Convergence of Smooth Fictitious Play in Population Network
+Games
+The finding on belief homogenization is non-trivial and also technically important. One implication is
+that the fixed points of systems with initially heterogeneous beliefs are the same as in systems with
+homogeneous beliefs. Thus, it follows from the belief dynamics for homogeneous systems (Proposition 3)
+that the fixed points of systems have the following property.
+Theorem 2 (Fixed Points of System Dynamics). For any system that initially have homogeneous
+or heterogeneous beliefs, the fixed points of the system dynamics is a pair pµ˚, x˚q that satisfy x˚
+i “ µ˚
+i
+for each population i P V and are the solutions of the system of equations
+x˚
+isi “
+exp
+´
+β ř
+jPVi eJ
+siAijx˚
+j
+¯
+ř
+s1
+iPSi exp
+´
+β ř
+jPVi eJ
+s1
+iAijx˚
+j
+¯
+(13)
+for every strategy si P Si and population i P V . Such fixed points always exist and coincide with the
+Quantal Response Equilibria (QRE) [33] of the population network game Γ.
+Note that the above theorem applies for all population network games.
+We study the convergence of SFP to the QRE under the both cases of network competition and
+network coordination. Due to space limits, in the following, we mainly focus on network competition and
+present only the main result on network coordination.
+6
+
+4.1
+Network Competition
+Consider a competitive population network game Γ. Note that in competitive network games, the Nash
+equilibrium payoffs need not to be unique (which is in clear contrast to two-player settings), and it
+generally allows for infinitely many Nash equilibria. In the following theorem, focusing on homogeneous
+systems, we establish the convergence of the belief dynamics to a unique QRE, regardless of the number
+of Nash equilibria in the underlying game.
+Theorem 3 (Convergence in Homogeneous Network Competition). Given a competitive Γ, for
+any system that has homogeneous beliefs, the belief dynamics (Equation 11) converges to a unique QRE
+which is globally asymptotically stable.
+Proof of Sketch. We proof this theorem by showing that the “distance” between xi and µi is strictly
+decreasing until the QRE is reached. In particular, we measure the distance in terms of the perturbed
+payoff and construct a strict Lyapunov function
+L :“
+ÿ
+iPV
+ωi
+“
+πi
+`
+xi, tµjujPVi
+˘
+´ πi
+`
+µi, tµjujPVi
+˘‰
+(14)
+where ω1 . . . ωn are the positive weights given by Γ, and πi is a perturbed payoff function defined as
+πi
+`
+xi, tµjujPVi
+˘ :“ xJ
+i
+ř
+jPVi Aijµj ´ 1
+β
+ř
+siPSi xisi lnpxisiq.
+Next, we turn to systems with initially heterogeneous beliefs. Leveraging that the variance of beliefs
+eventually goes to zero, we establish the following lemma.
+Lemma 1. For a system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10) is
+asymptotically autonomous [31] with the limit equation dµi
+dt “ xi ´ µi, which after time-reparmeterization
+is equivalent to the belief dynamics for homogeneous systems (Equation 11).
+For ease of presentation, we follow the convention to denote the solution flows of an asymptotically
+autonomous system and its limit equation by φ and Θ, respectively. Thieme [40] provides the following
+seminal result that connects the limit behaviors of φ and Θ.
+Lemma 2 (Thieme [40] Theorem 4.2). Given a metric space pX, dq. Assume that the equilibria of Θ
+are isolated compact Θ-invariant subsets of X. The ω-Θ-limit set of any pre-compact Θ-orbit contains a
+Θ-equilibrium. The point ps, xq, s ě t0, x P X, have a pre-compact φ-orbit. Then the following alternative
+holds: 1) φpt, s, xq Ñ e, t Ñ 8, for some Θ-equilibrium e, and 2) the ω-φ-limit set of ps, xq contains
+finitely many Θ-equilibria which are chained to each other in a cyclic way.
+Combining the above results, we prove the convergence for initially heterogeneous systems.
+Theorem 4 (Convergence in Initially Heterogeneous Network Competition). Given a compet-
+itive Γ, for any system that initially has heterogeneous beliefs, the mean belief dynamics (Equation 10)
+converges to a unique QRE.
+The following corollary immediately follows as the result of belief homogenization.
+Corollary 2. For any competitive Γ, under smooth fictitious play, the choice distributions and beliefs of
+every individual converges to a unique QRE (given in Theorem 2), regardless of belief initialization and
+the number of Nash equilibria in Γ.
+4.2
+Network Coordination
+We delegate most of the results on coordination network games to the supplementary, and summarize
+only the main result here.
+Theorem 5 (Convergence in Network Coordination with Star Structure). Given a coordination
+Γ where the network structure consists of a single or disconnected multiple stars, each orbit of the belief
+dynamics (Equation 11) for homogeneous systems as well as each orbit of the mean belief dynamics
+(Equation 10) for initially heterogeneous systems converges to the set of QRE.
+Note that this theorem applies to all 2-population coordination games, as network games with or
+without star structure are essentially the same when there are only two vertices. We also remark that
+pure or mixed Nash equilibria in coordination network games are complex; as reported in recent works
+[5, 4, 1], finding a pure Nash equilibrium is PLS-complete. Hence, learning in the general case of network
+coordination is difficult and generally requires some conditions for theoretical analysis [34, 35].
+7
+
+H
+S
+H
+(1, 1)
+(2, 0)
+S
+(0, 2)
+(4, 4)
+Table 1: Stag Hunt.
+Figure 2: Asymmetric Matching Pennies.
+Figure 3: Belief heterogeneity helps select the payoff dominant equilibrium pS, Sq (yellow: the equilibrium
+pS, Sq, blue: the equilibrium pH, Hq). As the variance of initial beliefs increases (from the left to right
+panel), a larger range of initial mean beliefs will approximately reach the equilibrium pS, Sq in the limit.
+For each panel, the initial variances of two populations σ2pµ1Hq and σ2pµ2Hq are the same.
+5
+Experiments:
+Equilibrium Selection in Population Network
+Games
+In this section, we complement our theory and present an empirical study of SFP in a two-population
+coordination (stag hunt) game and a five-population zero-sum (asymmetric matching pennies) game.
+Importantly, these two games both have multiple Nash equilibria, which naturally raises the problem of
+equilibrium selection.
+5.1
+Two-Population Stag Hunt Games
+We have shown in Figure 1 (in the introduction) that given the same initial mean belief, changing the
+variances of initial beliefs can result in different limit behaviors. In the following, we systematically study
+the effect of initial belief heterogeneity by visualizing how it affects the regions of attraction to different
+equilibria.
+Game Description. We consider a two-population stag hunt game, where each player in populations
+1 and 2 has two actions tH, Su. As shown in the payoff bi-matrices (Table 1), there are two pure strategy
+Nash equilibria in this game: pH, Hq and pS, Sq. While pH, Hq is risk dominant, pS, Sq is indeed more
+desirable as it is payoff dominant as well as Pareto optimal.
+Results. In this game, population 1 forms beliefs about population 2 and vice versa. We denote
+the initial mean beliefs by a pair p¯µ2H, ¯µ1Hq. We numerically solve the mean belief dynamics for a large
+range of initial mean beliefs, given different variances of initial beliefs. In Figure 3, for each pair of initial
+mean beliefs, we color the corresponding data point based on which QRE the system eventually converges
+to. We observe that as the variance of initial beliefs increases (from the left to right panel), a larger range
+of initial mean beliefs results in the convergence to the QRE that approximates the payoff dominant
+equilibrium pS, Sq. Put differently, a higher degree of initial belief heterogeneity leads to a larger region
+of attraction to pS, Sq. Hence, belief heterogeneity eventually vanishes though, it provides an approach to
+equilibrium selection, as it helps select the highly desirable equilibrium.
+5.2
+Five-Population Asymmetric Matching Pennies Games
+We have shown in Corollary 2 that SFP converges to a unique QRE even if there are multiple Nash
+equilibria in a competitive Γ. In the following, we corroborate this by providing empirical evidence in
+8
+
++1
++1
++1
+1
+Population 1
+Population 2
+Population 3
+Population 4
+Population 5
+H
+(L H)
+{H, T}
+[H, T}
+Match
+Match
+Match
+MatchBasins of Attraction of (S, S) and (H, H) under Different Variances of Initial Beliefs
+g?(μ1H)
+= 0
+g2(μ1H) = 0.02
+g?(μH)
+= 0.05
+g2(μ1H) = 0.1
+(S,S)
+j12H
+H
+2H
+0.8
+0.8
+0.8
+0.8
+Initial
+Initial
+Initial
+Initial
+0.6
+0.6
+0.6
+0.6
+(H,H)
+0.6
+0.8
+1
+0.6
+0.8
+1
+0.6
+0.8
+1
+0.6
+0.8
+1
+Initial jiH
+Initial jiiH
+Initial jiH
+InitialjiiHFigure 4: With different belief initialization, SFP selects a unique equilibrium where all agents in
+population 3 play strategy H with probability 0.5. We run 100 simulation runs for each initialization. The
+thin lines represent the mean mixed strategy (the choice probability of H) and the shaded areas represent
+the variance of the mixed strategies in the population. In the legends, B denotes Beta distribution;
+the two Beta distributions correspond to the initial beliefs about the neighbor populations 2 and 4,
+respectively.
+agent-based simulations with different belief initialization (the details of simulations are summarized in
+the supplementary).
+Game Description. Consider a five-population asymmetric matching pennies game [28], where
+the network structure is a line (depicted in Figure 2). Each agent has two actions tH, Tu. Agents in
+populations 1 and 5 do not learn; they always play strategies H and T, respectively. For agents in
+populations 2 to 4, they receive `1 if they match the strategy of the opponent in the next population,
+and receive ´1 if they mismatch. On the contrary, they receive `1 if they mismatch the strategy of the
+opponent in the previous population, and receive ´1 if they match. Hence, this game has infinitely many
+Nash equilibria of the form: agents in populations 2 and 4 play strategy T, whereas agents in population
+3 are indifferent between strategies H and T.
+Results. In this game, agents in each population form two beliefs (one for the previous population
+and one for the next population). We are mainly interested in the strategies of population 3, as the
+Nash equilibria differ in the strategies in population 3. For validation, we vary population 3’s beliefs
+about the neighbor populations 2 and 4, and fix population 3’s beliefs about the other populations. As
+shown in Figure 4, given differential initialization of beliefs, agents in population 3 converge to the same
+equilibrium where they all take strategy H with probability 0.5. Therefore, even when the underlying
+zero-sum game has many Nash equilibria, SFP with different initial belief heterogeneity selects a unique
+equilibria, addressing the problem of equilibrium selection.
+6
+Conclusions
+We study a heterogeneous beliefs model of SFP in network games. Representing the system state with a
+distribution over beliefs, we prove that beliefs eventually become homogeneous in all network games. We
+establish the convergence of SFP to Quantal Response Equilibria in general competitive network games
+as well as coordination network games with star structure. We experimentally show that although the
+initial belief heterogeneity vanishes in the limit, it plays a crucial role in equilibrium selection and helps
+select highly desirable equilibria.
+Appendix A: Corollaries and Proofs omitted in Section 3
+Proof of Proposition 1
+It follows from Equation 7 in the main paper that the change in µi
+jpk, tq between two discrete time steps
+is
+µi
+jpk, t ` 1q “ µi
+jpk, tq `
+¯xjptq ´ µi
+jpk, tq
+λ ` t ` 1
+.
+(15)
+9
+
+Probability of Playing H in Population 3
+Vary the inital Mean, Fix the initial Variance
+Vary the initial Variance, Fix the initial Mean
+B(5, 10), B(8, 2)
+B(10, 20), B(16, 4)
+B(5, 10), B(2, 8)
+B(2.5, 5), B(4,1)
+B(10, 5), B(8, 2)
+B(50, 100),B(80,20)
+0.5
+0
+0
+100
+200
+300
+400
+100
+200
+300
+400
+Time t
+Time tLemma 3. Under Assumption 1 (in the main paper), for an arbitrary agent k in population i, its belief
+µi
+jpk, tq about a neighbor population j will never reach the extreme belief (i.e., the boundary of the simplex
+∆i).
+Proof. Assumption 1 ensures that ¯xjp0q is in the interior of the simplex ∆j. Moreover, the logit choice
+function (Equation 5 in the main paper) also ensures that ¯xjptq stays in the interior of ∆j afterwards for
+a finite temperature β. Hence, from Equation 15, one can see that µi
+jpk, tq for every time step t will stay
+in the interior of ∆j.
+In the following, for notation convenience, we sometimes drop the agent index k and the time index
+t depending on the context. Consider a population i. We rewrite the change in the beliefs about this
+population as follows.
+µipt ` 1q “ µiptq ` ¯xiptq ´ µiptq
+λ ` t ` 1
+.
+(16)
+Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s. We
+rewrite the above equation as
+µipt ` δq “ µiptq ` δ ¯xiptq ´ µiptq
+λ ` t ` 1
+.
+(17)
+Next, we consider a test function θpµiq. Define
+Y “ Erθpµipt ` δqqs ´ Erθpµiptqqs
+δ
+.
+(18)
+Applying Taylor series for θpµipt ` δqq at µiptq, we obtain
+θpµipt ` δqq “ θpµiptqq `
+δ
+λ ` t ` 1Bµiθpµiq r¯xiptq ´ µiptqs
+`
+δ2
+2pλ ` t ` 1q2 r¯xiptq ´ µiptqsJ Hθpµiq r¯xiptq ´ µiptqs
+` o
+˜„
+δ ¯xiptq ´ µiptq
+λ ` t ` 1
+ȷ2¸
+(19)
+where H denotes the Hessian matrix. Hence, the expectation Erθpµipt ` δqqs is
+Erθpµipt ` δqqs “ Erθpµiptqqs `
+δ
+λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs
+`
+δ2
+2pλ ` t ` 1q2 E
+“
+r¯xiptq ´ µiptqsJHθpµiq r¯xiptq ´ µiptqs
+‰
+`
+δ2
+2pλ ` t ` 1q2 Eropr¯xiptq ´ µiptqs2qs
+(20)
+Moving the term Erθpµiptqqs to the left hand side and dividing both sides by δ, we recover the quantity
+Y , i.e.,
+Y “
+1
+λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs
+`
+δ
+2pλ ` t ` 1q2 Err¯xiptq ´ µiptqsJHθpµiptqqr¯xiptq ´ µiptqs ` o
+`
+p¯xiptq ´ µiptqq2˘
+s
+(21)
+Taking the limit of Y with δ Ñ 0, the contribution of the second term on the right hand side vanishes,
+yielding
+lim
+δÑ0 Y “
+1
+λ ` t ` 1ErBµiθpµiptqqp¯xiptq ´ µiptqqs
+(22)
+“
+1
+λ ` t ` 1
+ż
+ppµiptq, tq
+“
+Bµiθpµiptqqp¯xiptq ´ µiptqq
+‰
+dµiptq.
+(23)
+Apply integration by parts. We obtain
+lim
+δÑ0 Y “ 0 ´
+1
+λ ` t ` 1
+ż
+θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq
+(24)
+10
+
+where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero as a result
+of Lemma 1. On the other hand, according to the definition of Y ,
+lim
+δÑ0 Y “ lim
+δÑ0
+ż
+θpµiptqqppµi, t ` δq ´ ppµi, tq
+δ
+dµi “
+ż
+θpµiptqqBtppµi, tqdµi.
+(25)
+Therefore, we have the equality
+ż
+θpµiptqqBtppµi, tqdµi “ ´
+1
+λ ` t ` 1
+ż
+θpµiptqq∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs dµiptq.
+(26)
+As θ is a test function, this leads to
+Btppµi, tq “ ´
+1
+λ ` t ` 1∇ ¨ rppµiptq, tqp¯xiptq ´ µiptqqs .
+(27)
+Rearranging the terms, we obtain Equation 8 in the main paper. By the definition of expectation given a
+probability distribution, it is straightforward to obtain Equation 9 in the main paper. Q.E.D.
+Remarks: The PDEs we derived are akin to the continuity equation commonly encountered in physics
+in the study of conserved quantities.The continuity equation describes the transport phenomena (e.g., of
+mass or energy) in a physical system. This renders a physical interpretation for our PDE model: under
+SFP, the belief dynamics of a heterogeneous system is analogously the transport of the agent mass in the
+simplex ∆ “ ś
+iPV ∆i.
+Corollaries of Proposition 1
+Corollary 3. For any population i P V , the system beliefs about this population never go to extremes.
+Proof. This is a straightforward result of Lemma 1.
+Corollary 4. For any population i P V , the total probability mass ppµi, tq always remains conserved.
+Proof. Consider the time derivative of the total probability mass
+d
+dt
+ż
+ppµi, tqdµi.
+(28)
+Apply the Leibniz rule to interchange differentiation and integration,
+d
+dt
+ż
+ppµi, tqdµi “
+ż Bppµi, tq
+Bt
+dµi.
+(29)
+Substitute Bppµi,tq
+Bt
+with Equation 8 in the main paper,
+d
+dt
+ż
+ppµi, tqdµi
+“ ´
+ż
+∇ ¨
+ˆ
+ppµi, tq ¯xi ´ µi
+λ ` t ` 1
+˙
+dµi
+(30)
+“ ´
+ż
+ÿ
+siPSi
+Bµisi
+ˆ
+ppµi, tq ¯xisi ´ µisi
+λ ` t ` 1
+˙
+dµi
+(31)
+“ ´
+1
+λ ` t ` 1
+«ż
+ÿ
+siPSi
+Bµisi ppµi, tq p¯xisi ´ µisiq dµi `
+ż
+ppµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi
+ff
+(32)
+Apply integration by parts,
+ż
+ÿ
+siPSi
+Bµisi ppµi, tq p¯xisi ´ µisiq dµi “ 0 ´
+ż
+ppµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi.
+(33)
+where we have leveraged that the probability mass ppµi, tq at the boundary B∆i remains zero. Hence, the
+terms within the bracket of Equation 32 cancel out, and
+d
+dt
+ż
+ppµi, tqdµi “ 0.
+(34)
+11
+
+Proof of Proposition 2
+Lemma 4. The dynamics of the mean belief ¯µi about each population i P V is governed by a differential
+equation
+d¯µisi
+dt
+“ ¯xisi ´ ¯µisi
+λ ` t ` 1 ,
+@si P Si.
+(35)
+Proof. The time derivative of the mean belief about strategy si is
+d¯µisi
+dt
+“ d
+dt
+ż
+µisippµi, tqdµi.
+(36)
+We apply the Leibniz rule to interchange differentiation and integration, and then substitute Bppµi,tq
+Bt
+with
+Equation 8 in the main paper.
+d
+dt
+ż
+µisippµi, tqdµi
+(37)
+“
+ż
+µisi
+Bppµi, tq
+Bt
+dµi
+(38)
+“ ´
+ż
+µisi∇ ¨
+ˆ
+ppµi, tq ¯xi ´ µi
+λ ` t ` 1
+˙
+dµi
+(39)
+“ ´
+ż
+µisi
+ÿ
+siPSi
+Bµisi
+ˆ
+ppµi, tq ¯xisi ´ µisi
+λ ` t ` 1
+˙
+dµi
+(40)
+“ γ
+«ż
+µisi
+ÿ
+siPSi
+`
+Bµisi ppµi, tq
+˘
+p¯xisi ´ µisiq dµi `
+ż
+µisippµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi
+ff
+(41)
+where γ :“ ´
+1
+λ`t`1. Apply integration by parts to the first term in Equation 41.
+ż
+µisi
+ÿ
+siPSi
+`
+Bµisi ppµi, tq
+˘
+p¯xisi ´ µisiq dµi
+“ ´
+ż
+µisippµi, tq
+»
+– ÿ
+s1
+iPSi
+Bµis1
+i p¯xis1
+i ´ µis1
+iq
+fi
+fl ` ppµi, tqBµisi rµisip¯xisi ´ µisiqs dµi
+(42)
+where we have leveraged that the probability mass at the boundary remains zero. Hence, it follows from
+Equation 41 that
+d
+dt
+ż
+µisippµi, tqdµi
+(43)
+“ ´γ
+ż
+µisippµi, tq
+ÿ
+s1
+iPSi
+Bµis1
+i p¯xis1
+i ´ µis1
+iqdµi ´ γ
+ż
+ppµi, tqBµisi rµisip¯xisi ´ µisiqs dµi
+` γ
+ż
+µisippµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi
+(44)
+“ γ
+ż
+ppµi, tq
+“
+µisiBµisi p¯xisi ´ µisiq ´ Bµisi rµisip¯xisi ´ µisiqs
+‰
+dµi
+(45)
+“ γ
+ż
+ppµi, tqµisidµi ´
+ż
+ppµi, tq¯xisidµi
+(46)
+“ ¯xisi ´ ¯µisi
+λ ` t ` 1
+(47)
+We repeat the mean probability ¯xisi, which has been given in Equation 9 in the main paper, as follows:
+¯xisi “
+ż
+exp pβuisiq
+ř
+s1
+iPSi exp pβuis1
+iq
+ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+(48)
+12
+
+where uisi “ ř
+jPVi eJ
+siAijµj. Define ¯µ :“ t¯µjujPVi and
+fsiptµjujPViq :“
+exp pβ ř
+jPVi eJ
+siAijµjq
+ř
+s1
+iPSi exp pβ ř
+jPVi eJ
+s1
+iAijµjq.
+(49)
+Applying the Taylor expansion to approximate this function at the mean belief ¯µ, we have
+fsiptµjujPViq « fsip¯µq ` ∇fsip¯µq ¨ pµ ´ ¯µq ` 1
+2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq ` Op||µ ´ ¯µ||3q
+(50)
+where H denotes the Hessian matrix. Hence, we can rewrite Equation 48 as
+¯xisi “
+ż
+fsiptµjujPViq
+ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+(51)
+« fsip¯µq `
+ż
+∇fsip¯µq ¨ µ
+ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+´ ∇fsip¯µq ¨ ¯µ
+`
+ż 1
+2pµ ´ ¯µqJHfsip¯µqpµ ´ ¯µq
+ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+`
+ż
+Op||µ ´ ¯µ||q3 ź
+jPVi
+ppµj, tq
+˜ ź
+jPVi
+dµj
+¸
+(52)
+Observe that in Equation 52, the second and the third term can be canceled out. Moreover, for any
+two neighbor populations j, k P Vi, the beliefs µj, µk about these two populations are separate and
+independent. Hence, the covariance of these beliefs are zero. We apply the moment closure approximation
+[32, 13] with the second order and obtain
+¯xisi « fsip¯µq ` 1
+2
+ÿ
+jPVi
+ÿ
+sjPSj
+B2fsip¯µq
+pBµjsjq2 Varpµjsjq.
+(53)
+Hence, substituting ¯xisi in Lemma 4 with the above approximation, we have the mean belief dynamics
+d¯µisi
+dt
+« fsip¯µq ´ ¯µisi
+λ ` t ` 1
+`
+ř
+jPVi
+ř
+sjPSj
+B2fsip¯µq
+pBµjsj q2 Varpµjsjq
+2pλ ` t ` 1q
+.
+(54)
+Q.E.D.
+Remarks: the use of the moment closure approximation (considering only the first and the second
+moments) is for obtaining more conclusive results. Strictly speaking, the mean belief dynamics also depend
+on the third and higher moments. However, we observe in the experiments that these moments in general
+have little effects on the mean belief dynamics. To be more specific, given the same initial mean beliefs,
+while the variance of initial beliefs sometimes can change the limit behaviors of a system, we do not
+observe similar phenomena for the third and higher moments.
+Proof of Proposition 3
+Consider a population i. It follows from Equation 7 in the main paper that the change in the beliefs
+about this population can be written as follows.
+µipt ` 1q “ µiptq ` xiptq ´ µiptq
+λ ` t ` 1
+.
+(55)
+Suppose that the amount of time that passes between two successive time steps is δ P p0, 1s. We rewrite
+the above equation as
+µipt ` δq “ µiptq ` δ xiptq ´ µiptq
+λ ` t ` 1
+.
+(56)
+Move the term µiptq to the right hand side and divide both sides by δ,
+µipt ` δq ´ µiptq
+δ
+“ xiptq ´ µiptq
+λ ` t ` 1
+.
+(57)
+13
+
+Assume that the amount of time δ between two successive time steps goes to zero. we have
+dµi
+dt “ lim
+δÑ0
+µipt ` δq ´ µiptq
+δ
+“ xiptq ´ µiptq
+λ ` t ` 1
+.
+(58)
+Note that for continuous-time dynamics, we usually drop the time index in the bracket, yielding the belief
+dynamics (Equation 11) in Proposition 3. Q.E.D.
+Proof of Theorem 1
+Without loss of generality, we consider the variance of the belief µisi about strategy si of population i.
+Note that
+Varpµisiq “ Erpµisiq2s ´ p¯µisiq2.
+(59)
+Hence, we have
+dVarpµisiq
+dt
+“ dErpµisiq2s
+dt
+´ 2¯µisi
+d¯µisi
+dt .
+(60)
+Consider the first term on the right hand side. We apply the Leibniz rule to interchange differentiation
+and integration, and then substitute Bppµi,tq
+Bt
+with Equation 8 in the main paper.
+dErpµisiq2s
+dt
+“
+ż
+pµisiq2 Bppµi, tq
+Bt
+dµi
+(61)
+“ ´
+ż
+pµisiq2∇ ¨
+ˆ
+ppµi, tq ¯xi ´ µi
+λ ` t ` 1
+˙
+dµi
+(62)
+“ ´
+ż
+pµisiq2 ÿ
+siPSi
+Bµisi
+ˆ
+ppµi, tq ¯xisi ´ µisi
+λ ` t ` 1
+˙
+dµi
+(63)
+“ γ
+ż
+pµisiq2 ÿ
+siPSi
+Bµisi ppµi, tq p¯xisi ´ µisiq dµi ` γ
+ż
+pµisiq2ppµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi
+(64)
+where γ :“ ´
+1
+λ`t`1. Applying integration by parts to the first term in Equation 64 yields
+ż
+pµisiq2 ÿ
+siPSi
+Bµisi ppµi, tq p¯xisi ´ µisiq dµi
+“ ´
+ż
+pµisiq2ppµi, tq
+»
+– ÿ
+s1
+iPSi
+Bµis1
+i p¯xis1
+i ´ µis1
+iq
+fi
+fl ` ppµi, tqBµisi
+“
+pµisiq2p¯xisi ´ µisiq
+‰
+dµi
+(65)
+where we have leveraged that the probability mass at the boundary remains zero. Combining the above
+two equations, we obtain
+dErpµisiq2s
+dt
+“ ´γ
+ż
+pµisiq2ppµi, tq
+»
+– ÿ
+s1
+iPSi
+Bµis1
+i p¯xis1
+i ´ µis1
+iq
+fi
+fl ` ppµi, tqBµisi
+“
+pµisiq2p¯xisi ´ µisiq
+‰
+dµi
+` γ
+ż
+pµisiq2ppµi, tq
+ÿ
+siPSi
+Bµisi p¯xisi ´ µisiq dµi
+(66)
+“ γ
+ż “
+´ppµi, tqBµisi
+“
+pµisiq2p¯xisi ´ µisiq
+‰‰
+` pµisiq2ppµi, tqBµisi p¯xisi ´ µisiq dµi
+(67)
+“ γ
+ż
+2pµisiq2ppµi, tqdµi ´ γ
+ż
+2¯xisiµisippµi, tqdµi
+(68)
+“ ´2Erpµisiq2s ´ 2¯xisi ¯µisi
+λ ` t ` 1
+.
+(69)
+14
+
+Next, we consider the second term in Equation 60. By Lemma 4, we have
+2¯µisi
+d¯µisi
+dt
+“ 2¯µisip¯xisi ´ ¯µisiq
+λ ` t ` 1
+.
+(70)
+Combining Equations 69 and 70, the dynamics of the variance is
+dVarpµisiq
+dt
+“ ´2Erpµisiq2s ´ 2¯xisi ¯µisi
+λ ` t ` 1
+´ 2¯µisip¯xisi ´ ¯µisiq
+λ ` t ` 1
+(71)
+“ 2p¯µisiq2 ´ 2Erpµisiq2s
+λ ` t ` 1
+(72)
+“ ´2Varpµisiq
+λ ` t ` 1 .
+(73)
+Q.E.D.
+Remarks: We believe that the rationale behind such a phenomenon is twofold: 1) agents apply smooth
+fictitious play, and 2) agents respond to the mean strategy play of other populations rather than the
+strategy play of some fixed agents. Regarding the former, we notice that under a similar setting, population
+homogenization may not occur if agents apply other learning methods, e.g., Q-learning and Cross learning.
+Regarding the latter, imagine that agents adjust their beliefs in response to the strategies of some fixed
+agents. For example, consider two populations; one contains agents A and C, and the other one contains
+agents B and D. Suppose that agents A and B form a fixed pair such that they adjust their beliefs only in
+response to each other; the same applies to agents C and D. Belief homogenization may not happen.
+Appendix B: Proofs omitted in Section 4.1
+Proof of Theorem 2
+Belief homogenization implies that the fixed points of systems with initially heterogeneous beliefs are the
+same as in systems with homogeneous beliefs. Thus, we focus on homogeneous systems to analyze the
+fixed points. It is straightforward to see that
+dµi
+dt “ xi ´ µi
+λ ` t ` 1 “ 0 ùñ xi “ µi.
+(74)
+Denote the fixed points of the system dynamics, which satisfies the above equation, by px˚
+i , µ˚
+i q for each
+population i. By the logit choice function (Equation 5 in the main paper), we have
+x˚
+isi “
+exp pβuisiq
+ř
+s1
+iPSi exp pβuis1
+iq “
+exp pβ ř
+jPVi eJ
+siAijµ˚
+j q
+ř
+s1
+iPSi exp pβ ř
+jPVi eJ
+s1
+iAijµ˚
+j q.
+(75)
+Leveraging that x˚
+i “ µ˚
+i , @i P V at the fixed points, we can replace µ˚
+j with x˚
+j . Q.E.D.
+Proof of Theorem 3
+Consider a population i. The set of neighbor populations is Vi, the set of beliefs about the neighbor
+populations is tµjujPVi, and the choice distribution is xi. Given a population network game Γ, the
+expected payoff is given by xJ
+i
+ř
+pi,jqPE Aijµj. Define a perturbed payoff function
+πi
+`
+xi, tµjujPVi
+˘ :“ xJ
+i
+ÿ
+jPVi
+Aijµj ` vpxiq
+(76)
+where vpxiq “ ´ 1
+β
+ř
+siPSi xisi lnpxisiq. Under this form of vpxiq, the maximization of πi yields the choice
+distribution xi from the logit choice function [8]. Based on this, we establish the following lemma.
+Lemma 5. For a choice distribution xi of SFP in a population network game,
+Bxiπi
+`
+xi, tµjujPVi
+˘
+“ 0
+and
+ÿ
+jPVi
+`
+Aijµj
+˘J “ ´Bxivpxiq.
+(77)
+Proof. This lemma immediately follows from the fact that the maximization of πi will yield the choice
+distribution xi from the logit choice function [8].
+15
+
+The belief dynamics of a homogeneous populations can be simplified after time-reparameterization.
+Lemma 6. Given τ “ ln λ`t`1
+λ`1 , the belief dynamics of homogeneous systems (given in Equation 11 in
+the main paper) is equivalent to
+dµi
+dτ “ xi ´ µi.
+(78)
+Proof. From τ “ ln λ`t`1
+λ`1 , we have
+t “ pλ ` 1qpexp pτq ´ 1q.
+(79)
+By the chain rule, for each dimension si,
+dµisi
+dτ
+“ dµisi
+dt
+dt
+dτ
+(80)
+“ xisi ´ µisi
+λ ` t ` 1
+d ppλ ` 1qpexp pτq ´ 1qq
+dτ
+(81)
+“
+xisi ´ µisi
+λ ` pλ ` 1qpexp pτq ´ 1q ` 1pλ ` 1q exp pτq
+(82)
+“ xisi ´ µisi.
+(83)
+Next, we define the Lyapunov function L as
+L :“
+ÿ
+iPV
+ωiLi
+s.t.
+Li :“ πi
+`
+xi, tµjujPVi
+˘
+´ πi
+`
+µi, tµjujPVi
+˘
+.
+(84)
+where tωiuiPV is the set of positive weights defined in the weighted zero-sum Γ. The function L is
+non-negative because for every i P V , xi maximizes the function πi. When for every i P V , xi “ µi, the
+function L reaches the minimum value 0.
+Rewrite L as
+L “
+ÿ
+iPV
+«
+ωiπi
+`
+xi, tµjujPVi
+˘
+´ ωiµJ
+i
+ÿ
+jPVi
+Aijµj ´ ωivpµiq
+ff
+.
+(85)
+We observe that πi
+`
+xi, tµjujPVi
+˘
+is convex in µj, j P Vi by Danskin’s theorem, and ´vpµiq is strictly
+convex in µi. Moreover, by the weighted zero-sum property given in Equation 2 in the main paper, we
+have
+ÿ
+iPV
+˜
+ωiµJ
+i
+ÿ
+jPVi
+Aijµj
+¸
+“ 0
+(86)
+since µi P ∆i, µj P ∆j for every i, j P V. Therefore, the function L is a strictly convex function and attains
+its minimum value 0 at a unique point xi “ µi, @i P V.
+Consider the function Li. Its time derivative is
+9Li “ Bxiπi
+`
+xi, tµjujPVi
+˘
+9xi `
+ÿ
+jPVi
+”
+Bµjπi
+`
+xi, tµjujPVi
+˘ 9µj
+ı
+´ Bµiπi
+`
+µi, tµjujPVi
+˘ 9µi ´
+ÿ
+jPVi
+”
+Bµjπi
+`
+µi, tµjujPVi
+˘ 9µj
+ı
+.
+(87)
+Note that the partial derivative Bxiπi equals 0 by Lemma 5. Thus, we can rewrite this as
+9Li “ Bµiπi
+`
+µi, tµjujPVi
+˘ 9µi `
+ÿ
+jPVi
+”
+Bµjπi
+`
+xi, tµjujPVi
+˘
+´ Bµjπi
+`
+µi, tµjujPVi
+˘ı
+9µj
+(88)
+“ ´
+« ÿ
+jPVi
+`
+Aijµj
+˘J ` Bµivpµiq
+ff
+pxi ´ µiq `
+ÿ
+jPVi
+`
+xJ
+i Aij ´ µJ
+i Aij
+˘
+pxj ´ µjq
+(89)
+“ rBxivpxiq ´ Bµivpµiqs pxi ´ µiq `
+ÿ
+jPVi
+`
+xJ
+i Aijxj ´ µJ
+i Aijxj ´ xJ
+i Aijµj ` µJ
+i Aijµj
+˘
+.
+(90)
+16
+
+where from Equation 89 to 90, we apply Lemma 5 to substitute ř
+jPVi
+`
+Aijµj
+˘J with ´Bxivpxiq. Hence,
+summing over all the populations, the time derivative of L is
+9L “
+ÿ
+iPV
+ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq
+`
+ÿ
+iPV
+ÿ
+jPVi
+ωi
+`
+xJ
+i Aijxj ´ µJ
+i Aijxj ´ xJ
+i Aijµj ` µJ
+i Aijµj
+˘
+.
+(91)
+The summation in the second line is equivalent to
+ÿ
+pi,jqPE
+pωixJ
+i Aijxj ` ωjxJ
+j Ajixiq ´ pωiµJ
+i Aijxj ` ωjxJ
+j Ajiµiq
+(92)
+´ pωixJ
+i Aijµj ` ωjµJ
+j Ajixiq ` pωiµJ
+i Aijµj ` ωjµJ
+j Ajiµiq.
+(93)
+By the weighted zero-sum property given in Equation 2 in the main paper, this summation equals 0,
+yielding
+9L “
+ÿ
+iPV
+ωi rBxivpxiq ´ Bµivpµiqs pxi ´ µiq.
+(94)
+Note that the function v is strictly concave such that its second derivative is negative definite. By this
+property, 9L ď 0 with equality only if xi “ µi, @i P V , which corresponds to the QRE. Therefore, L is a
+strict Lyapunov function, and the global asymptotic stability of the QRE follows. Q.E.D.
+Remarks: Intuitively, the Lyapunov function defined above measures the distance between the QRE
+and a given set of beliefs. The idea of measuring the distance in terms of entropy-regularized payoffs is
+inspired from the seminal work [19]. However, different from the network games considered in this paper,
+Hofbauer and Hopkins [19] consider SFP in two-player games. To our knowledge, so far there has been
+no systematic study on SFP in network games.
+Proof of Theorem 4
+The proof of Theorem 4 leverages the seminal results of the asymptotically autonomous dynamical system
+[31, 40, 41] which conventionally is defined as follows.
+Definition 1. A nonautonomous system of differential equations in Rn
+x1 “ fpt, xq
+(95)
+is said to be asymptotically autonomous with limit equation
+y1 “ gpyq,
+(96)
+if fpt, xq Ñ gpxq, t Ñ 8, where the convergence is uniform on each compact subset of Rn. Conventionally,
+the solution flow of Eq. 95 is called the asymptotically autonomous semiflow (denoted by φ) and the
+solution flow of Eq. 96 is called the limit semiflow (denoted by Θ).
+Based on this definition, we establish Lemma 1 in the main paper, which is repeated as follows.
+Lemma 7. For a system that initially has heterogeneous beliefs, the mean belief dynamics is asymptotically
+autonomous [31] with the limit equation
+dµi
+dt “ xi ´ µi
+(97)
+which after time-reparameterization is equivalent to the belief dynamics for homogeneous systems.
+Proof. We first time-reparameterize the mean belief dynamics of heterogeneous systems. Assume τ “
+17
+
+Figure 5: Population network games where the underlying network consists of star structure.
+ln λ`t`1
+λ`1 . By the chain rule and Equation 54, for each dimension si,
+d¯µisi
+dτ
+“ d¯µisi
+dt
+dt
+dτ
+(98)
+“
+»
+—–fsip¯µq ´ ¯µisi
+λ ` t ` 1
+`
+ř
+jPVi
+ř
+sjPSj
+B2fsip¯µq
+pBµjsj q2 Varpµjsjq
+2pλ ` t ` 1q
+fi
+ffifl d ppλ ` 1qpexp pτq ´ 1qq
+dτ
+(99)
+“
+fsip¯µq ´ ¯µisi ` 1
+2
+ř
+jPVi
+ř
+sjPSj
+B2fsip¯µq
+pBµjsj q2
+´
+λ`1
+λ`t`1
+¯2
+σ2pµjsjq
+λ ` pλ ` 1qpexp pτq ´ 1q ` 1
+pλ ` 1q exp pτq
+(100)
+“ fsip¯µq ´ ¯µisi ` 1
+2
+ÿ
+jPVi
+ÿ
+sjPSj
+B2fsip¯µq
+pBµjsjq2 σ2pµjsjq exp p´2τq.
+(101)
+Observe that exp p´2τq decays to zero exponentially fast and that both σ2pµjsjq and
+B2fsip¯µq
+pBµjsj q2 are bounded
+for every µ in the simplex ś
+jPVi ∆j. Hence, Equation 101 converges locally and uniformly to the following
+equation:
+d¯µisi
+dτ
+“ fsip¯µq ´ ¯µisi.
+(102)
+Note that xisi “ fsip¯µq for homogeneous systems, and the above equation is algebraically equivalent to
+Equation 97. Hence, by Definition 1, Equation 101 is asymptotically autonomous with the limit equation
+being Equation 97.
+By the above lemma, we can formally connect the limit behaviors of initially heterogeneous systems
+and those of homogeneous systems. Recall that Theorem 3 in the main paper states that under SFP,
+there is a unique rest point (QRE) for the belief dynamics in a weighted zero-sum network game Γ;
+this excludes the case where there are finitely many equilibria that are chained to each other. Hence,
+combining Lemma 2 in the main paper, we prove that the mean belief dynamics of initially heterogeneous
+systems converges to a unique QRE. Q.E.D.
+Appendix C: Results and Proofs omitted in Section 4.2
+For the case of network coordination, we consider networks that consist of a star or disconnected multiple
+stars due to technical reasons. In Figure 1, we present examples of the considered network structure with
+different numbers of nodes (populations).
+In the following theorem, focusing on homogeneous systems, we establish the convergence of the belief
+dynamics to the set of QRE.
+Theorem 6 (Convergence in Homogeneous Network Coordination with Star Structure).
+Given a coordination Γ where the network structure consists of a single or disconnected multiple stars,
+each orbit of the belief dynamics for homogeneous systems converges to the set of QRE.
+Proof. Consider a root population j of a star structure. Its set of leaf (neighbor) populations is Vj, the
+set of beliefs about the leaf populations is tµiuiPVj, and the choice distribution is xj. Given the game Γ,
+18
+
+Five Populations
+Five Populations
+(Two Disconnected
+Stars)
+Three Populations
+Two Populations
+P2
+P1
+P3the expected payoff is xJ
+j
+ř
+iPVj Ajiµi. Define a perturbed payoff function
+πj
+`
+xj, tµiuiPVj
+˘ :“ xJ
+j
+ÿ
+iPVj
+Ajiµi ` vpxjq
+(103)
+where vpxjq “ ´ 1
+β
+ř
+sjPSj xjsj lnpxjsjq. Under this form of vpxjq, the maximization of πj yields the choice
+distribution xj from the logit choice function [8].
+Consider a leaf population i of the root population j. It has only one neighbor population, which
+is population j. Thus, given the game Γ, the expected payoff is xJ
+i Aijµj. Define a perturbed payoff
+function
+πi
+`
+xi, µj
+˘ :“ xJ
+i Aijµj ` vpxiq
+(104)
+where vpxiq “ ´ 1
+β
+ř
+siPSi xisi lnpxisiq. Similarly, the maximization of πi yields the choice distribution xi
+from the logit choice function [8]. Based on this, we establish the following lemma.
+Lemma 8. For choice distributions of SFP in a population network game with start structure,
+Bxjπj
+`
+xj, tµiuiPVj
+˘
+“ 0
+and
+ÿ
+iPVj
+pAjiµiqJ “ ´Bxjvpxjq
+if j is a root population,
+(105)
+Bxiπi
+`
+xi, µj
+˘
+“ 0
+and
+`
+Aijµj
+˘J “ ´Bxivpxiq
+if i is a leaf population.
+(106)
+Proof. This lemma immediately follows from the fact that the maximization of πj and πi , respectively,
+yield the choice distributions xj and xi from the logit choice function [8].
+For readability, we repeat the belief dynamics of a homogeneous population after time-reparameterization,
+which has been proved in Lemma 4 in Appendix B, as follows:
+dµi
+dτ “ xi ´ µi.
+(107)
+Let R Ă V be the set of all root populations. We define
+L :“
+ÿ
+jPR
+Lj
+s.t.
+Lj :“ µJ
+j
+ÿ
+iPVj
+Ajiµi ` vpµjq `
+ÿ
+iPVj
+vpµiq.
+(108)
+Consider the function Lj. Its time derivative 9Lj is
+9Lj “
+»
+–BµjpµJ
+j
+ÿ
+iPVj
+Ajiµiq 9µj `
+ÿ
+iPVj
+BµipµJ
+j
+ÿ
+iPVj
+Ajiµiq 9µi
+fi
+fl ` Bµjvpµjq 9µj `
+ÿ
+iPVj
+Bµivpµiq 9µi
+(109)
+“
+ÿ
+iPVj
+pAjiµiqJpxj ´ µjq `
+»
+– ÿ
+iPVj
+µJ
+j Ajipxi ´ µiq
+fi
+fl ` Bµjvpµjqpxj ´ µjq `
+ÿ
+iPVj
+Bµivpµiqpxi ´ µiq. (110)
+Since Γ is a coordination game, we have
+`
+Aijµj
+˘J “ µJ
+j AJ
+ij “ µJ
+j Aji. Hence, applying Lemma 8, we can
+substitute ř
+iPVjpAjiµiqJ with ´v1pxjq, and µJ
+j Aji with ´v1pxiq, yielding
+9Lj “ ´Bxjvpxjqpxj ´ µjq `
+»
+– ÿ
+iPVj
+p´Bxivpxiqqpxi ´ µiq
+fi
+fl ` Bµjvpµjqpxj ´ µjq `
+ÿ
+iPVj
+Bµivpµiqpxi ´ µiq
+(111)
+“ pBµjvpµjq ´ Bxjvpxjqqpxj ´ µjq `
+ÿ
+iPVj
+pBµivpµiq ´ Bxivpxiqqpxi ´ µiq
+(112)
+Note that the function v is strictly concave such that its second derivative is negative definite. By this
+property, 9Lj ě 0 with equality only if xi “ µi, @i P Vj and xj “ µj. Thus, the time derivative of the
+function L, i.e., 9L “ ř
+jPR 9Lj ě 0 with equality only if xi “ µi, @i P Vj, xj “ µj, @j P R.
+We generalize the convergence result to initially heterogeneous systems in the following theorem.
+19
+
+Theorem 7 (Convergence in Initially Heterogeneous Network Coordination with Star Struc-
+ture). Given a coordination Γ where the network structure consists of a single or disconnected multiple
+stars, each orbit of the mean belief dynamics for initially heterogeneous systems converges to the set of
+QRE.
+Proof. The proof technique is similar to that for initially heterogeneous competitive network games. By
+Lemma 1 in the main paper, we show that the mean belief dynamics of initially heterogeneous systems is
+asymptotically autonomous with the belief dynamics of homogeneous systems. Therefore, it follows from
+Lemma 2 in the main paper that the convergence result for homogeneous systems can be carried over to
+the initially heterogeneous systems.
+Remarks: The convergence of SFP in coordination games and potential games has been established
+under the 2-player settings [19] as well as some n-player settings [20, 39]. Our work differs from the
+previous works in two aspects. First, our work allows for heterogeneous beliefs. Moreover, we consider that
+agents maintain separate beliefs about other agents, while in the previous works agents do not distinguish
+between other agents. Thus, even when the system beliefs are homogeneous, our setting is still different
+from (and more complicated) than the previous settings.
+Appendix D: Omitted Experimental Details
+Numerical Method for the PDE model.
+PDEs are notoriously difficult to solve, and only limited
+types of PDEs allow analytic solutions. Hence, similar to previous research [23], we resort to numerical
+method for PDEs; in particular, we consider the finite difference method [38].
+Agent-based Simulations.
+The presented simulation results are averaged over 100 independent
+simulation runs to smooth out the randomness. For each simulation run, there are 1, 000 agents in each
+population. For each agent, the initial beliefs are sampled from the given initial probability distribution.
+Detailed Experimental Setups for Figure 1.
+In the case of small initial variance, the initial
+beliefs µ1H and µ2H are distributed according to the distribution Betap280, 120q. On the contrary,
+in the case of large initial variance, the initial beliefs µ1H and µ2H are distributed according to the
+distribution Betap14, 6q. Thus, initially, the mean beliefs in these two cases are both ¯µ1H “ ¯µ2H “ 0.7
+and ¯µ1S “ ¯µ2S “ 0.3. In both cases, the initial sum of weights λ “ 10 and the temperature β “ 10.
+Detailed Experimental Setups for Figure 3.
+We visualize the regions of attraction of different
+equilibria in stag hunt games by numerically solving the mean belief dynamics (Equation 10 in the main
+paper). The initial variances have been given in the title of each panel. In all cases, the initial sum of
+weights λ “ 0 and the temperature β “ 5.
+Detailed Experimental Setups for Figure 4.
+We let the initial beliefs about populations 1, 3 and 5
+remain unchanged across different cases, and vary the initial beliefs about populations 2 and 4. The initial
+beliefs about populations 1, 3 and 5, denoted by µ1H, µ3H and µ5H, are distributed according to the
+distributions Betap20, 10q, Betap6, 4q, and Betap10, 5q, respectively. The initial beliefs about populations
+2 and 4 have been given in the legends of Figure 4. In all cases, the initial sum of weights λ “ 10 and the
+temperature β “ 10. Note that µiT “ 1 ´ µiH for all populations i “ 1, 2, 3, 4, 5.
+Source Code and Computing Resource.
+We have attached the source code for reproducing our
+main experiments. The Matlab script finitedifference.m numerically solves our PDE model presented
+in Proposition 1 in the main paper. The Matlab script regionofattraction.m visualizes the region of
+attraction of different equilibria in stag hunt games, which are presented in Figure 3. The Python scripts
+simulation(staghunt).py and simulation(matchingpennies).py correspond to the agent-based simulations
+in two-population stag hunt games and five-population asymmetric matching pennies games, respectively.
+We use a laptop (CPU: AMD Ryzen 7 5800H) to run all the experiments.
+20
+
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